Applied mathematical analysis: theory, methods, and applications 9783319999173, 9783319999180

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Studies in Systems, Decision and Control 177

Hemen Dutta · James F. Peters   Editors

Applied Mathematical Analysis: Theory, Methods, and Applications

Studies in Systems, Decision and Control Volume 177

Series editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected]

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink.

More information about this series at http://www.springer.com/series/13304

Hemen Dutta James F. Peters •

Editors

Applied Mathematical Analysis: Theory, Methods, and Applications

123

Editors Hemen Dutta Department of Mathematics Gauhati University Guwahati, India

James F. Peters Computational Intelligence Laboratory Department of Electrical & Computer Engineering University of Manitoba Winnipeg, MB, Canada Department of Mathematics Adıyaman University Adıyaman, Adıyaman, Turkey

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-319-99917-3 ISBN 978-3-319-99918-0 (eBook) https://doi.org/10.1007/978-3-319-99918-0 Library of Congress Control Number: 2018968111 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The book is designed for graduate students, researchers, educators, engineers and scientists requiring various theories, methods and applications of mathematical analysis. It brings forward several important aspects of recent developments in mathematical analysis, its use and applications in diverse areas of scientific research and education. Each chapter provides an enrichment of the understanding of the research discussed along with a balanced effort in theories, methods and applications, and also contains the most recent advances made in an area of study. There are 23 chapters in this book, and they are organized as follows: The chapter “On Equivalent Properties of Hardy-Type Integral Inequality with the General Nonhomogeneous Kernel and Parameters” investigates some equivalent conditions of Hardy-type integral inequality with the general nonhomogeneous kernel, related to other inequalities, as well as the parameters and the integrals of the kernel. A few equivalent conditions of Hardy-type integral inequality with the general homogeneous kernel are also deduced as well as considers the other kind of integral inequality, the operator expressions, some corollaries and a few particular examples. The chapter “Fundamental Stabilities of Various Forms of Complex Valued Functional Equations” contains the solution and examination of fundamental stabilities of various forms of complex valued additive, quadratic, cubic and quartic functional equations in the vicinity of complex Banach spaces using direct and fixed point methods. The chapter “Statistical Summability of Double Sequences by the Weighted Mean and Associated Approximation Results” aims to extend various summability concepts and summability techniques by the weighted mean method with respect to the generalized difference operator involving (p, q)-Gamma function. It also investigates some inclusion relations among the new methods and presents some illustrative examples to connect with the existing literature. Further, some approximation theorems and their weighted statistical forms for functions of two variables are established and derived related approximation results associated with the (p, q)-analogue of generalized bivariate Bleimann-Butzer-Hahn operators.

v

vi

Preface

The rate of convergence of approximating positive linear operators in terms of the modulus of continuity is also estimated. The chapter “A Survey on a Conjecture of Rainer Brück” surveys the development of works done by several authors on a conjecture proposed by Rainer Brück on the single value sharing by an entire function with its first derivative. In 1996, Rainer Brück considered the uniqueness problem of an entire function that shares one value with its derivative. The conjecture of Brück is not completely resolved in its full generality till date. The chapter “Nonlinear Magneto-Elasticity: Direct and Inverse Problems” investigates a nonlinear problem which arises in magneto-elasticity. The equations of magneto-elasticity describe the behavior of, and the coupling between, an electrically conducting elastic solid and electromagnetic fields. The chapter considers a direct problem and an inverse problem and studies the existence and uniqueness of solutions. The chapter “Note on Periodic and Asymptotically Periodic Solutions of Fractional Differential Equations” presents several results on the existence or nonexistence of periodic solutions for fractional differential equations (FDEs) on arbitrary dimensional spaces involving Caputo fractional derivatives. The existence of S-asymptotically periodic solutions as well as periodic boundary value problems is also investigated. In fact, it considers a broad variety of FDEs by covering both finite dimensional FDEs and evolution FDEs in infinite dimensional spaces containing either single order or mixed orders of Caputo fractional derivatives with either finite or infinite lower limits of Caputo fractional derivatives. Qualitative results of different kinds are also derived for particular types of FDEs studied and several examples are incorporated to illustrate theoretical results. The chapter “Mathematics of Wavefields” presents the basic mathematical theory of certain aspects of wavefields, that is, waves and fields, as they occur under various physical situations. The basic mathematical concepts, tools and techniques, necessary for the presentation, are summarized in the beginning of the chapter. It is also shown that mathematical analyses reveal many subtleties hidden in the wavefields that would otherwise have gone unnoticed. The chapter “A Variational Technique to the Homogenization of Maxwell Equations” summarizes some homogenization results on the Maxwell equations in the stationary and nonstationary regime, coupled with linear and power law constitutive relations, which are obtained by a variational technique based on the Cconvergence of associated sequences of energies. The theory of homogenization focuses on finding the effective macroscopic behaviour of composite materials with a heterogeneous periodic microstructure. The Maxwell equations usually appear in many fields of Engineering, Mechanics and Physics. The chapter “The Narimanov-Moiseev Multimodal Analysis of Nonlinear Sloshing in Circular Conical Tanks” reports mathematical aspects of the Narimanov-Moiseev multimodal modelling for the liquid sloshing in rigid circular conical tanks, which perform small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. In order to derive the corresponding nonlinear modal system of ordinary differential equations, the

Preface

vii

chapter introduces an infinite set of the sloshing-related generalised coordinates governing the free-surface elevation but the velocity potential is posed as a Fourier series by the natural sloshing modes where the time-depending coefficients are treated as the generalized velocities. The Lukovsky non-conformal mapping technique is applied to transform the inner tank domain to an artificial upright circular cylinder, for which the single-valued representation of the free surface is possible. The occurrence of secondary resonances for the V-shaped truncated conical tanks is also evaluated. The Narimanov-Moiseev modal equations are further used for deriving an analytical steady-state (periodic) solution and its stability is also studied. The latter procedure is illustrated for longitudinal harmonic excitations. Standing (planar) waves and swirling as well as irregular sloshing (chaos) are established in certain frequency ranges. The corresponding amplitude response curves are drawn and discussed as well. The chapter “The Lengyel-Epstein Reaction Diffusion System” aims to review some relevant studies related to the Lengyel-Epstein system. It starts with a summary of the necessary theory behind each of the findings and examines different characteristics of the model including the local and global asymptotic stability, the existence of Turing patterns, and the Hopf-bifurcation. It also lists a number of modifications made to the original system with a particular focus on the CDIMA reaction model. Further, some numerical examples are incorporated to illustrate the behavior of the model and the types of possible patterns. The chapter “Prediction and Control of Buckling: The Inverse Bifurcation Problems for von Karman Equations” presents approaches to predict and control buckling of thin-walled structures. From mathematical point of view, these approaches are formalized as the first and second inverse bifurcation problems for von Karman equations. The approach considered was then applied to several problems, viz., for the first inverse problem, to the problems of optimal thickness distribution and optimal external pressure distribution for a cylindrical shell, optimal curvature for a cylindrical panel as well; for the second inverse problem, to the problem to predict buckling of a cylindrical shell under an external pressure. The chapter “Numerical Solution with Special Layer Adapted Meshes for Singularly Perturbed Boundary Value Problems” presents a comparative study of simple upwind finite difference method on various non-uniform meshes existing in the literature for resolving the boundary layer of two-point singularly perturbed problems. It begins with the Bakhvalov mesh and its modification using Padé approximation, and then continues with the piecewise uniform Shishkin-type meshes to the most recent W-grid using Lambert W-function. A new kind of mesh of Shishkin type has also been proposed as the method gives better results as compared to the results by using the standard Shishkin mesh. It is shown that the computed solution is uniformly convergent with respect to the small perturbation parameter for various meshes. Further, numerical results on a test problem are also presented to validate the theoretical considerations. The chapter “Use of Galerkin Technique in Some Water Wave Scattering Problems Involving Plane Vertical Barriers” used multi term Galerkin approximations to solve the classical problem of scattering of obliquely incident water

viii

Preface

waves by a vertical barrier present in deep water. The basis functions in multiterm Galerkin approximation are chosen as simple polynomial with suitable weight functions and quite accurate estimates of the physical quantities are obtained for two/three term approximations. The chapter “Dynamics of a Class of Leslie–Gower Predation Models with a Non-Differentiable Functional Response” presents an analysis of modified models of Leslie-Gower type, well-known in mathematical biology. In fact, two classes of Leslie-Gower type predator-prey models are analyzed, considering a non-usual functional response, called Rosenzweig or power functional responses. The chapter “Entire Solutions of a Nonlinear Diffusion System” analyses a diffusion model system of equations and entire solutions are established under certain conditions on its nonlinearity. In fact, the chapter aims to find answer to the question “Can one establish new results related to the existence and asymptotic behaviour of solutions for a system as considered in this chapter?” The chapter “Goal Programming Models for Managerial Strategic Decision Making” aims to present the state-of-the-art of Goal Programming (GP) models and highlight its applications to strategic decision making in portfolio investments, marketing decisions and media campaign. The GP model is an important Multiple Objective Programming technique that has been widely utilized for strategic decision making in presence of competing and conflicting objectives. The chapter “Modeling Highly Random Dynamical Infectious Systems” exhibits compartmental random dynamical models involving stochastic systems of differential equations, Markov processes, and random walk processes etc. to investigate random dynamical processes of infectious systems such as infectious diseases of humans or animals, the spread of rumours in social networks, and the spread of malicious signals on wireless sensory networks etc. The chapter provides an approach to identify, and represent the various constituents of random dynamic processes in infectious systems. A method to derive two independent environmental white noise processes, general nonlinear incidence rates, and multiple random delays in infectious systems is also presented. The ideas, mathematical modeling techniques and analysis, and the examples are delivered through original research on the modeling of vector-borne diseases of human beings or other species. A method to investigate the impacts of the strengths of the noises on the overall outcome of the infectious system is presented. Numerical simulation results are presented to validate the results of the chapter. The chapter “On Weighted Convergence of Double Singular Integral Operators Involving Summation” presents the conditions under which double singular integral operators involving summation are well-defined in the space of Lebesgue measurable functions defined on different sets, and then discusses Fatou type convergences on different domains of the operators. Further, it establishes the rate of convergences with respect to approximations obtained. The chapter “Circular-Like and Circular Elements in Free Product Banach
Algebras Induced by p-Adic Number Fields Qp Over Primes p” studies weighted-circular, and circular elements in a certain free product Banach *-probability space induced by measurable functions on p-adic number fields Qp , for

Preface

ix

primes p. It first constructs and considers weighted semicircular, and semicircular elements in free product Banach *-probability space induced by measurable functions on p-adic number fields Qp , for primes p. From the (weighted-) semicircular elements, it establishes (weighted-) circular elements and investigates their free distributions by computing joint free moments of them and their adjoints. The circular law is re-characterized by joint free moments of the circular elements and their adjoints. Further, the weighted-circularity dictated by p-adic analysis is fully characterized by weights of weighted-semicircular elements containing number-theoretic data obtained from fixed primes p. The chapter “On Statistical Deferred Weighted B-Convergence” discusses the deferred weighted B-mean method together with regularity condition. A theorem is given based on the regular methods showing the relation between convergence and summability via the deferred weighted B-mean, and established a Korovkin-type approximation theorem for the functions of two variables defined on a Banach space C (D). Further, another result on the rate of the deferred weighted B-statistical convergence for the same set of functions via the modulus of continuity has been established. Finally, a number of special cases and illustrative examples are incorporated. The chapter “Multi Poly-Bernoulli and multi Poly-Euler polynomials” introduces a certain variation of Bernoulli and Euler numbers and polynomials by means of polylogarithm, particularly, the poly-Bernoulli and Euler numbers and polynomials. Further, it presents generalizations of poly-Bernoulli and poly-Euler numbers and polynomials by means of multiple polylogarithm. Common properties shared by the family of Bernoulli and Euler numbers and polynomials are discussed including recurrence relations, explicit formulas and several identities expressing these generalizations in terms of the other special numbers and functions. The chapter “Geometric Properties of NormalizedWright Functions” investigates some subclasses of analytic functions in the open unit disk in the complex plane, and derives characteristic properties of the normalized Wright functions belonging to these classes and finds upper bound estimate for these functions belonging to the subclasses studied. Several sufficient conditions are obtained for the parameters of the normalized form of the Wright functions to be in this class, and some geometric properties of the integral transforms represented with the normalized Wright functions are also studied. Further, it gives some sufficient conditions for the integral operators involving normalized Wright functions to be univalent in the open unit disk. Finally, it introduces a Poisson distribution series, whose construction is alike Wright functions, and obtain necessary and sufficient conditions for this series belonging to the class S*C(a, b; c), and necessary and sufficient conditions for those belonging to the class TS*C(a, b; c), and also introduces two integral operators related to this series and investigate various geometric properties of these integral operators. The chapter “On the Spectra of Difference Operators Over Some Banach Spaces” aims to provide a literature review on spectral subdivisions of difference operators and compute the spectrum and the fine spectrum of third order difference operator D3 over the sequence space c. The operator D3 represents a lower forth tuple banded

x

Preface

infinite matrix. Finally, it finds the estimates for the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the operator over the Banach spaces c, c0 and l1. The editors are grateful to the contributors for their timely cooperation and patience while the chapters were being reviewed and processed. The editors also express their gratitude to the reviewers for their sincere efforts and time. The editors also thank the editors and supporting staff at Springer for their timely cooperation in bringing out this book. Guwahati, India Manitoba, Canada February, 2019

Hemen Dutta James F. Peters

Contents

On Equivalent Properties of Hardy-Type Integral Inequality with the General Nonhomogeneous Kernel and Parameters . . . . . . . . . . Bicheng Yang

1

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beri Venkatachalapathy Senthil Kumar and Hemen Dutta

29

Statistical Summability of Double Sequences by the Weighted Mean and Associated Approximation Results . . . . . . . . . . . . . . . . . . . . . Uğur Kadak

61

A Survey on a Conjecture of Rainer Brück . . . . . . . . . . . . . . . . . . . . . . Indrajit Lahiri

87

Nonlinear Magneto-Elasticity: Direct and Inverse Problems . . . . . . . . . 123 Viatcheslav Priimenko, Mikhail Vishnevskii and Adolfo Pires Note on Periodic and Asymptotically Periodic Solutions of Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Michal Fečkan Mathematics of Wavefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 D. N. Ghosh Roy A Variational Technique to the Homogenization of Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Hélia Serrano The Narimanov–Moiseev Multimodal Analysis of Nonlinear Sloshing in Circular Conical Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 A. V. Solodun and A. N. Timokha The Lengyel–Epstein Reaction Diffusion System . . . . . . . . . . . . . . . . . . 311 Salem Abdelmalek and Samir Bendoukha

xi

xii

Contents

Prediction and Control of Buckling: The Inverse Bifurcation Problems for von Karman Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Natalia I. Obodan, Victor J. Adlucky and Vasilii A. Gromov Numerical Solution with Special Layer Adapted Meshes for Singularly Perturbed Boundary Value Problems . . . . . . . . . . . . . . . 383 Deepti Kaur and Vivek Kumar Use of Galerkin Technique in Some Water Wave Scattering Problems Involving Plane Vertical Barriers . . . . . . . . . . . . . . . . . . . . . . 405 B. N. Mandal and Soumen De Dynamics of a Class of Leslie–Gower Predation Models with a Non-Differentiable Functional Response . . . . . . . . . . . . . . . . . . . 433 Viviana Rivera-Estay, Eduardo González-Olivares, Alejandro Rojas-Palma and Karina Vilches-Ponce Entire Solutions of a Nonlinear Diffusion System . . . . . . . . . . . . . . . . . . 459 Dragoş-Pătru Covei Goal Programming Models for Managerial Strategic Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Cinzia Colapinto, Raja Jayaraman and Davide La Torre Modeling Highly Random Dynamical Infectious Systems . . . . . . . . . . . . 509 Divine Wanduku On Weighted Convergence of Double Singular Integral Operators Involving Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Gümrah Uysal and Hemen Dutta Circular-Like and Circular Elements in Free Product Banach *-Algebras Induced by p-Adic Number Fields Qp Over Primes p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Ilwoo Cho On Statistical Deferred Weighted B-Convergence . . . . . . . . . . . . . . . . . 655 S. K. Paikray and Hemen Dutta Multi Poly-Bernoulli and Multi Poly-Euler Polynomials . . . . . . . . . . . . . 679 Roberto B. Corcino Geometric Properties of Normalized Wright Functions . . . . . . . . . . . . . 723 Nizami Mustafa, Veysel Nezir and Hemen Dutta On the Spectra of Difference Operators Over Some Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 Hemen Dutta and P. Baliarsingh

On Equivalent Properties of Hardy-Type Integral Inequality with the General Nonhomogeneous Kernel and Parameters Bicheng Yang

Abstract By the use of the way of real analysis and weight functions, we study some equivalent statements of Hardy-type integral inequality with the general nonhomogeneous kernel, related to another inequalities, as well as the parameters and the integrals of the kernel. As applications, a few equivalent stayements of Hardy-type integral inequality with the general homogeneous kernel are deduced. We also consider the other kind of integral inequality, the operator expressions, some corollaries and a few particular examples. Keywords Hardy-type integral inequality · Weight function · Equivalent form · Operator · Norm 2000 Mathematics Subject Classification 26D15 · 47A05

1 Introduction ∞ ∞ If 0 < 0 f 2 (x)d x < ∞ and 0 < 0 g 2 (y)dy < ∞, then we have the following Hilbert’s integral inequality (cf. [1]):  0

∞



∞ 0

f (x)g(y) d xd y < π x+y



∞ 0



∞

f (x)d x 2

g (y)dy 2

 21

,

(1)

0

where, the constant factor π is the best possible. In 1925, by introducing one pair of conjugate exponents ( p, q), Hardy [2] gave an extension of (1) as follows: For p > 1, 1p + q1 = 1, f (x), g(y) ≥ 0,

B. Yang (B) Department of Mathematics, Guangdong University of Education, Guangzhou 510303, Guangdong, People’s Republic of China e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_1

1

2

B. Yang

 0
0, such that  ∞ φ(σ) = h(u)u σ−1 du ∈ R+ , 0

then we have 

∞



0

1 < φ( ) p

∞

h(x y) f (x)g(y)d xd y

0



∞

x 0

p−2

f (x)d x p

 1p 

∞

g (y)dy q

 q1

,

(4)

0

where, the constant factor φ( 1p ) is the best possible (cf. [3], Theorem 350). In 1998, By introducing an independent parameter λ > 0, Yang gave an extension of (1) as follows (cf. [5, 6]):

On Equivalent Properties of Hardy-Type Integral Inequality …



∞



3

∞

f (x)g(y) d xd y (x + y)λ 0 0  ∞  21  ∞ λ λ 1−λ 2 1−λ 2 x f (x)d x y g (y)dy , < B( , ) 2 2 0 0

(5)

where, the constant factor B( λ2 , λ2 ) is the best possible (B(u, v) is the beta function). In 2004, by introducing another pair conjugate exponents (r, s), Yang [7] gave an extension of (2) as follows: If λ > 0, r > 1, r1 + 1s = 1, f (x), g(y) ≥ 0,  0
0), 

and

∞

k(λ1 ) :=

kλ (u, 1)u λ1 −1 du ∈ R+ ,

0

then we have 

∞



0

∞

kλ (x, y) f (x)g(y)d xd y

0



∞

< k(λ1 )

p(1−λ1 )−1

x

f (x)d x p

 1p 

∞

y

0

q(1−λ2 )−1 q

g (y)dy

 q1

,

(9)

0

where, the constant factor k(λ1 ) is the best possible. For λ = 1, λ1 = q1 , λ2 = 1p , (9) reduces to (3); for λ > 0, λ1 = λr , λ2 = λs , (9) reduces to (8). Also an extension of (4) was given as follows: 

∞



0

∞

h(x y) f (x)g(y)d xd y

0



∞

< φ(σ)

x

p(1−σ)−1

f (x)d x p

 1p 

∞

y

0

g (y)dy

q(1−σ)−1 q

 q1

,

(10)

0

where, the constant factor φ(σ) is the best possible (cf. [17]). For σ = 1p , (10) reduces to (4). Some equivalent inequalities of (9) and (10) were considered by [16]. In 2013, Yang [17] also studied the equivalency of (9) and (10) under a adding condition. In 2017, Hong [18] studied an equivalent condition between (9) and the parameters. Remark 1 (cf. [17]) If h(x y) = 0, for x y > 1, then 

1

φ(σ) =

h(u)u σ−1 du = φ1 (σ) ∈ R+ ,

0

and (10) reduces to the following first kind of Hardy-type integral inequality with the non-homogeneous kernel: 

∞

 g(y)

1 y

h(x y) f (x)d x dy

0

0

 < φ1 (σ) 0

∞

x p(1−σ)−1 f p (x)d x

 1p 

∞

y q(1−σ)−1 g q (y)dy

 q1

,

(11)

0

where, the constant factor φ1 (σ) is the best possible. If h(x y) = 0, for x y < 1, then

On Equivalent Properties of Hardy-Type Integral Inequality …

 φ(σ) =

∞

5

h(u)u σ−1 du = φ2 (σ) ∈ R+ ,

1

and (10) reduces to the following second kind of Hardy-type integral inequality with the nonhomogeneous kernel: 

∞



∞

g(y) 1 y

0



∞

< φ2 (σ)

x

h(x y) f (x)d x dy f (x)d x

p(1−σ)−1

p

 1p 

∞

y

0

g (y)dy

q(1−σ)−1 q

 q1

,

(12)

0

where, the constant factor φ2 (σ) is the best possible In this paper, by the use of the way of real analysis and the weight functions, we consider a few equivalent conditions of (11), (12) related to another Hardy-type inequality, as well as the parameters ς, σ and the integrals of h(u). As applications, some equivalent conditions of Hardy-type integral inequalities with the homogeneous kernel, as well as the parameters μ, σ, λ and the integrals of kλ (u, 1) are deduced. We also consider the other kind of integral inequality, the operator expressions, some corollaries and a few particular examples.

2 Two Lemmas In what follows, we assume that p > 1, 1p + measurable function in (0, ∞), such that 

1



∞

k1 (σ) : = 0

k2 (σ) : =

1 q

= 1, ς, σ ∈ R, h(u) is a nonnegative

h(u)u σ−1 du (≥ 0), h(u)u σ−1 du (≥ 0).

(13)

1

Lemma 1 If k1 (σ) > 0, there exists a constant M1 , such that for any nonnegative measurable functions f (x) and g(y) in (0, ∞), the following inequality 

∞

 g(y)

1 y

h(x y) f (x)d x dy

0

0



∞

≤ M1

x 0

p(1−σ)−1

f (x)d x p

 1p 

∞

y

g (y)dy

q(1−ς)−1 q

 q1 (14)

0

holds true, then we have ς = σ and k1 (σ) ≤ M1 < ∞. Proof Since k1 (σ) > 0, it follows that h(u) > 0 a.e. in an interval (a, b)(⊂ (0, 1)).

6

B. Yang

If ς > σ, then for n ≥

(n ∈ N), we set the following two functions:

1 ς−σ



x σ+ pn −1 , 0 < x ≤ 1 , 0, x > 1 1

f n (x) : =

0, 0 < y < 1 1 . y ς− qn −1 , y ≥ 1

gn (y) : = We find 

∞

J1 := 0



1

=

x p(1−σ)−1 f np (x)d x

1

=

1

x n −1 d x 1

y q(1−ς)−1 gnq (y)dy

 1p 

∞

 q1

y q(1−ς)−1 y q(ς− qn −1) dy 1

 q1

1

 1p 

0

∞ 0

x p(1−σ)−1 x p(σ+ pn −1) d x

0



 1p 

∞

y − n −1 dy 1

 q1

= n.

1

Setting u = x y, we obtain 

∞

I1 := 0



∞

=



1 y

gn (y)

h(x y) f n (x)d x dy

0



1 y

h(x y)x 1



0 ∞

=

1 σ+ pn −1

y (ς−σ)− n −1 dy 1



1

1

dx

y ς− qn −1 dy 1

h(u)u σ+ pn −1 du, 1

0

and then by (14) we have 

∞

y (ς−σ)− n −1 dy 1

1



1

h(u)u σ+ pn −1 du 1

0

= I1 ≤ M1 J1 = M1 n < ∞. 1 ≥ 0, n 1 σ+ pn −1

Since (ς − σ) −

it follows that

∞ 1

(15)

y (ς−σ)− n −1 dy = ∞. By (15), in 1

> 0 a.e. in an interval (a, b) ⊂ (0, 1), we find that view of h(u)u 1 1 σ+ pn −1 h(u)u du > 0 and then ∞ ≤ M1 n < ∞, which is a contradiction. 0 1 (n ∈ N), we set the following two functions: If ς < σ, then for n ≥ σ−ς  f n (x) : =



0, 0 < x < 1 1 , x σ− pn −1 , x ≥ 1

On Equivalent Properties of Hardy-Type Integral Inequality …

 gn (y) : =

7

y ς+ qn −1 , 0 < y ≤ 1 , 0, y > 1 1

and find J1 :=



∞

x

 f np (x)d x

p(1−σ)−1

0



∞

=

x

=

∞

x

x

− n1 −1

 1p 

y q(1−ς)−1 gnq (y)dy

 1p 

1

dx

y

 q1

1 −1) q(1−ς)−1 q(ς+ qn

y

 q1 dy

0

1

dx

1 n −1

y

1

∞ 0

1 −1) p(1−σ)−1 p(σ− pn

1



 1p 

 q1 dy

= n.

0

Setting u = x y, we obtain 

 I1 :=

∞

0



∞

=

 f n (x)



1 x

h(x y) gn (y)dy d x

0





1 x

h(x y)y 0

1



∞

=

1 ς+ qn −1

x

(σ−ς)− n1 −1



1

dx

1

dy x σ− pn −1 d x 1

h(u)u ς+ qn −1 du, 1

0

and then by Fubini theorem (cf. [19]) and (14), we have 

∞

x 1

= I1 =

(σ−ς)− n1 −1



∞

 dx 

 gn (y)

0

1 0 1 y

h(u)u ς+ qn −1 du h(x y)  f n (x)d x dy 1

0

≤ M1 J1 = M1 n. 1 ≥ 0, n 1 ς+ qn −1

Since (σ − ς) −

(16)

it follows that

∞ 1

x (σ−ς)− n −1 d x = ∞. By (16), in 1

> 0 a.e. in and interval (a, b) ⊂ (0, 1), we find that view of h(u)u 1 1 ς+ qn −1 du > 0, and then ∞ ≤ M1 n < ∞, which is a contradiction. 0 h(u)u Hence, we conclude that ς = σ. For ς = σ, we reduce (16) as follows: 

1

h(u)u σ+ qn −1 du ≤ M1 . 1

(17)

0

Since {h(u)u σ+ qn −1 }∞ n=1 is nonnegative and increasing in (0, 1], by Levi theorem (cf. [19]), we have 1

8

B. Yang

 k1 (σ) =

1

lim h(u)u σ+ qn −1 du 1

0 n→∞  1

h(u)u σ+ qn −1 du ≤ M1 < ∞. 1

= lim

n→∞ 0



The lemma is proved.

Lemma 2 If k2 (σ) > 0, there exists a constant M2 , such that for any non-negative measurable functions f (x) and g(y) in (0, ∞), the following inequality 

∞



∞

g(y) 1 y

0



∞

≤ M2

x

h(x y) f (x)d x dy

p(1−σ)−1

f (x)d x p

 1p 

∞

y

0

g (y)dy

q(1−ς)−1 q

 q1 (18)

0

holds true, then we have ς = σ and k2 (σ) ≤ M2 < ∞. Proof Since k2 (σ) > 0, it follows that h(u) > 0 a.e. in an interval (c, d)(⊂ (1, ∞)). 1 (n ∈ N), we set two functions  f n (x) and  gn (y) as in If ς < σ, then for n ≥ σ−ς Lemma 1. We find J1 =



∞

x

p(1−σ)−1

0

 f np (x)d x

 1p 

∞ 0

y q(1−ς)−1 gnq (y)dy

 q1

= n.

Setting u = x y, we obtain  I2 :=



∞

0



1

=

  gn (y)



∞ 1 y

 h(x y) f n (x)d x dy

∞

h(x y)x 1 y

0



1

=

y (ς−σ)+ n −1 dy 1

1 σ− pn −1



0

∞

dx

y ς+ qn −1 dy 1

h(u)u σ− pn −1 du, 1

1

and then by (18) we have 

1

y (ς−σ)+ n −1 dy 1



0

∞

h(u)u σ− pn −1 du 1

1

= I2 ≤ M2 J1 = M2 n < ∞. 1 ≤ 0, n 1 σ− pn −1

Since (ς − σ) +

it follows that

1 0

(19)

y (ς−σ)+ n −1 dy = ∞. By (19), in 1

> 0 a.e. in an interval (c, d) ⊂ [1, ∞), we find that view of h(u)u ∞ 1 σ− pn −1 du > 0, and then ∞ ≤ M2 n < ∞, which is a contradiction. 1 h(u)u

On Equivalent Properties of Hardy-Type Integral Inequality …

If ς > σ, then for n ≥ Lemma 1. We find 

∞

J1 =

x

9

(n ∈ N), we set two functions f n (x) and gn (y) as in

1 ς−σ

f np (x)d x

p(1−σ)−1

0

 1p 

∞ 0

y q(1−ς)−1 gnq (y)dy

 q1

= n.

Setting u = x y, we obtain 



∞

I2 := 0





1

=

f n (x)



∞

h(x y)gn (y)dy d x

1 x



∞

h(x y)y

1 ς− qn −1

1 x

0



1

=

x

(σ−ς)+ n1 −1



∞

dx

0

dy x σ+ pn −1 d x 1

h(u)u ς− qn −1 du, 1

1

and then by Fubini theorem (cf. [19]) and (18), we have 

1 0

 ∞ 1 1 x (σ−ς)+ n −1 d x h(u)u ς− qn −1 du  1  ∞

= I2 =

∞

gn (y)

1 y

0

h(x y) f n (x)d x dy

≤ M2 J1 = M2 n < ∞.

(20)

1

x (σ−ς)+ n −1 d x = ∞. By (20 ), in view of ∞ 1 1 h(u)u ς− qn −1 > 0 a.e. in interval (c, d) ⊂ (1, ∞), we find that 1 h(u)u ς− qn −1 du > 0, and then ∞ ≤ M2 n < ∞, which is a contradiction. Hence, we conclude the fact that ς = σ. For ς = σ, we reduce (20) as follows:

Since (σ − ς) +

1 n

≤ 0, it follows that



∞

1

0

h(u)u σ− qn −1 du ≤ M2 . 1

(21)

1

Since {h(u)u σ− qn −1 }∞ n=1 is nonnegative and increasing in [1, ∞), still by Levi theorem (cf. [19]), we have 1



∞

lim h(u)u σ− qn −1 du 1 n→∞  ∞ 1 = lim h(u)u σ− qn −1 du ≤ M2 < ∞.

k2 (σ) =

1

n→∞ 1

The lemma is proved.



10

B. Yang

3 Main Results and Applications Theorem 1 If k1 (σ) > 0, then the following statements are equivalent: (i) There exists a constant M1 , such that for any f (x) ≥ 0, satisfying 

∞

0
(i). Setting u = x y, we obtain the following weight function:  1 y ω1 (σ, y) := y σ h(x y)x σ−1 d x 0  1 h(u)u σ−1 du = k1 (σ)(y > 0). =

(25)

0

By Hölder’s inequality with weight and (25), for y ∈ (0, ∞), we have  

p h(x y) f (x)d x

0 1 y

= 0



1 y



y (σ−1)/ p h(x y) (σ−1)/q f (x) x

≤

h(x y) 0



f (x)d x

 q(1−σ)−1 p−1

= (k1 (σ)) p−1 y − pσ+1





h(x y) 0

h(x y) 0

h(x y) 0

y (σ−1)q/ p

y σ−1

1 y

1 y

x σ−1

1 y

p

x (σ−1) p/q

= ω1 (σ, y)y

 p x (σ−1)/q dx y (σ−1)/ p



y σ−1

1 y



x (σ−1) p/q

y σ−1 x (σ−1) p/q

p−1 dx

f p (x)d x

f p (x)d x.

(26)

If (26) takes the form of equality for a y ∈ (0, ∞), then (cf. [20]), there exists constants A and B, such that they are not all zero, and A

y σ−1 x (σ−1) p/q

f p (x) = B

x σ−1 y (σ−1)q/ p

a.e. in R+ .

We suppose that A = 0 (otherwise B = A = 0). It follows that x p(1−σ)−1 f p (x) = y q(1−σ) which contradicts the fact that 0 < the form of strict inequality.

∞ 0

B a.e. in R+ , Ax

x p(1−σ)−1 f p (x)d x < ∞. Hence, (26) takes

12

B. Yang

For ς = σ, by Fubini theorem (cf. [19]), we have J < (k1 (σ))

1 q



∞



h(x y) 0

= (k1 (σ))

1 q



0 ∞



h(x y)

= (k1 (σ))



∞

ω1 (σ, x)x

∞

= k1 (σ)

x

p(1−σ)−1

x (σ−1)( p−1)

f (x)d x p

f (x)d x dy

y σ−1

p(1−σ)−1

 1p

 1p

p

x (σ−1) p/q

0



y σ−1

1 x

0

0 1 q

1 y

f (x)d x

dy

f (x)d x p

 1p p

 1p

.

0

Setting M1 ≥ k1 (σ), then for k1 (σ) < ∞, (22) follows. Therefore, statements (i), (ii) and (iii) are equivalent. When statement (iii) holds true, if there exists a constant M1 < k1 (σ), such that (23) is valid, then we still have k1 (σ) ≤ M1 . This contradiction follows that the constant factor M1 = k1 (σ) in (23) is the best possible. The constant factor M1 = k1 (σ)(∈ R+ ) in (22) is still the best possible. Otherwise, by (24) (for ς = σ), we can conclude that the constant factor M1 = k1 (σ) in (23) is not the best possible.  Remark 2 If k1 (σ) = 0, then h(u) = 0 a.e. in (0, 1], we still can show that statement (i) is equivalent to statement (ii), but statement (ii) does not deduce to ς = σ in statement (iii). In particular, for σ = ς =

1 p

in Theorem 1, we have

Corollary 1 If k1 ( 1p ) > 0, then the following statements are equivalent: (i) There exists a constant M1 , such that for any f (x) ≥ 0, satisfying  0
0, we derive      φ(2q+ p a) φ(2 p a)   1  φ(2q · 2 p a) p     − φ(2 a)  2( p+q) − 2 p  = 2 p   2q 1  ϑ(2k+ p a) ≤ 2 p=0 2k+ p q−1

∞

≤

1  ϑ(2k+ p a) 2 p=0 2k+ p

→ 0 as p → ∞  p for all a ∈ X . Hence the sequence φ(22 p a) is a Cauchy sequence. Since Y is complete, there exists a mapping A : X −→ Y such that φ(2 p a) , for all a ∈ X . p→∞ 2p

A(a) = lim

Now, letting p → ∞ in (35), we see that (30) holds for all a ∈ X . We, claim that A satisfies (1). In fact, replacing (a, b) by (2 p a, 2 p b) and dividing by 2 p in (29), we obtain  1   Dφ (2 p a, 2 p b) ≤ 1 υ(2 p a, 2 p b) 2p 2p for all a, b ∈ X . Letting p → ∞ in the above inequality and using the definition of A implies that DA (a, b) = 0. Hence A satisfies (1) for all a, b ∈ X . To show A is  unique, let A be another complex valued additive mapping satisfying (1) and (30), then    1        A(a) − A (a) = p A(2 p a) − A (2 p a) 2    1     ≤ p A(2 p a) − φ(2 p a) + φ(2 p a) − A (2 p a) 2 ∞ 1  ϑ(2k+ p a) ≤ 2 k=0 2k+ p → 0 as p → ∞ 

for all a ∈ X . Hence A = A . Similarly, the theorem can be proved for the case  = −1 and hence we omit it.  The following corollary is an direct outcome of Theorem 5 concerning the stability of complex valued functional equation (1).

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

37

Corollary 1 Let μ and α be nonnegative real numbers. Let an odd function φ : X −→ Y with φ(ia) = iφ(a) satisfies the inequality    Dφ (a, b) ≤

⎧ μ, ⎪ ⎪ ⎨ μ (aα + bα ) ,

α = 1; α α a b , α = 21 ; μ ⎪ ⎪   ⎩  μ aα bα + a2α + b2α , α = 21 ;

(36)

for all a, b ∈ X . Then there persists an inimitable complex valued additive function A : X −→ Y such that ⎧ μ, ⎪ ⎪ ⎪ μ [1 + (−i)α ] ⎪ ⎪ aα , ⎪ ⎪ ⎨ 2|1 − 2α−1 | μ(−i)α φ(a) − A(a) ≤ a2α , ⎪ ⎪ 2|1 − 22α−1 | ⎪   ⎪ ⎪ ⎪ μ 1 + (−i)α + (−i)2α ⎪ ⎩ a2α 2|1 − 22α−1 |

(37)

for all a ∈ X .

4 Stability of the Quadratic Functional Equation (2) In this section, the generalized Ulam–Hyers stability of the quadratic functional equation (2) is investigated in the framework of complex Banach spaces using direct method. For the sake of easy manipulation, let us define a mapping Dq : X × X −→ Y by Dq (a, b) = q(2a + ib) + q(2a − ib) − q(a + ib) − q(a − ib) − 6q(a) for all a, b ∈ X . Theorem 6 Let  ∈ {−1, 1} and υ, ϑ : X 2 → [0, ∞) be a function such that   υ 3k a, 3k b lim =0 k→∞ 9k

(38)

for all a, b ∈ X . Let q : X −→ Y be a function satisfying the inequality    Dq (a, b) ≤ υ(a, b)

(39)

for all a, b ∈ X . Then there exists a unique complex valued quadratic mapping Q : X −→ Y which satisfies (2) and

38

B. V. Senthil Kumar and H. Dutta

q(a) − Q(a) ≤

∞ 1  ϑ(3 p a) 9 1− 9 p k=

(40)

2

where ϑ(3 p a) and Q(a) are defined by the following two formulae, respectively: 1 ϑ(3 p a) = υ(3 p a, −i3 p a) + υ(3 p a, 0) 2 and

q(3k a) k→∞ 9k

Q(a) = lim

(41)

(42)

for all a ∈ X . Proof Let us assume  = 1. Replacing (a, b) by (a, −ia) in (39), we get q(3a) − q(2a) − 5q(a) ≤ υ(a, −ia)

(43)

for all a ∈ X . Again, replacing (a, b) by (a, 0) in (39), we obtain q(2a) − 4q(a) ≤

1 υ(a, 0) 2

(44)

for all a ∈ X . It follows from (43), (44) and triangle inequality that q(3a) − 9q(a) ≤ q(3a) − q(2a) − 5q(a) + q(2a) − 4q(a) 1 ≤ υ(a, −ia) + υ(a, 0) 2

(45)

for all a ∈ X . Dividing the above inequality by 9, we obtain    ϑ(a)  q(3a)    9 − q(a) ≤ 9 where

(46)

1 ϑ(a) = υ(a, −ia) + υ(a, 0) 2

for all a ∈ X . Now, replacing a by 3a and dividing by 9 in (46), we get    q(32 a) q(3a)  ϑ(3a)    92 − 9  ≤ 9 · 9 for all a ∈ X . From (47) and (46), we obtain

(47)

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

        q(32 a) q(3a)   q(32 a)   q(3a)        92 − q(a) ≤  9 − q(a) +  92 − 9    1 ϑ(3a) ≤ ϑ(a) + 9 9

39

(48)

for all a ∈ X . Proceeding further and using induction on a positive integer k, we get   k−1  q(3k a)  1 ϑ(3 p a)  ≤ − q(a)  9k  9 9p p=0 ∞

≤

1  ϑ(3 p a) 9 p=0 9 p

(49)

 p for all a ∈ X . In order to prove the convergence of the sequence q(39 p a) , replacing a by 3 p a and then dividing by 9 p in (49), for any p, q > 0, and thus we deduce      q(3q+ p a) q(3 p a)   1  q(3q · 3 p a) p     − q(3 a)  9 p+q − 9 p  = 9 p  9q 1  ϑ(3k+ p a) ≤ 9 p=0 9k+ p q−1

∞

1  ϑ(3k+ p a) ≤ 9 p=0 9k+ p → 0 as p → ∞  p for all a ∈ X . Hence the sequence q(39 p a) is a Cauchy sequence. Since Y is complete, there exists a mapping Q : X −→ Y such that q(3 p a) , for all a ∈ X . p→∞ 9p

Q(a) = lim

Now, letting k → ∞ in (49), we see that (40) holds for all a ∈ X . We now claim that Q satisfies (2). In fact, replacing (a, b) by (3 p a, 3 p b) and dividing by 9 p in (39), we obtain  1 1  p p  (3 a, 3 b) D  ≤ p υ(3 p a, 3 p b) q 9p 9 for all a, b ∈ X . Letting p → ∞ in the above inequality and using the definition of Q implies that DQ (a, b) = 0. Hence Q satisfies (2) for all a, b ∈ X . To show Q is  unique, let Q be another quadratic mapping satisfying (2) and (40), then

40

B. V. Senthil Kumar and H. Dutta

   1        Q(a) − Q (a) = p Q(3 p a) − Q (3 p a) 9    1     ≤ p Q(3 p a) − q(3 p a) + q(3 p a) − Q (3 p a) 9 ∞ 1  ϑ(3k+ p a) ≤ 9 k=0 9k+ p → 0 as p → ∞ 

for all a ∈ X . Hence Q = Q . Similar proof can be given for the case  = −1 and hence we omit the proof.  The following corollary is an immediate consequence of Theorem 6 concerning the stability of Eq. (2). Corollary 2 Suppose μ and α be nonnegative real numbers. Let an even function q : X −→ Y satisfies the inequality ⎧ ⎪ ⎪ μ,   ⎨ μ (aα + bα ) , α = 2;  Dq (a, b) ≤ α α b a , 2α = 2; μ ⎪ ⎪    ⎩ μ aα bα + a2α + b2α , 2α = 2;

(50)

for all a, b ∈ X . Then one can acquire a unique complex valued quadratic function Q : X −→ Y such that ⎧ 3μ ⎪ ⎪ , ⎪ ⎪ 16 ⎪ ⎪ 5μ ⎪ ⎪ ⎨ aα , 2|9 − 3α | q(a) − Q(a) ≤ μ 2α ⎪ ⎪ ⎪ |9 − 32α | a , ⎪ ⎪ ⎪ ⎪ 7μ ⎪ ⎩ a2α 2|9 − 32α |

(51)

for all a ∈ X .

5 Stability of the Cubic Functional Equation (3) In this section, we prove stability results are valid for the cubic functional equation (3) in complex Banach spaces using direct method. For the purpose of proving stability results in simple manner, let us demarcate a mapping Dc : X × X −→ Y by Dc (a, b) = c(2a + ib) + c(2a − ib) − 2c(a + ib) − 2c(a − ib) − 12c(a) for all a, b ∈ X .

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

41

Theorem 7 Let  ∈ {−1, 1} and υ, ϑ : X 2 → [0, ∞) be a function such that   υ 2k a, 2k b lim =0 k→∞ 8k

(52)

for all a, b ∈ X . Let c : X −→ Y be a function satisfying the inequality Dc (a, b) ≤ υ(a, b)

(53)

for all a, b ∈ X . Then there exists a unique complex valued cubic mapping C : X −→ Y which satisfies (3) and c(a) − C(a) ≤

∞ 1  ϑ(2 p a) 16 1− 8 p k=

(54)

2

where ϑ(2 p a) and C(a) are defined by the following two formulae, respectively:

and

ϑ(2 p a) = υ(2 p a, 0)

(55)

c(2k a) k→∞ 8k

(56)

C(a) = lim for all a ∈ X .

Proof Let us assume  = 1. Replacing (a, b) by (a, 0) in (53), we get     c(2a) 1    8 − c(a) ≤ 16 υ(a, 0)

(57)

for all a ∈ X . Now, replacing a by 2a in (57), dividing by 8 and then summing the resultant with (57), we obtain  2   c(2 a)  1 ϑ(a)    82 − c(a) ≤ 16 8 where ϑ(a) = υ(a, 0) for all a ∈ X . From (58) and (57), we obtain

(58)

42

B. V. Senthil Kumar and H. Dutta

  2   2     c(2 a) c(2a)   c(2 a)   c(2a)        82 − c(a) ≤  8 − c(a) +  82 − 8    1 ϑ(2a) ≤ ϑ(a) + 8 8

(59)

for all a ∈ X . Proceeding further and using induction on a positive integer k, we get  k  k−1  c(2 a)  1 ϑ(2 p a)  ≤ − c(a)  8k  8 8p p=0 ∞

≤

1  ϑ(2 p a) 8 p=0 8 p

(60)

 p for all a ∈ X . In order to prove the convergence of the sequence c(28 p a) , replacing a by 2 p a and then divide by 8 p in (60), for any p, q > 0, and thus we deduce  q+ p   q p   c(2 a) c(2 p a)   1  c(3 · 3 a) p     − c(3 a)  8 p+q − 8 p  = 8 p  8q 1  ϑ(2k+ p a) ≤ 8 p=0 8k+ p q−1

∞

1  ϑ(2k+ p a) ≤ 8 p=0 8k+ p → 0 as p → ∞  p for all a ∈ X . Hence the sequence c(28 p a) is a Cauchy sequence. Since Y is complete, there exists a mapping C : X −→ Y such that c(2 p a) , for all a ∈ X . p→∞ 8p

C(a) = lim

Now, letting k → ∞ in (60), we see that (54) holds for all a ∈ X . We, claim that Q satisfies (3). Now, replacing (a, b) by (3 p a, 3 p b) and dividing by 9 p in (53), we obtain  1 1  p p  (2 a, 2 b) D  ≤ p υ(2 p a, 2 p b) c 8p 8 for all a, b ∈ X . Letting p → ∞ in the above inequality and using the definition of C implies that DC (a, b) = 0. Hence C satisfies (3) for all a, b ∈ X . To show that C  is unique, let C be another cubic mapping satisfying (3) and (54), then

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

43

   1        C(a) − C (a) = p C(2 p a) − C (2 p a) 8    1     ≤ p C(2 p a) − c(2 p a) + c(3 p a) − C (2 p a) 8 ∞ 1  ϑ(2k+ p a) ≤ 8 k=0 8k+ p → 0 as p → ∞ 

for all a ∈ X . Hence C = C . Here, we leave out the proof for the case  = −1 since it is similar to the case  = 1.  Using the result of Theorem 7, we can prove the following corollary related to the stability of Eq. (3). Corollary 3 Assume μ and α to be nonnegative real numbers. Let an odd function c : X −→ Y satisfies the inequality  Dc (a, b) ≤

μ, μ (aα + bα ) , α = 3;

(61)

for all a, b ∈ X . Then there exists a unique complex valued cubic function C : X −→ Y such that ⎧μ ⎨ , c(a) − C(a) ≤ 7 μ (62) ⎩ aα , α−3 8|1 − 2 | for all a ∈ X .

6 Stability of Quartic Functional Equation (4) In this section, we accomplish the generalized Ulam–Hyers stability of the quartic functional equation (4) in the setting of complex Banach spaces using direct method. For the sake of convenience, let us delineate the difference operator D Q : X × X −→ Y by D Q (a, b) = q(2a + ib) + q(2a − ib) − q(a + ib) − q(a − ib) − 6q(a) for all a, b ∈ X . Theorem 8 Let  ∈ {−1, 1} and υ, ϑ : X 2 → [0, ∞) be a function such that   υ 3 p a, 3 p b lim =0 p→∞ 81 p

(63)

44

B. V. Senthil Kumar and H. Dutta

for all a, b ∈ X . Let Q : X −→ Y be a function satisfying the inequality    D Q (a, b) ≤ υ(a, b)

(64)

for all a, b ∈ X . Then it is possible to obtain a unique complex valued quartic mapping Q : X −→ Y which satisfies (4) and Q(a) − Q(a) ≤

∞ 1  ϑ(3k a) 81 1− 81k k=

(65)

2

where ϑ(3k a) and Q(a) are defined by the following two formulae, respectively: ϑ(3k a) = υ(3k a, −i3k a) + 2υ(3k a, 0) and Q(a) = lim

p→∞

Q(3 p a) 81 p

(66)

(67)

for all a ∈ X . Proof Let us presume  = 1. Plugging (a, b) into (a, −ia) in (64), we get Q(3a) − 4Q(2a) − 17Q(a) ≤ υ(a, −ia)

(68)

for all a ∈ X . Now, replacing (a, b) by (a, 0) in (64),we obtain 2Q(2a) − 32Q(a) ≤ υ(a, 0)

(69)

for all a ∈ X . It follows from (68) and (69) and applying triangle inequality that Q(3a) − 9Q(a) ≤ Q(3a) − 4Q(2a) − 17Q(a) + 2 2Q(2a) − 32Q(a) ≤ υ(a, −ia) + 2υ(a, 0)

(70)

for all a ∈ X . Dividing the above inequality by 81, we obtain    ϑ(a)  Q(3a)    81 − Q(a) ≤ 81

(71)

where ϑ(a) = υ(a, −ia) + 2υ(a, 0) for all a ∈ X . Now, replacing a by 3a and dividing by 81 in (71), we get    Q(32 a) ϑ(3a) Q(3a)     812 − 81  ≤ 81 · 81

(72)

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

45

for all a ∈ X . From (71) and (72), we obtain         Q(32 a)  Q(32 a)   Q(3a) Q(3a)         812 − Q(a) ≤  81 − Q(a) +  812 − 81    1 ϑ(3a) ≤ ϑ(a) + 81 81

(73)

for all a ∈ X . Proceeding further and using induction on a positive integer p, we get   p−1    Q(3 p a) ϑ(3k a) ≤ 1  − Q(a)  81  81 p 81k k=0 ∞

≤

1  ϑ(3k a) 81 k=0 81k

(74)

 p a) , replacing for all a ∈ X . In order to prove the convergence of the sequence Q(3 81 p q q a by 3 a and then dividing by 81 in (74), for any p, q > 0, we find     p q  Q(3q+ p a)   f (3q a)    = 1  Q(3 · 3 a) − Q(3q a) −  81(q+ p)    q q p 81 81 81 ≤

1  ϑ(3k+q a) 81 k=0 81k+q

≤

1  ϑ(3k+q a) 81 k=0 81k+q

p−1

∞

→ 0 as q → ∞  p a) is a Cauchy sequence. Since Y is comfor all a ∈ X . Hence the sequence Q(3 81 p plete, there exists a mapping Q : X −→ Y such that Q(a) = lim

p→∞

Q(3 p a) , for all a ∈ X . 9p

Letting k → ∞ in (74), we see that (65) holds for all a ∈ X . Claim that Q satisfies (4). In fact, replacing (a, b) by (3 p a, 3 p b) and dividing by 81 p in (64), we obtain  1   D Q (3 p a, 3 p b) ≤ 1 υ(3 p a, 3 p b) p 81 81 p for all a, b ∈ X . Letting p → ∞ in the above inequality and using the definition of Q, we see that DQ (a, b) = 0. Hence Q satisfies (4) for all a, b ∈ X . To show Q is  unique, let Q be another quartic mapping satisfying (4) and (65), then

46

B. V. Senthil Kumar and H. Dutta

   1        Q(a) − Q (a) = p Q(3 p a) − Q (3 p a) 81    1     ≤ p Q(3 p a) − Q(3 p a) + Q(3 p a) − Q (3 p a) 81 ∞ 1  ϑ(3k+ p a) ≤ 81 k=0 81(k+ p) → 0 as p → ∞ 

for all a ∈ X . Hence Q = Q is unique. Similar result can be obtained for the case  = −1.  The following corollary is an immediate consequence of Theorem 8 concerning the stability of Eq. (4). Corollary 4 Let μ and α be nonnegative real numbers. Let a function Q : X −→ Y satisfies the inequality ⎧ ⎪ ⎪ μ,   ⎨ μ {aα + bα } , α = 4;  D Q (a, b) ≤ α α b a μ , 2α = 4; ⎪ ⎪    ⎩ μ aα bα + a2α + b2α , 2α = 4;

(75)

for all a, b ∈ X . Then there exists a unique complex valued quartic function Q : X −→ Y such that ⎧ 3μ ⎪ ⎪ , ⎪ ⎪ 80 ⎪ ⎪ 4μ ⎪ ⎪ aα , ⎨ |81 − 3α | Q(a) − Q(a) ≤ 9μ ⎪ ⎪ a2α , ⎪ 2α | ⎪ |81 − 3 ⎪ ⎪ ⎪ 4μ ⎪ ⎩ a2α |81 − 32α |

(76)

for all a ∈ X .

7 Stability of Functional Equations (1), (2), (3) and (4) Using Fixed Point Method In this section, we prove the generalized Ulam–Hyers stability of complex valued additive, quadratic, cubic and quartic functional equations (1), (2), (3) and (4) in complex Banach spaces using fixed point method. Let us recall some fundamental results in fixed point theory.

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

47

Theorem 9 (Banach’s Contraction Principle) Let (Y, d) be a complete metric space and consider a mapping J : Y −→ Y which is strictly contractive mapping, that is (A1 ) d(T a, T b) ≤ Ld(a, b) for some (Lipschitz constant) L < 1. Then, (i) The mapping J has one and only fixed point a ∗ = J (a ∗ ); (ii) The fixed point for each given element a ∗ is globally attractive, that is (A2 ) lim J n a = a ∗ , for any starting point a ∈ Y ; n→∞

(iii) One has the following estimation inequalities: (A3 ) d(J n a, a ∗ ) ≤ (A4 ) d(a, a ∗ ) ≤

1 d(J n a, 1−L

1 d(a, a ∗ ), 1−L

J n+1 a), for all n ≥ 0, for all a ∈ Y ;

for all a ∈ Y .

Theorem 10 ([13], The alternative of Fixed Point Theorem) Suppose that for a complete generalized metric space (A, d) and a strictly contractive mapping J : A −→ A with Lipschitz constant L. Then, for each given element a ∈ A, either d(J n a, J n+1 a) = ∞ for all n ≥ 0, (B1 ) or (B2 ) there exists a natural number n 0 such that: (i) d(J n a, J n+1 a) < ∞ for all n ≥ n 0 ; (ii) The sequence (J n a) is convergent to a fixed point b∗ of J (iii) b∗ is the unique fixed point of J in the set B = {b ∈ A : d(J n 0 a, b) < ∞}; 1 d(b, T b) for all b ∈ B. (iv) d(b∗ , b) ≤ 1−L In the sequel, we assume that U as a vector space and B as a complex Banach space respectively. Theorem 11 Let φ : U −→ B be an odd mapping with φ(ia) = iφ(a) for which there exists functions ζ, ϑ,  : U 2 → [0, ∞) with the condition p

lim

p

ζ(λ j a, λ j b) p

λj

p→∞

= 0,

(77)

where  λj =

2, j = 0, 1 , j =1 2

satisfying the functional inequality    Dφ (a, b) ≤ ζ(a, b) for all a, b ∈ U . If there exists an L = L( j) < 1 such that the function x → (a) = ϑ

a  2

,

(78)

48

B. V. Senthil Kumar and H. Dutta

has the property



a (a) = Lλ j  λj

 (79)

for all a ∈ U . Then there exists a unique complex valued additive function A : U −→ B satisfying the functional equation (1) and φ(a) − A(a) ≤

L 1− j (a) 1−L

(80)

holds for all a ∈ U . Proof Consider the set T = {g/g : U −→ B, g(0) = 0} and introduce a generalized metric on D, as follows: d(g, h) = inf{β ∈ (0, ∞) : g(a) − h(a) ≤ β(a), a ∈ U }. It can be easily shown that (T, d) is complete. Define J : T −→ T by J g(a) =

1 g(λ j a), for all a ∈ U . λj

Now p, q ∈ X , d(g, h) ≤ β =⇒ g(a) − h(a) ≤ β(a), a ∈ U    1 1 1  ≤ β(λ j a), a ∈ U =⇒  g(λ j a) − h(λ j a)  λj λj λj    1 1  =⇒   λ h(λ j a) − λ h(λ j a) ≤ Lβ(a), a ∈ U , j j =⇒ J g(a) − J h(a) ≤ Lβ(a), a ∈ U , =⇒d(J g, J h) ≤ Lβ. This implies d(J g, J h) ≤ Ld(g, h), for all g, h ∈ T That is, T is a strictly contractive mapping on T with Lipschitz constant L. From (78), we have    φ(2a)  ϑ(a)  ≤ − φ(a)  2  2 where ϑ(a) = ζ(a, −ia)

(81)

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

49

for all a ∈ U . Using (78) for the case j = 0, it reduces to    1  φ(2a) − φ(a) ≤ 1 (a)  2 2 for all a ∈ U , which implies d(J q, q) ≤ Now, substituting a =

a 2

1 = L = L 1−0 = L 1− j < ∞. 2

in (78), we get  a   a    ≤ϑ φ(a) − 2φ 2 2

for all a ∈ U . Using (78) for the case j = 1, it reduces to   a     ≤ (a) φ(a) − 2φ 2 for all a ∈ U , which implies d(q, J q) ≤ 1 = L 0 = L 1−1 = L 1− j < ∞. In the above cases, we arrive d(q, J q) ≤ L 1− j . Therefore (B2 (i)) holds. By (B2 (ii)), it follows that there exists a fixed point A of J in T such that p

A(a) = lim

p→∞

φ(λ j a) p λj

, for all a ∈ U .

(82)

  p p Claim that A : U −→ B is additive. Replacing (a, b) by λ j a, λ j b in (78) and p

dividing by λ j , it follows from (77) and (81), A satisfies (1) for all a, b ∈ U . By (B2 (iii)), A is a unique fixed point of J in the set U = {g ∈ T : d(J φ, A) < ∞}, using the fixed point alternative result A is the unique function such that φ(a) − A(a) ≤ β(a) for all a ∈ U and β > 0. Finally by (B2 (iv)), we obtain d(φ, A) ≤

1 d(φ, J φ) 1−L

50

B. V. Senthil Kumar and H. Dutta

implying d(φ, A) ≤

L 1− j . 1−L

Hence we conclude that φ(a) − A(a) ≤

L 1− j (a) 1−L

for all a ∈ U , which completes the proof.



The following corollary can be proved by using Theorem 11 associated with various stabilities of Eq. (1). Corollary 5 Let φ : U −→ B be a mapping satisfying φ(ia) = iφ(a). Then there exist real numbers θ and β such that ⎧ θ, ⎪ ⎪  ⎪    ⎨ θ aβ + bβ , β = 1;  Dφ (a, b) ≤ (83) β β b a θ , β = 21 ; ⎪ ⎪ ⎪    ⎩ θ aβ bβ + a2β + b2β , β = 21 for all a, b ∈ U . Then one can easily derive a unique complex valued additive function A : U −→ B such that ⎧ θ, ⎪  ⎪ ⎪ ⎪ θ 1 + (−i)β ⎪ ⎪ aβ , ⎪ β−1 | ⎪ 2|1 − 2 ⎪ ⎨ θ(−i)β φ(a) − A(a) ≤ ⎪ a2β , ⎪ ⎪ 2|1 − 22β−1 | ⎪ ⎪   ⎪ ⎪ ⎪ θ 1 + (−i)β + (−i)2β ⎪ ⎩ a2β 2|1 − 22β−1 |

(84)

for all a ∈ U . Theorem 12 Let q : U −→ B be a function. Suppose ζ, ϑ,  : U 2 → [0, ∞) are functions with the following condition p

lim

p

ζ(λ j a, λ j b)

p→∞

where

2p

λj 

λj =

= 0,

3, j = 0, 1 ,j=1 3

(85)

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

51

satisfying the functional inequality    Dq (a, b) ≤ ζ(a, b)

(86)

for all a, b ∈ U . If there exists an L = L( j) < 1 such that the function x → (a) = ϑ has the property

a  3

 (a) =

Lλ2j 

a λj

,

 (87)

for all a ∈ U . Then one can choose a distinctive complex valued quadratic function Q : U −→ B satisfying the functional equation (2) and q(a) − Q(a) ≤

L 1− j (a) 1−L

(88)

holds for all a ∈ U . Proof Consider the set T = {g/g : U −→ B, g(0) = 0} and introduce a generalized metric on D, as follows: d(g, h) = inf{β ∈ (0, ∞) : g(a) − h(a) ≤ β(a), a ∈ U }. It can be easily shown that (T, d) is complete. Define J : T −→ T by 1 J g(a) = 2 g(λ j a), for all a ∈ U . λj Now p, q ∈ X , d(g, h) ≤ β =⇒ g(a) − h(a) ≤ β(a), a ∈ U    1 1 1   =⇒  2 g(λ j a) − 2 h(λ j a) ≤ 2 β(λ j a), a ∈ U  λj λj λj   1  1   =⇒  2 h(λ j a) − 2 h(λ j a) ≤ Lβ(a), a ∈ U , λj  λj =⇒ J g(a) − J h(a) ≤ Lβ(a), a ∈ U , =⇒d(J g, J h) ≤ Lβ. This implies d(J g, J h) ≤ Ld(g, h), for all g, h ∈ T That is, T is a strictly contractive mapping on T with Lipschitz constant L.

52

B. V. Senthil Kumar and H. Dutta

From (86), we have    q(3a)  ϑ(a)  ≤ − q(a)  9  9 where

(89)

1 ϑ(a) = ζ(a, −ia) + ζ(a, 0) 2

for all a ∈ U . Using (89) for the case j = 0, it reduces to   1   q(3a) − q(a) ≤ 1 (a) 9  9 for all a ∈ U , which implies d(J q, q) ≤ Now, substituting a =

a 3

1 = L = L 1−0 = L 1− j < ∞. 9

in (86), we get  a   a    ≤ϑ q(a) − 9q 3 3

for all a ∈ U . Using (86) for the case j = 1, it reduces to   a    q(a) − 3q  ≤ (a) 3 for all a ∈ U , which implies d(q, J q) ≤ 1 = L 0 = L 1−1 = L 1− j < ∞. In the above cases, we arrive d(q, J q) ≤ L 1− j . Therefore (B2 (i)) holds. By (B2 (ii)), it follows that there exists a fixed point Q of J in T such that p

Q(a) = lim

p→∞

q(λ j a) 2p

λj

, for all a ∈ U .

(90)

  p p Claim that Q : U −→ B is quadratic. Replacing (a, b) by λ j a, λ j b in (86) 2p

and dividing by λ j , it follows from (85) and (86), Q satisfies (2) for all a, b ∈ U .

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

53

By (B2 (iii)), Q is a unique fixed point of J in the set U = {g ∈ T : d(J q, Q) < ∞}, using the fixed point alternative result Q is the unique function such that q(a) − Q(a) ≤ β(a) for all a ∈ U and β > 0. Finally by (B2 (iv)), we obtain d(q, Q) ≤

1 d(q, J q) 1−L

implying d(q, Q) ≤

L 1− j . 1−L

Hence we conclude that q(a) − Q(a) ≤

L 1− j (a) 1−L

for all a ∈ U , which completes the proof.



From Theorem 12, we obtain the following corollary concerning the stability of the Eq. (2). Corollary 6 Let q : U −→ B be a mapping satisfying q(ia) = −q(a). Then there exist real numbers θ and β such that ⎧ ⎪ ⎪ θ,    ⎨ θ aβ + bβ , β = 2;  Dq (a, b) ≤ β β b , ⎪ θ a ⎪  β = 1;  ⎩  θ aβ bβ + a2β + b2β , β = 1

(91)

for all a, b ∈ U . Then there persists an inimitable complex valued quadratic function Q : U −→ B such that ⎧ 3θ ⎪ ⎪ , ⎪ ⎪ ⎪ 16 ⎪ ⎪ 5θ ⎪ ⎪ aβ , ⎨ 2|9 − 3β | q(a) − Q(a) ≤ (92) θ 2β ⎪ ⎪ a , ⎪ ⎪ |9 − 32β | ⎪ ⎪ ⎪ 5θ ⎪ ⎪ a2β ⎩ 2|9 − 32β | for all a ∈ U .

54

B. V. Senthil Kumar and H. Dutta

Proof Let us define ⎧ θ, ⎪ ⎪ ⎨ aβ bβ  θ , + ζ(a, b) = β β b a θ , ⎪ ⎪   ⎩  θ aβ bβ + a2β + b2β for all a, b ∈ U . Now ⎧ θ ⎪ , ⎪ ⎪ ⎪ λ2j p ⎪ ⎪ ⎪  ⎪     ⎪ θ ⎪  p β  p β ⎪ ⎪ , a + b λ  λ ⎪ j j  p p ζ(λ j a, λ j b) ⎨ λ2j p =    2p ⎪ θ   p β  p β λj ⎪ ⎪ a b λ λ   , ⎪ j j ⎪ λ2j p ⎪ ⎪ ⎪ ⎪   ⎪        ⎪ θ ⎪  p β  p  β  p 2β  p 2β ⎪ ⎩ 2 λ j b λ j b + λ j a  + λ j b λj p ⎧ → 0 as p → ∞, ⎪ ⎪ ⎨ → 0 as p → ∞, = → 0 as p → ∞, ⎪ ⎪ ⎩ → 0 as p → ∞. That is, (85) holds. But, we have     a −ia 1 a  = ζ + ζ (a) = ϑ , ,0 . 3 3 3 2 3 a 

Hence ⎧ 3θ ⎪ ⎪ , ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 5θ ⎪   ⎪   aβ , ⎨ 1 a  a −ia 2.3β , + ζ ,0 = (a) = ζ ⎪ 3 3 2 6 ⎪ θ a2β , ⎪ ⎪ ⎪ 32β ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 7θ a2β . 2.32β

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

55

Also, ⎧ ⎧ 3θ 3θ ⎪ ⎪ ⎪ , ⎪ ⎪ λ−2 , 2 ⎪ ⎪ ⎧ −1 j ⎪ ⎪ λ · 2 2 ⎪ ⎪ λ j (a), ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ β  5θ ⎪ ⎪ ⎪ 5θ ⎪ ⎪ ⎪ β−2   β ⎪ ⎪ ⎪ a ⎨ λj ⎨ λ2 · 2 · 3β λ j a ⎨ λβ−1 (a) β 1 j 2 · 3 j . (λ j a) = = =   2 2β−1 θ ⎪ ⎪ ⎪ λj 2β−2 θ λ j a 2β (a) λ ⎪ ⎪ ⎪ 2β j ⎪ ⎪ ⎪ a λj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ2j · 32β 32β ⎪ ⎪ ⎩ 2β−1 ⎪ ⎪ ⎪ ⎪ λ j (a)   5θ ⎪ ⎪ 2β 7θ ⎪ ⎪   2β−2 2β ⎩ ⎪ λ a j ⎩ 2 a λ j λ j · 2 · 32β 2 · 32β Now, from (88), we prove the following conditions: Case: 1 L = 9−2 , if j = 0  −2 1−0 9 L 1− j 3θ 3θ q(a) − Q(a) ≤ (a) = = . · −2 1−L 1 − (9) 2 16 Case: 2 L = 32 , if j = 1 q(a) − Q(a) ≤

−3θ L 1− j (9)1−1 3θ (a) = · = . 1−L 1−9 2 16

Case: 1 L = 3β−2 , if j = 0 q(a) − Q(a)

 β−2 1−0 3 L 1− j 5θ 5θ 3β 5θ aβ β β a a ≤ (a) = . = = 1−L 1 − 3β−2 2 · 3β 9 − 3β 2 · 3β 2(9 − 3β )

Case: 2 L =

1 , 3β−2

if j = 1

q(a) − Q(a)

 1 1−1 5θ 5θ 3β 5θ aβ L 1− j β−2 β β a a (a) = 3 . = = ≤ 1 2 · 3β 1−L 3β − 9 2 · 3β 2(3β − 9) 1 − 3β−2

Case: 1 L = 32β−2 if j = 0 q(a) − Q(a)

 2β−2 1−0 3 L 1− j θ θ 32β θ a2β 2β 2β a a ≤ = = . (a) = 1−L 1 − 32β−2 32β 9 − 32β 32β 9 − 32β

56

B. V. Senthil Kumar and H. Dutta

Case: 2 L =

1 , 32β−2

if j = 1

q(a) − Q(a)

 1 1−1 θ θ L 1− j 32β θ a2β 2β−2 2β 2beta a a ≤ = = (a) = 3 . 1 1−L 32β 32β − 9 32β 32β − 9 1 − 32β−2

Case: 1 L = 32β−2 , if j = 0 q(a) − Q(a) ≤

L 1− j 1−L



(a) =

Case: 2 L =

1 , 32β−2

32β−2

1−0 7θ

1 − 32β−2 2 · 32β

a2β =

7θ 32β 7θ a2β a2β = . 2β 2β 9−3 2·3 2(9 − 32β )

if j = 1

q(a) − Q(a)

 1 1−1 7θ 32β 7θ 7θ a2β L 1− j 2β−2 a3β = 2β a2β = (a) = 3 . ≤ 1 2β 2β 1−L 3 −93 2(32β − 9) 1 − 32β−1 2 · 3 

Hence the proof.

In the subsequent theorem, we justify that the stability results of cubic functional equation (3) are legitimate using fixed point method. We also skip over the proof since it is akin to Theorem 7. Theorem 13 Let c : U −→ B be a mapping. Suppose ζ, ϑ,  : U 2 −→ [0, ∞) are functions with the following condition p

lim

p

ζ(λ j a, λ j b) 3p

λj

p→∞

= 0,

(93)

where  λj =

2, 1 , 2

j = 0, j=1

satisfying the functional inequality Dc (a, b) ≤ ζ(a, b) for all a, b ∈ U . If there exists an L = L( j) < 1 such that the function a → (a) = ϑ

a  2

,

(94)

Fundamental Stabilities of Various Forms of Complex Valued Functional Equations

has the property

 (a) =

Lλ3j 

a λj

57

 (95)

for all a ∈ U . Then there exists a unique complex valued cubic function C : U −→ B satisfying the functional equation (3) and c(a) − C (a) ≤

L 1− j (a) 1−L

(96)

holds for all a ∈ U . Corollary 7 Let c : U −→ B be a mapping satisfying c(ia) = −ic(a) then there exist real numbers θ and β such that  Dc (a, b) ≤

θ,  θ aβ + bβ , β = 3;

(97)

for all a, b ∈ U . Then there exists a unique complex valued cubic function C : U −→ B such that ⎧ θ ⎪ ⎨ , 7 c(a) − C (a) ≤ (98) θ β ⎪ ⎩ a , 8|1 − 2β−3 | for all a ∈ U . The following theorem contains the stability result of quartic functional equation (4) using fixed point method. Since the proof is similar to that of Theorem 12, we provide statement only. Theorem 14 Let Q : U −→ B be a mapping. Assume ζ, ϑ,  : U 2 −→ [0, ∞) are functions with the ensuing condition p

lim

p

ζ(λ j a, λ j b) 4p

λj

p→∞

= 0,

(99)

where  λj =

3, j = 0, 1 , j =1 3

satisfying the functional inequality    D Q (a, b) ≤ ζ(a, b)

(100)

58

B. V. Senthil Kumar and H. Dutta

for all a, b ∈ U . If there exists an L = L( j) < 1 such that the function a → (a) = ϑ has the property

a  3

 (a) =

Lλ2j 

a λj

,

 (101)

for all a ∈ U . Then there exists a unique complex valued quartic function Q : U −→ B satisfying the functional equation (4) and Q(a) − Q(a) ≤

L 1− j (a) 1−L

(102)

holds for all a ∈ U . Corollary 8 Let Q : U −→ B be a mapping satisfying Q(ia) = Q(a) then there exist real numbers θ and β such that    D Q (a, b) ⎧ θ, ⎪ ⎪  ⎨  θ aβ + bβ , β = 4; ≤ β β b a θ , ⎪ ⎪  β = 2;  ⎩  θ aβ bβ + a2β + b2β , β = 2;

(103)

for all a, b ∈ U . Then one can approximately estimate a distinctive complex valued quartic function Q : U −→ B such that ⎧ 3θ ⎪ ⎪ , ⎪ ⎪ ⎪ 80 ⎪ ⎪ 4θ ⎪ ⎪ aβ , ⎨ |81 − 3β | Q(a) − Q(a) ≤ 9θ ⎪ ⎪ a2β , ⎪ ⎪ |81 − 32β | ⎪ ⎪ ⎪ 4θ ⎪ ⎪ a2β ⎩ |81 − 32β |

(104)

for all a ∈ U .

References 1. Aczel, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989) 2. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

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3. Arunkumar, M.: Solution and stability of arun-additive functional equations. Int. J. Math. Sci. Engg. Appl. 4(3), 33–46 (2010) 4. Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge (2002) 5. Eskandani, G.Z., Gavrut ˇ a, ˇ P., Rassias, J.M., Zarghami, R.: Generalized Hyers–Ulam stability for a general mixed functional equation in quasi-β-normed spaces. Mediterr. J. Math. 8, 331– 348 (2011) 6. Gavrut ˇ a, ˇ P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) 7. Gavrut ˇ a, ˇ P.: An answer to a question of J.M. Rassias concerning the stability of Cauchy functional equation. Advances in Equations and Inequalities, Hadronic Mathematics Series, pp. 67–71 (1999) 8. Gavrut ˇ a, ˇ P.: On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 261, 543–553 (2001) 9. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941) 10. Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables. Birkhauser, Basel (1998) 11. Jung, S.M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001) 12. Lee, D.O.: Hyers–Ulam stability of an additive type functional equation. J. Appl. Math. Comput. 13(1–2), 471–477 (2003) 13. Margolis, B., Diaz, J.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968) 14. Mondal, P., Samanta, T.K.: Stability of a functional equation in complex Banach spaces. TWMS J. Appl. Eng. Math. 6(2), 307–314 (2016) 15. Rassias, J.M.: On approximately of approximately linear mappings by linear mappings. J. Funct. Anal. USA 46, 126–130 (1982) 16. Rassias, J.M.: On approximately of approximately linear mappings by linear mappings. Bull. Sc. Math. 108, 445–446 (1984) 17. Rassias, J.M., Kim, H.M.: Generalized Hyers–Ulam stability for general additive functional equations in quasi-β-normed spaces. J. Math. Anal. Appl. 356, 302–309 (2009) 18. Rassias, J.M., Jun, K.W., Kim, H.M.: Approximate (m, n)-Cauchy–Jensen additive mappings in C∗ -algebras. Acta Math. Sin. English Series 27(10), 1907–1922 (2011) 19. Rassias, J.M., Thandapani, E., Ravi, K., Senthil Kumar, B.V.: Functional Equations and Inequalities: Solution and Stability Results. World Scientific Publishing Company, Singapore (2017) 20. Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) 21. Ravi, K., Arunkumar, M.: On a n- dimensional additive functional equation with fixed point alternative. In: Proceedings of the International Conference on Mathematics and Science, Malaysia (2007) 22. Ravi, K., Arunkumar, M., Rassias, J.M.: On the Ulam stability for the orthogonally general Euler–Lagrange type functional equation. Int. J. Math. Sci. 3(08), 36–47 (2008) 23. Ravi, K., Rassias, J.M., Senthil Kumar, B.V.: Ulam stability of generalized reciprocal functional equation in several variables. Int. J. Appl. Math. Stat. 19(D10), 1–19 (2010) 24. Ravi, K., Rassias, J.M., Senthil Kumar, B.V.: Ulam stability of reciprocal difference and adjoint functional equations. Aust. J. Math. Anal. Appl. 8(1), 1–18 (2011). Art. 13 25. Ravi, K., Thandapani, E., Senthil Kumar, B.V.: Stability of reciprocal type functional equations. Pan-Am. Math. J. 21(1), 59–70 (2011) 26. Saadati, R., Sadeghi, G.: Nonlinear stability of a quadratic functional equation with complex involution. Arch. Math. (BRNO) 47, 111–117 (2011) 27. Ulam, S.M.: Problems in Modern Mathematics. Science Edn., Wiley, New York (1964). (Chapter VI, Some Questions in Analysis: 1, Stability)

Statistical Summability of Double Sequences by the Weighted Mean and Associated Approximation Results U˘gur Kadak

Abstract The goal of this chapter is to extend various summability concepts and summability techniques by the weighted mean method with respect to the generalized difference operator involving ( p, q)-Gamma function. We also obtain some inclusion relations between newly proposed methods and present some illustrative examples to show that these non-trivial generalizations are more powerfull than the existing literature on this topic. Furthermore, some approximation theorems and their weighted statistical forms for functions of two variables are proved. As application, related approximation results associated with the ( p, q)-analogue of generalized bivariate Bleimann-Butzer-Hahn operators are derived. Finally, we estimate the rate of convergence of approximating positive linear operators in terms of the modulus of continuity. Keywords Statistically weighted summability · Weighted statistical convergence of double sequences · ( p, q)-analogue of Gamma function · Double diference operator · Korovkin type approximations for functions of two variables · Bivariate Bleimann–Butzer–Hahn operators · The rate of convergence MSC (2010) Primary: 40B05 · 40G15 · 41A36; Secondary: 47B39 · 41A25

1 Introduction and Preliminaries The concept of infinity, which was sparked by the set theory of Cantor in the 1870s, plays a fundamental role in classical and modern mathematics. Cantor presented an infinite method to measure the size of infinite sets using the cardinality of the real number system. In the early decades of the nineteenth century, the infinite processes became an indispensable tool for development and research in science and engineering. The current way of teaching calculus involving the rigorous epsilon-delta ( − δ) U. Kadak (B) Institute of Natural And Applied Sciences, Gazi University, TR-06500 Ankara, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_3

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definitions of limits, continuity and differentiation is indeed the simplest approach to this general trend. At the end of the nineteenth century, many researchers focused their attention directly on alternative ways of understanding infinite series. These different ways of summation may be termed Summability Methods which have been used efficiently throughout analysis and applied mathematics. Further, the summability methods not only help us to generalize the limit of a series but also provide an alternative formulation of convergence of such series which is divergent in the usual sense. Based on such characteristic feature, sequence spaces can be considered as a powerful tool to obtain prominent results through summability methods. Due to the rapid development of sequence spaces many authors have focused on the concept of statistical convergence, which was introduced by Fast [16] in 1951 using the asymptotic density of a set of natural numbers. In particular, if K ⊆ N then the asymptotic density (or natural density) of K , denoted by δ(K ), is given by  1  {k ≤ n : k ∈ K }  n→∞ n

δ(K ) = lim

provided that the limit exists, where |B| denotes the cardinality of the set B. A real (or complex) valued sequence x = (xk ) is called statistically convergent (st−convergent) to the number L if   δ k ∈ N : |xk − L|   = 0 for every  > 0. In this case we write it st − lim x = L and call L the statistical limit of x. During the last decades, the concepts of statistical convergence and statistical summability have also been generalized, extended, and revisited in a variety of ways. For instance, Karakaya and Chishti [26] have introduced the notion of weighted statistical convergence for single sequences of real numbers and also Mursaleen et al. [40] applied this idea to Korovkin type approximation theorem. More recently, Kadak has introduced the weighted statistical convergence based on ( p, q)-integer in [20]. For further related results and generalizations of weighted statistical convergence, see, for instance [8, 9, 25, 30, 41, 46, 47]. Very recently, Aktu˘glu [3] has proposed the (α, β)-statistical convergence via two sequences {α(k)}k∈N and {β(k)}k∈N of positive numbers which satisy the conditions: α(k) and β(k) are both non-decreasing sequences, β(k)  α(k) for all k ∈ N and β(k) − α(k) → ∞ as k → ∞. In what follows, we denote by  for the set of pairs (α, β) satisfying the above conditions. A sequence x = (xn ) is said to be (α, β)statistically convergent to L if, for each ε > 0, lim

n→∞

  1  k ∈ [α(n), β(n)] : |xk − L|    = 0, (α, β) ∈ . β(n) − α(n) + 1

A double sequence is a real-valued function whose domain is the set N2 := N × N. The convergence for double sequences was introduced by Pringsheim [42]

Statistical Summability of Double Sequences by the Weighted Mean …

63

in 1900. A double sequence x = (x jk ) j,k∈N is said to converge to the number L, denoted by P − lim x = L, provided that given  > 0 there exists an integer N such that |x jk − L| <  whenever j, k ≥ N . We shall describe such a convergent double sequence x = (x jk ) more briefly as P-convergent and can easily verify that L is always unique. A double sequence x = (x jk ) of real or complex numbers is called bounded if x∞ = sup j,k |x jk | < ∞. We recall that a convergent double sequence may not be bounded in contrast to the case for single real sequences. The idea of statistical convergence for double sequences was introduced and studied by Mursaleen and Edely [33]. Let K (m, n) = {( j, k) ∈ N2 : j  m and k  n}. Then the two-dimensional analogue of natural density (shortly, double natural density) is given by δ2 (K ) = P − lim

m,n→∞

1 |K (m, n)|, mn

if it exists. A double sequence x = (xmn ) of real numbers is said to be statistically convergent to L in the Pringsheim’s sense, if for every  > 0,   δ2 ( j, k), j  m and k  n : |x jk − L|   = 0. In this case we write st2 − lim x = L and call L the double statistical limit of x in Pringsheim’s sense. Note that every P-convergent double sequence is statistically convergent to the same limit point but not conversely. More informations regarding the double sequences and the double series can be found in [4, 11, 34, 43, 48]. In recent years there have been significant developments in the theory of infinite series as well as further developments in the sequence spaces. The concept of difference sequence space, which was firstly introduced by Kızmaz [28], can be regarded as an extended version of the classical sequence spaces. Further this concept was examined and extended by several authors (see, for instance, [5, 7, 14, 23, 24, 37]). Recently, Baliarsingh [6] has introduced certain difference sequence spaces based on fractional difference operator involving Euler Gamma function. In the year 2017, Kadak [21] introduced weighted statistical convergence based on generalized difference operator involving ( p, q)-gamma function. Our goal is to extend the notions of statistical convergence and statistical summability of double sequences by the weighted mean method using a new generalization of double difference operator involving ( p, q)-gamma function. We establish a Korovkin type approximation theorem associated with the statistically weighted  (α,β) -summability which will improve the corresponding results in the existing N  literature. By using the ( p, q)-analogue of generalized bivariate Bleimann–Butzer– Hahn operators, we give an example which shows that proposed methods successfully  (α,β) -summability work. Furthermore, we obtain the rate of statistically weighted N  of double sequences of positive linear operators in terms of the modulus of continuity.

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 (α,β) -Summability Based on 2 Statistically Weighted N  Generalized ( p, q)-Difference Operator In this section, first we recall the concept of ( p, q)-integer and related formulas and we give a definition of the notion of generalized double difference operator involving ( p, q)-Gamma function. Secondly, we give the definitions of statistically weighted  (α,β) -summability, double weighted a,,bc N p,q -statistical convergence and weighted  (α,β)  strongly [N ]r -summability of double sequences for 0 < r < ∞.  For any p > 0 and q > 0, the ( p, q)-integers [n] p,q are defined by [n] p,q := p n−1 + p n−2 q + p n−3 q 2 + · · · + pq n−2 + q n−1 ⎧ pn −q n ( p = q = 1) ⎪ ⎪ ⎨ p−q n−1 ( p = q = 1) , = np ⎪ ( p = 1) [n]q ⎪ ⎩ n ( p = q = 1) where [n]q denotes the q-integer and n = 0, 1, 2, . . .. By simple calculations we can find that, [n] p,q = p n−1 [n] qp . The ( p, q)-factorial is defined by [0] p,q ! := 1 and [n]! p,q = [1] p,q [2] p,q · · · [n] p,q if n ≥ 1. The ( p, q)-binomial coefficient satisfies [n] p,q ! n , = [r ] p,q ! [n − r ] p,q ! r p,q

0 ≤ r ≤ n.

Also, the formula for ( p, q)-binomial expansion is as follows: (cx +

dy)np,q

=

n  n k=0

k

p

(n−k)(n−k−1) 2

q

k(k−1) 2

cn−k d k x n−k y k .

p,q

For a non-negative integer n, the ( p, q)-analogue of gamma function (see [44]) is defined as p,q (x) = ( p − q)1−x where (a; b)∞ p,q =

∞  ( p; q)∞ p n+1 − q n+1 p,q 1−x = ( p − q) n+x − q n+x x ; q x )∞ p ( p p,q n=0

∞  n=0

(ap n − bq n ),

a, b ∈ (0, 1].

Statistical Summability of Double Sequences by the Weighted Mean …

65

Note that for p = 1, the ( p, q)-gamma function turns out to be q-gamma function. Moreover, for a non-negative integer n, p,q (n + 1) = [n]! p,q for 0 < q < p. Subsequently, some interesting results on ( p, q)-calculus in approximation theory (see [1, 2, 10, 18, 27, 32, 35] and the references therein). Now we define the following double difference operator which includes the ( p, q)analogue of gamma function. Let ω2 be the space of all real double sequences of the form x = (xm,n ), where m, n ∈ N0 , the set of all nonnegative integers. Let x = (xm,n )∞ m,n=0 be a double real number sequence in ω2 , and let 0 < q1 , q2 < p1 , p2 ≤ 1. For non-negative integers a1 , a2 , b1 , b2 , c1 and c2 , we define a generalized double difference operator as 2 ,b1 ,b2 ,c1 ,c2 (x ap11,a mn ) = , p2 ,q1 ,q2

k l k l ∞  ∞  [a1 ] p1 ,q1 [a2 ] p2 ,q2 [b1 ] p1 ,q1 [b2 ] p2 ,q2 (−1)k+l xm−k,n−l k l [k] p1 ,q1 ! [l] p2 ,q2 ![c1 ] p1 ,q1 [c2 ] p2 ,q2 k=0 l=0

(1) i

where [r ] p,q is ( p, q)-shifted factorial of nonnegative integer r which is being defined ( p, q)-gamma function as  i

[r ] p,q :=

1

, (r = 0 or i = 0)

p,q (r +1) p,q (r −i+1) = [r ] p,q [r − 1] p,q [r − 2] p,q . . . [r − i + 1] p,q ,

k

(r ∈ N)

l

and [c1 ] p1 ,q1 = 0, [c2 ] p2 ,q2 = 0 for all k, l ∈ N. Throughout, it is being presumed that the double series defined in (1) is convergent in Pringsheim’s sense and xk,l = 0 for any negative integers of k and l. Using ( p, q)shifted factorial in the infinite sum of (1), for any m, n ∈ N, ai ∈ N and bi = ci for i = 1, 2, one can observe that 2 ,b1 ,b2 ,c1 ,c2 (x ap11,a mn ) ,q1 , p2 ,q2

=

 ∞  k l ∞  [a1 ] p1 ,q1  [a2 ] p2 ,q2 (−1)k (−1)l xm−k,n−l [k] p1 ,q1 ! [l] p2 ,q2 ! k=0

l=0

= xm,n − [a2 ] p2 ,q2 xm,n−1 +

[a2 ] p2 ,q2 [a2 − 1] p2 ,q2 xm,n−2 − [a1 ] p1 ,q1 xm−1,n + · · · [2] p2 ,q2 !

+ [a1 ] p1 ,q1 [a2 ] p2 ,q2 xm−1,n−1 − −

[a1 ] p1 ,q1 [a2 ] p2 ,q2 [a2 − 1] p2 ,q2 xm−1,n−2 + · · · [2] p2 ,q2 !

[a1 ] p1 ,q1 [a1 − 1] p1 ,q1 [a2 ] p2 ,q2 [a1 ] p1 ,q1 [a1 − 1] p1 ,q1 [a2 ] p2 ,q2 [a2 − 1] p2 ,q2 xm−2,n−1 + xm−2,n−2 + · · · [2] p1 ,q1 ! [2] p1 ,q1 ! [2] p2 ,q2 !

= xm,n −

p2a2 − q2a2 ( pa2 − q2a2 )( p2a2 −1 − q2a2 −1 ) pa1 − q1a1 xm,n−1 + 2 xm,n−2 − 1 xm−1,n + · · · 2 p2 − q2 ( p2 + q2 )( p2 − q2 ) p1 − q1

+

( p1a1 − q1a1 )( p2a2 − q2a2 ) ( pa1 − q1a1 )( p2a2 − q2a2 )( p2a2 −1 − q2a2 −1 ) xm−1,n−2 + · · · xm−1,n−1 − 1 ( p1 − q1 )( p2 − q2 ) ( p1 − q1 )( p2 + q2 )( p2 − q2 )2

+

( p1a1 − q1a1 )( p1a1 −1 − q1a1 −1 ) ( pa1 − q1a1 )( p1a1 −1 − q1a1 −1 )( p2a2 − q2a2 ) xm−2,n − 1 xm−2,n−1 + · · · 2 ( p1 + q1 )( p1 − q1 ) ( p1 + q1 )( p2 − q2 )( p1 − q1 )2

+

( p1a1 − q1a1 )( p1a1 −1 − q1a1 −1 )( p2a2 − q2a2 )( p2a2 −1 − q2a2 −1 ) xm−2,n−2 + · · · . ( p1 + q1 )( p2 + q2 )( p1 − q1 )2 ( p2 − q2 )2

66

U. Kadak

In particular, for ai = 1, bi = ci and pi = qi = 1, i.e., [n i ] pi ,qi = n i (n i ∈ N) for (xm,n ) is reduced to the ordinary double i = 1, 2, the difference operator 1,1,b,b,b,b 1,1,1,1 difference operator 1 (xmn ) defined by 1 (xm,n ) := xm,n − xm,n−1 − xm−1,n + xm−1,n−1 . The following example is given to illustrate the role of ( p, q)-integers for the proposed double difference operator in (1). Example 1 Define the double sequence (xm,n ) by xm,n = m − n for all m, n ∈ N. It is clear that the double sequence (xm,n ) is not P-convergent. On the other hand the ordinary double difference sequence 2 (xm,n ) of second order converges to zero in Pringsheim’s sense, that is, P − limm,n→∞ 2 (xmn ) = 0. Let a1 = 2, a2 = 1, bi = ci for i = 1, 2 and 0 < q1 , q2 < p1 , p2 ≤ 1. Since 2,1,b,b,b,b p1 ,q1 , p2 ,q2 (x mn ) = x m,n − x m,n−1 − [2] p1 ,q1 x m−1,n + [2] p1 ,q1 x m−1,n−1 + x m−2,n − x m−2,n−1

then P − lim 2,1,b,b,b,b p1 ,q1 , p2 ,q2 (x m,n ) = p1 + q1 − 2. m,n→∞

Thus, depending on the choice of the parameters p1 and q1 , the double sequence (2,1,b,b,b,b p1 ,q1 , p2 ,q2 (x m,n )) has infinitely many limit points. This problem is closely related to the formal definition of ( p, q)-integers. In order to reach to convergence results of the operator, it would be necessary to determine the appropriate approach. To do this, we replace p1 , p2 , q1 , q2 by real sequences ( pm ), ( pn ), (qm ), (qn ) respectively, satisfying lim pm = P1 ,

m→∞

lim qm = Q 1 ,

m→∞

lim pn = P2 ,

lim qn = Q 2

n→∞

n→∞

1 , lim (qm )m = Q 1 , lim ( pn )n = P 2 , lim ( pm )m = P

m→∞

m→∞

n→∞

2 lim (qn )n = Q

n→∞

(2) (3)

i , Q i ∈ (0, 1] for i = 1, 2. For examwhere 0 < qm , qn < pm , pn ≤ 1 and Pi , Q i , P m m ple, take qm = ( m+x ) < ( m+y ) = pm for all m ∈ N and for 0 < y < x. Clearly, lim pm = 1, lim qm = 1, lim (qm )m = e−x , lim ( pm )m = e−y .

m→∞

m→∞

m→∞

m→∞

Thus, we have P − lim 2,1,b,b,b,b pm ,qm , pn ,qn (x m,n ) = P − lim m,n→∞

m,n→∞



 pm + qm − 2 = 0.

Compared with the existing literature on this topic, our generalizations based on the corresponding parameters not only provide much more flexibility but also give us a new approach to considerations of the convergence conditions (in any manner).

Statistical Summability of Double Sequences by the Weighted Mean …

67

Definition 1 ([22]) Let K 2 ⊂ N2 be a two-dimensional subset of positive integers and (α, β) ∈ . Let (sm )m∈N0 and (tn )n∈N0 be two sequences of non-negative real numbers such that s0 , t0 > 0, Sm(α,β)

β(m)



=

sk → ∞ as m → ∞

k=α(m)

and

β(n) 

Tn(α,β) =

tl → ∞ as n → ∞.

l=α(n)

The lower and upper weighted double αβ-densities of the set K 2 are defined by (α,β)

δ2

(K 2 ) = P − lim inf

m,n→∞

     k ≤ S (α,β) and l ≤ T (α,β) : (k, l) ∈ K 2  m n  (α,β) 

1 (α,β)

Sm

Tn

and (α,β) δ 2 (K 2 )

     k ≤ S (α,β) and l ≤ T (α,β) : (k, l) ∈ K 2  m n  (α,β) 

1

= P − lim sup

(α,β)

Sm

m,n→∞

Tn

(α,β)

(α,β)

respectively, provided that the limits exist. If δ 2 (K 2 ) = δ 2 (α,β) K 2 has weighted double αβ-density δ2 (K 2 ) defined by (α,β) δ2 (K 2 )

= P − lim

m,n→∞

     k ≤ S (α,β) and l ≤ T (α,β) : (k, l) ∈ K 2 , m n  (α,β) 

1 (α,β)

Sm

(K 2 ), we say that

Tn

provided that the P-limit exists. Throughout this paper, we assume that ai , bi and ci are non-negative integers for i = 1, 2 and ( pm ), (qm ), ( pn ), (qn ) are sequences satisfying the conditions (2) and (3). Definition 2 Let (α, β) ∈ .  (α,β) (a) A double sequence (xm,n ) of real numbers is said to be weighted N  summable to a number L, if  (α,β) (xm,n ) = L P − lim N  m,n→∞

where  (α,β) (xm,n ) = N 



1 (α,β)

Sm

(α,β)

Tn

(α,β)

k∈Im

 (α,β)

l∈In

sk tl ap1m,a,q2m,b,1p,bn ,q2 ,cn 1 ,c2 (xk,l )

68

U. Kadak (α,β)

and Ir = [α(r ), β(r )] is a closed interval for (α, β) ∈  and r ∈ N. By  (α,β) -summable double  (α,β) ; sk , tl ] we denote the set of all weighted N [N    (α,β) ; sk , tl ] − limmn x = L. sequences and we shall write [N  In particular, if we take sn = tn = 1, α(n) = 1, β(n) = n for all n ∈ N, ai = 1, bi = ci for i = 1, 2 and pm , pn → 1, qm , qn → 1 as m, n → ∞, then weighted  (α,β) -summability is reduced to the classical summability of double sequences N  (see [31]).  (α,β) -summable to L if, for (b) A double sequence (xm,n ) is statistically weighted N  every  > 0,  (α,β)    N (xm,n ) − L    = 0. δ2 (m, n) :  That is,    1   (α,β) (xk,l ) − L|    = 0. (k, l); k  m, l  n : | N   m,n→∞ mn 

P − lim

 (α,β) ] − lim xmn = L. In this case we write st2 [N  Definition 3 Let (α, β) ∈ . (a) A double sequence (xm,n ) is said to be double weighted a,b,c p,q -statistically convergent to L, if for every  > 0, (α,β) 

 (m, n) : sm tn |ap1m,a,q2m,b,1p,bn ,q2 ,cn 1 ,c2 (xmn ) − L|   = 0.

δ2

Equivalently, we may write that P − lim m,n

1 (α,β)

Sm

     (k, l); k ≤ S (α,β) , l ≤ T (α,β) : sk tl |a1 ,a2 ,b1 ,b2 ,c1 ,c2 (xkl ) − L|    = 0. m n p ,q , p ,q   m m n n (α,β)

Tn

a,b,c By [st2 (a,b,c p,q ); sk , tl ] we denote the set of all double weighted  p,q -statistically convergent sequences and we denote it by [st2 (a,b,c p,q ); sk , tl ] − lim x mn = L. Again, if we take sn = tn = 1, α(n) = 1, β(n) = n for all n ∈ N, and ai = 1, bi = ci for i ∈ {1, 2}, pm , pn → 1, qm , qn → 1 as m, n → ∞, then double weighted a,b,c p,q -statistical convergence reduces to ordinary statistical convergence of double sequences (see [33, 48]).  (α,β) ]r -summable (b) A double sequence (xm,n ) is said to be weighted strongly [N  (0 < r < ∞) to a number L, if

P − lim

m,n→∞

1 (α,β) (α,β) Sm Tn

 (α,β)

k∈Im



  sk tl ap1m,a,q2m,b,1p,bn ,q2 ,cn 1 ,c2 (xkl ) − L  = 0.

(α,β)

l∈In

r (a,b,c We write it as [N p,q ); sk , tl ] − lim x mn = L.

Statistical Summability of Double Sequences by the Weighted Mean …

69

Remark 1 Now, we shall give the following special cases to show the effectiveness of proposed method. (a) Assume that θ = (u n ) is an increasing sequence of positive integers such that k0 = 0, 0 < u n < u n+1 and h n = (u n − u n−1 ) → ∞ as n → ∞. Under these conditions, θ is called a lacunary sequence. For, α(n) = u n−1 + 1, β(n) = u n , sn = tn = 1 for all n ∈ N, ai = 1, bi = ci for i = 1, 2, and pm , pn → 1,  (α,β) -summability is reduced qm , qn → 1 as m, n → ∞, statistically weighted N  to the lacunary statistical summability for double sequences. Moreover, the double weighted a,b,c p,q -statistical convergence turns out to be lacunary statistical convergence of double sequences (see [15, 36]). (b) Assume that (λn ) is a strictly increasing sequence of positive numbers such that limn λn = ∞, λn+1 ≤ λn + 1 and λ1 = 1. Assume further that α(n) = n − λn + 1, β(n) = n, sn = tn = 1 for all n ∈ N, ai = 1, bi = ci for i = 1, 2, and  (α,β) pm , pn → 1, qm , qn → 1 as m, n → ∞. Then, the statistically weighted N  summability is reduced to the λ-statistical summability and also the double weighted a,b,c p,q -statistical convergence is reduced to λ-statistical convergence of double sequences (cf. [8, 38]).

3 Some Inclusion Relations In this section we first prove some inclusion relations between statistically weighted  (α,β) -summability, double weighted a,b,c N p,q -statistical convergence and weighted   (α,β) ]r -summability for 0 < r < ∞. strongly [N  For simplicity of notation we shall use [(xk,l )] instead of ap1m,a,q2m,b,1p,bn ,q2 ,cn 1 ,c2 (xk,l ). Theorem 1 Let ai , bi and ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Let ( pm ), (qm ), ( pn ) and (qn ) be sequences satisfying (2) and (3) as m, n → ∞. Assume that   sk tl [(xk,l )] − L  ≤ M for all k, l ∈ N. If the sequence (xmn ) is double weighted a,b,c p,q -statistical conver (α,β) -summable to the same limit L. But gence to L then it is statistically weighted N  the converse may not be true. Proof Let ai , bi and ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Further, let us assume that  ( pm ), (qm ), (pn ) and (qn ) are sequences satisfying (2) and (3). Assume that sk tl [(xk,l )] − L  ≤ M holds for all k, l ∈ N. Setting     E  = (k, l); k ≤ Sm(α,β) , l ≤ Tn(α,β) : sk tl [(xk,l )] − L    and

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U. Kadak

    (α,β) (α,β)   = (k, l); k ≤ Sm , l ≤ Tn : sk tl [(xk,l )] − L < 

E C we see that

lim

m,n→∞

   E  (α,β)

Sm

(α,β)

Tn

= 0.

Then, considering our hypothesis, we get (α,β)

 |N    = 

≤ ≤

≤

(xm,n ) − L| 

1 (α,β)

Sm

(α,β)

Tn

(α,β)

k∈Im

(α,β)

,l∈In



1 Sm

(α,β)

  sk tl [(xk,l )] − L 

(α,β)

Tn

(α,β)

k∈Im

1 (α,β) (α,β) Sm Tn

M |E  | (α,β) (α,β) Sm Tn

(α,β)

,l∈In



(α,β)

    sk tl [(xk,l )] − L  + |L|    sk tl [(xk,l )] − L  +

(α,β)

k∈Im ,l∈In ((k,l)∈E  )

+

 |E C | (α,β) (α,β) Sm Tn



1 (α,β)

Sm

(α,β)

Tn

1 (α,β) (α,β) Sm Tn

(α,β)

k∈Im



(α,β)

(α,β)

  sk tl − 1

,l∈In

  sk tl [(xk,l )] − L  (α,β)

k∈Im ,l∈In ((k,l)∈E C )

→ 0 +  · 1 =  (m, n → ∞).

 (α,β) -summable to L. This implies that (xm,n ) is Therefore, (xm,n ) is weighted N  (α,β)  statistically weighted N -summable to L.  For the converse part, we will give the following example.  Example 2 Let ai , bi and ci be non-negative integers for i = 1, 2 and let ( pm ), (qm ), ( pn ) and (qn ) be sequences satisfying (2) and (3) as m, n → ∞. For each m, n ∈ N, define the double difference sequence ym,n = [(xmn )] by  m = u 4 − u 2 , u 4 − u 2 + 1, . . . , u 4 − 1; u = 2, 3, 4, . . . n = v 4 − v 2 , v 4 − v 2 + 1, . . . , v 4 − 1; v = 2, 3, 4, . . .  = m = u 4 , u = 2, 3, 4, . . . 2 2 ⎪ v ) −(u ; ⎪ ⎪ n = v 4 , v = 2, 3, 4, . . . ⎪ ⎪ ⎪ ⎩ 0 (otherwise); ⎧ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨

ymn

; .

Let us take sn = tn = 1, α(n) = 1, β(n) = n for all n ∈ N. It is not hard to see that (ymn ) is divergent. However, m  n   (α,β) (xm,n ) = 1 ykl N  mn k=1 l=1

Statistical Summability of Double Sequences by the Weighted Mean …

71

 ⎧ ⎪ m = u 4 − u 2 + A; u = 2, 3, 4, . . . ; A = 0, 1, 2, . . . , u 2 − 1, ⎨ (A+1)(B+1) mn = n = v 4 − v 2 + B; v = 2, 3, 4, . . . ; B = 0, 1, 2, . . . , v 2 − 1, ⎪ ⎩ 0 (otherwise). Therefore

 (α,β) (xm,n ) = 0 P − lim N  m,n→∞

which implies that

 (α,β) (xm,n ) = 0. st2 − lim N 

On the other hand, since      1    (k, l); k ≤ m, l ≤ n : [(xk,l )]    = 0, lim m,n→∞ mn  then the double difference sequence (ymn ) is not double weighted a,b,c p,q -statistically convergent to zero. Theorem 2 Let ai , bi and ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Let ( pm ), (qm ), ( pn ) and (qn ) be sequences satisfying (2) and (3) as m, n → ∞.  (a) Assume that (xm,n ) is weighted strongly [N 

(α,β)

]r -summable to L. If

  0 < r < 1 and 0 ≤ [(xk,l )] − L  < 1 or

  1 ≤ r < ∞ and 1 ≤ [(xk,l )] − L  < ∞

for all k, l ∈ N, then (xm,n ) is double weighted a,b,c p,q -statistically convergent to L. (b) Assume that (xm,n ) is double weighted a,b,c p,q -statistically convergent to L.     Assume further that sk tl [(xk,l )] − L ≤ M for all k, l ∈ N. If 0 < r < 1 and M ∈ [1, ∞)

or

1 ≤ r < ∞ and M ∈ [0, 1),

 (α,β) ]r -summable to L. then (xmn ) is weighted strongly [N  r (a,b,c Proof (a) Let [N p,q ); sk , tl ] − lim x m,n = L. Under the above conditions, we have

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U. Kadak

1 (α,β)

Sm

   E  = (α,β)

Tn

≤

≤ ≤



1 (α,β)

 Sm

(α,β)

Tn



(α,β) (α,β) Sm Tn



(α,β) (α,β) Sm Tn

(α,β)

  sk tl [(xk,l )] − L 

(α,β)

k∈Im ,l∈In ((k,l)∈E  )



1

(α,β)

k∈Im

1 (α,β)

(α,β)



1

 Sm



(α,β)

k∈Im ,l∈In ((k,l)∈E  )

(α,β)

Tn

  sk tl [(xk,l )] − L 

(α,β)

,l∈In



(α,β)

k∈Im

r  sk tl [(xk,l )] − L  → 0,

(α,β)

,l∈In

as m, n → ∞ in the Pringsheim’s sense. Hence, (xmn ) is double weighted a,b,c p,q statistically convergent to L. (b) Let (xm,n ) be double weighted a,b,c p,q -statistically convergent to L. From the hypothesis, we deduce that 

1 (α,β)

Sm =

(α,β)

Tn

(α,β)

k∈Im

1 (α,β)

Sm

(α,β)

Tn

r  sk tl [(xkl )] − L  (α,β)

,l∈In



(α,β)

 r sk tl [(xkl )] − L  + (α,β)

k∈Im ,l∈In ((k,l)∈E  )



1 (α,β)

Sm

(α,β)

Tn

(α,β)

 r sk tl [(xkl )] − L  (α,β)

k∈Im ,l∈In ((k,l)∈E C )

= 1 (m, n) + 2 (m, n)

where

1 (m, n) =

and

2 (m, n) =

1 (α,β) (α,β) Sm Tn

1 (α,β) (α,β) Sm Tn

For the case (k, l) ∈ E C , we get

 (α,β)

r  sk tl [(xkl )] − L  (α,β)

k∈Im ,l∈In ((k,l)∈E  )

 (α,β)

r  sk tl [(xkl )] − L  . (α,β)

k∈Im ,l∈In ((k,l)∈E C )

Statistical Summability of Double Sequences by the Weighted Mean …

2 (m, n) =

≤

=



1 (α,β) (α,β) Sm Tn

(α,β)

r  sk tl [(xkl )] − L  (α,β)

k∈Im ,l∈In ((k,l)∈E C )



1 (α,β) (α,β) Sm Tn

 (α,β) (α,β) Sm Tn

(α,β)

73

  sk tl [(xkl )] − L 

(α,β)

k∈Im ,l∈In ((k,l)∈E C )

|E C | → 

(m, n → ∞).

Also, for (k, l) ∈ E  , we obtain

1 (m, n) =

≤

=



1 (α,β)

Sm

(α,β)

Tn

(α,β)

r  sk tl [(xkl )] − L  (α,β)

k∈Im ,l∈In ((k,l)∈E  )



1 (α,β) (α,β) Sm Tn

M (α,β) (α,β) Sm Tn

(α,β)

  sk tl [(xkl )] − L 

(α,β)

k∈Im ,l∈In ((k,l)∈E  )

|E  | → 0

(m, n → ∞).

 r -summable to L. Therefore, (xmn ) is weighted strongly [N] This completes the proof.



 (α,β) -summable douIn the next theorem we characterize statistically weighted N  ble sequences in terms of its double subsequences. Theorem 3 Assume that ai , bi and ci are non-negative integers for i = 1, 2 and α(n) = 1, β(n) = n for all n ∈ N. Assume further that ( pm ), (qm ), ( pn ) and (qn ) are sequences with the properties (2) and (3) as m, n → ∞. A double sequence  (α,β) -summable to L if and only if there exists a set (xm,n ) is statistically weighted N    H 2 (m, n) = (km , ln ) : k1 < k2 < · · · < km < · · · and l1 < l2 < · · · < ln < · · · such that δ2 (H 2 (m, n)) = 1

and

 (α,β) ] − lim xkm ,ln = L , st2 [N 

where δ2 (H 2 ) denotes the double natural density of the set H 2 ⊆ N2 . Proof Let us suppose that there exists a two-dimensional set H 2 (m, n) ⊆ N2 such  (α,β) -summable to L. that δ2 (H 2 (m, n)) = 1 and (xkm ,ln ) is statistically weighted N  Then there exist two positive integers m 0 , n 0 ∈ N such that, for all m > m 0 and n > n 0 , we get

74

U. Kadak

 |N 

(α,β)

  (xkm ,ln ) − L| = 

where Skm =

 ln km   1  sk tl [(xkl )] − L  <  Skm Tln k=1 l=1

km 

sk → ∞ as km → ∞

k=1

and Tln =

ln 

tl → ∞ as ln → ∞.

l=0

Let us set    (α,β) (xkm ,ln ) − L|   H2 := (m, n) : m, n ∈ N and |N  and

  2 := km 0 +1 , km 0 +2 , . . . , ln 0 +1 , ln 0 +2 , . . . . H

2 ) = 1 and H2 ⊂ N2 \ H 2 which yields that δ2 (H2 ) = 0 in PringThen, δ2 ( H  (α,β) sheim’s sense. Therefore, the double sequence (xmn ) is statistically weighted N  summable to L.  (α,β) -summable to L. For ν ∈ N, Conversely, let (xmn ) be statistically weighted N  put   1 (α,β) 2 2  K (ν) = (m, n) ∈ N : |N (xkm ,ln ) − L| ≥ ν and

  1 (α,β) 2  . M (ν) = (m, n) ∈ N : |N (xkm ,ln ) − L| < ν 2

Then δ2 (K 2 (ν)) = 0 and M 2 (1) ⊇ M 2 (2) ⊇ · · · M 2 (i) ⊇ M 2 (i + 1) ⊇ · · ·

(4)

and δ2 (M 2 (ν)) = 1

(ν ∈ N).

(5)

 We will now show that st2 [N ] − lim xkm ,ln = L for (m, n) ∈ M 2 (ν). Suppose   (α,β) -summable that the double subsequence (xkm ,ln ) is not statistically weighted N  to L. Therefore, there exists  > 0 such that (α,β)

 |N 

(α,β)

(xkm ,ln ) − L| ≥ 

Statistical Summability of Double Sequences by the Weighted Mean …

75

for infinitely many terms. Let us set   (α,β) 2  M () := (m, n) ∈ N : |N (xkm ,ln ) − L| <  and  > 1/ν (ν ∈ N). 2

It is obvious that δ2 (M 2 ()) = 0 and, by (4), M 2 (ν) ⊂ M 2 (). Hence δ2 (M 2 (ν)) = 0, which contradicts (5). As a  (α,β) -summable to L, as consequence, we have (xkm ,ln ) is statistically weighted N  asserted by Theorem 3. 

4 Korovkin-Type Approximation Theorem During the last two centuries, the idea of approximating functions by simpler functions has attracted great attention of thousands researchers. The classical approximation theorem of Korovkin [29] plays a fundamental role in development of approximation theory and appears in a very natural way in many problems dealing with certain restricted classes of continuous linear operators on locally convex spaces. This famous theorem asserts that for a given sequence L n of positive linear operators acting on C[0, 1], L n ( f ) → f uniformly on [0, 1] for every f ∈ C[0, 1]. In fact, it is sufficient to verify that L n ( f ) uniformly converges to f on [0, 1] only for the test function ei (x) = x i , (i = 0, 1, 2). Moreover, other finite classes of test functions have been considered, in both one- and multi-dimensional case. Starting with this result, during the last two decades, several mathematicians have worked on extending Korovkins theorems in many fields such as functional analysis, harmonic analysis, measure theory, probability theory, summability theory, Banach lattices, Banach algebras and so on. In recent years, using the new classes of summability methods, statistical convergence and some of its extended versions have been used in positive approximation processes (see for examples [12, 13, 17, 19, 45, 49] and the references therein), and the present approximation result is a continuation of [21]. Now, we shall obtain a Korovkin type approximation theorem for functions of two  (α,β) -summability of double sequences of variables through statistically weighted N  positive linear operators. Let R2+ = [0, ∞) × [0, ∞). By C B (R2+ ), we denote the space of all bounded and continuous functions on R2+ which is linear normed space with  f C B (R2+ ) = sup | f (x, y)| x,y≥0



 f ∈ C B (R2+ ) .

Let Hω (R2+ ) denote the space of all real-valued functions f defined on R2+ such that

76

U. Kadak

⎛ | f (u, v) − f (x, y)|  ω ⎝ f ;



u x − 1+u 1+x



2 +

v y − 1+v 1+y

2

⎞ ⎠

where ω( f ; δ), δ > 0 is the modulus of continuity of the function f defined by ω( f ; δ) =

   | f (u, v) − f (x, y)| : (u − x)2 + (v − y)2  δ .

sup

(6)

u,v,x,y∈R+

Moreover we have ⎛  u ⎜ 1+u − | f (u, v) − f (x, y)|  ω( f ; δ) ⎜ ⎝

 x 2 1+x

+



v 1+v

−

y 1+y

δ

2

⎞ ⎟ + 1⎟ ⎠ . (7)

Note that Hω (R2+ ) ⊂ C B (R2+ ) for the bounded and continuous function f on R2+ , and the necessary and sufficient condition for f ∈ Hω (R2+ ) is lim ω( f ; δ) = 0.

δ→0

Then as usual, we say that T is positive linear operator provided that f ≥ 0 implies T f ≥ 0. Also, we denote the value of T ( f ) at a point (x, y) ∈ R2+ by T ( f (u, v); x, y) or, briefly T ( f ; x, y). Theorem 4 Let ai , bi , ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Let ( pm ), (qm ), ( pn ) and (qn ) be sequences satisfying st − lim pm , pn = 1, st − lim qm , qn = 1

(8)

st − lim ( pm )m = μ1 ,

st − lim ( pn )n = μ2

(9)

st − lim (qm )m = μ˜ 1 ,

st − lim (qn )n = μ˜ 2

(10)

m,n→∞

m,n→∞

m→∞

n→∞

m→∞

n→∞

where qr ∈ (0, 1), pr ∈ (qr , 1] for all r ∈ N and μi , μ˜ i ∈ (0, 1]. Assume that (Tm,n ) is a double double sequence of positive linear operators acting from Hω (R2+ ) into C B (R2+ ). Assume further that Jm,n : Hω (R2+ ) → C B (R2+ ) is a double sequence of positive linear operators defined by Jm,n (·; x, y) =



1 (α,β)

Sm

(α,β)

Tn

(α,β)

k∈Im

 (α,β)

l∈In

2 ,c1 ,c2 sk tl ap1m,a, p2 n,b,q1 ,b (Tk,l (·; x, y)) m ,qn

Statistical Summability of Double Sequences by the Weighted Mean …

77

satisfying the following conditions: st2 −

Jm,n (1; x, y) − 1C (R2 ) = 0, (11) B + " "   " x " u " Jm,n ; x, y − st2 − lim " = 0, (12) m,n→∞ " 1+u 1 + x "C B (R2 ) + " "   " v y " " Jm,n ; x, y − st2 − lim " = 0, (13) m,n→∞ " 1+v 1 + y "C B (R2 ) + " 2  2 2  2 "    " " u v x y " J st2 − lim " + ; x, y − + = 0. m,n " m,n→∞ " 1+u 1+v 1+x 1+y C B (R2 ) lim

m,n→∞

+

(14)

Then, for all f ∈ Hω (R2+ ), st2 − lim Jm,n ( f ; x, y) − f C B (R2+ ) = 0.

(15)

m,n→∞

Proof First we assume that conditions (11)–(14) hold. Due to the continuity of f on R2+ , for a given  > 0, there exists a number δ = δ() such that for all u, v, x, y ∈ R+ we have | f (u, v) − f (x, y)| < 

(16)

whenever      v  u x  y     1 + u − 1 + x  < δ and  1 + v − 1 + y  < δ. u Also we obtain for all x, y ∈ R+ satisfying | 1+u − that

| f (u, v) − f (x, y)| ≤

v  δ and | 1+v −

y | 1+y

 2M  ψu (x) + ψv (y) 2 δ

where M := sup(x,y)∈R2+ | f (x, y)| and ψu (x) = and (17), we deduce that | f (u, v) − f (x, y)|
0, we choose a number  > 0 such δ2 that  < γ . Further, let us define the sets by putting   J := (m, n) : Jm,n ( f ; x, y) − f C B (R2+ )  γ ,   γ − J0 := (m, n) : Jm,n (1; x, y) − 1C B (R2+ )  , 4K  " "   " u x " " " ; x, y − J1 := (m, n) : " Jm,n  1+u 1 + x "C B (R2+ )  " "   " v y " " J ; x, y −  J2 := (m, n) : " " m,n 1 + v 1 + y "C B (R2+ )

γ − 4K γ − 4K

# , #

and "  2  2   " u v " + ; x, y J3 := (m, n) : " Jm,n 1+u 1+v 2  2 "   " x y γ − " . − +  " 1+x 1+y 4K C B (R2+ ) 3 Then the inclusion J ⊂ ∪i=0 Ji holds true and hence the following relation is valid:

δ2 (J) ≤ δ2 (J0 ) + δ2 (J1 ) + δ2 (J2 ) + δ2 (J3 ). Passing to the statistical limit in the Pringsheim’s sense as m, n → ∞ and using the hypothesis, we get

Statistical Summability of Double Sequences by the Weighted Mean …

79

st2 − lim Jm,n ( f ; x, y) − f C B (R2+ ) = 0, m,n→∞



which completes the proof.

We now present an example associated with the ( p, q)-analogue of generalized bivariate Bleimann–Butzer–Hahn (BBH) operators (see [39]) to demonstrate how the Theorem 4 works. Example 3 Let 0 < qm , qn < pm , pn ≤ 1 and f : R+ → R. The ( p, q)-analogue of generalized bivariate BBH operators is defined by pm , pn ),(qm ,qn ) ( L (m,n

f ; x, y) =

m,n 

 f

k,l=0

pmm−k+1 [k] pm ,qm pnn−l+1 [l] pn ,qn , [m − k + 1] pm ,qm qmk [n − l + 1] pn ,qn qnl

(m−k)(m−k−1) k(k−1) (n−l)(n−l−1) l(l−1) pm 2 qm 2 pn 2 qn 2 m k n x yk . × $ $n−1 i2 i1 i1 i2 l k pm ,qm im−1 ( p + q x) ( p + q y) p ,q m m n n n n =0 i =0 1



(19)

2

It is easy to check that (19) is linear and positive. If we put pm = pn = 1, then we ( p , p ),(q ,q ) obtain q-bivariate BBH operators. Then, for the operators L m,nm n m n ( f ; x, y) defined by (19), we have pm , pn ),(qm ,qn ) (1; x, y) L (m,n   u ( pm , pn ),(qm ,qn ) ; x, y L m,n 1 + u  v pm , pn ),(qm ,qn ) L (m,n ; x, y 1+v   u2 ( pm , pn ),(qm ,qn ) L m,n ; x, y (1 + u)2

= 1; = = = +

pm , pn ),(qm ,qn ) L (m,n



v2 ; x, y (1 + v)2

 = +

[m] pm ,qm x ; [m + 1] pm ,qm 1 + x [n] pn ,qn y ; [n + 1] pn ,qn 1 + y pm qm2 [m] pm ,qm [m − 1] pm ,qm x2 [m + 1]2pm ,qm (1 + x)2 ( pm + qm x) m+1 pm [m] pm ,qm x ; [m + 1]2pm ,qm 1 + x pn qn2 [n] pn ,qn [n − 1] pn ,qn y2 [n + 1]2pn ,qn (1 + y)2 ( pn + qn y) n+1 pn [n] pn ,qn y . [n + 1]2pn ,qn 1 + y

Now we consider the following positive linear operators Bm,n : Hω (R2+ ) → C(R2+ ) such that pm , pn ),(qm ,qn ) ( f ; x, y) Bm,n ( f ; x, y) = (1 + h m,n ) L (m,n

(20)

where the sequence (h m,n ) is defined as in Example 2. By taking into account the  (α,β) (xm,n ) = 0, we deduce that properties (8)–(10) and st2 − lim N 

80

U. Kadak

st2 − lim Bm,n (1; x, y) − 1C B (R2 ) = 0, + m,n→∞ " "   " u x " " st2 − lim " = 0, B ; x, y − m,n m,n→∞ " 1+u 1 + x "C B (R2+ ) " "   " y " v " st2 − lim " = 0, ; x, y − Bm,n " m,n→∞ 1+v 1 + y "C B (R2+ ) "    2  2 2  2 " " " v x y u " + ; x, y − + = 0. st2 − lim " B m,n " m,n→∞ " 1+u 1+v 1+x 1+y C B (R2+ )

Thus, all the conditions of Theorem 4 hod true and hence st2 − lim Bm,n ( f ; x, y) − f C B (R2+ ) = 0 m,n→∞

for all f ∈ Hω (R2+ ). On the other hand, clearly, Theorem 4 does not work for the double sequence (Bm,n ) of positive linear operators in classical, statistical and weighted statistical versions of Korovkin type approximation theorems.

 (α,β) -Summability 5 Rate of Statistically Weighted N  of Double Sequences  In this section, we estimate the rates of statistically weighted N -summability of  double sequences of positive linear operators defined from Hω (R2+ ) into C B (R2+ ). We first give the following definition. (α,β)

Definition 4 Let ai , bi , ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Let ( pm ), (qm ), ( pn ) and (qn ) be sequences of nonnegative real numbers satisfying (2) and (3). Let (θm,n ) be a positive nonincreasing double sequence in the Pringsheim’s  (α,β) sense. A double sequence x = (xm,n ) is said to be statistically weighted N  summable to a number L with the rate o(θm,n ) if for every  > 0,     1 (α,β)   (k, l); k  m and l  n : |N (xkl ) − L|    = 0 P − lim m,n→∞ mnθm,n  (21) where  (α,β) (xkl ) = N 

1





si t j (α,β) (α,β) Sk Tl (α,β) (α,β) i∈Ik j∈Il

 In this case we denote it by xm,n − L = st2 [N 

(α,β)

2 ,c1 ,c2 ap1k,a,q2k,b, p1l,b (xi j ). ,ql

] − o(θm,n ) as m, n → ∞.

Lemma 1 Let ai , bi , ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Suppose that ( pm ), (qm ), ( pn ) and (qn ) are sequences satisfying (2) and (3). Suppose

Statistical Summability of Double Sequences by the Weighted Mean …

81

also that (u m,n ), (vm,n ) are two positive non-increasing double sequences and (xm,n ), (ym,n ) are double sequences satisying  (α,β) ] − o(u m,n ) and ym,n − L 2 = st2 [N  (α,β) ] − o(vm,n ) xm,n − L 1 = st2 [N   as m, n → ∞. Then  (α,β) ] − o(wm,n ) as m, n → ∞, (1) (xm,n − L 1 ) ± (ym,n − L 2 ) = st2 [N  (α,β)  (2) (xm,n − L 1 )(ym,n − L 2 ) = st2 [N ] − o(u m,n vm,n ) as m, n → ∞,  (α,β)  (3) λ(xm,n − L 1 ) = st2 [N ] − o(u m,n ), for any scalar λ, as m, n → ∞, where wm,n := max{u m,n , vm,n }. Proof We prove only (1), and proofs of other statements may be obtained similarly. Let  xm,n − L 1 = st2 [N 

(α,β)

 ] − o(u m,n ) and ym,n − L 2 = st2 [N 

(α,β)

] − o(vm,n )

as m, n → ∞. Also, for  > 0, let us set     (α,β)  (α,β) (ykl ) − (L 1 + L 2 ) ≥  , N (xkl ) + N H : = (k, l); k  m and l  n :     (α,β)   H0 : = (k, l); k  m and l  n :  N (xkl ) − L 1  ≥ , 2   (α,β)   H1 : = (k, l); k  m and l  n :  N (ykl ) − L 2  ≥ . 2 We then observe that H ⊂ H0 ∪ H1 , which yields, for m, n ∈ N, that |H0 | |H1 | |H| ≤ + . mnwm,n mnu m,n mnvm,n

(22)

where wm,n := max{u m,n , vm,n }. Letting m, n → ∞ on both sides of (22), we get P − lim

m,n→∞

Therefore, the proof is completed.

|H| = 0. mn wm,n 

 Now, we present the next theorem to get the rates of statistically weighted Nsummability by means of the modulus of continuity in (6). Theorem 5 Let ai , bi , ci be non-negative integers for i = 1, 2 and (α, β) ∈ . Let ( pm ), (qm ), ( pn ) and (qn ) be sequences satisfying (8)–(10). Assume that (Tm,n ) is a double sequence of positive linear operators acting from Hω (R2+ ) into C(R2+ ).  ) be positive non-increasing double sequences. Assume further that ( θm,n ) and (θm,n Then, for all f ∈ Hω (R2+ ),

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 Tm,n ( f ; x, y) − f C B (R2+ ) = st2 [N 

(α,β)

] − o(θm,n ) as m, n → ∞,

 with θm,n := max{ θm,n , θm,n }, provided that the following conditions hold:

 (α,β) ] − o( θm,n ) as m, n → ∞, (i) Tm,n (1; x, y) − 1C B (R2+ ) = st2 [N  (α,β)   ] − o(θm,n ) as m, n → ∞, where (ii) ω( f ; δm,n ) = st2 [N 

δm,n := with

 ϕ(u, v) =



Tm,n (ϕ; x, y)C B (R2+ )

x u − 1+u 1+x

2

 +

y v − 1+v 1+y

2 .

Proof To see this, we first assume that f ∈ Hω (R2+ ) and (x, y) ∈ R2+ be fixed, and that (i) and (ii) hold. Let δ be a positive number. Using the linearity and the positivity of the double operators Tm,n and the inequality in (7), we have |Tm,n ( f ; x, y) − f (x, y)| = Tm,n (| f (u, v) − f (x, y)|; x, y) + | f (x, y)| |Tm,n (1; x, y) − 1|   ϕ(u, v) + 1; x, y + | f (x, y)| |Tm,n (1; x, y) − 1| ≤ ω( f ; δ)Tm,n δ2   1 ≤ ω( f ; δ) 2 Tm,n (ϕ; x, y) + Tm,n (1; x, y) + | f (x, y)| |Tm,n (1; x, y) − 1|. δ Taking the supremum over (x, y) ∈ R2+ on the both sides of the above inequality, we obtain Tm,n ( f ; x, y) − f C B (R2+ )   1 2 2 ≤ ω( f ; δ) 2 Tm,n (ϕ; x, y)C B (R+ ) + Tm,n (1; x, y) − 1C B (R+ ) + 1 δ + ETm,n (1; x, y) − 1C B (R2+ ) where E =  f C B (R2+ ) . Now, if we take δ := δm,n :=

 Tm,n (ϕ; x, y)C B (R2+ )

in the last relation, we deduce that Tm,n ( f ; x, y) − f C B (R2+ )   ≤ ω( f ; δm,n ) Tm,n (1; x, y) − 1C B (R2+ ) + 2 + ETm,n (1; x, y) − 1C B (R2+ ) Hence, we get

Statistical Summability of Double Sequences by the Weighted Mean …

83

Tm,n ( f ; x, y) − f C B (R2 ) +    ω( f ; δm,n )Tm,n (1; x, y) − 1 ≤E + ω( f ; δ ) + T (1; x, y) − 1 m,n m,n C B (R2 ) C B (R2 ) +

+

 = max{E, 2}. Replacing Tm,n (·; x, y) by where E Jm,n (·; x, y) =

1 (α,β) (α,β) Sm Tn

 (α,β)

k∈Im



2 ,c1 ,c2 sk tl ap1m,a, p2 n,b,q1 ,b (Tk,l (·; x, y)) m ,qn

(α,β)

l∈In

on both sides of the last inequality, we find that Jm,n ( f ; x, y) − f C B (R2 ) (23) +    ω( f ; δm,n )Jm,n (1; x, y) − 1 ≤K C B (R2 ) + ω( f ; δm,n ) + Jm,n (1; x, y) − 1C B (R2 ) . +

+

For a given  > 0, we consider the following sets:   A := (m, n) ∈ N2 : Jm,n ( f ; x, y) − f C B (R2+ ) ≥  ,    , A0 := (m, n) ∈ N2 : ω( f ; δm,n )Jm,n (1; x, y) − 1C B (R2+ )   3E    , A1 := (m, n) ∈ N2 : ω( f ; δm,n )   3E    . A2 := (m, n) ∈ N2 : Jm,n (1; x, y) − 1C B (R2+ )   3E 2 Then, it follows from (23) that A ⊂ ∪i=0 Ai . By taking Lemma 1 into account, and hence by letting m, n → ∞ (in any manner), we are led to the fact that

 (α,β) ] − o(θm,n ) as m, n → ∞, Tm,n ( f ; x, y) − f C B (R2+ ) = st2 [N  as asserted by Theorem 5.



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30. Moricz, F., Orhan, C.: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Studia Scientiarum Mathematicarum Hungarica 41(4), 391–403 (2004) 31. Moricz, F., Stadtmüller, U.: Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim’s sense. Int. J. Math. Math. Sci. 65, 3499–3511 (2004) 32. Mursaleen, M., Ansari, K.J., Khan, A.: On ( p, q)-analogue of Bernstein Operators. Appl. Math. Comput. 266, 874-882 (2015). (Erratum to On ( p, q)-analogue of Bernstein Operators’ Appl. Math. Comput. 266, 874–882 (2015)) 33. Mursaleen, M., Edely, O.H.H.: Statistical convergence of double sequences. J. Math. Anal. Appl. 288, 223–231 (2003) 34. Mursaleen, M., Mohiuddine, S.A.: Convergence Methods for Double Sequences and Applications, pp. 1–171 (2014) 35. Mursaleen, M., Nasiruzzaman, M., Ansari, K.J., Alotaibi, A.: Generalized (p,q)-BleimannButzer-Hahn operators and some approximation results. J. Inequal. Appl. 310 (2017). https:// doi.org/10.1186/s13660-017-1582-x 36. Mursaleen, M.: On statistical lacunary summability of double sequences Mursaleen, M., Belen, C. Filomat 28(2), 231-239 (2014) 37. Mursaleen, M.: Generalized spaces of difference sequences. J. Math. Anal. Appl. 203(3), 738– 745 (1996) 38. Mursaleen, M.: λ-statistical convergence. Math. Slovaca 50, 111–115 (2000) 39. Mursaleen, M., Nasiruzzaman, Md: Some approximation properties of bivariate BleimannButzer-Hahn operators based on (p, q)-integers. Boll. Unione Mat. Ital. 10, 271–289 (2017) 40. Mursaleen, M., Karakaya, V., Ertürk, M., Gürsoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012) 41. Pradhan, T., Paikray, S.K., Jena, B.B., Dutta, H.: Statistical deferred weighted B-summability and its applications to associated approximation theorems. J. Inequalities Appl. https://doi.org/ 10.1186/s13660-018-1650-x 42. Pringsheim, A.: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53, 289–321 (1900) 43. Robison, G.M.: Divergent double sequences and series. Trans. Am. Math. Soc. 28(1), 50–73 (1926) 44. Sadjang, P.N.: On the (p,q)-gamma and the (p,q)-beta functions. arXiv:1506.07394v1 45. Sahin, P.O., Dirik, F.: A Korovkin-type theorem for double sequences of positive linear operators via power series method. Positivity 22(1), 209–218 (2018) 46. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: A certain class of weighted statistical convergence and associated Korovkin-type approximation theorems involving trigonometric functions. Math. Methods Appl. Sci. (2017). https://doi.org/10.1002/mma.4636 47. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: Generalized equi-statistical convergence of the deferred Norlund summability and its applications to associated approximation theorems, Revista de la Real Academia de Ciencias Exactas. Fsicas y Naturales. Serie A. Matematicas (2017). https://doi.org/10.1007/s13398-017-0442-3 48. Tripathy, B.C., Sarma, B.: Statistically convergent difference double sequence spaces. Acta Mathematica Sinica, English Series 24(5), 737–742 (2008) 49. Yavuz, E., Talo, O.: Approximation of continuous periodic functions of two variables via power series methods of summability. Bull. Malays. Math. Sci. (2017). https://doi.org/10. 1007/s40840-017-0577-6

A Survey on a Conjecture of Rainer Brück Indrajit Lahiri

Abstract In 1996 Rainer Brück considered the uniqueness problem of an entire function that shares one value with its derivative. He proposed a conjecture on the single value sharing by an entire function with its first derivative[14]. Till date the conjecture of Brück is not completely resolved in its full generality. However it initiated a stream of research on a new branch of uniqueness theory. In the survey we intend to present the development of works done by several authors on the conjecture. Keywords Entire function · Meromorphic function · Uniqueness 2010 Mathematics Subject Classification 30D35

1 Preliminaries So far two major theoretical tools are used to tackle the conjecture of R. Brück: The Nevanlinna Theory and the Wiman-Valiron Theory. At the outset we discuss some basic ideas of the two theories in order to make the reader familiar with the theoretical instruments to be used. Nevanlinna’s theory of meromorphic functions occupies the central position of Complex Analysis, specially of the Value Distribution Theory. In early nineteenth century a famous Finnish mathematician Rolf Nevanlinna developed a systematic theory of meromorphic functions that made a revolutionary change in the approach of the Value distribution Theory. We, therefore, first discuss Nevanlinna’s theory in brief. An analytic function that does not have any essential singularity in the open complex plane C is called a meromorphic function in C. The focal theme of the value distribution theory is to study the distribution of roots of f (z) − a = 0 for a meromorphic function f and a complex number a. Its origin goes back to the I. Lahiri (B) Department of Mathematics, University of Kalyani, West Bengal 741235, India e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_4

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classical works of nineteenth century like Sokhotskii’s theorem, Weierstrass-Casorati theorem, Picard’s theorem and many others. At the end of nineteenth and at the beginning of twentieth century further works were carried out by the French school of scholars like Hadamard, Borel and Valiron. All these contributions are regarded as classical whereas the method developed by Rolf Nevanlinna, is considered as modern and is dominating till date. Since the theory developed by Nevanlinna plays an effective role in the research on Brück’s conjecture, it is a good idea to explain the basics of the theory. The books of Hayman [25] and of Yang and Yi [45] are good sources of the preliminaries of Nevanlinna theory apart from the monograph [40] written by Rolf Nevanlinna himself.   1 the nonnegative increasing function Let us denote by n(r, a; f ) = n r, f −a that counts the number of zeros of f − a lying in | z |≤ r , counted with multiplicities, for a meromorphic function f and a ∈ C. By n(r, ∞; f ) = n(r, f ) we denote the number of poles of f lying in | z |≤ r , counted withmultiplicities.  1 In a similar fashion we denote by n(r, a; f ) = n r, and n(r, ∞; f ) = f −a n(r, f ) the number of zeros of f − a and poles respectively, lying in | z |≤ r , counted ignoring multiplicities. A major problem of using the counting functions n(r, a; f ) and n(r, a; f ) is their discontinuities at countable number of points. In order to overcome this difficulty, the integrated counting functions are introduced as follows: 

r

n(t, a; f ) − n(0, a; f ) dt + n(0, a; f ) log r t

r

n(t, a; f ) − n(0, a; f ) dt + n(0, a; f ) log r, t

N (r, a; f ) = 0



and N (r, a; f ) =

0

with N (r, ∞; f ) = N (r, f ) and N (r, ∞; f ) = N (r, f ). For a meromorphic function f and a ∈ C another component is proved to be useful, called the proximity function, and is defined as follows  m(r, a; f ) = m r,

1 f −a

 =

1 2π



2π 0

log+

1 dθ, | f (r eiθ ) − a |

where log+ x = log x if x > 1 and log+ x = 0 if 0 ≤ x ≤ 1. We use m(r, f ) to denote  2π 1 m(r, ∞; f ) = log+ | f (r eiθ ) | dθ. 2π 0 The proximity function measures the average magnitude of positive logarithm of 1 and | f (z) |, whichever is the case. | f (z) − a |

A Survey on a Conjecture of Rainer Brück

89

For a meromorphic function f , Rolf Nevanlinna introduced a function T (r, f ), which is a positive convex function of log r and plays almost a similar role as that of the maximum modulus function for an entire function. The function T (r, f ) = m(r, f ) + N (r, f ) is called the characteristic function of the meromorphic function f. Another important quantity in Nevanlinna theory is the so called error term S(r, f ). For a meromorphic function f , S(r, f ) denotes any quantity that satisS(r, f ) → 0 as r → ∞ possibly outside an exceptional fies the following property: T (r, f ) set of finite Lebesgue measure. Since S(r, f ) is ultimately negligible in comparison to T (r, f ), it is sometimes called an unimportant error term. It is to be noted that S(r, f ) does not denote any particular quantity, rather it denotes any quantity that satisfies the above property. The two path breaking results of Rolf Nevanlinna changed the dimension of the value distribution theory: The First Fundamental Theorem and the Second Fundamental Theorem. Actually the first fundamental theorem can be taken as a restatement of PoissonJensen theorem and can be stated as follows. Theorem 1 (First Fundamental Theorem) Let f be a nonconstant meromorphic function in C and a ∈ C ∪ {∞}, then T (r, f ) = m(r, a; f ) + N (r, a; f ) + O(1), where O(1) denotes a suitable bounded quantity as r → ∞. The second fundamental theorem is probably one of the most outstanding results in the history of the value distribution theory, if not in the history of the complex function theory. We now state the following form of the theorem. Theorem 2 (Second Fundamental Theorem) Let f be a nonconstant meromorphic function in C and a1 , a2 , . . . , aq (q ≥ 2) be distinct finite complex numbers. Then m(r, ∞; f ) +

q 

m(r, aν ; f ) ≤ 2T (r, f ) − N1 (r, f ) + S(r, f ),

ν=1

where N1 (r, f ) = N (r, 0; f  ) + 2N (r, ∞; f ) − N (r, ∞; f  ). A combination of Theorems 1 and 2 gives the following more useful version of the second fundamental theorem. Theorem 3 Let f be a nonconstant meromorphic function in C. Let a1 , a2 , . . . , aq (q ≥ 2) be distinct finite complex numbers. Then (q − 1)T (r, f ) ≤

q  ν=1

N (r, aν ; f ) + N (r, ∞; f ) + S(r, f ).

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We shall mention other definitions, notations and results of Nevanlinna’s theory as and when required. Let us now discuss some preliminaries of the Wiman-Valiron theory as available in [32]. The Wiman-Valiron theory has been proved to be an essential tool to study the value distribution of entire solutions of complex differential equations. Let f be an ∞  an z n valid in C. entire function with Taylor expansion f (z) = Since the power series

∞ 

n=0

an r n converges for every r > 0, for a given r > 0

n=0

we have lim | an | r n = 0. So the maximum term μ(r, f ) = max | an | r n is wellr →∞

n≥0

defined. Also another concept, known as the central index, plays a vital role in the theory and is defined as ν(r, f ) = max{m :| am | r m = μ(r, f )}. If f is a polynomial of degree n with leading coefficient an , then clearly μ(r, f ) = | an | r n and ν(r, f ) = n for all sufficiently large values of r . For a transcendental entire function we can note the following properties of the maximum term and the central index: (1) | an | r n ≤ μ(r, f ) for n ≥ 0, (2) | an | r n < μ(r, f ) for n > ν(r, f ), (3) μ(r, f ) is strictly increasing for all sufficiently large values of f , is continuous and tends to +∞ as r → ∞, (4) ν(r, f ) is increasing, piecewise constant, right-continuous and tends to +∞ as r → ∞. For an entire function f the order and hyper-order are defined as σ( f ) = lim sup r →∞

log log M(r, f ) log log log M(r, f ) and σ2 ( f ) = lim sup , log r log r r →∞

where M(r, f ) = max | f (z) |. |z|=r

If we consider limit inferior instead of limit superior, then the corresponding quantities are respectively called the lower order λ( f ) and lower hyper-order λ2 ( f ). For a meromorphic function f , we shall replace log M(r, f ) by T (r, f ) in the above definitions. If, in particular, f becomes entire, then the definitions of order, lower order, hyper-order and lower hyper-order expressed in terms of M(r, f ) become equivalent to those expressed in terms of T (r, f ). For an entire function the above notions can also be expressed in terms of μ(r, f ) and ν(r, f ) (see for example [32, p. 51]), [18, 44]. Let us now state one of the most useful results of Wiman-Valiron theory. Theorem 4 ([32, p. 51]) Let f be a transcendental entire function, let 0 < δ < and z be such that | z |= r and that | f (z) |> M(r, f )ν(r, f )− 4 +δ 1

1 4

A Survey on a Conjecture of Rainer Brück

holds. Then there exists a set E ⊂ R+ of finite logarithmic measure i.e., such that   ν(r, f ) m (1 + o(1)) f (z) f (m) (z) = z

91



dt E t

< +∞

holds for m = 0, 1, 2, . . . and for all r ∈ / E, where o(1) denotes a quantity that tends to zero as r → +∞. Before concluding the section we recall the definitions of small function and value sharing. Given two meromorphic functions f and a, the function a is called a small function of f if T (r, a) = S(r, f ). Suppose that f , g and a be three meromorphic functions. If f − a and g − a have the same set of zeros, with counting multiplicities, then f and g are said to share the function a CM. Also if f − a and g − a have the same set of zeros, with ignoring multiplicities, then f and g are said to share the function a IM. If, in particular, a is a constant, then f and g are said to share the value a CM or IM.

2 Background of the Conjecture and the Work of Rainer Brück In this section we draw attention to a tiny paper of Rainer Brück [14], which opened a new stream of research on the uniqueness theory. The basic objective of the uniqueness theory is to study the relation between two entire or meromorphic functions, when they share certain values. We sometimes gets striking results, if instead of considering two functions, we consider one entire or meromorphic function and its derivative. Thus the study of such problem has become a prominent sub-field of the uniqueness theory. The following result of Rubel and Yang [42] is the first of this kind. However, one may find similar results in [21, 39]. Theorem 5 ([42]) Let f be a nonconstant entire function. If f and f  share two distinct values a, b ∈ C CM, then f  = f . Theorem 6 ([21]) Let f be a nonconstant entire function. If f and f  share two distinct values a, b ∈ C\{0} IM, then f  = f .  z z t e−e (1 − et )dt one may verify that Theorems 5 and Considering f (z) = ee 0

6 are not valid for a single value sharing. In fact we have f  (z) − 1 = e z ( f (z) − 1) and so f and f  share the value 1 CM but f  = f . Also we note that σ2 ( f ) = 1. Further counter examples are given by the solutions of the following differential n z equations f  (z) − 1 = e z ( f (z) − 1) and f  (z) − 1 = ee ( f (z) − 1). In this case we note respectively that σ2 ( f ) = n and σ2 ( f ) = ∞. Under this backdrop Rainer Brück [14] asked the following question: What conclusion can be made, if f and f  share only one value, and if an appropriate growth restriction on f is imposed? Accordingly he [14] proposed the following conjecture.

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Brück’s Conjecture: Let f be a nonconstant entire function. If σ2 ( f ) < ∞ and σ2 ( f ) is not a positive integer and if f and f  share a value a ∈ C CM, then f  − a = c( f − a) for some constant c ∈ C\{0}. R. Brück himself established the conjecture for a = 0. We now reproduce the corresponding result of Brück [14] and its proof, as the proof is simple and short. Theorem 7 ([14]) Let f be a nonconstant entire function with σ2 ( f ) < ∞ and σ2 ( f ) is not a positive integer. If f and f  share the value 0 CM, then f  = c f for some c ∈ C\{0}. Proof The hypothesis implies that f and f  have no zero. This implies f = eg for some entire function g and σ(g) = σ2 ( f ). In particular, we have σ(g) < ∞ and σ(g) is not a positive integer and g is not a constant. Since σ(g  ) = σ(g), Hadamard factorization theorem gives g  = c for some constant c ∈ C\{0}. Hence f  = c f and the theorem is proved. The problematic part is the case a = 0. Till date the conjecture is not resolved in its full generality for the case a = 0. The first attempt was made by R. Brück himself but under some hypothesis. His [14] result is the following Theorem 8 ([14]) Let f be a nonconstant entire function such that N (r, 0; f  ) = S(r, f ). If f and f  share the value 1 CM, then f  − 1 = c( f − 1) for some c ∈ C\{0}. f If for some nonzero constant a we apply Theorem 8 to the function , then we a get the general case. We now reproduce the proof of Theorem 8 from [14]. Proof For a nonconstant entire function f we have   f (1) N (r, 0; f (2) ) = N r, (2) + N1 (r, 0; f (1) ) f   f (2) ≤ T r, (1) + N1 (r, 0; f (1) ) + O(1) f ≤ N (r, 0; f (1) ) + S(r, f ), where N1 (r, 0; f (1) ) denotes the counting function of the zeros of f (1) , where a zero of multiplicity p is counted p − 1 times.   f (2) (2) Hence by the hypothesis we get N (r, 0; f ) = S(r, f ) and T r, (1) = f S(r, f ). f (3) 2 f (1) f (2) 2 f (2) Let F = (2) − (1) − (1) + . Then F is meromorphic and f f f −1 f −1 m(r, F) = S(r, f ). Since the poles of F are simple and are contributed only by the zeros of f (1) and f (2) , we have N (r, F) ≤ N (r, 0; f (1) ) + N (r, 0; f (2) ) = S(r, f ). Thus we have T (r, F) = S(r, f ). Let F ≡ 0. If f (z 0 ) = 1 for some z 0 ∈ C, then

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f (1) (z 0 ) = 1 and f (2) (z 0 ) = 0. Also it is easy to verify that F is regular at z 0 and F(z 0 ) = 0. Hence N (r, 1; f ) ≤ N (r, 0; F) = S(r, f ). Since f and f  share the value 1 CM, we get by the second fundamental theorem applied to f  T (r, f  ) ≤ N (r, 0; f  ) + N (r, 1; f  ) + N (r, ∞; f  ) + S(r, f  ) = S(r, f ).  Therefore m(r, 1; f ) ≤ m r,

   f 1 + m r,  = S(r, f ). So by the first funf −1 f damental theorem we obtain T (r, f ) = m(r, 1; f ) + N (r, 1; f ) + O(1) = S(r, f ), a contradiction. Therefore F ≡ 0 and on integration we get f (2) c (1) = f



f (1) − 1 f −1

2

for some constant c ∈ C\{0}. f (2) (z) f (2) (z 0 )

≡ c. If f (z 0 ) = 1 for some z 0 ∈ C, then (1) = c. We assume that (1) f (z) f (z 0 ) Hence     f (2) f (2) N (r, 1; f ) ≤ N r, c; (1) ≤ T r, (1) = S(r, f ). f f So as above we get T (r, f ) = S(r, f ), a contradiction. Therefore so

f −1 is a constant. This proves the theorem. f −1

f (2) (z) ≡ c and f (1) (z) 

Using the following two examples, Brück [14] remarked that his conjecture ceases to hold if f and f  share one value IM. Example 1 Let f (z) = z 2 . Then f  (z) = 2z and so f and f  share the value 0 IM. Also f  = c f does not hold for any c ∈ C\{∞}.  z  z2 t2 e− 2 (1 − t)dt + 1 . Then f (0) = 1 and f  (z) − Example 2 Let f (z) = e 2 0

1 = z( f (z) − 1). Hence f and f  share the value 1 IM because 0 is a simple 1-point of f but a double 1-point of f  . Also the inference of the conjecture does not hold. As we have noted that the conjecture of Brück is completely resolved for sharing the value 0, the attention of researchers are focused on the case of sharing a nonzero finite value. In the next sections we discuss the contributions of several authors on the topic.

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3 Brück’s Conjecture for Functions of Finite Order and Extensions The first attempt to resolve Brück’s conjecture for nonzero value sharing without any hypothesis on the zeros of f or f  was done by Gundersen and Yang [24]. They proved the conjecture for entire functions of finite order. It is to be noted that all entire functions of finite order have zero hyper-order. The following is the result of Gundersen and Yang [24]. Theorem 9 Let f be a nonconstant entire function of finite order. If f and f  share a nonzero finite value a CM, then f  − a = c( f − a) for some constant c ∈ C\{0}. In order to reproduce the proof of Theorem 9 we need the following lemmas. Lemma 1 ([22]) Let F be a nonconstant meromorphic function of finite order ρ, and let ε > 0 be a given constant. Then there exists a set E ⊂ [0, 2π) that has linear measure zero, such that if ψ0 ∈ [0, 2π)\E, then there is a constant  R 0 = R0 (ψ0 ) > 0  F (z)   ≤ |z|ρ−1+ε . such that for all z satisfying arg z = ψ0 and |z| ≥ R0 , we have  F(z)  Lemma 2 ([23]) Let F be an entire function and suppose that |F  (z)| is unbounded iφ on some ray arg z = φ. Then there exists an infinite  sequence of points z n = rn e   n)  where rn → +∞, such that F  (z n ) → ∞ and  FF(z  (z )  ≤ (1 + o(1))|z n | as z n → ∞. n Proof of Theorem 9. First we prove that if Q(z) is a nonconstant polynomial, then every solution F of the differential equation F  − e Q(z) F = 1

(1)

is an entire function of infinite order. Clearly every solution of (1) is entire. Let us suppose on the contrary that F is a solution of the equation (1) that has finite order ρ. From (1) we get F 1 − e Q(z) = . F F

(2)

Let ε > 0 be any given constant. Then from Lemma 1 there exists a set E ⊂ [0, 2π) with linear measure zero, such that for all z satisfying arg z = ψ0 and |z| ≥ R0 , we have     F (z)  ρ−1+ε   . (3)  F(z)  ≤ |z| Let θ ∈ [0, 2π)\E and for every α > 0    Q(r eiθ )   e rα

→ +∞

(4)

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as r → +∞. Then from (2), (3) and (4) it follows that F(r eiθ ) → 0

(5)

as r → +∞. We suppose that φ is a real number that satisfies φ ∈ [0, 2π) and for every β > 0 r β e Q(r e

iφ

)

→0

(6)

as r → +∞. We now show that |F  (z)| is bounded on the ray arg z = φ. On the contrary we suppose that |F  (z)| is not bounded on the ray arg z = φ. Then by Lemma 2 there exists an infinite sequence of points z n = rn eiφ , where rn → +∞, such that F  (z n ) → ∞ and    F(z n )    (7)  F  (z )  ≤ (1 + o(1))|z n | n as z n → ∞. Since F  (z n ) → ∞, it follows from (1) and (6) that F(z n ) → ∞. Then from (2), (6) and (7) we obtain that F  (z n ) → 1, which contradicts F  (z n ) → ∞. This contradiction proves that |F  (z)| must be bounded on the ray arg z = φ. By considering the formula 

z

F(z) = F(0) +

F  (w)dw,

0

we obtain |F(z)| ≤ |F(0)| + M|z|

(8)

for all z satisfying arg z = φ, where M = M(φ) > 0 is some constant. We see that (8) holds for any φ ∈ [0, 2π) with property (6) and that (5) holds for any θ ∈ [0, 2π)\E with property (4). Since Q(z) is a nonconstant polynomial, there exist only finitely many real number in [0, 2π) that do not satisfy either (6) or ( 4). We also note that the set E has linear measure zero. Therefore, since F has finite order, it can be deduced from (5), (8), the Phragm´en-Lindel¨of theorem [38, pp. 270–271] and Liouville’s theorem, that F must be a polynomial with deg F ≤ 1 . But this is impossible as Q(z) is nonconstant in (1). Therefore F must be of infinite order. Since f is of finite order and f and f  share the nonzero value a CM, by Hadamard factorization theorem we get f −a = e Q(z) , (9) f −a where Q(z) is a polynomial. We put F =

f − 1. Then from (9) we get a

F  − e Q(z) F = 1.

(10)

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If Q(z) is nonconstant, then from above and from (10) we see that F has infinite order. This is impossible as f has finite order. So Q(z) is a constant and the theorem is proved. z Considering a function f satisfying f  (z) − 1 = ee ( f (z) − 1) (see [14]) we can verify that the hypothesis on the order of f cannot be dropped from Theorem 9. Also 2e z + z + 1 Theorem 9 is not valid for meromorphic functions. For, let f (z) = (see ez + 1 [24]). Then f and f  share the value 1 CM but f  − 1 = c( f − 1) does not hold for any c ∈ C\{0}. This example further raises a question of investigation of Brück’s conjecture for a meromorphic function. We shall consider this problem at the later part of the section. Chen and Shon [16] improved Theorem 9 by relaxing the hypothesis on the order of f . They [16] proved the following theorem. Theorem 10 Let f be a nonconstant entire function with σ2 ( f ) < 21 . If f and f  share a nonzero finite value a CM, then f  − a = c( f − a), where c ∈ C\{0}. It is worthy to mention that Wiman-Valiron theory is an effective device to prove a number of results that we discuss in this section. For example, we see in the proof of the following theorem of Lahiri and Das [27], which improves Theorem 9. Theorem 11 Let f be a nonconstant entire function with λ( f ) < 1 and σ2 ( f ) < ∞. Suppose that a = a(z) is a polynomial. If f and f (k) share a CM, then f (k) − a = c( f − a), where c is a nonzero constant. We require the following lemmas. Lemma 3 ([32, p. 9]) Let P(z) = bn z n + bn−1 z n−1 + · · · + b0 (bn = 0) be a polynomial of degree n. Then for every (> 0) there exists R(> 0) such that for all |z| = r > R we get (1 − )|bn |r n ≤ |P(z)| ≤ (1 + )|bn |r n . Following is a particular case of Theorem 4. Lemma 4 Let f be a transcendental entire function. Then there exists a set E ⊂ (1, ∞) with finite logarithmic measure such that for |z| = r ∈ / [0, 1] ∪ E and | f (z)| = M(r, f ) we get   f (k) (z) ν(r, f ) . = (1 + o(1)) f (z) z Lemma 5 ([32, p. 5]) Let g : (0, +∞) → R and h : (0, +∞) → R be monotone increasing functions such that g(r ) ≤ h(r ) outside of an exceptional set E of finite logarithmic measure. Then for any α > 1, there exists R > 0 such that g(r ) ≤ h(r α ) holds for r > R.

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Lemma 6 ([13]) Let u(z) be a nonconstant subharmonic function in the open complex plane C of lower order λ, 0 ≤ λ < 1. If λ < α < 1, then logdens{r : A(r ) > (cos απ)B(r )} ≥ 1 −

λ , α

where A(r ) = inf u(z) and B(r ) = sup u(z) |z|=r

|z|=r

Note 1 Since for an entire function f , log | f (z)| is a subharmonic function in C [20, p. 394], we can apply Lemma 6 to the function u(z) = log | f (z)|. Proof of Theorem 11. Since f (k) − a and f − a share 0 CM, there exists an entire function Q such that f (k) − a = eQ. (11) f −a We now consider the following cases. Case 1. Let σ( f ) < ∞. Then from (11) we see that Q is a polynomial. If σ( f ) < 1, then (11) implies that Q is a constant. So we suppose that σ( f ) ≥ 1 and so f is transcendental. Now for any z with | f (z)| = M(r, f ), noting that f is transcendental, we get by Lemma 3 M(r, a) 2|β|r deg a a(z) |≤ ≤ →0 (12) | f (z) M(r, f ) M(r, f ) as r → ∞, where β is the leading coefficient of a = a(z). From (11) we get f (k) − af f Q e = . 1 − af

(13)

Now by Lemma 4 there exists E ⊂ (1, ∞) with finite logarithmic measure such that for all large |z| = r ∈ / [0, 1] ∪ E and | f (z)| = M(r, f ) we get in view of (12) and (13)   ν(r, f ) k e Q(z) = (1 + o(1)) . (14) z Now from (14) we get for all large |z| = r ∈ / [0, 1] ∪ E with | f (z)| = M(r, f ) |Q(z)| = | log e Q(z) |   ν(r, f ) k | + o(1) = | log z = |k log ν(r, f ) − k log z| + o(1) ≤ k log ν(r, f ) + k log r + 6kπ < 2k(σ( f ) + 1) log r + 6kπ.

(15)

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Also by Lemma 3 we obtain for all large |z| = r 1 |δ|r deg Q ≤ |Q(z)|, 2

(16)

where δ is the leading coefficient of Q. Now (15) and (16) together imply deg Q = 0 and so Q is a constant. Case 2. Let σ( f ) = ∞. We now consider the following subcases. Subcase 2.1. Let Q be a polynomial. Then from (14) we get for all large |z| = r ∈ / [0, 1] ∪ E with | f (z)| = M(r, f ) |Q(z)| ≤ k log ν(r, f ) + k log r + 6kπ.

(17)

We suppose that Q is nonconstant. Then from (16) and (17) we obtain for all large |z| = r ∈ / [0, 1] ∪ E with | f (z)| = M(r, f ) 1 |δ|r deg Q ≤ k log ν(r, f ) + k log r + 6kπ. 2 So for all large |z| = r ∈ / [0, 1] ∪ E we get 1 |δ|r deg Q ≤ k log ν(r, f ) + k log r + 6kπ. 2 Hence by Lemma 5 for given α, 1 < α < 23 , we get for all large values of r 1 |δ|r deg Q ≤ k log ν(r α , f ) + kα log r + 6kπ 2 

and so r deg Q

1 kα log r |δ| − deg Q 2 r



≤ k log ν(r α , f ) + 6kπ.

This gives deg Q ≤ αλ2 ( f ) = 0, because λ( f ) < 1 implies that λ2 ( f ) = 0. This is a contradiction. Therefore Q is a constant. Subcase 2.2. Let Q be a transcendental entire function. Suppose that H = {r : A(r ) > (cosαπ)B(r )}, where A(r ) = inf log | f (z)|, B(r ) = sup log | f (z)| and |z|=r

|z|=r

λ(Q) < α < We note by (11) that λ(Q) ≤ λ2 ( f ) = 0. Then by Lemma 6, H has infinite logarithmic measure. Also by Lemma 2, for |z| = r ∈ H \{[0, 1] ∪ E} with | f (z)| = M(r, f ) we get 1 . 2

  ν(r, f ) k f (k) (z) = (1 + o(1)) . f (z) z

(18)

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Now by (12), (13) and (18) we get for all large |z| = r ∈ H \{[0, 1] ∪ E} with | f (z)| = M(r, f )   ν(r, f ) k e Q(z) = (1 + o(1)) z and so |Q(z)| = | log e Q(z) |   ν(r, f ) k | + o(1) = | log z = |k log ν(r, f ) − k log z| + o(1) ≤ k log ν(r, f ) + k log r + 6kπ < 2kr σ2 ( f )+1 .

(19)

So by (19) and by Lemma 6 there exists a constant d, 0 < d ≤ 1, such that (M(r, Q))d ≤ 2kr σ2 ( f )+1 for all large values of |z| = r ∈ H \{[0, 1] ∪ E} and | f (z)| = M(r, f ). This is impossible because Q is transcendental and so (M(r, Q))d lim = ∞. This proves the theorem.  r →∞ r σ2 ( f )+1 Let us give our attention once again to Theorem 9. A natural and immediate extension of Theorem 9 is to prove it for the general order derivative of an entire function. Using a similar technique of [24], Yang [46] proved it as a consequence of the following theorem. Theorem 12 ([46]) Let Q be a nonconstant polynomial and k be a positive integer. Then every solution of the differential equation F (k) − e Q F = 1 is an entire function of infinite order. We now state the result of Yang [46] on Brück’s conjecture along with its proof. Theorem 13 Let f be a nonconstant entire function of finite order and a be a nonzero finite value. If f and f (k) share the value a CM, then f (k) − a = c( f − a) for some c ∈ C\{0}. Proof Since f has finite order and f , f (k) share the value a CM, by Hadamard factorization theorem we get f (k) − a = e Q ( f − a), where Q is a polynomial. We f put F = − 1. Then from above we get F (k) − e Q F = 1. If Q is nonconstant, then a by Theorem 12 we see that F and so f is of infinite order, a contradiction. Therefore Q is a constant and the theorem is proved.  In 2004 Wang [43] extended Theorem 13 by considering a polynomial sharing instead of a value sharing and proved the following result. Theorem 14 Let f be a nonconstant entire function of finite order, Q be a polynomial of degree ≥ 1 and let k be a positive integer. If f and f (k) share Q CM, then for some nonzero finite constant c, f (k) − Q = c( f − Q).

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The proof of Theorem 14 is in the line of Theorem 12 and Theorem 13 with necessary modifications. In 2008 Li and Cao [33] considered two shared polynomials and proved the following result. Theorem 15 Let Q 1 and Q 2 be two nonzero polynomials and P be a polynomial. If f is a nonconstant entire solution of the equation f (k) − Q 1 = ( f − Q 2 )e P , then σ2 ( f ) = deg P, where deg P denotes the degree of P. If, further, one assumes that σ2 ( f ) < ∞ and σ2 ( f ) is not a positive integer, then Theorem 15 implies that P is a constant and so f (k) − Q 1 = c( f − Q 2 ) for some c ∈ C\{0}. In 2010 Li and Yi [34] considered the problem of improvement of Theorem 14 with sharing of a rational function. The following result of Li and Yi [34] requires that the shared rational function must have a pole. However we note that a rational function without any pole is a polynomial and the case is already dealt in Theorem 14. Theorem 16 Let f be a nonconstant entire function of finite order and R( ≡ 0) be a rational function that has at least one pole. If f and f (k) share R CM, then f (k) = f , where k is a positive integer. We note that the conclusion of Theorem 16 is better than that of Theorem 14. The only reason behind this is the existence of at least one pole of R. For, if f (k) − R = c( f − R) for some constant c ∈ C\{0}, then f (k) − c f = (1 − c)R and so c = 1 because f is entire but R has a pole. The result of G. G. Gundersen and L. Z. Yang (Theorem 9) was the first attempt to establish Brück’s conjecture for nonzero value sharing without any further hypothesis on the zeros of f  . It is also noted that Theorem 9 is not valid for meromorphic functions. If we have a careful look on the counter example, we can notice that the function has infinitely many poles. So it becomes a natural query to examine the validity of Theorem 9 for a meromorphic function having finitely many poles. Also in Theorems 14 and 16 we see that the shared values has been replaced by a polynomial and a rational function respectively. So further research demands that one should consider a more general shared entire or a shared meromorphic function. In 2009 Chang and Zhu [15] addressed both the problems and proved the following two theorems. Theorem 17 Let f and a be two entire functions such that σ(a) < σ( f ) < ∞. If f and f  share the function a CM, then f  − a = c( f − a) for some constant c ∈ C\{0}. Theorem 18 Let f and a be two meromorphic functions such that σ(a) < σ( f ) < ∞. Suppose that f and a have only finite number of poles and they do not have any common pole. If f and f  share the function a CM, then f  − a = c( f − a) for some constant c ∈ C\{0}. The hypothesis on the order of the shared function is necessary as we see in the following example.

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Example 3 Let f (z) = e2z − (z − 1)e z and a = e2z − ze z . Then σ(a) = σ( f ) = 1 and f  − a = e z ( f − a) . In the year 2010, Li and Yi [34] extended Theorems 17 and 18 to the general order derivatives. We note that a linear differential polynomial is a natural extension of a derivative. Mao [37] considered the problem of extension of Brück’s conjecture to a linear differential polynomial. His [37] first result may be stated as follows. Theorem 19 Let P be a nonzero polynomial, Ak ( ≡ 0), . . . , A0 be polynomials and deg A j − deg Ak , 0} and σ2 ( f ) < f be an entire function with σ( f ) > 1 + max { 0≤ j≤k−1 k− j 1 . If f and L( f ) = Ak f (k) + · · · + A1 f (1) + A0 f share P CM, then L( f ) − P = 2 c( f − P) for some constant c ∈ C\{0}, where deg A j denotes the degree of A j and k is a positive integer. One can easily verify that Theorem 19 holds good for σ( f ) < 1. Following examdeg A j − deg Ak , 0} ples show that the condition σ( f ) < 1 or σ( f ) > 1 + max { 0≤ j≤k−1 k− j is crucial. Example 4 ([37]) Let f (z) = e−z + z and L( f ) = f (2) + 2 f (1) + f . Then L( f ) − z = 2e z ( f − z) and σ( f ) = 1. 1 z 1 z2 Example 5 ([37]) Let f (z) = e− 2 + z 2 and L( f ) = f (2) + f (1) + f . Then 3 3 3 deg A j − deg A2 2 z2 , 0}. L( f ) − z 2 = e 2 ( f − z 2 ) and σ( f ) = 1 + max { 0≤ j≤1 3 2− j Mao [37] also noticed that if the first derivative is dropped from L( f ), then one gets a better result. Theorem 20 Let Ak ( ≡ 0), . . . , A2 be polynomials and f be a nonconstant entire function with σ2 ( f ) < ∞, where σ2 ( f ) is not a positive integer. If f and L( f ) = Ak f (k) + · · · + A2 f (2) + f share z CM, then L( f ) − z = c( f − z) for some c ∈ C\{0}, where k(≥ 2) is an integer. Let f be an entire function of order σ = σ( f ), 0 < σ < ∞. Then the quantity τ ( f ) = lim sup r →∞

log M(r, f ) rσ

is called the type of f . Xu and Yang [44] considered the result of Mao [37], where they replaced the shared polynomial by an entire function a with 0 < σ(a) = σ( f ) < ∞ and proved the following result. Theorem 21 Let f and a be two nonconstant entire functions such that 0 < σ(a) = σ( f ) < ∞ and τ ( f ) > τ (a). Further let P be a polynomial such that

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deg A j − deg Ak , 0}, where L( f ) = Ak f (k) + · · · + 0≤ j≤k−1 k− j A1 f (1) + A0 f and Ak ( ≡ 0), . . . , A1 , A0 are polynomials. If f is a nonconstant entire solution of L( f ) − a = ( f − a)e P , then P is a constant.

σ( f ) > deg P + max {

Recently another extension of a result of J. Chang and Y. Zhu (Theorem 17) is done by Li and Yi [35]. They [35] proved the following theorem. Theorem 22 Let f be a nonconstant entire function such that σ( f ) < ∞ and a( ≡ 0) be an entire function such that σ(a) < σ( f ). Further suppose that L( f ) = f (k) + ak−1 f (k−1) + · · · + a1 f (1) + a0 f , where k is a positive integer and a0 , a1 , . . . , ak−1 are complex numbers. If f and L( f ) share the function a CM, then σ( f ) = 1 and one of the following two cases occurs: (i) L( f ) − a = c( f − a) for some c ∈ C\{0}, (ii) f is a solution of the equation L( f ) − a = ( f − a)eαz+β such that σ( f ) = μ( f ) = 1, where a0 , a1 , . . . , ak−1 are not all zero and α( = 0), β are complex numbers. The attempt to extend the result of J. Chang and Y. Z. Zhu on meromorphic functions (Theorem 18) has not yet been found in the literature. Lahiri and Das [28] extensively used the Wiman-Valiron theory to extend Theorem 18 to a linear differential polynomial with constant coefficients. Their result (see [28]) may be stated as follows. Theorem 23 Let f and a be meromorphic functions, both having finite number of poles. Suppose that f and a do not have any common pole and σ(a) < σ( f ) < ∞. Let L( f ) = a0 f + a1 f (1) + · · · + ak f (k) , where k(≥ 1) is an integer and a0 , a1 , . . . ak ( = 0) are complex numbers. If f and L( f ) share the function a CM, then L( f ) − a = c( f − a), where c ∈ C\{0}, provided that one of the following holds: (i) σ( f ) = 1 or (ii) σ( f ) = 1 and a0 = a1 = · · · = ak−1 = 0. Example 4 shows that the condition (ii) of Theorem 23 is essential. Before concluding the section this is worthy to draw attention of the reader to the fact that all the results discussed here are somewhat inspired by the first theorem of Brück (Theorem 7) but for a nonzero shared value or a nonzero shared function. In the next section we discuss the research that is stimulated by the second theorem of R. Brück (Theorem 8).

4 Brück’s conjecture and Nevanlinna theory We now discuss the development of research on Brück’s conjecture using Nevanlinna theory as the main tool. At this point we need the ideas of deficiency and ramification index.

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Let f be a meromorphic function and a ∈ C ∪ {∞}. Then the quantities δ(a; f ) = 1 − lim sup r →∞

N (r, a; f ) N (r, a; f ) and (a; f ) = 1 − lim sup T (r, f ) T (r, f ) r →∞

are respectively called the deficiency and the ramification index of the value a for the function f . Using the definitions and the second fundamental theorem it can be proved that the set {a ∈ C ∪ {∞} : (a; f ) > 0} is countable and 0≤

 a∈C∪{∞}

δ(a; f ) ≤



(a; f ) ≤ 2.

a∈C∪{∞}

We start the section with a result of Yu [47], where he used the deficiency of the value zero for an entire function to obtain a result on sharing of a single function. Theorem 24 Let f be a nonconstant entire function and a( ≡ 0, ∞) be a meromorphic function such that T (r, a) = o(T (r, f )) as r → ∞. If for a positive integer k, 3 f and f (k) share the function a CM and δ(0; f ) > , then f = f (k) . 4 He [47] also proved the following theorem for meromorphic functions having at least one pole. Theorem 25 Let f be a nonconstant nonentire meromorphic function and a( ≡ 0, ∞) be a meromorphic function such that f and a do not have any common pole and T (r, a) = o(T (r, f )) as r → ∞. If for a positive integer k, f and f (k) share the function a CM and 4δ(0; f ) + 2(8 + k)(∞; f ) > 19 + 2k, then f = f (k) . In 2004 Liu and Gu [36] improved the results of Yu [47] and answered one of his questions. We now state the result of Liu and Gu [36]. Theorem 26 Let k be a positive integer and f be a nonconstant meromorphic function, a( ≡ 0, ∞) be a meromorphic small function of f . If f (k) and a do not have any common pole of same multiplicity and f and f (k) share the function a CM and 2δ(0; f ) + 4(∞; f ) > 5, then f = f (k) . Since for an entire function (∞; f ) = 1, from Theorem 26 we obtain the following theorem that answers a question of Yu [47]. 1 Theorem 27 Let f be a nonconstant entire function with δ(0; f ) > and a( ≡ 2 0, ∞) be a meromorphic small function of f . If k is a positive integer and f and f (k) share the function a CM, then f = f (k) . We now give a sketch of proof of Theorem 26 considering it as a sample, also as it answers a question of K. W. Yu. For the proof we need the following lemmas. Lemma 7 ([45]) Let f 1 and f 2 be two nonconstant meromorphic functions and let c1 , c2 , c3 be nonzero constants. If c1 f 1 + c2 f 2 = c3 , then

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T (r, f 1 ) ≤ N (r, 0; f 1 ) + N (r, 0; f 2 ) + N (r, ∞; f 1 ) + S(r, f 1 ). Lemma 8 ([45]) Let f i (i = 1, 2, . . . , n) be n linearly independent meromorphic n  f j ≡ 1, then for 1 ≤ j ≤ n we have functions. If j=1

T (r, f j ) ≤

n 

N (r, 0; f i ) + N (r, f j ) + N (r, D) −

i=1

n 

N (r, f i ) − N (r, 0; D) + S(r ),

i=1

where D is the Wronskian determinant W ( f 1 , f 2 , . . . , f n ), S(r ) = o(T (r )) as r → +∞ possibly outside a set of finite linear measure and T (r ) = max T (r, f j ). 1≤ j≤n

Lemma 9 ([36]) Let f be a meromorphic function in the complex plane. If for a positive integer k, f (k) ≡ 0, then N (r, 0; f (k) ) ≤ T (r, f (k) ) − T (r, f ) + N (r, 0; f ) + S(r, f ). Proof of Theorem 26. We suppose that f = f (k) and put f (k) − a = h. f −a

(20)

If h ≡ c( = 1), a constant, then f (k) cf − =1−c a a and so by Lemmas 7 and 9 we get T (r, f

(k)

  f (k) + S(r, f ) ) ≤ T r, a       a a f (k) ≤ N r, (k) + N r, + N r, + S(r, f ) f f a ≤ T (r, f (k) ) − T (r, f ) + 2N (r, 0; f ) + N (r, f ) + S(r, f ),

which implies T (r, f ) ≤ 2N (r, 0; f ) + N (r, f ) + S(r, f ). Hence 2δ(0; f ) + (∞; f ) ≤ 2, a contradiction. So we suppose that h is nonconstant. We put f (k) hf , f2 = − and f 3 = h. Then (20) implies f1 = a a f 1 + f 2 + f 3 = 1.

(21)

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If f 1 , f 2 , f 3 are linearly independent, then by Lemma 8 we have

  f (k) T (r, f (k) ) ≤ T r, + S(r, f ) a         a 1 hf a + N r, + N (r, D) − N r, − N (r, h) + S(r ). ≤ N r, (k) + N r, fh h a f

Also we note that

    hf hf N (r, D) − N r, − N (r, h) ≤ 2N r, + 2N (r, h). a a

By the hypothesis  that h has no zero and a pole of h must be a pole of f or  it follows hf ≤ N (r, f ) + S(r, f ) and N (r, h) ≤ N (r, f ) + S(r, f ). of a = a(z). So N r, a Hence from above we get T (r, f (k) ) ≤ N (r, 0; f (k) ) + N (r, 0; f ) + 2N (r, 0; h) + 2N (r, h) + 2N (r, f ) + S(r, f ),

which implies by Lemma 9, T (r, f ) ≤ 2N (r, 0; f ) + 4N (r, f ) + S(r, f ). This ultimately leads to a contradiction. Finally we suppose that f 1 , f 2 , f 3 are linearly dependent. So there exists constants c1 , c2 , c3 , not all zero, such that c1 f 1 + c2 f 2 + c3 f 3 ≡ 0.

(22)

ac3 , which is impossible c2 as a = a(z) is a small function of f . Hence we can deduce from (22) Clearly c1 = 0. For, otherwise we get from (22) f =

(c2 − c1 )

hf + (c1 − c3 )h = c1 . a

We suppose that c2 − c1 = 0 and c1 − c3 = 0. Then from (23) we get 1 c1 − c3 c1 − c2 f · + = . c1 a h c1 So by Lemma 7 we get   f T (r, f ) ≤ T r, + S(r, f ) a     a f + N (r, h) + N r, + S(r, f ) ≤ N r, f a ≤ N (r, 0; f ) + 2N (r, f ) + S(r, f ), which ultimately leads to a contradiction.

(23)

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If c2 − c1 = 0 and c1 − c3 = 0, then from (23) we get h =

c1 , which is a c1 − c3

contradiction as h is nonconstant. hf = c1 and so If c2 − c1 = 0 and c1 − c3 = 0, then from (23) we get (c2 − c1 ) a (21) implies c2 f (k) +h = . a c2 − c1 If c2 = 0, then by Lemmas 7 and 9 we can deduce that T (r, f ) ≤ N (r, 0; f ) + N (r, f ) + S(r, f ), which ultimately leads to a contradiction. If c2= 0,then we have f (k) + ah = 0 and so by (20) we get f f (k) = a 2 . We note that N r,

1 f

≤ N (r, 0; f f (k) ) and

          1 f f (k) 1 1 1 ≤ m r, 2 ≤ m r, + m r, = m r, + S(r, f ). m r, f f f2 f f (k) f f (k)

From these two inequalities we can deduce by the first fundamental theorem T (r, f ) ≤ T (r, f f (k) ) + S(r, f ) = T (r, a 2 ) + S(r, f ) = S(r, f ), a contradiction. This proves the theorem.  Afterwards a lot of work has been done on the paper of Yu [47]. Most of those aim on the relaxation of the nature of sharing and the hypothesis on the deficiency. A. H. H. Al-Khaladi contributed a number of significant results on Brück’s conjecture using Nevanlinna theory, which demand separate attention for their own merit. He [2] observed by the following example that the second result of Brück (Theorem 8) is not valid for a shared small function. ez . Then a is a small function ez − 1  of f , and f and f share the function a CM. Also N (r, 0; f  ) = 0 but f − a = e−z ( f  − a).

Example 6 Let f (z) = 1 + exp(e z ) and a(z) =

This counter example opened a new avenue of investigation of Brück’s conjecture for a shared small function. Following is the first result proved by A. H. H. Al-Khaladi [2] in this direction. Theorem 28 Let f be a nonconstant entire function satisfying N (r, 0; f  ) = S(r, f ) and a( ≡ 0, ∞) be a meromorphic small function of f . If f and f  share the function α α a CM, then f − a = (1 − )( f  − a), where 1 − = eβ , α is a constant and β is a a an entire function.

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α 1 − eβ and a ≡ 0, ∞ imply that β is a nonzero constant. So we obtain the following result from Theorem 28. If in Theorem 28 we assume a to be an entire small function of f , then a =

Theorem 29 Let f be a nonconstant entire function satisfying N (r, 0; f  ) = S(r, f ) and let a( ≡ 0, ∞) be an entire small function of f . If f and f  share the function a CM, then f  − a = c( f − a) for some c ∈ C\{0}. Therefore by virtue of Theorem 29 we see that Theorem 8 is valid for a shared entire small function. At this point it is relevant to note that the shared small function as considered in Example 6 is meromorphic having infinitely many poles. We now reproduce the proof of Theorem 28 as it is the beginning of a series of papers of A. H. H. Al-Khaladi on Brück’s conjecture. We require the following lemmas. Lemma 10 ([25, p. 50]) If f and g be two transcendental entire functions, then T (r, g) = 0. lim r →∞ T (r, f (g)) Lemma 11 ([45, p. 96] Let f j ( j = 1, 2, 3, 4) be meromorphic functions and fk (k = 4  1, 2) be nonconstant satisfying f j ≡ 1. If j=1 4 

N (r, 0; f j ) + 3

j=1

4 

N (r, f j ) < (λ + o(1))T (r, f k )

j=1

for r ∈ I and k = 1, 2, where 0 < λ < 1 and I is a set of infinite linear measure. Then f 3 ≡ 1, f 4 ≡ 1 or f 3 + f 4 ≡ 1. Lemma 12 ([25, p. 47] Let f be a nonconstant meromorphic function and a1 , a2 , a3 be distinct small functions of f , then T (r, f ) ≤

3 

N (r, 0; f − a j ) + S(r, f ).

j=1

Proof of Theorem 28. If a is a finite nonzero constant, then it is done by Theorem 8. So we suppose that a is a nonconstant meromorphic function. Since f and f  share the function a CM, there exists an entire function β such that (24) f − a = eβ ( f  − a). Differentiating (24) we obtain a

1  β  eβ  eβ   β β + 1 e + f −  f −  f = 1. a a a a

(25)

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We now consider the following cases. 

Case 1. Let aa β  + 1 eβ ≡ c, a constant. If c = 0, then on integration we get a = Ae−β , where A is a nonzero constant. So T (r, eβ ) = S(r, f ). Also if c = 0, then clearly by Lemma 10 we get T (r, eβ ) = S(r, f ).   1 1   = Case 2. Let a  f be a constant. Then T (r, f ) = S(r, f ). Hence N r, f −a   1 N r,  = S(r, f ) and f −a  m r,

1 f −a



    f  − a 1 = S(r, f ). ≤ m r, + m r,  f −a f − a

So by the first theorem we get T (r, f ) = S(r, f ), a contradiction.

fundamental  Case 3. Let aa β  + 1 eβ and a1 f  be nonconstant. We note that           1 f 1 f  1 N r,  ≤ N r,  + N r,  ≤ T r,  + N r,  + O(1) ≤ 2N (r, 0; f  ) + S(r, f ) = S(r, f ). f f f f f

We now apply Lemma 11 to (25) and consider the following sub-cases. Sub-case 3.1 Let −

β  eβ  f ≡ 1. Then from (25) we get a f  e−β f  − = 1. aβ  + a  aβ  + a 

From this and the second fundamental theorem for H =

f  we get aβ  + a 

T (r, H ) ≤ N (r, 0; H ) + N (r, 1; H ) + N (r, H ) + S(r, H )

≤ N (r, 0; f  ) + N (r, aβ  + a  ) + N (r, 0; f  ) + N (r, aβ  + a  ) + N (r, f  ) +N (r, 0; aβ  + a  ) + S(r, f )

= S(r, f )

and so T (r, f  ) = S(r, f ). Therefore     f f  T (r, f  ) ≤ T r,  + T (r, f  ) = T r,  + S(r, f ) = S(r, f ) f f and so as in Case 2 we see that T (r, f ) = S(r, f ), a contradiction. eβ Sub-case 3.2 Let −  f  ≡ 1. Similarly as Sub-case 3.1 we arrive at a contradiction. a eβ β  eβ Sub-case 3.3 Let −  f  −  f  ≡ 1. Then from (25) we get a a

A Survey on a Conjecture of Rainer Brück

f  = −(aβ  + a  )eβ .

109

(26)

Differentiating (26) we obtain f  = −eβ (a  + 2a  β  + aβ  + aβ 2 ).

(27)

Eliminating f  and f  from (26), (27) and the given identity we get e−2β =

2aβ 2 a  a   + 3β + β + . a a a

This implies T (r, eβ ) = S(r, f ). Hence we prove that in (24), eβ is a small function of f . Now rewriting (24) as f  = e−β ( f − b),

(28)

 where b = a(1 − eβ ) is a small function of f , we get N r,

 1 = N (r, 0; f  ) = f −b S(r, f ). Putting F = f − b and using the second fundamental theorem we get       1 1 1 + N r, + N (r, F) − N r,  + S(r, F) T (r, F) ≤ N r, F F −1 F     1 1 − N r,  + S(r, f ) ≤ N r, F −1 F   1 ≤ T (r, F) − N r,  + S(r, f ) F

 and so N r,

1  f − b

 = S(r, f ). Hence by Lemma 12 we get if b ≡ 0

T (r, f  ) ≤ N (r, 0; f  ) + N (r, 0; f  − b ) + N (r, f  ) + S(r, f  ) = S(r, f ), which contradicts (28). So b ≡ 0 and a(1 − eβ ) = α, where α is a constant. This proves the theorem.  In 2005 Al-Khaladi [3] extended Theorem 28 to the general order derivative of an entire function. Also he relaxed the hypothesis on the zeros of the derivative. We now state the result of Al-Khaladi [3]. Theorem 30 Let f be a nonconstant entire function satisfying N (r, 0; f (k) ) = (k) S(r, f ) and a( ≡ 0, ∞) be a meromorphic small  function of f . If f and f share  P Pk−1 k−1 ( f (k) − a), where 1 − = eβ , the function a CM, then f − a = 1 − a a Pk−1 is a polynomial of degree at most k − 1 and β is an entire function.

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Pk−1 and 1 − eβ a ≡ 0, ∞ imply that β is a nonzero constant and so a is a polynomial (including a constant). Hence from Theorem 30 we get f (k) − a = c( f − a) for some constant c ∈ C\{0}. In 1998 Zhang [48] extended the second theorem of R. Brück (Theorem 8) to meromorphic functions and proved the following result. Again if we suppose that a is an entire small function of f , then a =

Theorem 31 Let f be a nonconstant meromorphic function. If f and f  share a value a( = 0, ∞) CM and if N (r, f ) + N (r, 0; f  ) < λT (r, f  ) + S(r, f  ) for some 1 real constant λ, 0 < λ < , then f  − a = c( f − a) for some c ∈ C\{0}. 2 Al-Khaladi [4] also extended Theorem 8 to meromorphic functions. Theorem 32 Let f be a nonconstant meromorphic function satisfying N (r, 0; f  ) = S(r, f ). If f and f  share a value a( = 0, ∞) CM, then f  − a = c( f − a) for some c ∈ C\{0}. Al-Khaladi [4] exhibited by the following two counter examples that the CM shared value cannot be replaced by an IM shared value and the hypothesis N (r, 0; f  ) = S(r, f ) is essential. Example 7 Let f (z) = 1 + tan z. In this case, we have f  (z) = sec2 z and f  − 1 = ( f − 1)2 , so that f and f  share the value 1 IM and N (r, 0; f  ) = 0 but we do not have the conclusion of Theorem 32. ze z f (z) − 1 Example 8 Let f (z) = . Then f  (z) − 1 = and N (r, 0; f  ) = z 1+e 1 + ez N (r, 0; e z + z + 1) = S(r, f ). Hence f and f  share the value 1 CM but the conclusion of Theorem 32 does not hold. Let us exhibit an outline of the proof of Theorem 32 as it presents the meromorphic version of the second theorem of Brück (Theorem 8). We need the following lemmas. Lemma 13 ([4]) Let f be a nonconstant meromorphic function satisfying N (r, 0; f  ) + N (2 (r, f ) = S(r, f ), where we denote by N (2 (r, f ) the reduced count ing function  of the  multiple poles of f . If f and f sharethe value 1 CM, then 1 1 1 = S(r, f ), where N(2 r,  denotes the counting N(2 r,  + m r,  f f −1 f function of multiple zeros of f  . Lemma 14 ([4]) Let f be a meromorphic function such that f  is not a constant. If f (z) = 1 whenever f  (z) = 1, then either  N1) r,

1  f −1



≤ m(r, f ) + N (r, 0; f  ) + S(r, f )

 or f  − 1 = c( f − 1), where c( = 0) is a constant and N1) r, counting function of simple zeros of f  − 1.

 1 f  −1

denotes the

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Lemma 15 ([4]) Let f be a meromorphic function such that f  is not a constant. c2 + c3 , where c1 , c2 ( = 0), c3 are constants, or Then either f (z) = z + c1     1 1 N1) (r, f ) ≤ N (2 (r, f ) + N1) r,  + N r,  + S(r, f ). f f Proof of Theorem 32. Without loss of generality we take a = 1. Suppose that Theorem 32 does not hold. Then by Theorem 31 we get T (r, f  ) ≤ 2N (r, 0; f  ) + 2N (r, f ) + S(r, f ).

(29)

Since N (r, 0; f  ) = S(r, f ), we have N (r, f ) = N1) (r, f ) + S(r, f ).

(30)

      f  f  f  2 f f − + − 2 and F = 2 . f f f −1 f −1 f By the hypothesis and the theorem on logarithmic derivative we get in view of (30) that T (r, W ) + T (r, F) = S(r, f ). F 2 (z) + 2F  (z) + W (z) . Then we have T (r, a) = S(r, f ). Also Let a(z) = 2 F (z) + 2F  (z) − W (z) using the hypothesis it can be proved that if z 0 is a simple zero of f  and f  (z 0 ) = 0, then f  (z 0 ) = a(z 0 ). Let z 1 be a zero of f − 1. Then using Taylor expansion of f about z 1 we can deduce that (31) f  (z 1 ) − 3F(z 1 ) f  (z 1 ) + 2W (z 1 ) = 0. 

We now define W =

f  − 3F f  + 2W f  . We note that  is f  ( f  − 1) analytic at every simple pole of f . Hence by the hypothesis, (30) and (31) we get N (r, ) = S(r, f ). Also by Lemma 13 we have m(r, ) = S(r, f ) and so T (r, ) = S(r, f ). Now eliminating f  from the definitions of W and  we get We now consider the function  =

2f 2 ( f  − 1) = 3 f 2 + 3W f 2 − 6F f  f  . Since f  (z 0 ) = a(z 0 ), we have from (32), (z 0 )= by the hypothesis and Lemma 13 we obtain

(32)

3W 3W (z 0 ) . If  ≡ , 2[a(z 0 ) − 1] 2(a−1)

      1 1 1  N r,  ≤ N r, + N (r, 0; f r, = S(r, f ). ) + N (2 3W f f   − 2(a−1)

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Thus by (29), (30) and Lemma 15 we get T (r, f  ) = S(r, f ). Hence     1 1 + m r, + O(1) T (r, f ) = N r, f −1 f −1     1 1 + m r,  + S(r, f ) ≤ N r,  f −1 f ≤ 2T (r, f  ) + S(r, f ) = S(r, f ), a contradiction. Therefore  ≡

3W and (32) becomes 2(a − 1)

W f 2 ( f  − a) = f  ( f  − 2F f  )(a − 1).

(33)

Differentiating (33) and using f  (z 0 ) = a(z 0 ) we obtain a  (z 0 )a(z 0 )W (z 0 ) = 2[a(z 0 ) − 1]F(z 0 ) f  (z 0 ).

(34)

2 f  (z 0 ) and so from f  (z 0 ) = a(z 0 ) we get f  (z 0 ) 1 a  (z 0 ) = −F(z 0 ), f  (z 0 ) = − W (z 0 )a(z 0 ). Substituting this in (34) we obtain 2 a(z 0 ) − 1 because a(z 0 )W (z 0 ) = 0. a

≡ −F, then similarly as above we arrive at a contradiction. If a−1  a Let ≡ −F. Then by integration we get a−1

1 1 1 a−1 , (35) + = ( f − 1)2 c f  − 1 ( f  − 1)2 We also note that W (z 0 ) = −

where c( = 0) is a constant.

 1 = f −1 S(r, f ). Finally from this we can deduce by the hypothesis and Lemma 14 that  Using (35) and T (r, a) = S(r, f ) we get by Lemma 13 that m r,   1 + S(r, f ) T (r, f ) = N r, f −1   1 + S(r, f ) = N r,  f −1   1 + S(r, f ) = N1) r,  f −1 ≤ m(r, f ) + S(r, f ).

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This implies N (r, f ) = S(r, f ) and so by (29) we get T (r, f  ) = S(r, f ). This ultimately implies T (r, f ) = S(r, f ), a contradiction. This proves the theorem. As a natural extension of a derivative of a meromorphic function is a linear differential polynomial generated by the function, Zhang and Yang [49] extended and improved the result of Liu and Gu [36] (Theorem 26) in the following manner. Theorem 33 Let k(≥ 1) be an integer and f be a nonconstant meromorphic function and let a( ≡ 0, ∞) be a small meromorphic function of f . Suppose that L( f ) = f (k) + ak−1 f (k−1) + · · · + a0 f , where a0 , a1 , . . . , ak−1 are polynomials. If f and L( f ) share the function a IM and 5δ(0; f ) + (2k + 6)(∞; f ) > 2k + 10, then f = L( f ). Theorem 34 If in Theorem 33 f and L( f ) share the function a CM and 2δ(0; f ) + 3(∞; f ) > 4, then f = L( f ). If we stick to entire functions only, then in 2002 Qiu [41] extended the second theorem of Brück (Theorem 8) to a linear differential polynomial and proved the following result. Theorem 35 Let f be a nonconstant entire function satisfying N (r, 0; f  ) = S(r, f ) and L( f ) = an f (n) + an−1 f (n−1) + · · · + a1 f (1) , where n is a positive integer and an ( = 0), an−1 , . . . , a1 are constants. If f and L( f ) share a value a( = 0, ∞) CM, then f − a = c(L( f ) − 1) for some c ∈ C\{0}. It is already mentioned that Al-Khaladi [2] proved a Brück type result for a small function a = a(z) shared by an entire function f and its first derivative (Theorem 28). We observe there that f − a is not, in general, a constant multiple of f  − a. In 2010 Chen and Wu [17] extended Theorem 28 to a linear differential polynomial with small functions coefficients. Theorem 36 Let f be a nonconstant entire function satisfying N (r, 0; f  ) = S(r, f ), a( ≡ 0, ∞) be a small meromorphic function of f and L( f ) = an f (n) +an−1 f (n−1) + · · · + a1 f (1) , where n is a positive integer and an ( = 0), an−1 , . . . , a1 are small entire functions of f . If f and L( f ) share the function a CM, then f − a = α α (1 − )(L( f ) − 1), where 1 − = eβ and α is a constant and β is an entire funca a tion. If in Theorem 36 we assume the small function a to be entire, then since α a( ≡ 0, ∞), we have by a = that eβ is a nonzero constant and so f − a = 1 − eβ c(L( f ) − a) for some c ∈ C\{0}. In 2010 Al-Khaladi [6] improved Theorems 30 and 31 and proved the following result. Theorem 37 Let f be a nonconstant meromorphic function and let a( ≡ 0, ∞) be a meromorphic small function of f . If f and f (k) share the function a CM, and if N (r,∞; f ) + N (r, 0; f (k) ) < λT (r, f (k) ) + S(r, f (k) ) for some constant λ ∈  1 0, , then k+1

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  Pk−1 ( f (k) − a), f −a = 1− a Pk−1

≡ 0. a Let k be a positive integer and f be a meromorphic function. We denote by Nk (r, 0; f ) the integrated counting function of zeros of f , where a zero of multiplicNk (r, 0; f ) . ity p is counted min( p, k) times. Also we put δk (0; f ) = 1 − lim sup T (r, f ) r →∞ Clearly δ(0; f ) ≤ δk (0; f ) ≤ δk−1 (0; f ) ≤ · · · ≤ δ1 (0; f ) = (0; f ). Al-Khaladi [7] also improved the result of Liu and Gu [36] (Theorem 26) by relaxing the hypothesis on the deficiency of zero. where Pk−1 is a polynomial of degree at most k − 1 and 1 −

Theorem 38 Let f be a nonconstant meromorphic function and let a( ≡ 0, ∞) be a meromorphic small function of f . If f and f (k) share the function a CM and δ2 (0; f ) + δk+1 (0; f ) + 3(∞; f ) > 4, then f = f (k) . Al-Khaladi [1] proved an important result on Brück conjecture. He [1] proved that Theorem 30 is valid for meromorphic functions also, with an obvious modification. Theorem 39 Let f be a nonconstant meromorphic function satisfying N (r, 0; small f (k) ) = S(r, f ) and let a( ≡ 0, ∞) be a meromorphic  function of f . If f and  P k−1 ( f (k) − a), where Pk−1 is f (k) share the function a CM, then f − a = 1 − a Pk−1

≡ 0. a polynomial of degree at most k − 1 and 1 − a Al-Khaladi [5, 8] considered a single nonzero finite value sharing by a meromorphic function with its first and higher order derivatives using some hypothesis on the zeros of the function itself. First we state two results from [5] for the first derivative and then we state two results from [8] for higher order derivatives. We note that for higher derivatives the results are more precise than that for the first derivative. Theorem 40 Let f be a nonconstant meromorphic function. If f and f  share a value a( = 0, ∞) CM and if N (r, 0; f ) = S(r, f ), then either f = f  or f (z) = a(z − c) , where A( = 0) and c are constants. 1 + Ae−z Theorem 41 Let f be a nonconstant meromorphic function. If f and f  share a value a( = 0, ∞) IM and N (r, 0; f ) + N (r, 0; f  ) = S(r, f ), then either f = f  or 2a f (z) = , where A( = 0) is a constant. 1 − Ae−z Theorem 42 Let f be a nonconstant meromorphic function. If f and f (k) (k ≥ 2) share a value a( = 0, ∞) CM and N (r, 0; f ) = S(r, f ), then f = f (k) . Theorem 43 Theorem 42 also holds if f and f (k) share the value a IM and N (r, 0; f ) + N (r, 0; f (k) ) = S(r, f ).

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Let f , g and a be meromorphic functions in the open complex plane C. If f − a and g − a have the same zeros but with different multiplicities, then we say that f and g share the function a DM. If, in particular, a is a constant, then we say that f and g share the value a DM. In 2013 Al-Khaladi [9] considered the problem of sharing a single nonzero finite value DM by a meromorphic function with its first derivative. Later he [10] extended the result to a shared small function. Theorem 44 Let f be a nonconstant meromorphic function. If f and f  share a 2ae2z value a( = 0, ∞) DM and if N (r, 0; f ) = S(r, f ), then f (z) = 2z , where c is e +c a nonzero constant. Theorem 45 Let f be a nonconstant meromorphic function. If f and f  share the value a( = 0, ∞) DM and if N (r, 0; f  ) = S(r, f ), then f (z) = a{1 + b + (1 − b)ce2blz } , where b, c, l are nonzero constants and b2 l = −1. 1 + ce2blz Theorem 46 Let f be a nonconstant meromorphic function and β( ≡ 0, ∞) be a small meromorphic function of f . If f and f  share β DM and if N (r, 0; f ) = z e2z e2t S(r, f ), then f (z) = , where c is a constant and α(z) = dt. α(z) + c 0 β(t) Theorem 47 Let f be a nonconstant meromorphic function and β( ≡ 0, ∞) be a small function of f such that β ≡ β  . If f and f  share β DM and N (r, 0; f  ) = S(r, f ), then β is a constant and f is given as in Theorem 45 with β = a. Al-Khaladi [11] proved several results on the uniqueness of meromorphic functions sharing a small function with derivative under hypothesis on the deficiencies. We now state one typical result that improves a particular case of Theorem 34 when L( f ) = f (k) . Theorem 48 Let k(≥ 1) be an integer and f be a nonconstant meromorphic function, a( ≡ 0, ∞) be a small meromorphic function of f . If f and f (k) share the function a CM and if δ(0; f ) + (∞; f ) > 1, then f = f (k) . If we consider Example 6, then we see that f and f  share the function a CM and δ(0; f ) + (∞; f ) = 0 + 1 = 1 but f = f  . So the hypothesis δ(0; f ) + (∞; f ) > 1 of Theorem 48 is sharp. Recently Al-Khaladi [12] extended his own results (Theorems 40 and 41) to a shared small function. Theorem 49 Let f be a nonconstant meromorphic function and β( ≡ 0, ∞) be a small meromorphic function of f and let k(≥ 1) be an integer. If f and f (k) share the function β CM and if N (r, 0; f ) = S(r, f ), then either f = f (k) or k = 1 and z α(z) + b f (z) = , where b, c(

= 0) are constants and α(z) = β(t)dt. 1 + ce−z 0

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Theorem 50 Let f be a nonconstant meromorphic function and β( ≡ 0, ∞) be a small meromorphic function of f and let k(≥ 1) be an integer. If f and f (k) share the function β IM and if N (r, 0; f ) + N (r, 0; f (k) ) = S(r, f ), then either f = f (k) 2β or k = 1, β is a constant and f (z) = , where c is a nonzero constant. 1 + ce−2z Recently Lahiri and Sarkar [31] proved a uniqueness theorem for a meromorphic function that shares a small meromorphic function with a linear differential polynomial. The result is a sort of improvement of Theorem 37. Theorem 51 Let f be a nonconstant meromorphic function such that L( f ) is nonconstant, where L( f ) = a1 f (1) + a2 f (2) + · · · + ak f (k) and a1 , a2 , . . . , ak ( = 0) are constants. Suppose that a( ≡ 0, ∞) is a small meromorphic function of f . If f and L( f )   share the function a CM and (k + 1)N (r, ∞; f ) + N  (r, 0;c f ) + Nk (r, 0; f ) < (L( f ) − a), where λT (r, f  ) + S(r, f  ) for some λ ∈ (0, 1), then f − a = 1 + a c c is a constant and 1 + ≡ 0. a Lahiri and Pal [29, 30] proved two more results in 2017 on Brück’s conjecture for differential polynomials. Theorem 52 Let f be a transcendental meromorphic function and L( f (k) ) = a0 f (k) + a1 f (k+1) + · · · + a p f (k+ p) be nonconstant, where a0 , a1 , . . . , a p are constants and k(≥ 1), p(≥ 0) are integers such that p = 0 if k = 1 and 0 ≤ p ≤ k − 2 if k ≥ 2. function a( ≡ 0, ∞) of f Suppose that f and L( f (k) ) share a small meromorphic   Pk−1 (k) (k) (L( f ) − a), where CM and N (r, 0; f ) = S(r, f ), then f − a = 1 + a Pk−1 Pk−1 is a polynomial of degree at most k − 1 and 1 +

≡ 0. a Following example [29] shows that the hypothesis N (r, 0; f (k) ) = S(r, f ) is essential for Theorem 52. P(z)e z . Then Example 9 Let P be a nonconstant polynomial and let f (z) = 1 + ez f  (z) =

e z (P(z) + P  (z) + P  (z)e z ) (1 + e z )2

and hence N (r, 0; f  ) = S(r, f ). Also f and f  share the small function P  CM but 1 f  − P = ( f − P  ). 1 + ez

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Theorem 53 Let f be a transcendental entire function such that p k   ψ(z) = bi ( f ( j) )li j is nonconstant, where bi ’s are constants and li j ’s are noni=1

j=1

negative integers such that

k 

li j = n for i = 1, 2, . . . , p.

j=1

Suppose that a( ≡ 0, ∞) be a small meromorphic function of f . If f n and ψ( f ) (r, 0; f ) < λT (r, ( f n ) ) + S(r, ( f n ) ) share the small function a CM and (kn  + 3)N c n (ψ( f ) − a), where c is a constant and for some λ ∈ (0, 1), then f − a = 1 + a c 1 + = eγ for some entire function γ. a c If in Theorem 53 we assume a to be an entire small function, then a = γ e −1 and a( ≡ 0, ∞) imply that γ is a constant. Hence in this case we have f n − a = c(ψ( f ) − a) for some c ∈ C\{0}. We conclude the article with the proof of Theorem 52. We require the following lemmas. Lemma 16 ([29]) Let f be a nonconstant meromorphic function and L = L( f (k) ), given by Theorem 52, be nonconstant. If f − a and L − a share 0 CM, where a = a(z)( ≡ 0, ∞) is a small function of f , then one of the following holds:  (L − a),where Pk−1 is a polynomial of degree at most k − 1 (i) f − a = 1 + Pk−1 a Pk−1

≡ 0, and 1 + a (ii) T (r, f (k) ) ≤ (k + p + 1)N (r, ∞; f ) + N (r, 0; f (k) ) + N (r, 0; f (k) ) + S(r, f ). Lemma 17 ([4]) Let k be a positive integer and f be a meromorphic function such k+1



k+2 that f (k) is not constant. Then either f (k+1) = c f (k) − λ for some nonzero constant c or k N1) (r, ∞; f ) ≤ N (2 (r, ∞; f ) + N1) (r, λ; f (k) ) + N (r, 0; f (k+1) ) + S(r, f ), where λ is a constant. Lemma 18 ([26]) Given a transcendental meromorphic function f and a constant K > 1. Then there exists a set M(K ) whose upper logarithmic density is at most δ(K ) = min{(2e K −1 − 1)−1 , (1 + e(K − 1) exp(e(1 − K )))} such that for every positive integer k, lim sup

r →∞,r ∈M(K / )

T (r, f ) ≤ 3eK . T (r, f (k) )

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Lemma 19 ([29]) Let f be a transcendental meromorphic function such that N (r, 0; f (1) ) = S(r, f ). If f − a and a1 f (1) − a share 0 CM, where a = a(z)( ≡ 0, ∞) is a small function of f and a1 is a nonzero constant, then N1) (r, 0; f (2) ) ≤ N (2 (r, ∞; f ) + S(r, f ). Lemma 20 ([19]) Let f be a transcendental meromorphic function and k be a positive integer. Then k N (r, ∞; f ) ≤ N (r, 0; f (k) ) + (1 + ε)N (r, ∞; f ) + S(r, f ), where ε is any fixed positive number. Proof of Theorem 52. First we verify that

f (k+1)

k+1

k+2

= c f (k) ,

(36)

where c( = 0) is a constant. If (36) does not hold, then we get 

f (k+1) f (k)

k+1

= c f (k) .

(37)

Differentiating (37) and then using (37) we obtain 

f (k+1) f (k)

−2 

f (k+1) f (k)

 =

1 . k+1

Integrating twice we get f (k) =

1 , {C z + D(k + 1)}k+1

where C( = 0) and D are constants. This is impossible because f is transcendental. Let k ≥ 2. We suppose that T (r, f (k) ) ≤ (k + p + 1)N (r, ∞; f ) + N (r, 0; f (k) ) + N (r, 0; f (k) ) + S(r, f ). Since N (r, 0; f (k) ) = S(r, f ), we get from above T (r, f (k) ) ≤ (k + p + 1)N (r, ∞; f ) + S(r, f ). Also from Lemma 20 we obtain for 0 < ε
0 is some fixed time; ρ, λ, μ are, respectively, the density of the elastic body Ω and the Lamé coefficients, ν = 1/σ μ0 is the magnetic viscosity, μ0 is the magnetic permeability, σ is the electric conductivity, f, g are, respectively, the elastic and electromagnetic external sources and n = (n 1 , n 2 , n 3 ) is the outer unit normal at x ∈ ∂Ω. We assume that ρ, λ, μ, ν : Ω → R+ , f, g : Ω × (0, T ) → R3 , u0 , u1 , h0 : Ω → 3 R are given and sufficiently smooth functions, μ0 is a known positive constant and 0 < m 0 ≤ ρ(x), λ(x), μ(x), ν(x) ≤ m 1 < ∞, x ∈ Ω. The solvability of Direct Problem 1 was proved in [8] in the case of constant coefficients ρ, λ, μ, σ, μ0 . In our work we consider variable coefficients ρ, λ, μ, ν, which is essential in many applications. The 1D-case was studied in [9–11].

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Preliminaries and Notation

Let us briefly describe the function spaces, which we shall use in this work (c.f. Temam [12] for more details). We denote by Ω an open bounded set of the Euclidean space R3 with a smooth boundary ∂Ω. With Ω we associate the usual spaces L m (Ω) (m ∈ N) and H s (Ω) (s ∈ R); H0s (Ω) is the subspace of H s (Ω) made of functions vanishing on ∂Ω (we shall also use the notations Lm (Ω) = [L m (Ω)]3 , Hs (Ω) = [H s (Ω)]3 , Hs0 (Ω) = [H0s (Ω)]3 ). Equation (1) justify the introduction of the space S(Ω) = {u ∈ [C0∞ (Ω)]3 : ∇ · u = 0, x ∈ Ω; n · u = 0, x ∈ ∂Ω}. V (Ω) is the closure of S(Ω) in L2 (Ω) and V1 (Ω) is the closure of S(Ω) in H1 (Ω). We equip V (Ω) with the scalar product (u, v) =

3   i=1

Ω

u i vi dx, u, v ∈ V (Ω).

 1/2 If u ∈ V (Ω), we define uV (Ω) = u, u . We equip the space V1 (Ω) with the norm  · H1 (Ω) . If u ∈ V1 (Ω), then ∇ · u = 0 and         ∇ × r ∇ × u = ∇ r ∇ · u − ∇ · r ∇u = −∇ · r ∇u , for any smooth scalar function r = r (x). Introduce the following bilinear forms  a(u, v) =

Ω

    μ ∇ × u · ∇ × v dx +



and b(u, v) =

Ω

 Ω

   (λ + μ) ∇ · u ∇ · v dx, u, v ∈ H1 (Ω).

    ν ∇ × u · ∇ × v dx, u, v ∈ H1 (Ω).

From Theorem 6.1 of [13, Chap. 7, §6] it follows that b(u, u) ≥ Cu2H1 (Ω) , u ∈ V1 (Ω).

Thus, the norms u = b(u, u)1/2 and uH1 (Ω) are equivalent on V1 (Ω). Remark 1 Throughout Sect. 2.1, C stands for a generic positive constant computed in terms of known quantities, and may change from line to line.

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A similar estimate we can obtain for the first form too a(u, u) ≥ Cu2H1 (Ω) , u ∈ H1 (Ω). Let X be a real Banach space with norm  ·  X . The space L p (0, T ; X ) is a set of all strongly measurable functions u : (0, T ) → X , such that u L p (0,T ;X ) < ∞, where   1/ p T p u(t) X dt , 1 ≤ p < ∞; 0 u L p (0,T ;X ) = ess supt∈(0,T ) u(t) X , p = ∞. We define now the following form by setting  c(u, v, w) =

Ω

    u × v · ∇ × w dx, u, v, w ∈ [C 1 (Ω)]3 .

We recall (c.f. Temam [12]) that if n i ≥ 0, i = 1, 2, 3, and satisfy 1. n 1 + n 2 + n 3 > 3/2, or 2. n 1 + n 2 + n 3 = 3/2 and at least two n i ’s are = 0, then c(u, v, w) is a trilinear continuous form on Hn 1 (Ω) × Hn 2 (Ω) × Hn 3 +1 (Ω). Note the following inequality |c(u, v, w)| ≤ CuHn1 (Ω) · vHn2 (Ω) · wHn3 +1 (Ω) .

(2)

Further consider the space H 3/2 (Ω) and define H3/2 (Ω) equipped with the scalar product 3      u, v = u i , vi H 3/2 (Ω) . i=1

The space V3/2 (Ω) is the closure of S(Ω) in H3/2 (Ω). If u ∈ V3/2 (Ω), we define uV3/2 (Ω) =

 1/2 u, u ,

for more on this space see [14, Chap. 1, §6].

2.1.2

Solving Direct Problem (1)

In this section we prove an existence and uniqueness (under some additional conditions) result of the solution to Direct Problem (1).

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Weak Formulation In order to obtain our result we have to define a weak formulation of the Direct Problem (1). We multiply the first equation in (1) by a test function φ ∈ H10 (Ω) and the second one – by a test function ψ ∈ V3/2 (Ω). Integrating these results over Ω, we arrive at   ρutt , φ + a(u, φ) − μ0 c(h, φ, h) = (f, φ), (ht , ψ) + b(h, ψ) + c(h, ut , ψ) = (g, ψ), valid for a.a. t ∈ [0, T ). This suggests the following formulation: Definition 1 For given   f, g ∈ L 2 0, T, L2 (Ω) , u0 ∈ H10 (Ω), u1 ∈ L2 (Ω), h0 ∈ V1 (Ω),

(3)

find u, h satisfying     u ∈ L ∞ 0, T ; H10 (Ω) , ut ∈ L ∞ 0, T ; L2 (Ω) ,     h ∈ L 2 0, T ; V1 (Ω) ∩ L ∞ 0, T ; V (Ω) ,

(4)

and   ρutt , φ + a(u, φ) − μ0 c(h, φ, h) = (f, φ), ∀φ ∈ H10 (Ω), (ht , ψ) + b(h, ψ) + c(h, ut , ψ) = (g, ψ) , ∀ψ ∈ V3/2 (Ω),

(5)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x), h(x, 0) = h0 (x)

(6)

for a.a. t ∈ [0, T ). The pair (u, h) is called the weak solution of problem (1). Observe, that from (5) follows (the prime denotes the dual space) utt ∈ L 2 (0, T ; H−1 (Ω)) ⇒ u, ut ∈ C(0, T ; H−1 (Ω)),



ht ∈ L 2 (0, T ; V3/2 (Ω)) ⇒ h ∈ C(0, T ; V3/2 (Ω)).

Thus, conditions (6) hold. Indeed, let us prove that, for example, the condition h(x, 0) = h0 (x) makes sense. For any ψ ∈ V3/2 (Ω) the bilinear form b(h, ψ) can be represented as follows:   b(h, ψ) = B(h), ψ , where B ∈ L (V1 (Ω); V1 (Ω)) and B(h) ∈ L 2 (0, T ; V1 (Ω)) for any h ∈ L 2 (0, T ; V1 (Ω)). For any ψ ∈ V3/2 (Ω) the trilinear form c(h, ut , ψ) can be represented as follows:   c(h, ut , ψ) = ξ (h, ut ), ψ , where ξ (h, ut ) ∈ L 2 (0, T ; V3/2 (Ω)) for ut ∈ L ∞ (0, T ; L2 (Ω)) and h ∈ L 2 (0, T ;

V1 (Ω)). Hence ht = g − B(h) − ξ (h, ut ) and using V1 (Ω) ⊂ V3/2 (Ω) we con-

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clude that ht ∈ L 2 (0, T ; V3/2 (Ω)) ⇒ h ∈ C(0, T ; V3/2 (Ω)). Therefore, the condition h(x, 0) = h0 (x) makes sense.

Existence Concerning the existence of solution of our problem, we shall prove the following result. Theorem 1 Given f, g, u0 , u1 , h0 , which satisfy (3). There exists at least one pair (u, h) of functions, which satisfy (4)–(6). Proof To show the existence of a suitable weak solution we use the Faedo-Galerkin ∞ be a sequence of functions in H10 (Ω), orthogonal in H10 (Ω) method [14]. Let {φ i }i=1 2 and orthonormal in L (Ω), such that for all m φ 1 , φ 2 , . . . , φ m are linearly independent and the finite linear combinations of the φ i ’s are dense in H10 (Ω). The space H10 (Ω) is separable and therefore such sequences exist. ∞ be the special basis in V3/2 (Ω), see [14, pp. 85–88]: Let {ψ i }i=1     ψ k , v = ξk ψ k , v , ∀v ∈ V3/2 (Ω), ξk > 0, ψ k L2 (Ω) = 1, ψ k V3/2 (Ω) = ξk . Define Φm = span{φ 1 , φ 2 . . . , φ m }, Ψm = span{ψ 1 , ψ 2 , . . . , ψ m } with a fixed m ∈ N. Then the projector Pm : V → Ψm defined as Pm v =

m  (v, ψ i )ψ i , v ∈ V (Ω), i=1

has the following properties



Pm L (V3/2 (Ω);V3/2 (Ω)) ≤ 1, Pm L (V3/2 (Ω);V3/2 (Ω)) ≤ 1,

see [14, p. 89]. Fix a positive integer m, and write um (x, t) =

m  i=1

aim (t)φ i (x), hm (x, t) =

m 

bim (t)ψ i (x),

(7)

i=1

where we intend to select the coefficients aim (t), bim (t) (t ∈ [0, T ], i = 1, 2, . . . , m) to satisfy   m ρutt , φ i + a(um , φ i ) − μ0 c(hm , φ i , hm ) = (f, φ i ), 1 ≤ i ≤ m, m m m (hm t , ψ i ) + b(h , ψ i ) + c(h , ut , ψ i ) = (g, ψ i ), 1 ≤ i ≤ m, aim (0) = (u0 , φ i ), aitm (0) = (u1 , φ i ), bim (0) = (h0 , ψ i ), 1 ≤ i ≤ m.

(8)

Applying standard existence theory for ordinary differential equations we can prove that for each m ∈ N there exist functions um , hm of the form (7) satisfying (8) for t ∈ [0, tm ); the estimate (10), which we will prove later, show that we may take tm = T .

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Next, multiply the first equation in (8) by aitm (t) and the second one – by μ0 bim (t), sum the results obtained over all i from 1 to m, and then recall (7) to find  1 d √ m  ρut (·, t)2L2 (Ω) + μ0 hm (·, t)2L2 (Ω) + a(um , um ) 2 dt m + μ0 b(hm , hm ) = (f, um t ) + μ0 (g, h ). (9) In addition, the following inequalities are valid m m um 0 H10 (Ω) ≤ u0 H10 (Ω) , u1 L2 (Ω) ≤ u1 L2 (Ω) , h0 V1 (Ω) ≤ h0 V1 (Ω) , √ 2 m 2 m 2 m 0 um t L2 (Ω) ≤  ρut L2 (Ω) ≤ m 1 ut L2 (Ω) ,

|a(um , um )| ≥ Cum 2H1 (Ω) , |b(hm , hm )| ≥ Chm 2V1 (Ω) , 0

|(f, um t )|

1 1 2 m 2 g2L2 (Ω) + εhm ≤ f2L2 (Ω) + εum t L2 (Ω) , |(g, ht )| ≤ t L2 (Ω) 4ε 4ε

with ε > 0. Integrating (9) over (0, t), t ≤ T , using the estimates obtained above, initial data in (8), and choosing ε small enough, we get the first a priori estimate um  L ∞ (0,T ;H10 (Ω)) + um t  L ∞ (0,T ;L2 (Ω)) + hm  L ∞ (0,T ;V (Ω)) + hm  L 2 (0,T ;V1 (Ω)) ≤ C (10) with C independent of m.

2 Next we show that hm t , m = 1, 2, 3, . . . , are limited in L (0, T ; V3/2 (Ω)). Indeed, using (10) we obtain m



ξ (hm , um t ) L 2 (0,T ;V3/2 (Ω)) + B(h ) L 2 (0,T ;V1 (Ω)) ≤ C

with C independent of m. From the second equation in (8) we deduce that   m m m hm t = Pm g − B(h ) − ξ (h , ut ) ,



2 where g − B(hm ) − ξ (hm , um t ) ∈ L (0, T ; V3/2 (Ω)) (thanks to V1 (Ω) ⊂ V3/2 (Ω)). m



Recalling that P L (V3/2 (Ω),V3/2 (Ω)) ≤ 1, we finally obtain the second a priori estimate

(11) hm t V3/2 (Ω) ≤ C

with C independent of m. Next we pass to limits as m → ∞, to build a weak solution of our initial boundary value problem (1). According to estimates (10) and (11), we see that the sequence 1 ∞ m ∞ ∞ 2 {um }∞ m=1 is bounded in L (0, T ; H0 (Ω)), {ut }m=1 is bounded in L (0, T ; L (Ω)), m ∞ ∞ 2 m ∞ {h }m=1 is bounded in L (0, T ; V (Ω)) and L (0, T ; V1 (Ω)), and {ht }m=1 is

(Ω)). As a consequence there exist subsequences (which bounded in L 2 (0, T ; V3/2

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m ∞ for simplicity we call again {um }∞ m=1 and {h }m=1 ), such that

um → u weakly * in L ∞ (0, T ; H10 (Ω)), ∞ 2 um t → ut weakly * in L (0, T ; L (Ω))

(12a) (12b)

hm → h weakly in L 2 (0, T ; V1 (Ω)), hm → h weakly * in L ∞ (0, T ; V (Ω)),

(13a) (13b)

2

hm t → ht weakly in L (0, T ; V3/2 (Ω)).

(13c)

and

Observe that (12a) and (12b) imply that um (x, 0) → u0 (x) weakly in L2 (Ω) and u(x, 0) = u0 (x). Applying Theorem 5.1 from [14, Chap. 1, §5] (with B0 = V, B1 =

, B = V1 ) we conclude that there exist a subsequence (which for simplicity we V3/2 call again {hm }∞ m=1 ), such that hm → h strongly in L 2 (0, T ; V (Ω)) and a.e. in Ω × (0, T ).

(Ω) and Using (13a) and (13c) we obtain that hm (x, 0) → h0 (x) weakly in V3/2 h(x, 0) = h0 (x). ∞ Now let us show that the sequence {c(hm , um t , ψ)}m=1 , ψ ∈ Ψm , converges weakly 2 * in L (0, T ). According to (2) we have

c(hm , um t , ψ) L 2 (0,T ) ≤ C with C independent of m. Consequently, there exists a subsequence, which for sim∞ ∞ m m plicity we call again {c(hm , um t , ψ)}m=1 , such that {c(h , ut , ψ)}m=1 converges weakly in L 2 (0, T ) to a function β(t). Let us show that β(t) = c(h, ut , ψ). Indeed, m m m c(hm , um t , ψ) − c(h, ut , ψ) = c(h, ut − ut , ψ) + c(h − h, ut , ψ).

According to (12b) and (13b), and estimate (2) (with n 1 = 1, n 2 = 0, n 3 = 1/2), both the first and second terms on the right-hand side of this equality tend to zero. 2 Similarly we can show that {c(hm , φ, hm )}∞ m=1 converges weakly in L (0, T ) to c(h, φ, h) for any φ ∈ Φm . Thus, after passing to the limit we obtain functions u, h, which satisfy Eq. (5) in ∞ 2 −1 the weak sense. At the same time {um tt }m=1 converges weakly in L (0, T ; H (Ω)) ∞ m −1 to utt , therefore {ut (x, 0)}m=1 converges weakly in H (Ω) to u1 (x) and ut (x, 0) =  u1 (x). Uniqueness To our knowledge, the question of uniqueness remains open under the conditions of Theorem 1. For this reason, a natural question arises: We know that there exists at

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least one solution under conditions (4). What additional properties of the solution guarantee its uniqueness? In this regard, we prove the following result: Theorem 2 Let (u, h) be a pair of functions satisfying (4)–(6), and, in addition, the following conditions ut ∈ L 2 (0, T ; L6 (Ω)), h ∈ L ∞ (0, T ; V3/2 (Ω)).

(14)

Then such solution is unique. Proof Let u(k) , h(k) , k = 1, 2, be two solutions satisfying (4)–(6), (14). Setting u = u(1) − u(2) , h = h(1) − h(2) we obtain   ρutt , φ + a(u, φ) − μ0 c(h, φ, h(1) ) − μ0 c(h(2) , φ, h) = 0, ∀φ ∈ H10 (Ω), (2) (ht , ψ) + b(h, ψ) + c(h, u(1) t , ψ) + c(h , ut , ψ) = 0, ∀ψ ∈ V3/2 (Ω), (15) u(x, 0) = 0, ut (x, 0) = 0, h(x, 0) = 0

for a.a. t ∈ [0, T ). (2) Next, we show that in terms c(h, u(1) t , ψ) and c(h , ut , ψ) h can be used instead of ψ and estimate the resulting forms. We estimate only one of the terms (2) in c(h, u(1) t , h), c(h , ut , h), whereas all other are evaluated analogously. Consider, for instance, the following integral  Ω

h 2 u (1) 3t h 3x2 dx,

(1) where h 2 is the second component of h, u (1) 3t is the third component of ut , and h 3x2 is the derivative with respect of x2 variable of the third component of h. If h ∈ V3/2 (Ω) then ∂h i /∂ x j ∈ L 3 (Ω) (i, j = 1, 2, 3) and



∂h i (·, t) L 3 (Ω) ≤ h(·, t)V3/2 (Ω) , ∂x j

see [14, Chap. 1, §6.4] for details. According to Lemma 6.1 in [14, Chap. 1, §6.1] (with n = 3) we have H 1 (Ω) ⊂ L 6 (Ω), moreover, if ξ ∈ H 1 (Ω) then ξ  L 6 (Ω) ≤ ξ  H 1 (Ω) . Finally, Hölder’s inequality and our assumptions imply  |

Ω

(1) h 2 u (1) 3t h 3x2 dx| ≤ h 2 (·, t) L 6 (Ω) · u 3t (·, t) L 2 (Ω) · h 3x2 (·, t) L 3 (Ω) ≤ C.

Now consider c(h, φ, h(1) ), c(h(2) , φ, h). Let us show that we can use ut ∈ L (0, T ; L2 (Ω)) as a test function in these forms. It is enough to estimate only ∞

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one term because the other ones are estimated similarly. Using the Hölder inequality and our assumptions we obtain  |

Ω

(1) h 2 u 3t h (1) 3x2 dx| ≤ h 2 (·, t) L 6 (Ω) · u 3t (·, t) L 2 (Ω) · h 3x2 (·, t) L 3 (Ω) ≤ C.

Next, setting in (15) ψ = h and φ = ut , multiplying the first line by μ−1 0 and summing the result obtained with the second one, we deduce  1 d −1 √ 2 μ0  ρut (·, t)2L2 (Ω) + μ−1 a(u, u) + h(·, t) 2 0 L (Ω) 2 dt (1) + b(h, h) + c(h, u(1) t , h) = c(h, ut , h ). (1) Let us estimate c(h, u(1) t , h), c(h, ut , h ). Using the Hölder inequality we obtain (1) |c(h, u(1) t , h)| ≤ C∇h(·, t)L2 (Ω) · h(·, t)L3 (Ω) · ut (·, t)L6 (Ω) ,

(16)

where ∇h(·, t)L2 (Ω) = max1≤k≤3 ∇h k (·, t)L2 (Ω) . Define M(t) = u(1) t (·, t) L6 (Ω) , then M(t) is a positive function in L 2 (0, T ) and for this reason M(t) ∈ L 1 (0, T ). According to Theorem 1.2 in [15, Chap. I, §1] and Theorem 2.2 and Remark 2.1 in [16, Chap. II, §2], we have 1/2

1/2

h(·, t)L3 (Ω) ≤ C∇h(·, t)L2 (Ω) · h(·, t)L2 (Ω) . Substituting the last inequality into (16), we obtain |c(h, u(1) t , h)| ≤ M1 (t)∇h(·, t)L2 (Ω) · h(·, t)L2 (Ω) , 3/2

1/2

where M1 (t) = C M(t). Applying Young’s inequality with ε, we arrive at 2 2 |c(h, u(1) t , h)| ≤ ε∇h(·, t)L2 (Ω) + M2 (t)h(·, t)L2 (Ω) ,

where M2 (t) is a positive function in L 1 (0, T ) and ε > 0. Now let us estimate c(h, ut , h(1) ). Since h(1) ∈ V3/2 (Ω), we have |c(h, ut , h(1) )| ≤ Ch(·, t)L3 (Ω) · ut (·, t)L2 (Ω) . As in our early treatments we obtain 

|c(h, ut , h(1) )| ≤ ε∇h(·, t)2L2 (Ω) + C ut (·, t)2L2 (Ω) + h(·, t)2L2 (Ω) , where ε > 0. Thus,

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(1)

− b(h, h) − c(h, ut , h) + c(h, ut , h(1) ) ≤ −b(h, h) + |c(h, ut , h)| + |c(h, ut , h(1) )| 

1 √ ≤ −b(h, h) + 2ε∇h(·, t)2L2 (Ω) + M3 (t) μ−1  ρut (·, t)2L2 (Ω) + h(·, t)2L2 (Ω) , 0 2

where M3 (t) is a positive function in L 1 (0, T ). Choosing ε small enough, we obtain −b(h, h) + 2ε∇h(·, t)2L2 (Ω) ≤ 0. Write N (t) = Then we have

 1 √  ρut (·, t)2L2 (Ω) + a(u, u) + μ0 h(·, t)2L2 (Ω) . 2μ0 d N (t) ≤ M3 (t) · N (t), N (0) = 0. dt

Thus N (t) ≡ 0 for a.a. t ∈ [0, T ]. For this reason u(x, t) ≡ 0, h(x, t) ≡ 0. The latter proves our uniqueness result.



2.2 Inverse Problem In this section we study one of the possible inverse problems associated with Direct Problem 1, namely, the so-called source inverse problem. The results obtained in Sect. 2.1 will be used. We assume that the elastic body density force f of Eq. (1) admits the following representation f(x, t) = α(t)β(x, t), (17) where β ∈ L ∞ (0, T ; L2 (Ω)) is known, while α ∈ L 2 (0, T ) is unknown. The coefficients ρ, λ, μ, ν, μ0 , the free member g, and the initial data u0 , u1 , h0 are assumed to be known and enjoy the properties formulated in Sect. 2.1. In order to recover the function α we need to prescribe additional information, usually some physical measurement. We can now exactly state our identification problem. Inverse Problem 1 Determine a triplet (u, h, α) of functions u, h : Ω × [0, T ] → R3 α : [0, T ] → R

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satisfying Eqs. (1), (17) and the additional condition  (η, ρu) =

Ω

η(x)ρ(x)u(x, t)dx = γ (t), t ∈ [0, T ],

(18)

where η : Ω → R3 is a prescribed smooth vector function and γ : [0, T ] → R is given and sufficiently smooth. Remark 2 Equation (18) imply that the data u0 , u1 , γ must satisfy the following consistency conditions γ (0) = (ρu0 , η), γ (0) = (ρu1 , η).

2.2.1

Several Estimates

To solve Inverse Problem 1 we need some additional estimates. Using (9) with m → ∞, we obtain the following estimate: Cu(·, t)2H1 (Ω) + m 0 ut (·, t)2L2 (Ω) + μ0 h(·, t)2L2 (Ω) + μ0 Ch2L 2 (0,t;V1 (Ω)) 0  t ≤ C1 + 2 |(f, ut )|dτ for a.a. t ∈ (0, T ] 0

(19) with C1 = C u0 2H1 (Ω) + m 1 u1 2L2 (Ω) + μ0 h0 2L2 (Ω) + 0

1 g2L 2 (0,T ;L2 (Ω)) , μ0 C

where C denotes a positive constant. Here we have used the inequalities:  0

t

μ0 C 1 h2L 2 (0,t;L2 (Ω)) + g2L 2 (0,t;L2 (Ω)) , 2 2μ0 C g L 2 (0,t;L2 (Ω)) ≤ g L 2 (0,T ;L2 (Ω)) for t ∈ (0, T ].

|(g, h)|dτ ≤

In particular,  m 0 ut (·, t)2L2 (Ω))

t

≤ C1 + 2

|(f, ut )|dτ for a.a t ∈ (0, T ].

0

Applying Cauchy’s inequality with ε to (f, ut ) in the last inequality and integrating the result over (0, t), we obtain m 0 ut 2L 2 (0,t;L2 (Ω)) ≤ tC1 + 2tεut 2L 2 (0,t;L2 (Ω)) +

t f2L 2 (0,t;L2 (Ω)) for a.a t ∈ (0, T ]. 2ε

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Taking ε = m 0 /4t we arrive at 2t 2 m0 f2L 2 (0,t;L2 (Ω)) for a.a t ∈ (0, T ], ut 2L 2 (0,t;L2 (Ω)) ≤ tC1 + 2 m0 and



t

|(f, ut )|dτ ≤

0

1 2t C1 + f2L 2 (0,t;L2 (Ω)) for a.a t ∈ (0, T ]. 2 m0

Thus, from (19) we have Cu(·, t)2H1 (Ω) + m 0 ut (·, t)2L2 (Ω) + μ0 h(·, t)2L2 (Ω) + μ0 Ch2L 2 (0,t;V1 (Ω)) 0  t ≤ C1 + 2 |(f, ut )|dτ ≤ 2C1 + tC2 α2L 2 (0,T ) , ∀t ∈ (0, T ] (20) 0

2 with C2 = 4m −1 0 β L ∞ (0,T ;L2 (Ω)) . Here we have used the following inequality

f L 2 (0,t;L2 (Ω)) ≤ α L 2 (0,T ) β L ∞ (0,T ;L2 (Ω)) . 2.2.2

Solving Inverse Problem 1

In Sect. 2.1 we have proved that under conditions (3) there exists at least one weak solution of Direct Problem (1) in the class of functions (4). Under additional restrictions (14) such solution will be unique. We assume that for any α Direct Problem (1) has the unique solution. In what follows, we write u(α) = u(x, t; α), h(α) = h(x, t; α) to show the dependence of u, h of the function α. Replacing u, h in (5) by u(α), h(α) and φ by η, and using (18) we obtain α=

γ

+ a(u(α), η) − μ0 c(h(α), η, h(α)) := F(α). (β, η)

(21)

Remark 3 In order to be able to rewrite our problem as fixed-point Eq. (21) we have to assume that   β(·, t), η(·) ≥ η0 , for a.e. t ∈ (0, T ), where η0 is a known positive constant. Using the representation (21), Inverse Problem 1 can be reduced to the following: define a function α ∈ L 2 (0, T ) such that α = F(α), ∀t ∈ [0, T ].

(22)

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It is easy to show that if α is a fixed-point of the operator F(α), i.e., α = F(α), then the triplet u(α), h(α), α  will be a solution of Inverse Problem (1). The reciprocal  statement is also true: if u(α), h(α), α is a solution of Inverse Problem (1), then α = F(α). We can now state our result concerning the identification problem under consideration. Theorem 3 There T ∗ > 0 for which Inverse Problem (1) admits a  exists a number  unique solution u(α), h(α), α for any T ∈ (0, T ∗ ). Proof Introduce the family X (M, T ) of complete metric subspaces in L 2 (0, T ) depending on the two positive constants M, T : X (M, T ) = {α ∈ L 2 (0, T ) : α L 2 (0,T ) ≤ M}. If we assume that α ∈ X (M, T ) is a solution to the operator Eq. (18), we can rewrite it in the fixed-point form (22). Our task consists in showing that Eq. (22) is locally solvable. First of all, let us prove that F(α) ∈ X (T, M). Based on the properties of the forms a and c, introduced in Sect. 2.1.1, we have |a(u(α), η)| ≤ C3 u(α)H10 (Ω) , |c(h(α),η, h(α))| ≤ C4 h(α)V (Ω) · h(α)V1 (Ω) ,

(23)

where C3 and C4 are some positive constants. Using (20) and (23) we easily obtain 

t

|a(u(α), η)|2 dτ ≤ tC5 + t 2 C6 M 2 ,

0



t

2

|c(h(α), η, h(α))|2 dτ ≤ C7 + tC8 M 2 ,

0

where Ck , k = 5, 6, 7, 8, are some positive constants independent of α. Thus, F(α)2L 2 (0,T ) ≤ C9 (T0 ) + C10 (T0 , M)T M 2 , where T0 : T < T0 , and C9 and C10 are some positive constants. Choosing the pair (M, T1 ) in the following way M 2 = 2C9 , 2T1 C10 < 1, we deduce that F(α) maps X (T, M) into itself for any T ∈ (0, T1 ]. We now estimate the difference F(α1 ) − F(α2 ) for any α1 , α2 ∈ X (M, T ), T ∈ (0, T1 ]. For this purpose we consider the equation

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a(u, η) − μ0 c(h, η, h(2) ) − μ0 c(h(1) , η, h) , (β, η)

F(α1 ) − F(α2 ) =

(24)

where u = u(α1 ) − u(α2 ) := u(1) − u(2) , h = h(α1 ) − h(α2 ) := h(1) − h(2) . Let us estimate the functions u, h. Performing similar computations as before, we easily obtain  1 d √  ρut (·, t)2L2 (Ω) + a(u, u) + μ0 h(·, t)2L2 (Ω) + μ0 b(h, h) 2 dt = μ0 c(h, ut , h(1) ) − μ0 c(h, u(1) t , h) + (αβ, ut ), (25) where α = α1 − α2 . After integration (25) over (0, t), we arrive at  1

Cu(·, t)2H1 (Ω) + m 0 ut (·, t)2L2 (Ω) + μ0 h(·, t)2L2 (Ω) 0 2  t  t 2 + μ0 Ch L 2 (0,t;V1 (Ω)) ≤ |c(τ )|dτ + |(αβ, ut )|dτ, 0

0

(26) where c(t) = μ0 c(h, ut , h(1) ) − μ0 c(h, u(1) t , h). In particular, using the Cauchy inequality with ε, we obtain m0 ut 2L 2 (0,t;L2 (Ω)) ≤ 2

 t 0



≤t

τ

|c(η)|dηdτ +

0 t

0

 t 0

τ

|(αβ, ut )|dηdτ

0

|c(τ )|dτ + tεut 2L 2 (0,t;L2 (Ω)) +

t αβ2L 2 (0,t;L2 (Ω)) . 4ε

Taking ε = m 0 /4t, we have m0 ut 2L 2 (0,t;L2 (Ω)) ≤ t 4



t

|c(τ )|dτ +

0

t2 αβ2L 2 (0,t;L2 (Ω)) . m0

(27)

Using estimate (27) in (26), we arrive at  1

Cu(·, t)2H1 (Ω) + m 0 ut (·, t)2L2 (Ω) + μ0 h(·, t)2L2 (Ω) 0 2  t 2t + μ0 Ch L 2 (0,t;V1 (Ω)) ≤ 2 |c(τ )|dτ + αβ2L 2 (0,t;L2 (Ω)) . m 0 0 (28)

Nonlinear Magneto-Elasticity: Direct and Inverse Problems

Let us now estimate 

t

t 0

139

|c(τ )|dτ . We have

|c(h, u(1) t , h)|dτ ≤ C 11



0

t 0

∇h(·, τ )L2 (Ω) · h(·, τ )L3 (Ω) · u(1) t (·, τ )L6 (Ω) dτ

with some positive constant C11 . Applying to hL3 (Ω) the multiplicative inequality (2.9) of Theorem 2.2 from [16, Chap. 2, §2] (with q = 3, r = 2, α = 1/2, β = m = 2), and taking into account that the mean value of h is equal to zero and u(1) t ∈ L ∞ (0, T ; L6 (Ω)), we obtain 

t

|c(h, u(1) t , h)|dτ ≤ C 12

0



t 0

3/2

1/2

∇h(·, τ )L2 (Ω) · h(·, τ )L2 (Ω) dτ,

where C12 is a positive constant. Applying Young’s inequality (1.3) from [16, Chap. 2, §1] (with m = n = 4/3), we arrive at 

t

 t 4/3  3ε 4 ∇h(·, τ )2L2 (Ω) + 4 h(·, τ )2L2 (Ω) dτ 4 ε 0 3C12 ε4/3 4C 12 ≤ h2L 2 (0,t;V1 (Ω)) + 4 h2L 2 (0,t;V (Ω)) . (29) 4 ε

|c(h, u(1) t , h)|dτ ≤C 12

0

Next, we have 

t

|c(h, ut , h(1) )|dτ ≤ C13

0

 0

t

h(·, τ )L3 (Ω) · ut (·, τ )L2 (Ω) · h(1) (·, τ )L6 (Ω) dτ

with some positive constant C13 . Owing to V3/2 (Ω) ⊂ L6 (Ω) we get  t  t (1) |c(h, ut , h )|dτ ≤ C14 h(·, τ )L3 (Ω) · ut (·, τ )L2 (Ω) dτ 0 0   C14 t

h(·, τ )2L3 (Ω) + ut (·, τ )2L2 (Ω) dτ ≤ 2 0 with some positive constant C14 . Again, applying Theorem 2.2 from [16, Chap. 2, §2] (with q = 3, r = 2, α = 1/2, β = m = 2) and the Cauchy inequality with ε1 , we obtain h(·, τ )2L3 (Ω) ≤ 4∇h(·, τ )L2 (Ω) · h(·, τ )L2 (Ω) 1 1 ≤ 4ε1 ∇h(·, τ )2L2 (Ω) + h(·, τ )2L2 (Ω) ≤ 4ε1 h(·, τ )2V1 (Ω) + h(·, τ )2L2 (Ω) . ε1 ε1 For this reason  t 0

|c(h, ut , h(1) )|dτ ≤

 C14

1 4ε1 h2 2 + h2 2 + ut 2 2 . L (0,t;V1 (Ω)) L (0,t;L2 (Ω)) L (0,t;L2 (Ω)) 2 ε1

(30)

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Using (29) and (30), we obtain 

t

|c(τ )|dτ ≤

0



3C ε4/3 12 + 2C14 ε1 h2L 2 (0,t;V1 (Ω)) 4

4C C14  C14 12 ut 2L 2 (0,t;L2 (Ω)) . h2L 2 (0,t;V (Ω)) + + + 4 ε 2ε1 2

Using (28) and the last inequality with ε, ε1 : 3C12 ε4/3 + 8C14 ε1 ≤ 2μ0 C, we arrive at Cu(·, t)2H1 (Ω) + m 0 ut (·, t)2L2 (Ω) + μ0 h(·, t)2V (Ω) + μ0 Ch2L 2 (0,t;V1 (Ω)) 0

≤ C14 ut 2L 2 (0,t;L2 (Ω)) + C15 h2L 2 (0,t;V (Ω)) +

4t αβ2L 2 (0,t;L2 (Ω)) , m0

where C15 is a positive constant and t ∈ (0, T ]. Integrating the last inequality over (0, t), we obtain Cu2L 2 (0,t;H1 (Ω)) + m 0 ut 2L 2 (0,t;L2 (Ω)) + μ0 h2L 2 (0,t;V (Ω)) + tμ0 Ch2L 2 (0,t;V1 (Ω)) 0

≤ tC14 ut 2L 2 (0,t;L2 (Ω)) + tC15 h2L 2 (0,t;V (Ω)) +

2t 2 αβ2L 2 (0,t;L2 (Ω)) . m0

−1 −1 Taking T2 : 2T2 = min{m 0 C14 , μ0 C15 }, we finally arrive at

 1

m 0 ut 2L 2 (0,t;L2 (Ω)) + μ0 h2L 2 (0,t;V (Ω)) 2 + Cu2L 2 (0,t;H1 (Ω)) + tμ0 Ch2L 2 (0,t;V1 (Ω)) ≤ 0

2t 2 αβ2L 2 (0,t;L2 (Ω)) , (31) m0

valid for any t ∈ (0, T ], where T < T2 . Using (23), (24) and (31) we can estimate

 F(α1 ) − F(α2 )2L 2 (0,T ) ≤ C16 u2L 2 (0,T ;H1 (Ω)) + h2L 2 (0,T ;V (Ω)) + h2L 2 (0,T ;L2 (Ω)) 1 0

 2 ≤ C17 (T0 , M)T + C18 (T0 , M)T α2L 2 (0,T ) ,

where Ck , k = 16, 17, 18, are some positive constants independent of α and T . Finally, taking T3 > 0 : C17 T32 + C18 T3 ≤ q 2 < 1 and T ∗ = min{T0 , T1 , T2 , T3 } we arrive at F(α1 ) − F(α2 ) L 2 (0,T ) ≤ qα1 − α2  L 2 (0,T ) valid for any T ∈ (0, T ∗ ]. Application of the Banach contraction principle (cf. [17, p. 103]) guarantees that Eq. (22) admits a unique solution α in X (M, T ) for any  T ∈ (0, T ∗ ].

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3 2D-Case In this section we study 2D analogue of the direct and inverse problems considered in Sect. 2.

3.1 Direct Problem Assume that an isotropic non-homogeneous conductive elastic medium is associated with the domain Ω = Ω × R, where Ω ⊂ R2 with the smooth boundary ∂Ω. We will assume that (in cartesian coordinates (x, y, z)) all physical quantities are independent of z, and the solid displacement and the magnetic field have a special form, namely, (u 1 (x, t), u 2 (x, t), 0) and (0, 0, h(x, t)), x = (x, y). In this case, we can formulate the following 2D analogue of Direct Problem 1, studied in Sect. 2.1. Direct Problem 2 Determine the state u : Ω × [0, T ] → R2 , h : Ω × [0, T ] → R of the elastic and electrically conducting body Ω satisfying the following equations ρutt = ∇ · τ (u) − μ0 h∇h + f, (x, t) ∈ Ω × (0, T ), h t = ∇ · (ν∇h) − ∇ · (hut ) + g, (x, t) ∈ Ω × (0, T ), u(x, 0) = u0 (x), ut (x, 0) = u1 (x), h(x, 0) = h 0 (x), x ∈ Ω,

(32)

u = 0, n · ∇h = 0, (x, t) ∈ ∂Ω × (0, T ). Here u = (u 1 , u 2 ), ∇ = (∂x , ∂ y ), τ (u) = λtrε(u) · I + 2με(u), ε(u) = (∇u + ∇u∗ ) /2 are the stress tensor and the tensor of deformations, T > 0 is some fixed time; I is the 2 × 2 identity matrix, and n = (n 1 , n 2 ) is the outer unit normal at x ∈ ∂Ω. We assume that μ0 is a known positive constant, ρ, λ, μ, ν : Ω → R+ , f : Ω × (0, T ) → R2 , g : Ω × (0, T ) → R, u0 , u1 : Ω → R2 , h 0 : Ω → R are given and sufficiently smooth functions and 0 < m 0 ≤ ρ(x), λ(x), μ(x), ν(x) ≤ m 1 < ∞, x ∈ Ω. The existence and uniqueness result for Direct Problem 2 was proved in [18] in the case of constant coefficients ρ, λ, μ, ν, μ0 . We consider this problem with variable coefficients ρ, λ, μ, ν, which is important in many applications.

3.1.1

Preliminaries and Notation

Let Ω ⊂ R2 be an open bounded set with a smooth boundary ∂Ω. Due to the use of vector-valued functions with two components, we shall use the notations Lm (Ω) = [L m (Ω)]2 , Hs (Ω) = [H s (Ω)]2 , Hs0 (Ω) = [H0s (Ω)]2 .

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Direct Problem 2 justifies the introduction of the space S(Ω) = {u ∈ C0∞ (Ω) : n · ∇u = 0, x ∈ ∂Ω}. V (Ω) is the closure of S(Ω) in L 2 (Ω). We equip V (Ω) with the scalar product  (u, v) =

Ω

uvdx, u, v ∈ V (Ω).

 1/2 If u ∈ V (Ω), we define uV (Ω) = u, u . V1 (Ω) is the closure of S(Ω) in H 1 (Ω). The space V1 (Ω) we equip with the norm  ·  H 1 (Ω) . Let A(u) = ∇ · τ (u), u ∈ D(A) = {u ∈ H2 (Ω) ∩ H10 (Ω)}, be an unbounded operator in L2 (Ω) associated with the following bilinear form a(u, v) =



Ω

λ(∇ · u)(∇ · v) + 2μ

2 

 εi j (u)εi j (v) dx, u, v ∈ H10 (Ω).

i, j=1



The norms u = a(u, u)1/2 and uH1 (Ω) are equivalent on H10 (Ω). It is easy to show   ∇ · τ (u) = ∇ · (μ∇u) + ∇ (λ + μ)∇ · u , u ∈ D(A). Let B(u) = ∇ · (ν∇u), u ∈ D(B) = {u ∈ H 2 (Ω) : n · ∇u = 0 on ∂Ω}, be unbounded operator in L 2 (Ω) associated with the following bilinear form  b(u, v) =

Ω

    ν ∇u · ∇v dx, u, v ∈ H 1 (Ω).



The norms u = b(u, u)1/2 and u H 1 (Ω) are equivalent on V1 (Ω). We define the fractional powers of the operators A, B in the standard way. According to [19], we have D(As ) = H2s (Ω) for s < 3/4, and D(As ) = {u ∈ H2s (Ω) : u = 0 on ∂Ω} for 3/4 < s ≤ 1; D(B s ) = H 2s (Ω) for s < 3/4, and D(B s ) = {u ∈ H 2s (Ω) : n · ∇u = 0 on ∂Ω} for 3/4 < s ≤ 1. Define  c(u, v, w) = uv · ∇wdx, u, w ∈ C 1 (Ω), v ∈ [C 1 (Ω]2 . Ω

We recall (c.f. Temam [12]) that if n i ≥ 0, i = 1, 2, 3, and satisfy:

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1. n 1 + n 2 + n 3 > 1, or 2. n 1 + n 2 + n 3 = 1 and at least two n i ’s are = 0, then c is a trilinear continuous form on H n 1 (Ω) × Hn 2 (Ω) × H n 3 +1 (Ω) and |c(u, v, w)| ≤ Cu H n1 (Ω) · vHn2 (Ω) · w H n3 +1 (Ω) .

(33)

Remark 4 Throughout Sect. 3.1, C stands for a generic positive constant computed in terms of known quantities, and may change from line to line. Now we formulate two statements, which were proved in [18]. Lemma 1 Suppose u ∈ H s (Ω), s ≤ 1, and v ∈ H 1 (Ω) ∩ L ∞ (Ω). Then   uv H s (Ω) ≤ Cu H s (Ω) v H 1 (Ω) + v L ∞ (Ω) . Lemma 2 Let m(u, v) =

 Ω

(34)

u∇ · vdx, u ∈ H 1 (Ω), v ∈ H10 (Ω). Then

|m(u, v)| ≤ u H s (Ω) vH1−s (Ω) , s ∈ [0, 1], s = 1/2.

(35)

In addition, we need the following result, see [20]. Lemma 3 (Brézis-Wainger inequality) Let u ∈ H 1+s (Ω), s > 0. Then

u H 1+s (Ω) 1/2 . u L ∞ (Ω) ≤ Cu H 1 (Ω) 1 + ln 1 + u H 1 (Ω) 3.1.2

Solving Direct Problem 2

Consider the following weak formulation of Direct Problem 2. Definition 2 For given     f ∈ L 2 0, T ; Hs (Ω) , g ∈ L 2 0, T ; H s (Ω) , s s u0 ∈H1+s 0 (Ω), u1 ∈ H (Ω), h 0 ∈ H (Ω), (0 < s < 1/2)

find a pair (u, h) of functions satisfying     ∞ 0, T ; Hs (Ω) , u ∈ L ∞ 0, T ; H1+s 0 (Ω) , ut ∈ L     h ∈ L 2 0, T ; H 1+s (Ω) ∩ L ∞ 0, T ; H s (Ω) ,

(36)

(37)

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and (ρutt , φ) + a(u, φ) + μ0 c(h, φ, h) = (f, φ), ∀φ ∈ H10 (Ω), (h t , ψ) + b(h, ψ) − c(h, ut , ψ) = (g, ψ), ∀ψ ∈ V1 (Ω), u(x, 0) = u0 (x), ut (x, 0) = u1 (x), h(x, 0) = h 0 (x)

(38)

(39)

for a.a. t ∈ [0, T ). The pair (u, h) is called the weak solution of Direct Problem 2. Existence Theorem 4 Suppose functions f, g, u1 , u0 , h 0 satisfy (36). Then there exists a pair (u, h) of functions, which satisfy (37)–(39). Proof To show the existence of a suitable weak solution we use the Faedo-Galerkin method [14]. Let φ i (x) ∈ H10 (Ω) and ψi (x) ∈ V1 (Ω) be eigenfunctions of the following spectral problems a(φ i , φ) = λi (φ i , φ), ∀φ ∈ H10 (Ω), such that (φ i , φ j ) = δi j , φ i H10 (Ω) = λi , and b(ψi , ψ) = μi (ψi , ψ), ∀ψ ∈ V1 (Ω), such that (ψi , ψ j ) = δi j , ψi V1 (Ω) = μi , where δi j is the Kronecker delta. Define Φm = span{φ 1 , φ 2 . . . , φ m } and Ψm = span{ψ1 , ψ2 , . . . , ψm }, m ∈ N. Fix a positive integer m, and write um (x, t) =

m 

aim (t)φ i (x), h m (x, t) =

i=1

m 

bim (t)ψi (x),

(40)

i=1

where we intend to select the coefficients aim (t), bim (t) (t ∈ [0, T ], i = 1, 2, . . . , m) to satisfy  m  ρutt , φ i + a(um , φ i ) + μ0 c(h m , φ i , h m ) = (f, φ i ), 1 ≤ i ≤ m, m m m (h m t , ψi ) + b(h , ψi ) − c(h , ut , ψi ) = (g, ψi ), 1 ≤ i ≤ m, aim (0)

= (u0 , φ i ),

aitm (0)

= (u1 , φ i ),

bim (0)

(41)

= (h 0 , ψi ), 1 ≤ i ≤ m.

The nonlinear problem (41) has a maximal solution um , h m of the form (40) defined on some interval [0, tm ); the first a priori estimate (43), which we will prove later, show that we may take tm = T . Next, multiply the first equation in (41) by aitm (t) and the second one – by μ0 bim (t), sum for i = 1, 2, . . . , m, and then recall (40) to find  m m m m m m ρutt , ut + a(um , um t ) + μ0 c(h , ut , h ) = (f, ut ), m m m m m m m μ0 (h m t , h ) + μ0 b(h , h ) − μ0 c(h , ut , h ) = μ0 (g, h ),

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for a.e. t ∈ [0, T ]. Adding the results obtained together, we arrive at  1 d √ m  ρut (·, t)2L2 (Ω) + a(um , um ) + μ0 h m (·, t)2L 2 (Ω) 2 dt m + μ0 b(h m , h m ) = (f, um t ) + μ0 (g, h ). (42) Integrating (42) over [0, t] and using the Bihari lemma, see [21], we get the first a priori estimate 

E 0m (t) ≤ C u0 2H1 (Ω) + u1 2L2 (Ω) + h 0 2L 2 (Ω)  t   f(·, τ )2L2 (Ω) + g(·, τ )2L 2 (Ω) dτ +C 0

(43) with C independent of m and  2 m 2 E 0m (t) = um (·, t)2H1 (Ω) + um t (·, t)L2 (Ω) + h (·, t) L 2 (Ω) +

t 0

∇h m (·, τ )2L2 (Ω) dτ.

Now, multiply the first equation in (41) (with φ i substituted by As φ i ) by aitm (t), and the second one (with ψi substituted by B s ψi ) – by μ0 bim (t), 0 < s < 1/2, sum for i = 1, 2, . . . , m, and integrate over [0, t]. By adding the results obtained together, we obtain s+1 s √ s m 2 2 2 2 m 2 E 1m (t) + I m (t) = A 2 um 0 L2 (Ω) +  ρ A u1 L2 (Ω) + μ0 B h 0  L 2 (Ω)  t

 s  s   s s 2 2 m A 2 f(·, τ ), A 2 um +2 t (·, τ ) + μ0 B g(·, τ ), B h (·, τ ) dτ,

0

where E 1m (t) = A

s+1 2

√ s 2 um (·, t)2L2 (Ω) +  ρ A 2 um t (·, t)L2 (Ω) s

+ μ0 B 2 h m (·, t)2L 2 (Ω) + 2μ0



t 0

B

s+1 2

h m (·, τ )2L 2 (Ω) dτ,

and I m (t) = 2μ0

 t 0

Ω

h m (x, τ )∇h m (x, τ ) · As um t (x, τ )dxdτ  t   ∇ · h m (x, τ )um (x, τ ) B s h m (x, τ )dxdτ. − 2μ0 0

Ω

(44)

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Using (34)–(35) to estimate the integral terms on the right-hand side of (44), we arrive at 

t

|I m (t)| ≤ C 0

h m (·, τ ) H 1 (Ω) h m (·, τ ) H 1+s (Ω) um t (·, τ )Hs (Ω)

 h m (·, τ ) H 1+s (Ω) 1/2  dτ · 1 + 1 + ln 1 + h m (·, τ ) H 1 (Ω)

with C independent of m. By applying the Young inequality with arbitrary ε > 0, we get |I m (t)| ≤ ε

 t 0

h m (·, τ )2

dτ H 1+s (Ω)

+ Cε −1

 t 0

h m (·, τ )2

H 1 (Ω)

2 m um t (·, τ )Hs (Ω) dτ + I3 (t),

where I3m (t)

= Cε

−1



h m (·, τ ) H 1+s (Ω)  2 dτ h m (·, τ )2H 1 (Ω) um t (·, τ )Hs (Ω) ln 1 + h m (·, τ ) H 1 (Ω) 0  t

h m (·, τ ) H 1+s (Ω) ≤ Cε−1 h m (·, τ )2H 1 (Ω) 1 + h m (·, τ ) H 1 (Ω) 0 

2 m 2 dτ. + um (·, τ ) ln 1 + u (·, τ ) s s t H (Ω) t H (Ω) t

To estimate I3m (t) we have used x y ≤ x ln(1 + x) + e y with

h m (·, τ ) H 1+s (Ω)  2 . (·, τ ) , y = ln 1 + x = um s t H (Ω) h m (·, τ ) H 1 (Ω) Choosing ε > 0 to be sufficiently small, and applying Young’s and Gronwall’s inequalities, we obtain E 1m (t)

≤ C exp{C

 t 0

  2 1 + h m (·, τ )2H 1 (Ω) 1 + ln 1 + um dτ }. t (·, τ )Hs (Ω)

  In terms of E 2m (t) = ln 1 + E 1m (t) the latter inequality can be represented as follows  E 2m (t) ≤ C + C

0

t

  h m (·, τ )2H 1 (Ω) 1 + E 2m (τ ) dτ.

Applying the Gronwall inequality we arrive at  t  t

 h m (·, τ )2H 1 (Ω) dτ exp{C h m (·, τ )2H 1 (Ω) dτ } E 2m (t) ≤ C 1 + 0

0

(45)

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with C independent of m. From (43) we have 

t

0

h m (·, τ )2H 1 (Ω) dτ ≤ C.

Using (45) we obtain E 2m (t) ≤ C ⇒ E 1m (t) ≤ C with C independent of m. Therefore, the second a priori estimate is 2 um (·, t)2H1+s (Ω) +um t (·, t)Hs (Ω)

+ h

m

(·, t)2H s (Ω)



t

+ 0

h m (·, τ )2H 1+s (Ω) dτ ≤ C, (46)

valid for any t ∈ [0, T ], m ∈ N. According to (43) and (46) there exist subsequences, which we call again {um }∞ m=1 , m ∞ {h }m=1 , such that um → u weakly* in L ∞ (0, T ; H10 (Ω)), um → u weakly* in L ∞ (0, T ; H1+s (Ω)) 2 2 um t → ut weakly in L (0, T ; L (Ω)),

(47)

∞ s um t → ut weakly* in L (0, T ; H (Ω)),

and h m → h weakly in L 2 (0, T ; V1 (Ω)), h m → h weakly* in L ∞ (0, T ; V (Ω)), h m → h weakly in L 2 (0, T ; H 1+s (Ω)), h m → h weakly* in L ∞ (0, T ; H s (Ω)).

(48)

Using (47) we obtain that um (x, 0) → u0 (x) weakly in L2 (Ω) and u(x, 0) = u0 (x). Applying Theorem 5.1 from [14, Chap. 1, §5] (with B0 = V1 (Ω), B = V (Ω), B1 = V1 (Ω)) we conclude that there exists a subsequence (which for simplicity we call m 2 again {h m }∞ m=1 ), such that h → h strongly in L (0, T ; V (Ω)) and a.e. in Q T . m Observe that (48) imply h (x, 0) → h 0 (x) strongly in H s (Ω) and h(x, 0) = h 0 (x). 2 Now let us show that c(h m , um t , ψ) → c(h, ut , ψ) weakly in L (0, T ) for any ψ ∈ Ψm . According to (33) we have c(h m , um t , ψ) L 2 (0,T ) ≤ C with C independent of m. Consequently there exists a subsequence, which for sim∞ m m plicity we call again {c(h m , um t , ψ)}m=1 , such that c(h , ut , ψ) → β(t) weakly in 2 L (0, T ). Using (47) and (48), estimate (33) (with n 1 = 1, n 2 = 0, n 3 = 1/2) we can prove that β = c(h, ut , ψ). Similarly we can show that c(h m , φ, h m ) → c(h, φ, h) weakly in L 2 (0, T ) for any φ ∈ Φm . Thus, after passing to the limit we obtain func-

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tions u, h, which satisfy Eq. (38) in the weak sense. At the same time um tt → utt −1 weakly in L 2 (0, T ; H−1 (Ω)), therefore um (x, 0) → u (x) weakly in H (Ω) and 1 t  ut (x, 0) = u1 (x). Uniqueness To our knowledge, the question of uniqueness in the 3D-case remains open. Uniqueness was only proved under certain additional assumptions, see Sect. 2.1.2 for details. In the 2D-case, we are able to prove the uniqueness result without additional a priori assumptions about the smoothness of the solutions obtained. Theorem 5 A weak solution of Direct Problem 2 is unique. Proof Let u(k) , h (k) , k = 1, 2, be two solutions of (36)–(39). Setting u = u(1) − u(2) , h = h (1) − h (2) we have (ρutt , φ) + a(u, φ) − μ0 c(h, φ, h (1) ) − μ0 c(h (2) , φ, h) = 0, (2) (h t , ψ) + b(h, ψ) + c(h, u(1) t , ψ) + c(h , ut , ψ) = 0, u(x, 0) = ut (x, 0) = 0, h(x, 0) = 0,

(49)

for a.a. t ∈ [0, T ). Next, setting φ = ut and ψ = μ0 h in (49), summing the first line with the second one, and integrating over [0, t] the result obtained, we arrive at:  t √ 2 2 b(h, h)dτ  ρut (·, t)L2 (Ω) + a(u, u) + μ0 h(·, t) L 2 (Ω) + 2μ0 0  t  (1) c(h, u(1) = 2μ0 t , h) − c(h, ut , h ) dτ. 0

(50) The bilinear forms a(u, u) and b(h, h) can be estimated in the following way Ca u(·, t)2H1 (Ω) ≤ a(u, u) ≤ Ca

u(·, t)2H1 (Ω) , Cb h(·, t)2H 1 (Ω) ≤ b(h, h) ≤ Cb

h(·, t)2H 1 (Ω) ,

(51)

where Ca < Ca and Cb < Cb

are some positive constants. Using (50) and (51), we obtain  t  t |c(h, u(1) , h)|dτ + 2μ |c(h, ut , h (1) )|dτ, (52) J (t) ≤ 2μ0 0 t 0

0

where we set J (t) := m 0 ut (·, t)2L2 (Ω) + Ca u(·, t)2H1 (Ω) + μ0 h(·, t)2L 2 (Ω) + 2μ0 Cb

 0

t

h(·, τ )2H 1 (Ω) dτ. (53)

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Let us write in more detail the derivation of the estimate of the first integral term on the right-hand side of (52). Using the general Hölder inequality, we have  (1) (1) |c(h, ut , h)| = h(x, t)ut (x, t) · ∇h(x, t)dx Ω

 s/2  (1−s)/2  1/2 (1) 2/s ≤ |h(x, t)| dx · |ut (x, t)|2/(1−s) dx · |∇h(x, t)|2 dx . Ω

Ω

Ω

Then, using the Cauchy inequality with ε1 , we obtain 

t

|c(h, u(1) t , h)|dτ

 ≤ ε1

0

t

h(·, τ )2H 1 (Ω) dτ  t 1 2 h(·, τ )2L 2/s (Ω) · u(1) + t (·, τ )L2/(1−s) (Ω) dτ. 4ε1 0 (54) 0

Next, using the limiting case (m → ∞) of a priori estimates (43) and (46), (51), (54), the embeddings H s (Ω) ⊂ L 2/(1−s) (Ω), H 1−s (Ω) ⊂ L 2/s (Ω), the interpolation s inequality h H 1−s (Ω) ≤ C1 h1−s H 1 (Ω) h L 2 (Ω) (C 1 > 0 is a known constant) and the Young inequality with ε2 ab ≤

1 m m m − 1 −m/(m−1) m/(m−1) ε a + ε b , m 2 m 2

2s where a = h2(1−s) H 1 (Ω) , b = h L 2 (Ω) , m = 1/(1 − s) (0 < s < 1/2), we obtain  t  t (1) 2μ0 |c(h, ut , h)|dτ ≤ ε3 J (t) + A1 (τ )J (τ )dτ, (55) 0

0

where we set ε3 :=

 1 2 1/(1−s) 4ε1 + ε2 CC1 ,

4ε1 Ca −1/s

A1 (τ ) :=

sC1 ε1 2ε

2 u(1) t (·, τ )Hs (Ω) .

Similarly, we can estimate   t (1) 2μ0 |c(h, ut , h )|dτ ≤ ε4 J (t) + 0

t

A2 (τ )J (τ )dτ,

(56)

0

where ε4 is a positive constant and A2 (t) ∈ L 1 [0, T ] is a known positive function. Summing together (55) and (56) with ε3 + ε4 ≤ 1/2, we arrive at  t A(τ )J (τ )dτ, (57) J (t) ≤ 0

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where A(t) ∈ L 1 [0, T ] is a known positive function. Finally, using (49), (53) and Gronwall’s inequality, we deduce from (57) that J (t) ≡ 0 for any t ∈ [0, T ], which proves our uniqueness result. 

3.2 Inverse Problem In this section we assume that the elastic body density force f in Eq. (32) admits the following representation f = α(t)β(x, t), where β ∈ L ∞ (0, T ; Hs (Ω)) is a known function, while α ∈ L 2 (0, T ) is unknown. The coefficients ρ, λ, μ, ν, μ0 , the free member g, and the initial data u0 , u1 , h 0 are assumed to be known and enjoy the properties formulated in Sect. 3.1. Inverse Problem 2 Find function α ∈ L 2 (0, T ) such that the solution of Direct Problem 2 satisfies the condition  (η, ρu) = η(x)ρ(x)u(x, t)dx = γ (t), t ∈ [0, T ], (58) Ω

where η : Ω → R2 and γ : [0, T ] → R are prescribed smooth functions. Remark 5 In order to solve Inverse Problem 2 we must make the following assumptions involving the data u0 , u1 , γ , and the kernel η defining the additional information (58)     γ (0) = η, ρu0 , γ (0) = η, ρu1 .

3.2.1

Solving Inverse Problem 2

The proof of the solvability of Inverse Problem 2 is similar to the proof of the solvability of Inverse Problem 1 and is based on the results obtained in solving Direct Problem 2. As in the previous case, in order to apply the Banach contraction mapping, we have to assume that β, η satisfy the condition (β(·, t), η(·)) ≥ η0 , for a.e. t ∈ (0, T ), where η0 is a known positive constant. We can now state (without demonstration) our result concerning the inverse problem under consideration. Theorem 6 There exists T ∗ > 0 for which Inverse Problem 2 has a unique solution α ∈ L 2 (0, T ) for any T ∈ (0, T ∗ ).

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Acknowledgements The first author is a member of the National Institute of Science and Technology of Petroleum Geophysics (INCTGP/CNPq/MEC), Brazil. This work was partially supported by the Carlos Chagas Filho Foundation for Research Support of the State of Rio de Janeiro–FAPERJ, Brazil, grant No. E-26/010.001037/2016.

References 1. Alvén, A.: Cosmical Electrodynamics. Clarendon, Oxford (1950) 2. Parton, V.Z., Kudrjavtsev, B.A.: Electromagnetoelasticity: Piezoelectric and Electrically Conductive Solids. Gordon and Breach Science Publishers, New York (1988) 3. Dunkin, J.W., Eringen, A.C.: On the propagation of waves in an electromagnetic elastic solid. Intern. J. Eng. Sci. 1, 461–495 (1963) 4. Paria, G.: Magneto-elasticity and magnetothermo-elasticity. Adv. Appl. Mech. 10(1), 73–112 (1967) 5. Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua, vol. I, II. Springer, Berlin (1990) 6. Pride, S.R.: Governing equations for the coupled electromagnetics and acoustics of porous media. Phys. Rev. B50(21), 15 678–15 698 (1994) 7. Priimenko, V., Vishnevskii, M.: On an initial boundary value problem in nonlinear 3Dmagnetoelasticity. Appl. Math. Lett. 50, 23–28 (2015) 8. Botsenyuk, O.M.: On the solvability of the initial- and boundary-value problem for the system of semilinear equations of magnetoelasticity. Ukr. Mat. Zb. 44(9), 1181–1185 (1992) 9. Priimenko, V., Vishnevskii, M.: An initial boundary-value problem for a model electromagnetoelasticity system. J. Diff. Equ. 235, 31–55 (2007) 10. Priimenko, V., Vishnevskii, M.: The first initial-boundary-value problem for a nonlinear model of the electrodynamics of vibrating elastic media. Nonlinear Anal. 68, 2913–2932 (2008) 11. Priimenko, V., Vishnevskii, M.: Direct problem for a nonlinear evolutionary system. Nonlinear Anal. 73, 1767–1782 (2010) 12. Temam, R.: Navier-Stokes Equations and Non-linear Functional Analysis. NSF/CBMS Regional Conferences Series in Applied Mathematics. SIAM, Philadelphia (1983) 13. Duvaut, G., Lions, J.-L.: Les inéquations em mécanique et en physique. Dunod, GauthierVillars, Paris (1972) 14. Lions, J.-L.: Quelques méthodes de résolution des problémes aux limites non linéaire. Dunod, Gauthier-Villars, Paris (1969) 15. Temam, R.: Navier-Stokes Equations. North-Holland Publishing Company, Amsterdam (1977) 16. Ladyzhenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1968) 17. Schwartz, L.: Cours d’Analyse. Hermann, Paris (1967) 18. Botsenyuk, O.M.: Global solutions of a two-dimensional initial boundary-value problem for a system of semilinear magnetoelasticity equations. Ukranian Math. J. 48(2), 181–188 (1996) 19. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary-Value Problems and Applications. Springer, New York (1972) 20. Brézis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Part. Differ. Equ. 5, 773–789 (1980) 21. Bihari, I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hungar. 7, 81–94 (1956)

Note on Periodic and Asymptotically Periodic Solutions of Fractional Differential Equations Michal Feˇckan

Abstract Several results are presented on the existence or nonexistence of periodic solutions for fractional differential equations (FDEs for short) on arbitrary dimensional spaces involving Caputo fractional derivatives. The existence of S-asymptotically periodic solutions as well as periodic boundary value problems are also investigated. A rather broad variety of FDEs is considered by covering both finite dimensional FDEs and evolution FDEs in infinite dimensional spaces containing either single order or mixed orders of Caputo fractional derivatives with either finite or infinite lower limits of Caputo fractional derivatives. Different qualitative results are derived for particular types of studied FDEs, for instance, a uniform upper bound for Lyapunov exponents of solutions. Several examples are presented to illustrate theoretical results, such as fractional Duffing equations or periodically forced nonlinear fractional wave equations. Keywords Fractional differential equations · Periodic solutions · Asymptotically periodic solutions · Periodic boundary value problems 2010 MSC 34A08 · 35R11

This work was supported by the Slovak Research and Development Agency (grant number APVV14-0378) and the Slovak Grant Agency VEGA (grant numbers 2/0153/16 and 1/0078/17). M. Feˇckan (B) Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia e-mail: [email protected] M. Feˇckan Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_6

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1 Introduction To show the existence or nonexistence of periodic solutions in systems modelled by either differential or difference equations is the first important task for understanding dynamics of those systems. For this reason, the corresponding theory is classical and well-developed [5]. On the other hand, in recent years, fractional differential equations FDEs have attracted increasing interest due to many applications [1, 4, 14, 17, 18, 22, 28, 33, 34]. This paper deals with problems overlapping these two nice theories, so we present several results on the existence or nonexistence of periodic solutions for FDEs, then we study S-asymptotically periodic solutions in FDEs on the nonnegative real half-line and also boundary value problems (BVPs for short) with periodic conditions on finite intervals. We consider Caputo fractional derivatives. The variety of studied FDEs is rather broad ranging from finite dimensional FDEs to evolution FDEs in infinite dimensional spaces, then with involving either single order or mixed orders Caputo fractional derivatives and with containing either finite or infinite lower limits of Caputo fractional derivatives. Depending on the types of studied FDEs, different qualitative results are derived. Finally, we refer the reader to other interesting results in the cited references.

2 No Periodics for FDEs with Finite Lower Limits Let X be a Banach space with a norm  · . Set J := [0, ∞). By C(J, X ) we denote the Fréchet space of all continuous functions f : J → X endowed with the topology of uniform convergence on any interval [0, n], n ∈ N. First, we recall the following definition. Definition 1 ([11, Definition 3.1]) A function f ∈ C(J, X ) is called S-asymptotically T -periodic if there exists T > 0 such that limt→∞ ( f (t + T ) − f (t)) = 0. In this case, we say that T is an asymptotic period of f . Now, let us consider

q

D0 u(t) = f (t), t ∈ J, u(0) = u 0 ,

(1)

q

where D0 is the Caputo fractional derivative of order q ∈ (0, 1) with the lower limit at zero and f ∈ C(J, X ). If f is T -periodic for some T > 0, then we know [31] that u is not T -periodic but S-asymptotically T -periodic. We refine this result as follows. Theorem 1 If f ∈ C(J, X ) is T -periodic in (1), then u is S-asymptotically T T periodic. It is bounded if and only if f¯ := T1 0 f (t)dt = 0 and then u(t) is asymptotic to a T -periodic function (see (4)). Proof It is well known [33] that the problem (1) is equivalent to the following integral equation

Periodic and Asymptotically Periodic Solutions of FDEs

1 (q)

u(t) = u 0 +



t

155

(t − s)q−1 f (s)ds.

(2)

0

T T We can write f (t) = f¯ +  f (t) with f¯ = T1 0 f (t)dt and 0  f (t)dt = 0. Then we have  t f¯ 1 q t + (t − s)q−1  f (s)ds. u(t) = u 0 + (q + 1) (q) 0 The function t → t q is S-asymptotically T -periodic, since by the mean value theorem, there is θ ∈ [t, t + T ] such that (t + T )q − t q =  = as t → ∞. Clearly, function F(t) as u(t) = u 0 +

t 0

Tq Tq ≤ 1−q → 0 1−q θ t  f (s)ds is T -periodic. Next, we rewrite (2)

 t−T  t f¯ 1 1 tq + (t − s)q−1  (t − s)q−1  f (s)ds + f (s)ds (q + 1) (q) 0 (q) t−T

(3)

for t ≥ T . Then we compute  t−T 0

 − T ) + (q − 1) = T q−1 F(t

(t − s)q−1  f (s)ds =  t−T

 t−T 0

 (s)ds (t − s)q−1 F

  + (q − 1) (t − s)q−2 F(s)ds = T q−1 F(t)

0

 t T

 − z)dz. z q−2 F(t

Next, we derive    

t+T

z



q−2 

t

F(t + T − z)dz −

T

 ∞ ≤  F

z T



t+T

 ∞ z q−2 dz =  F

t

since

     F(t − z)dz  =

q−2 

(t + T )q−1 − t q −1

t q−1

t+T

z ≤

(t + T )q−1 − t q−1 q−2 = T t0 ≤ T t q−2 q −1

 ∞ = maxt∈R  F(t).  for a t0 ∈ (t, t + T ) and  F We note 

t

(t − s)

q−1

t−T

 f (s)ds =



T

z q−1  f (t − z)dz

0

is continuous and T -periodic. Summarizing, (3) has the form

  F(t − z)dz  

q−2 

 ∞ T  F , 2−q t

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M. Feˇckan

u(t) =

f¯ t q + μ(t) + u 0 + ν(t) (q + 1)

(4)

for  ∞ 1  − s)ds, μ(t) = − s q−2 F(t (q − 1) t  T  ∞ T q−1  1 1 q−1   − s)ds, ν(t) = s s q−2 F(t f (t − s)ds + F(t) + (q) (q) 0 (q − 1) T where μ(t) → 0 as t → ∞ and ν(t) is T -periodic. Consequently, the proof is finished.  Remark 1 1. The improvement in Theorem 1 to previous results is the decomposition (4) and if u(t) is bounded then it is asymptotic to a T -periodic function. Of course, any function asymptotic to a T -periodic√function is S-asymptotically T -periodic. But opposite is not true: the function sin t is S-asymptotically T -periodic for any T > 0, it is bounded but is is not asymptotic to a periodic function. So this function cannot be a solution of (1) with a periodic f (t). 2. We note  ∞  F μ(t) ≤ . (q)t 1−q So we also know the rate of convergence of μ(t) → 0 as t → ∞. 3. If f is S-asymptotically T -periodic then u may be not. Indeed, taking h(t) = t r for r ∈ (0, 1) we get u(t) = u 0 + B(q, 1 + r )t q+r , which is not S-asymptotically T -periodic for r > 1 − q. For example, the solution of the following FDE q

D0 x(t) + βx(t) = γ cos(t + α), x(0) = x0

(5)

is given by [33]  x(t) = E q (−βt q )x0 + γ

t

(t − s)q−1 E q,q (−β(t − s)q ) cos(s + α)ds,

(6)

0

which cannot be integrated, so we cannot find its exact solution. By an exact solution we mean a solution which can be expressed as a formula by some known special functions. But using the expansions of the Mittag-Leffler functions E q,q (z) =

∞  k=0

zk , (qk + q)

E q (z) =

∞  k=0

zk , (qk + 1)

Periodic and Asymptotically Periodic Solutions of FDEs

157

it is possible to find approximate solutions for concrete values of q, β, α, . Moreover, even for β = 0, i.e. q

D0 x(t) = γ cos(t + α), x(0) = x0

(7)

we get a rather complicated exact solution given by  t γ (t − s)q−1 cos(s + α)ds x(t) = x0 + (q) 0   γ cos α t γ sin α t q−1 = x0 + (t − s) cos s ds − (t − s)q−1 sin s ds (q) 0 (q) 0   γ cos α q q 1 q 1 = x0 + t 1 F2 1; + , + 1; − 2 t 2 (q + 1) 2 2 2 4   q q 3 1 γ sin α t q+1 1 F2 1; + 1, + ; − 2 t 2 , − (q + 2) 2 2 2 4

(8)

where 1 F2 stands as the generalized hypergeometric function. Note (8) is nonperiodic. However, on the other side, we consider the Caputo derivative with the lower limit at −∞ q (9) D−∞ x(t) + βx(t) = γ cos(t + α), where [33] q

D−∞ x(t) =

1 (1 − q)



t −∞

(t − s)−q x  (s)ds, t ∈ R.

We are looking for a solution of (9) of the form x(t) = A cos t + B sin t.

(10)

Using the formulas [12, (24), (26)], we derive q

D−∞ x(t)x(t) =

 Aq cos

πq πq + Bq sin 2 2



  πq πq cos t + Bq cos − Aq sin sin t. 2 2

(11) Inserting (11) into (9), we obtain πq πq + Bq sin + β A = γ cos α, 2 2 πq πq − Aq sin + β B = −γ sin α, Bq cos 2 2 Aq cos

which has a solution

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M. Feˇckan



γ β cos α + q cos α − πq 2 A= , β 2 + 2βq cos πq + 2q 2

γ β sin α + q sin α − πq 2 B=− . 2q β 2 + 2βq cos πq +  2

(12)

Inserting (12) into (10), we obtain γ β cos(t + α) + q cos t + α − x(t) = β 2 + 2βq cos πq + 2q 2

πq

2

.

(13)

Hence (9) has a periodic solution (13). In particular, the following FDE q

D−∞ x(t) = γ cos(t + α)

(14)

 πq x(t) = x0 + γ−q cos t + α − 2

(15)

has a periodic solution

for any x0 . Theorem 2 Solutions (8) of (7) and (15) of (9) coincide asymptotically. Proof Using the asymptotic expansion from [24], for t → ∞ one obtains   q 1 q 1 γ cos α q t 1 F2 1; + , + 1; − 2 t 2 (q + 1) 2 2 2 4   q q 3 1 γ sin α t q+1 1 F2 1; + 1, + ; − 2 t 2 − (q + 2) 2 2 2 4 q+1 q+2 q  γ( 2 )( 2 )2 cos α πq = x0 + cos t − √ 2 π(q + 1)q q+2 q+3 q+1  γ( 2 )( 2 )2 sin α π πq − + O(t −1 ) − cos t − √ 2 2 π(q + 2)q    πq πq − sin α sin t − + O(t −1 ) = x0 + γ−q cos α cos t − 2 2  πq + O(t −1 ), = x0 + γ−q cos t + α − 2 x(t) = x0 +

where we applied the well-known duplication formula for the Gamma function. So, the solution of (7) tends asymptotically to the solution of (9) as t → ∞.  Furthermore, if we particularize α = 0 in (7), the obtained FDE q

D0 x(t) = γ cos t, x(0) = 0

Periodic and Asymptotically Periodic Solutions of FDEs

159

Fig. 1 The graph of (16) on [0, 20]

Fig. 2 The graph of x0 (t + 2π) − x0 (t) on [0, 20]

has the S-asymptotically x0 (t) =

2π -periodic 

and bounded solution x0 (t) on J given by

  q 1 q 1 γ t q 1 F2 1; + , + 1; − 2 t 2 . (q + 1) 2 2 2 4

The FDE (9) with α = 0

(16)

q

D−∞ x(t) = γ cos t

has a periodic solution  πq . x−∞ (t) = γ−q cos t − 2

(17)

This is also demonstrated with  = 1, γ = 1 and q = 1/2 on Figs. 1, 2 and 3. The above arguments cannot be applied to nonhomogeneous fractional evolution equations of the type q

D0 u(t) = Au(t) + f (t), 0 < q < 1, t ∈ J, u(0) = u 0 ,

(18)

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M. Feˇckan

Fig. 3 The graphs of (16) and (17) on [0, 20]

where A: D(A) ⊆ X → X is the generator of a C0 -semigroup {S(t), t ≥ 0} on X , and u and f are T -periodic L ∞ functions for a fixed T > 0. Denote ξq (θ) =

1 1 −1− q1 θ q (θ− q ), q

where q (θ) =

∞ (nq + 1) 1 sin(nπq), θ ∈ (0, ∞). (−1)n−1 θ−nq−1 π n=1 n!

Note that ξq (θ) is aprobability density function defined on (0, ∞), that is ξq (θ) ≥ ∞ 0, θ ∈ (0, ∞) and 0 ξq (θ)dθ = 1. Define T : X → X and S : X → X given by 

∞

T (t) = 0

 ξq (θ)S(t q θ)dθ, S (t) = q

∞

θξq (θ)S(t q θ)dθ.

0

Lemma 1 ([35, Lemmas 3.2 and 3.3]) The operators T (t) and S (t) have the following properties: (1) Suppose that supt≥0 S(t) ≤ M. For any fixed t ≥ 0, T (·) and S (·) are linear and bounded operators, i.e., for any u ∈ X , T (t)u ≤ Mu and S (t)u ≤

M u. (q)

(2) {T (t), t ≥ 0} and {S (t), t ≥ 0} are strongly continuous. (3) {T (t), t > 0} and {S (t), t > 0} are compact, if {S(t), t > 0} is compact. Definition 2 ([35, Lemma 3.1 and Definition 3.1]) By the mild solution of (18), we mean that u ∈ C(J, X ) satisfying

Periodic and Asymptotically Periodic Solutions of FDEs



t

u(t) = T (t)u 0 +

161

(t − s)q−1 S (t − s) f (s)ds, t ∈ J.

0

General integrals are considered in the sense of Bochner. Now we are ready to prove the following result. Theorem 3 There is no nonconstant T -periodic mild solution of (18) on J . Proof Suppose that u is a mild T -periodic solution of (18) with a T -periodic L ∞ function f . Without loss of generality, we may assume that A is exponentially stable, i.e., there exist positive numbers a, M such that S(t) ≤ Me−at .

(19)

Then by [21, Theorem 5.3, p. 20], it holds (−a, ∞) ⊂ ρ(A).

(20)

In particular A−1 ∈ L(X ). Moreover, we have the following useful result. Lemma 2 ([13, Proposition 2.1]) If the C0 -semigroup {S(t)}t≥0 generated by A is exponentially stable, then T (t) ≤

m m and S (t) ≤ , t∈J (1 + t)q (1 + t)2q

(21)

for a constant m > 0. Now we split f (t) = f¯ + f˜(t), Hence



T

1 f¯ = T



T

f (t)dt.

0

f˜(t)dt = 0.

0

Then setting

in (18), we get

u(t) ˜ = u(t) + A−1 f¯ q ˜ = Au(t) ˜ + f˜(t). D0 u(t)

So we may suppose for a while that 

T 0

f (t)dt = 0

(22)

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M. Feˇckan

in (18). Next, t → T (t)u 0 is bounded and continuous on J , t → t q−1 S (t) is continuous on (0, ∞) satisfying t q−1 S (t) ≤ mt q−1 , t

0 (t

and t →

− s)q−1 S (t − s) f (s)ds is continuous on J satisfying

 t   t   tq q−1  (t − s)q−1 S (t − s) f (s)ds  ≤ m f ∞ . (t − s) ds = m f  ∞   q 0

0

So these three functions have Laplace transforms on (0, ∞). Hence we can apply the Laplace operator for  u(t) = T (t)u 0 +

t

(t − s)q−1 S (t − s) f (s)ds

0

to obtain (see [35]) u(λ) ˆ = λq−1 (λq I − A)−1 u 0 + (λq I − A)−1 fˆ(λ), λ > 0.

(23)

By (20), (23) is well-defined. Note that u and f are T -periodic functions, one can show that T T u(t)e−λt dt f (t)e−λt dt ˆ(λ) = 0 , f . (24) u(λ) ˆ = 0 −λT 1−e 1 − e−λT Submitting (24) into (23), we have T

T

0

0

u(t)e−λt dt = λq−1 (λq I − A)−1 u 0 + (λq I − A)−1 1 − e−λT

f (t)e−λt dt , 1 − e−λT

which implies ∞  i=0

for

∞

λi uˆ i =

 1 − e−λT q q λ (λ I − A)−1 u 0 + λi (λq I − A)−1 fˆi λ i=0

(−1)i uˆ i = i!



T i

u(t)t dt, 0

(−1)i fˆi = i!



T

(25)

f (t)t i dt.

0

Letting λ → 0+ for both sides of (25), we derive uˆ 0 = −A−1 fˆ0 = 0 by (22). Then (25) reduces to

Periodic and Asymptotically Periodic Solutions of FDEs ∞ 

163 ∞

λi−q uˆ i =

i=1

 1 − e−λT q (λ I − A)−1 u 0 + λi−q (λq I − A)−1 fˆi . λ i=1

(26)

Letting λ → 0+ for both sides of (26), we derive −A−1 u 0 = 0, so u 0 = 0. Then (25) reduces to ∞ 

λi uˆ i+1 =

i=0

∞ 

λi (λq I − A)−1 fˆi+1 .

(27)

i=0

Letting λ → 0+ for both sides of (27), we derive uˆ 1 = −A−1 fˆ1 , and repeating this procedure, we obtain uˆ i = −A−1 fˆi , i ∈ N0 := {0} ∪ N.

(28)

Then (27) becomes ∞ 



λi (λq I − A)−1 + A−1 fˆi+1 = 0.

i=0

Using formula (2–5) of [29, Theorem 2.3, pp. 274] (λq I − A)−1 + A−1 = −λq A−2 + O(λ2q ), we arrive at ∞ 



λi −A−2 + O(λq ) fˆi+1 = 0.

(29)

i=0

which implies fˆi = 0, i ∈ N. Recalling (22), we obtain fˆ(λ) = 0, then by (28) u(λ) ˆ = 0, so f = 0 and u = 0. If (22) does not hold, then using the splitting of f at the beginning of this section, we see that (18) has a periodic solution only if f is constant, so f (t) = f¯ and then u = −A−1 f¯. So there is no nonconstant T -periodic mild solution of (18) on J . 

3 S-Asymptotically Periodics for FDEs with Finite Lower Limits There are many results on the existence and uniqueness of S-asymptotically periodic solutions of either ODEs, PDEs or FDEs. For instance, the following FDE is considered in [31]

164

M. Feˇckan q

D0 u(t) = f (t, u(t)), t ∈ J, u(0) = u 0 ,

(30)

where f ∈ C(J × R, R) satisfies | f (t, u 1 ) − f (t, u 2 )| ≤ a(t)|u 1 − u 2 |, ∀ t ∈ J, u 1 , u 2 ∈ R, | f (t + T, u) − f (t, u)| ≤ b(t)(|u| + 1), ∀ t ∈ J, u ∈ R

(31)

for some a, b ∈ C(J, J ). Theorem 4 ([31]) Assume (31) holds. If there are ψ, χ ∈ C(J, J ) such that

 t



(t − s)q−1 f (s, 0)ds ≤ ψ(t),

0

 t ∗ (a∞ ψ∗ + M0 ) (t + T )q − t q 1 ≤ χ(t) (t − s)q−1 (b(s)(ψ(s + T ) + 1)+a(s)χ(s)) ds + (q) 0 (q + 1) |u 0 | +

 t 1 1 (t − s)q−1 a(s)ψ(s)ds + (q) 0 (q)

∀ t ∈ J , where M0∗ = max | f (s, 0)|, a∗ = max |a(s)|, ψ∗ = max |ψ(s)|, s∈[0,T ]

s∈[0,T ]

s∈[0,T ]

then (30) has a unique solution in Bψ,χ = {u ∈ C(J, R) | |u(t)| ≤ ψ(t), |u(t + T ) − u(t)| ≤ χ(t), ∀ t ∈ J } . Theorem 1 together with the method used in the proof of Theorem 4 can be applied to FDE with different fractional derivatives, like q

D0 i xi = f i (t, x1 , x2 , x3 ), xi (0) = u i , i = 1, 2, 3,

(32)

where q1 , q2 , q3 ∈ (0, 1), u i ∈ R and f i ∈ C(J × R3 , R) are locally Lipschitz continuous in x = (x1 , x2 , x3 ). We may suppose q1 ≥ q2 ≥ q3 . Then xi (t) = Fi (x)(t) = u i +

 t 1 (t − s)qi −1 f i (s, x1 (s), x2 (s), x3 (s))ds, i = 1, 2, 3, (qi ) 0

which is locally solvable on [0, δ] for a δ > 0 by the standard Picard iteration method [33] starting with xi0 (t) = u i . If we assume | f i (t, x1 , x2 , x3 )| ≤ m i1 |x1 | + m i2 |x2 | + m i3 |x3 | + vi , ∀(t, x1 , x2 , x3 ) ∈ J × R3 (33) for some m i, j ≥ 0 and vi ≥ 0, then

Periodic and Asymptotically Periodic Solutions of FDEs

1 (qi )

|xi (t)| ≤ |u i | + = |u i | + +

1 (qi )

≤ |u i | + + ≤ |u i | +

1 

1 (qi )



t

t−1

t

(t − s)qi −1

0

(t − s)qi −1

(t − s)qi −1

t−1



m i j |x j (s)| + |vi | ds

t

t−1

3 



m i j |x j (s)| + |vi | ds

j=1

t−1

(t − s)q1 −1

0



3  j=1

t



3 

m i j |x j (s)| + |vi | ds j=1

0



1 (qi )

1 (qi )





165

3 

m i j |x j (s)| + |vi | ds j=1

(t − s)q3 −1

3 

m i j |x j (s)| + |vi | ds

j=1

3



(t − s)q1 −1 + (t − s)q3 −1 m i j |x j (s)| + |vi | ds

0

j=1

for i = 1, 2, 3 and  = mini=1,2,3 (qi ). Setting M = {m i j }i, j=1,2,3 , u = (u 1 , u 2 , u 3 ), v = (v1 , v2 , v3 ), we get ||x(t) ≤ u +

1 



t



(t − s)q1 −1 + (t − s)q3 −1 (Mx(s) + v) ds

0

for the norm x = max{|x1 |, |x2 |, |x3 |} on R3 . Now we are looking for K ≥ 0 and κ > 0 so that 

1 t (t − s)q1 −1 + (t − s)q3 −1 MK eκs + v ds u +  0 (34) κt ≤ K e , t ≥ 0, then we get

x(t) ≤ K eκt , t ≥ 0.

To prove (34), we derive 





(t − s)q1 −1 + (t − s)q3 −1 MK eκs + v ds 0     MK (q1 ) (q3 ) κt t q3 v t q1 + e , + + q ≤ u +  q1 q3  κq1 κ3

u +

1 

t

(35)

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M. Feˇckan

since 

t

(t − s)q−1 eκs ds =z=κ(t−s)

0

eκt κq



κt

z q−1 e−z dz ≤

0

(q)eκt , q > 0. κq

So to get (34), we need v u + 



t q1 t q3 + q1 q3

which holds if

and then we take



   M (q1 ) (q3 ) ≤ K 1− eκt , t ≥ 0, + q  κq1 κ3

M 



(q1 ) (q3 ) + q κq1 κ3

 (2M)1/q . If we assume | f i (t + T, x1 , x2 , x3 ) − f i (t, y1 , y2 , y3 )| ≤ ki1 |x1 − y1 | + ki2 |x2 − y2 | + ki3 |x3 − y3 | + gi (t), ∀t ∈ J, (x1 , x2 , x3 ), (y1 , y2 , y3 ) ∈ R3

(38) for some ki, j ≥ 0 and gi ∈ C(J, J ), then assuming xi ∈ Bψ,χ , i = 1, 2, 3 for some ψ, χ ∈ C(J, J ), we derive |Fi (x)(t + T ) − Fi (x)(t)| ≤

  t

1 t+T

q −1 q −1 i i (t + T − s) f i (s, x1 (s), x2 (s), x3 (s)) − (t − s) f i (s, x1 (s), x2 (s), x3 (s)) ds

(qi ) 0 0  t 1 ≤ (t − s)qi −1 | f i (s + T, x1 (s + T ), x2 (s + T ), x3 (s + T )) − f i (s, x1 (s), x2 (s), x3 (s))| ds (qi ) 0  T 1 + (t + T − s)qi −1 | f i (s, x1 (s), x2 (s), x3 (s))|ds (qi ) 0 3  t  t f ∗ ((t + T )qi − t qi ) 1 j=1 ki j (t − s)qi −1 χ(s)ds + (t − s)qi −1 gi (s)ds + i ≤ (qi ) (qi ) 0 (qi + 1) 0

for

f i∗ =

max

s∈[0,T ],x≤ψ(s)

| f i (s, x1 , x2 , x3 )|.

Summarizing, we arrive at the following Theorem 5 Suppose (33) and (38). Assume there is χ ∈ C(J, J ) such that 3

j=1 ki j

(qi )

 t 0

(t − s)qi −1 χ(s)ds +

 t f ∗ ((t + T )qi − t qi ) 1 (t − s)qi −1 gi (s)ds + i ≤ χ(t) (qi ) 0 (qi + 1)

(39) for all t ∈ J and i = 1, 2, 3. Then (32) has a unique solution x(t) on J which also satisfies xi ∈ Bψ,χ , i = 1, 2, 3 for ψ(t) = K eκt with (36) and (37). Proof Set F(x)(t) = (Fi (x)(t), F2 (x)(t), F3 (t)) for t ∈ J . Then like in [31], we get that F : C(J, R) → C(J, R) is continuous and F(Bψ,χ ) is pre-compact. By (34) and (39), we have F(Bψ,χ ) ⊂ Bψ,χ . So F has a fixed point in Bψ,χ by the Tikhonov fixed point theorem [15, 16, 20]. The uniqueness of x(t) follows from the above considerations.  Let Cb (J, X ) be the Banach space of all continuous and bounded functions from J into X with the norm of the uniform convergence denoted by  · ∞ . Let S A PT (X ) represent the space formed by all S-asymptotically T -periodic functions from Cb (J, X ). Then S A PT (X ) is a Banach space see [11, Proposition 3.5]). We have the following result.

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Theorem 6 ([31]) Assume (31) holds. If

 t

q−1

sup (t − s) f (s, 0)ds

< ∞, t∈J 0  t lim (t − s)q−1 b(s)ds = 0, t→∞ 0  t sup (t − s)q−1 a(s)ds < (q), t≥0

0

then (30) has a unique S-asymptotically T -periodic solution in S A PT (R). Theorem 6 can be extended to fractional evolution equations of the type q

D0 u(t) = Au(t) + f (t, u(t)), 0 < q < 1, t ∈ J, u(0) = u 0 ,

(40)

where A: D(A) ⊆ X → X is the generator of a C0 -semigroup {S(t), t ≥ 0} and f : J × X → X is continuous. Definition 3 ([35, Lemma 3.1 and Definition 3.1]) By the mild solution of (40), we mean that u ∈ C(J, X ) satisfying 

t

u(t) = T (t)u 0 +

(t − s)q−1 S (t − s) f (s, u(s))ds, t ∈ J.

0

Theorem 7 ([25]) Assume that (19) holds, and there exist two functions g, h ∈ C(J, J ) such that  f (t, u 1 ) − f (t, u 2 ) ≤ g(t)u 1 − u 2 , ∀ t ∈ J, u 1 , u 2 ∈ X,  f (t + T, u) − f (t, u) ≤ h(t)(u + 1), ∀ t ∈ J, u ∈ X. If

  M˜ := sup   

lim

t∈J t

t→∞ 0

(41)

  (t − s)q−1  < ∞, f (s, 0)ds  (1 + t − s)2q

t 0

(t − s)q−1 h(s)ds = 0, (1 + t − s)2q

and

 μ := sup t∈J

0

t

(t − s)q−1 1 g(s)ds < , 2q (1 + t − s) m

where m is given in Lemma 2. Then (40) has a unique u ∈ S A PT (X ). Moreover, it holds u 0  + M˜ u∞ ≤ m , 1 − mμ

Periodic and Asymptotically Periodic Solutions of FDEs

169

along with u(t + T ) − u(t) ≤ ρ(t) for the unique solution ρ ∈ Cb (J, J ) of  ρ(t) = ϕ(t) + m

t

0

(t − s)q−1 g(s)ρ(s)ds (1 + t − s)2q

with   t (t − s)q−1 m M¯ T q u 0  + M˜ 2mu 0  + 1 + + m m h(s)ds, ϕ(t) = 2q (1 + t)q q(1 + t)2q 1 − mμ 0 (t − s) where

M¯ :=

sup s∈[0,T ],v≤m

u 0 + M˜ 1−mμ

 f (s, v) < ∞.

Note ρ(t) → 0 as t → ∞. Corollary 1 ([25]) Suppose (19), and (41) holds with g(s) = L and limt→∞ h(t) = 0. If f 0 := supt∈J  f (t, 0) < ∞ and m L B(q, q) < 1, then (40) has a unique solution in S A PT (X ). Now we consider a finite dimensional case of (40) q

D0 u(t) = Au(t) + f (t, u(t)), 0 < q < 1, t ∈ J, u(0) = u 0 ,

(42)

where A : Rn → Rn is a matrix and f : J × X → X is continuous. We assume a hyperolicity of A [3] qπ , ∀λ ∈ σ(A), | arg λ| = 2 where σ(A) is the spectrum of A. First, we study q

D0 u(t) = Au(t) + f (t), t ∈ J, u(0) = u 0

(43)

for f ∈ Cb (J, Rn ). By changing coordinates in (43), we have A = diag(Au , As ) and q

D0 x(t) = Au x(t) + g(t), t ∈ J, x(0) = x0

(44)

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M. Feˇckan

and

q

D0 y(t) = As y(t) + h(t), t ∈ J, y(0) = y0

(45)

, ∀λ ∈ σ(As ) and | arg λ| < qπ , ∀λ ∈ σ(Au ). Then we know [3] where | arg λ| > qπ 2 2 that (44) has a unique solution x ∈ Cb (J, Rk ) and (45) has a family of solutions in Cb (J, Rl ). Note Rn = Ru × Rs . We need to show that if g and h are S-asymptotically T -periodic then these solutions are also S-asymptotically T -periodic. We follow the way of [2, 3], so we start by considering the scalar cases: q

D0 y(t) = λy(t) + h(t), t ∈ J,

(46)

y(0) = y0 with | arg λ| >

qπ 2

and y, h ∈ Cb (J, C). Then the solution is given by (see (6)) 

t

y(t) = E q (λt )y0 + q

(t − s)q−1 E q,q (λ(t − s)q )g(s)ds.

(47)

0

Note (47) is just the formula in Definition 2 with T (t) = E q (λt q ), S (t) = E q,q (λt q ). Using the asymptotic expansion of Mittag-Leffler functions [3, Lemma 2], we see that (21) holds also in this case, so we get the result for (46) from [25]. Now we study q

D0 x(t) = λx(t) + g(t), t ∈ J,

(48)

x(0) = x0

and x, g ∈ Cb (J, C). Then the bounded solution on J of (48) is with | arg λ| < qπ 2 given by (see [2])  x(t)=

t

(t − s)q−1 E q,q (λ(t − s)q )g(s)ds−λ q −1 E q (λt q ) 1

0



∞

1

exp(−λ q s)g(s)ds. 0

(49) So x0 is determined. Using again the asymptotic expansion of Mittag-Leffler functions [3, Lemma 2], we get 1 1 exp(λ q t) + θ1 (t), q 1 1 1 t q−1 E q,q (λt q ) = λ q −1 exp(λ q t) + θ2 (t), q

E q (λt q ) =

for functions θ1,2 ∈ Cb (J, C) with

Periodic and Asymptotically Periodic Solutions of FDEs

|θ1 (t)| ≤

171

m¯ m¯ , |θ2 (t)| ≤ , ∀t ∈ J q (1 + t) (1 + t)q+1

for a constant m¯ > 0. Then (49) has the form   t  ∞ 1 1 −1 1 1 1 −1 ∞ x(t) = − λ q exp(λ q (t − s))g(s)ds + θ2 (t − s)g(s)ds − λ q θ1 (t) exp(−λ q s)g(s)ds q t 0 0  t  ∞  1 1 −1 1 1 1 −1 ∞ exp(−λ q s)g(s + t)ds + θ2 (t − s)g(s)ds − λ q θ1 (t) exp(−λ q s)g(s)ds. = − λq q 0 0 0

  1 arg λ q | exp(−λ s)| ≤ exp −|λ| s cos q

Note

1 q

and cos argq λ > 0, so

∞ 0

 x∞ ≤

1

| exp(−λ q s)|ds ≤ 1 q|λ| cos argq λ

1 1

|λ| q cos

arg λ q

. Clearly, we have

m¯ m¯ + + q |λ| cos argq λ

 g∞ .

Suppose g ∈ S A PT (R). Then for any ε > 0 there is a tε > 0 so that |g(t + T ) − g(t)| ≤ ε, ∀t ≥ tε . Then for t ≥ tε , we compute ε mg ¯ ∞ + arg λ arg λ q|λ| cos q |λ|(cos q )(1 + t)q  T  tε  t + g∞ θ2 (t + T − s)ds + 2g∞ θ2 (t − s)ds + θ2 (t − s)|g(s + T ) − g(s)|ds |x(t + T ) − x(t)| ≤

0

0

tε

mT ¯ g∞ 2g∞ m¯ mε ¯ . + + + + ≤ arg λ arg λ q(1 + t − tε )q q (1 + t)1+q q|λ| cos q |λ|(cos q )(1 + t)q ε

mg ¯ ∞

We see that there is a t˜ε ≥ tε such that  |x(t + t) − x(t)| ≤ 1 +

1 q|λ| cos argq λ

m¯ + q

 ε, ∀t ≥ t˜ε .

So x ∈ S A PT (R) given by (49). Summarizing we have the following preliminary result. Lemma 3 Consider (43) with A = diag{λ1 , . . . , λu , λu+1 , . . . , λn } for Au = diag {λ1 , . . . , λu } and As = diag{λu+1 , . . . , λn }. Then for any f ∈ S A PT (Rn ), (43) may have a solution x ∈ S A PT (Rn ) if and only if u 0 ∈ Rs , i.e., u 0 = y0 . If u 0 ∈ Rs , then this solution u exits and it is unique. So there are maps ϒ1 : Rs → S A PT (Rn ) and ϒ2 : S A PT (Rn ) → S A PT (Rn ) given by u = ϒ1 y0 + ϒ2 f . ϒ1,2 are linear and bounded.

172

M. Feˇckan

Continuing with the approach of [3], we may suppose for a general hyperbolic A that A = D + δ N for D = diag{λ1 , . . . , λu , λu+1 , . . . , λn } and the nilponent matrix ⎡

0 1 ⎢ 0 0 ⎢ N := ⎢ ⎢··· ··· ⎣ 0 0 0 0

⎤ ··· 0 ··· 0 ⎥ ⎥ ··· ···⎥ ⎥ ··· 1 ⎦ ··· 0

and any δ > 0 but depending on coordinate changes. So we have q

D0 u(t) = Du(t) + δ N u(t) + f (t), t ∈ J, u(0) = y0 ∈ Rs ,

(50)

i.e., (I − δϒ2 N )u = ϒ1 y0 + ϒ2 f. Clearly ϒ2 N : S A PT (Rn ) → S A PT (Rn ). Then for a δ small, i.e. δ < ϒ2 N −1 , we obtain u = (I − δϒ2 N )−1 ϒ1 y0 + (I − δϒ2 N )−1 ϒ2 f ∈ S A PT (Rn ) and (I − δϒ2 N )−1 ϒ1 : Rs → S A PT (Rn ) and (I − δϒ2 N )−1 ϒ2 : S A PT (Rn ) → S A PT (Rn ) are linear and bounded. Thus Lemma 3 holds also for general hyperbolic A. Now we are ready to prove the following result. Theorem 8 Assume that A is hyperbolic in (42). In addition, we suppose that f ∈ C(J × Rn , Rn ) satisfies (i) There is an L > 0 such that  f (t, u 1 ) − f (t, u 2 ) ≤ Lu 1 − u 2 , ∀t ∈ J , u 1 , u 2 ∈ Rn . (ii) For any r > 0 and ε > 0, there is tr,ε > 0 such that  f (t + T, u) − f (t, u) ≤ ε, ∀t ≥ tr,ε , u ∈ Rn , u ≤ r . If ϒ2 L < 1, then (42) has a unique solution u(u 0 ) ∈ S A PT (Rn ) for any u 0 ∈ Rs . The map u 0 → u(u 0 ) is Lipschitz continuous. Proof We need to solve u = ϒ1 u 0 + ϒ2 F(u),

(51)

where F : S A PT (Rn ) → S A PT (Rn ) is the Nemitski operator F(u)(t) = f (t, u(t). It is easy to see from (ii) that really F(u) ∈ S A PT (Rn ) for any u ∈ S A PT (Rn ). Moreover, (i) implies F(u 1 ) − F(u 2 )∞ ≤ Lu 1 − u 2 ∞ , ∀u 1 , u 2 ∈ S A PT (Rn ). Then our assumption implies, that the mapping

Periodic and Asymptotically Periodic Solutions of FDEs

173

u → ϒ1 u 0 + ϒ2 F(u) is contractive on S A PT (Rn ) in u. So (51) has a unique solution u(u 0 ) by the Banach fixed point theorem. Moreover, we have u(u 0 ) − u(u 0 )∞ ≤

ϒ1  u  − u 0  1 − ϒ2 L 0

for any u 0 , u 0 ∈ Rs .



4 Periodic BVP for FDEs Since there are no periodic solutions of FDEs in the interval J with nonlinearities periodic in time. The another reasonable alternative for FDEs is to study periodic boundary conditions on finite intervals, that is, we consider a BVP for FDEs of the form: p (52) D0 x(t) = f (t, x(t)) for some p ∈ (0, 1), and subjected to the periodic boundary condition x(0) = x(T ),

(53)

t ∈ [0, T ], x : [0, T ] → D, f : G → Rn are continuous functions, G := [0, T ] × D and D ⊂ Rn is a closed and bounded domain. Assume that function f in the system (52) is bounded by a constant vector M = col(M1 , M2 , . . . , Mn ) ∈ Rn and it satisfies the Lipschitz condition with a non–negative real matrix K = (ki j )i,n j=1 , i.e., the following inequalities | f (t, x)| ≤ M, (54) | f (t, u) − f (t, v)| ≤ K |u − v|

(55)

are true for any t ∈ [0, T ], x, u, v ∈ D, where operations | · |, ≤ between matrixes and vectors are understood componentwise. Suppose that the set   Dβ = x0 ∈ D : {u ∈ Rn : |u − x0 | ≤ β} ⊂ D is non-empty, where β =

MT p , 22 p−1 ( p+1)

Q :=

(56)

and the spectral radius r (Q) of matrix KT p 22 p−1 ( p + 1)

(57)

satisfies the estimate: r (Q) < 1.

(58)

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M. Feˇckan

We intend to find a solution of FDEs (52) that satisfies periodic boundary condition (53) in the space of continuous vector–functions x : [0, T ] → D. For this purpose, let us connect with the BVP (52), (53), the following sequence of functions {xm }m∈N0 , N0 = {0, 1, 2, . . .} given by the iterative formula: xm (t, x0 ) := x0 +

1 ( p)

 t 0

(t − s) p−1 f (s, xm−1 (s, x0 ))ds−  −

t T

p  T 0

 (T − s) p−1 f (s, xm−1 (s, x0 ))ds ,

(59) where t ∈ [0, T ], x0 ∈ Dβ and x0 (t, x0 ) = x0 . Now we have the following results [6]. Theorem 9 Assume that conditions (54)–(58) hold for the BVP (52), (53). Then for all fixed x0 ∈ Dβ , it holds: 1. Functions of the sequence (59) are continuous and satisfy xm (0, x0 ) = xm (T, x0 ). 2. The sequence of functions (59) for t ∈ [0, T ] converges uniformly as m → ∞ to the limit function (60) x∞ (t, x0 ) = lim xm (t, x0 ). m→∞

3. The limit function x∞ satisfies x∞ (0, x0 ) = x∞ (T, x0 ). 4. The limit function (60) is a unique continuoussolution of the integral equation     x(t) := x0 +

1 ( p)

t

0

(t − s) p−1 f (s, x(s))ds −

t T

p

T

0

(T − s) p−1 f (s, x(s))ds ,

i.e., it is the unique solution on [0, T ] of the Cauchy problem for the modified system of FDEs p D0 x(t) = f (t, x(t)) + (x0 ), x(0) = x0 , where p (x0 ) := − p T



T

(T − s) p−1 f (s, x∞ (s, x0 ))ds.

(61)

0

5. The following error estimation holds |x∞ (t, x0 ) − xm (t, x0 )| ≤

Tp Q m (I − Q)−1 M, 22 p−1 ( p + 1)

where I is the unit n–dimension matrix. The connection of the limit function to BVP is as follows. Consider the Cauchy problem p D0 x(t) = f (t, x(t)) + μ, t ∈ [0, T ], (62) x(0) = x0 ,

Periodic and Asymptotically Periodic Solutions of FDEs

175

where μ ∈ Rn is a control parameter and x0 ∈ Dβ . Theorem 10 Let x0 ∈ Dβ and μ ∈ Rn be some given vectors. Suppose that all conditions of Theorem 9 hold for the system of FDEs (52). Then for the solution x = x(·, x0 , μ) of the initial–value problem (62) satisfies also boundary condition (53) if and only if (63) μ = (x0 ), where (x0 ) is given by (61). In that case x(t, x0 , μ) = x∞ (t, x0 ) for t ∈ [0, T ].

(64)

By Theorem 10, we derive: Theorem 11 Let the original BVP (52), (53) satisfy conditions (54)–(58). Then x∞ (·, x0∗ ) is a solution of FDEs (52) with periodic boundary condition (53) if and only if the point x 0∗ is a solution of the determining system of equations: (x0∗ ) = 0,

(65)

where  is given by (61). The above theory is applied in [6] to fractional Duffing equation 1/2

D0 u(t) = v(t),

1/2 D0 v(t) = −ε u(t) + u 3 (t) + v(t) + a cos t , u(0) = u(T ), v(0) = v(T ) on the domain D := {(u, v) : |u| ≤ 1, |v| ≤ 1} , where T > 0, a ∈ R are fixed and ε ∈ [0, 1] is a parameter. We refer the reader to that paper for more details. The above results are extended to periodic BVP for mixed FDEs in [9] (see also (32) and [7]). More precisely, it is studied FDEs of the form p

D0 x = f (t, x(t), y(t)), q D0 y = g(t, x(t), y(t))

(66)

for some p, q ∈ (0, 1], and subjected to periodic boundary conditions x(0) = x(T ),

y(0) = y(T ),

where f : G f → Rn 1 , g : G g → Rn 2 are continuous functions, G f := [0, T ] × D f , G g := [0, T ] × Dg and D f ⊂ Rn 1 , Dg ⊂ Rn 2 are closed and bounded domains.

176

M. Feˇckan

There are many interesting applications of (66) like to economic systems in [27]. In fact, (66) may formulate a dynamical macroeconomic model of two national economies. We consider in [9] a fractional Duffing equation c 1/2 0 Dt u(t) = v(t),  3

v (t) = λ −u (t) + 4at , u(0) = u(1/4), v(0) = v(1/4)

on the domain D f = Dg = [−1, 1], where λ ∈ [0, 1] is a parameter and a ∈ (0, 1] is fixed. The above numerical-analytical method of finding solutions of FDEs satisfying periodic BVP can be extended to higher order FDEs. Let us investigate solutions of FDEs p (67) D0 x(t) = f (t, x (t)) , p ∈ (m, m + 1), m ∈ N with periodic boundary conditions of the form x(0) = x(T ), x  (0) = x  (T ), ···

(68)

x (m) (0) = x (m) (T ), where t ∈ [0, T ], x ∈ C m ([0, T ], D), D ⊂ Rn , f ∈ C(G, Rn ), G := [0, T ] × D. Together with FDE (67) we consider a perturbed FDE p

D0 x(t) = f (t, x (t)) + , p ∈ (m, m + 1), k ≥ 1

(69)

for a parameter  ∈ Rn . Then the solution of (69) is as follows [33] x(t) =

m  tk k=0

k!

ξk +

1 ( p)



t

(t − s) p−1 f (s, x(s))ds +

0

t p . ( p + 1)

(70)

We find values of the unknown ξk , k = 1, m and parameter  by substituting (70) into periodic conditions (68). For this purpose we differentiate (70) according to the general formula below: x (i) (t) =

m  k=i

t k−i 1 ξk + (k − i)! ( p − i) t p−i + , ( p − i + 1)



t

(t − s) p−i−1 f (s, x(s))ds

0

(71) i = 0, m.

Periodic and Asymptotically Periodic Solutions of FDEs

177

By putting (71) into periodic conditions (68) we get a system of equations ξi =

 T m  T k−i 1 (T − s) p−i−1 f (s, x(s))ds ξk + (k − i)! ( p − i) 0 k=i T p−i + , ( p − i + 1)

(72)

i = 0, m.

From the m−th equation of system (72) we can define expression of the parameter   p−m T  = − p−m (T − s) p−m−1 f (s, x(s))ds. (73) T 0 After substitution (73) into (72) we get a system of linear non–homogenius equations with unknown value of variables ξi , i = 1, m  T T k−i 1 (T − s) p−i−1 f (s, x(s))ds ξk = − (k − i)! ( p − i) 0 k=i+1  ( p − m)T m−i T (T − s) p−m−1 f (s, x(s))ds, i = 0, m − 1. + ( p − i + 1) 0 m 

(74)

Let us put

⎡

T Im ⎢ 0m  := ⎢ ⎣ ··· 0m

T2 I 2 m

T Im ··· 0m

··· ··· ··· ···

Tm I m! m T m−1 I (m−1)! m

··· T Im

⎤ ⎥ ⎥ ⎦

for the m-dimensional identity matrix Im and the m-dimensional zero matrix 0m . As we see, the matrix  has a triangular structure, moreover, det  = T m = 0. This means, that algebraic system (74) has a unique solution ξ1 , ξ2 , . . ., ξm that can be calculated, using the Gauss method. On the other hand, we find the inverse of . First we note ⎤ ⎡ 0m Im · · · 0m ⎢ 0m 0m · · · 0m ⎥ ⎥ T2 ⎢ ⎢··· ··· ··· ···⎥  = T Im×m + ⎢ 2 ⎣0 0 ··· I ⎥ m m m ⎦ 0m 0m · · · 0m ⎡ ⎤ 0m 0m · · · Im m ∞ ⎥  Tm ⎢ T i i−1  T i i−1 ⎢ 0m 0m · · · 0m ⎥ = + ··· + = N N m! ⎣ · · · · · · · · · · · · ⎦ i=1 i! n i! n i=1 0m 0m · · · 0m

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M. Feˇckan

⎡

and

0m ⎢ 0m ⎢ Nm := ⎢ ⎢··· ⎣ 0m 0m

Im 0m ··· 0m 0m

··· ··· ··· ··· ···

⎤ 0m 0m ⎥ ⎥ ···⎥ ⎥. Im ⎦ 0m

Note Nmi = 0m×m for i ≥ m. Introducing the function g(x) =

∞  x i−1 , i! i=1

we have  = T g(T Nm ). But

∞

g(x) =

 Bi 1 x ex − 1 , G(x) = = x = xi x g(x) e −1 i! i=0

for the Bernoulli numbers Bi [26]. So ∞ m−1  Bi 1 1  Bi G(T Nm ) = (T Nm )i = T i−1 Nmi T T i=0 i! i! i=0 ⎡ ⎤ m−2 Bm−1 T −1 Im B1 Im · · · T (m−1)! Im ⎢ ⎥ m−3 T B −1 m−2 ⎢ 0m T Im · · · (m−2)! Im ⎥ =⎢ ⎥. ⎣ ··· ⎦ ··· ··· ··· −1 0m ··· T Im 0m

−1 =

Note B0 = 1, B1 = − 21 , B2i+1 = 0, B4i < 0, B4i+2 > 0 for i ≥ 1. Moreover |B2i | 2 2(1 − 4−i ) < < . 2i −i 2i (2π) (1 − 2.4 ) (2i)! (2π) (1 − 2.4−i ) Using the above formula, direct computations show, that the exact solutions of the system (74) are the following: ξk =

  T m  T j−k−1 B j−k 1 − (T − s) p− j f (s, x(s))ds ( j − k)! ( p − j + 1) 0 j=k   ( p − m)T m− j+1 T (T − s) p−m−1 f (s, x(s))ds , k = 1, m. + ( p − j + 2) 0

(75)

Periodic and Asymptotically Periodic Solutions of FDEs

179

By substituting (73) and (75) into (70) we get the general formula for the solution of (68) and (69)   T m k  m  T j−k−1 B j−k 1 t − (T − s) p− j f (s, x(s))ds k! ( j − k)! ( p − j + 1) 0 k=1 j=k   ( p − m)T m− j+1 T (T − s) p−m−1 f (s, x(s))ds + ( p − j + 2) 0  t  T ( p − m)t p 1 p−1 (t − s) f (s, x(s))ds − p−m (T − s) p−m−1 f (s, x(s))ds. + ( p) 0 T ( p + 1) 0

x(t) = ξ0 +

(76)

Setting b0 =

1 ( p)



t

(t − s) p−1 f (s, x(s))ds −

0

 T ( p − m)t p (T − s) p−m−1 f (s, x(s))ds, T + 1) 0  T (T − s) p− j f (s, x(s))ds p−m ( p

1 ( p − j + 1) 0  ( p − m)T m− j+1 T + (T − s) p−m−1 f (s, x(s))ds, i = 1, m, ( p − j + 2) 0 bj = −

(76) has the form x(t) = ξ0 + = ξ0 +

m m  t k  T j−k−1 B j−k b j + b0 k! j=k ( j − k)! k=1

j m   t k T j−k−1 B j−k b j + b0 k! ( j − k)! j=1 k=1

j m  bj 

j! T j−k−1 t k B j−k + b0 k!( j − k)! j=1 k=1   j     j−k m  b j T j−1  j t = ξ0 + Bk − B j + b0 k j! T j=1 k=0     m  b j T j−1 t Bj − B j + b0 , = ξ0 + j! T j=1 = ξ0 +

where

j!

j    j B j (z) = z j−k Bk , k

j = 1, m

k=0

are the Bernoulli polynomials [26]. Consequently, we get

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M. Feˇckan

x(t) = ξ0 +

     T m  T j−1 t 1 Bj − Bj − (T − s) p− j f (s, x(s))ds j! T ( p − j + 1) 0 j=1

  ( p − m)T m− j+1 T (T − s) p−m−1 f (s, x(s))ds ( p − j + 2) 0  t  T ( p − m)t p 1 (t − s) p−1 f (s, x(s))ds − p−m (T − s) p−m−1 f (s, x(s))ds. + ( p) 0 T ( p + 1) 0 +

(77)

So we apply an iteration approach to      T m  T j−1 t 1 Bj − Bj − (T − s) p− j f (s, xk (s))ds j! T ( p − j + 1) 0 j=1   ( p − m)T m− j+1 T + (T − s) p−m−1 f (s, xk (s))ds ( p − j + 2) 0  t  T 1 ( p − m)t p p−1 + (t − s) f (s, xk (s))ds − p−m (T − s) p−m−1 f (s, xk (s))ds, ( p) 0 T ( p + 1) 0

xk+1 (t) = ξ0 +

k = 0, 1, . . . .

(78) We need to find conditions like above for the convergence of (78). Then its limit x∞ (t, ξ0 ), a fixed point of (77), is plugged into (73) to get  (ξ0 ) =

T

(T − s) p−m−1 f (s, x∞ (s, ξ0 ))ds.

(79)

0

We do not go into details, we postpone them to another paper. Of course, there are more approaches for solving periodic BVP of FDEs. For instance, a functional method based on the Fredholm alternative theorem is used in [30].

5 Periodics for FDEs with Infinite Lower Limits A general form of (9) is as follows k1  i=1

qi Ai D−∞ x(t)

+ A0 x(t) +

k2  j=1

B j x(t + η j ) =

k3 

eır t wr

(80)

r =1

for linear, bounded operators Ai , B j : X → X , a linear, closed operator A0 : D (A0 ) → X with a dense definition domain D(A0 ), ηr ∈ R, r > 0, wr ∈ X and X is a complex Banach space. By looking for a solution in the form

Periodic and Asymptotically Periodic Solutions of FDEs

x(t) =

k3 

181

eır t vr , vr ∈ X,

r =1

we get a system of equations Mr vr = wr , r = 1, . . . , k3 ⎛

for

Mr = ⎝

k1 

rqi e

1 2 ıπqi

Ai + A 0 +

k2 

i=1

⎞ B j eır η j ⎠ .

j=1

Since the analysis of the invertibility of Mr is difficult, we do not go into details. Next, we try to extend the above approach to nonlinear FDEs in the form k1 

q

i Ai D−∞ x(t) + A0 x(t) +

i=1

k2 

B j x(t + η j ) + φ(x(t))x(t) = eıt w

(81)

j=1

for Ai , B j : X → X , A0 : D(A0 ) → X , ηr as in (80),  > 0, w ∈ X , φ ∈ C(X, C) and  ·  is the norm on X . We set ⎛ ⎞ k1 k2   1 M =⎝ qi e 2 ıπqi Ai + A0 + B j eıη j ⎠ . i=1

j=1

Now we can derive the following existence results. Theorem 12 Suppose that M −1 : X → X is continuous and compact. If there is a r0 such that (82) M −1 (|φ(r0 )|r0 + w) < r0 , then (81) has a solution x of the form x(t) = eıt v, v ∈ D(A)

(83)

with v < r0 . Proof Inserting (83) into (81), we get a fixed point problem v = F(v) := M −1 (−φ(v)v + w) .

(84)

Clearly F : X → X is continuous and compact. Furthermore, (82) implies that if v = λF(v) for some λ ∈ [0, 1] and v = r0 then r0 = v ≤ M −1 (|φ(r0 )|r0 + w) < r0 ,

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which is a contradiction. So by the Leray-Schauder fixed point principle, F has a  fixed point v with v < r0 . This completes the proof. Theorem 13 Suppose that M −1 : X → X is continuous and there are nonnegative constants K and L such that |φ(r )| ≤ K , |φ(r ) − φ(s)| ≤ L

(85)

M −1 K < 1,

(86)

for any r, s ∈ R. If

then (81) has a solution x a form (83) for any w ∈ X satisfying w < Proof Set r¯0 =

(1 − M −1 K )2 . M −1 2 L

(87)

wM −1  . 1 − M −1 K

By (85) and (86), for any v1 , v2 ∈ X with v1 , v2  ≤ r¯0 , we derive F(v1 ) ≤ M −1 (K r¯0 + w) = r¯0 , and F(v2 ) − F(v1 ) = M −1 (−φ(v2 )v2 + φ(v1 )v1 )  ≤ M −1 φ(v1 ) − φ(v2 ))v2 + φ(v1 )(v1 − v2 ) ≤ M −1  (|φ(v1 ) − φ(v2 )|v2  + |φ(v1 )|v1 − v2 ) ≤ M −1 (L r¯0 + K )v1 − v2 .

Using (87), we obtain ⎛   wM −1  −1 ⎝ + K < M M −1 (L r¯0 + K ) = M −1  L  L 1 − M −1 K

(1−M −1 K )2 M −1  M −1 2 L −1 1 − M K

⎞ + K ⎠ = 1.

Then F : B(¯r0 ) → B(¯r0 ) is a contraction for the ball B(¯r0 ) := {v ∈ X | v ≤ r¯0 }. So by the Banach fixed point theorem, F has a unique fixed point v with v ≤ r¯0 . This completes the proof.  Finally, we consider periodically forced nonlinear fractional wave equation 

1

q

D−∞,t x(t, y) − ax yy (t, y) +

2 |x(t, y)|2 dy

x(t, y) = eıt g(y),

y ∈ (0, 1), t ∈ R,

0

x(·, 0) = x(·, 1) = 0

(88)

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183

for a > 0, q ∈ (1, !2) and g ∈ C([0, 1], C). Then (88) has a form of (81) with X = 1 2    2 L (0, 1), x = 0 |x(y)| dy, Ax = −ax(y) and D(A) = {x ∈ X | x , x ∈ X }. Clearly φ(r ) = r 2 . Hence (82) has the form w 0, √ (H2) 53 < α2 27+4 749 , . (H4) a 2 > 3 and c > α3αb 2 −3 , (H5) 53 < α2 < 3, (H6) α2 > 3 and 0 < c < α3αb 2 −3 . The following theorem summarizes the findings of [43]. 3α2 −5 Theorem 13 Suppose that α, σ > 0 so that (H1) is satisfied, and let b0 = ( σα ) . The following holds:

(1) (u ∗ , v ∗ ) is locally asymptotically stable if b > b0 , and unstable if b < b0 . (2) System (u ∗ , v ∗ ) undergoes a Hopf–bifurcation at b = b0 with: (i) A subcritical direction and orbitally asymptotically stable bifurcating periodic solutions subject to (H2). (ii) A supercritical direction and unstable bifurcating periodic solutions subject to (H3).

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4.2.2

335

Hopf Bifurcation of the 1D Diffusion Case

In the one–dimensional interval  = (0, l) , l > 0, a steady state of the Lengyel– Epstein system (19) is any positive solution U = (u, v) , 0 < x < l, to the problem 

4vu u  + a − u − 1+u 2 = 0, 4vu  dv + u − 1+u 2 = 0,

(108)

subject to boundary conditions u  (0) = u  (l) = v  (0) = v  (l) = 0 . We will give a brief summary of the findings related to the one dimensional case as reported in [20]. Define the map F : (0, ∞) × E → Y as

 u + f (u, v) . (109) F (d, U ) = dv  + g (u, v) The solutions of (108) are exactly the zeros of this map. Defining U ∗ = (u ∗ , v ∗ ), we have F (d, U ∗ ) = 0 for all d > 0. If there exists a number τ > 0 such that every neighborhood of (τ , U ∗ ) contains zeros of F in (0, ∞) × E not lying on the curve, then (τ , U ∗ ) is a bifurcation point of the equation F = 0 with respect to this curve. According to [9],(τ , U ∗ ) is a bifurcation point subject to: (a) The existence and continuity of partial derivatives Fd , FU ,and FdU . (b) ker (τ , U ∗ ) and Y | R (FU (τ , U ∗ )) being one–dimensional. / R (FU (τ , U ∗ )) given ker (τ , U ∗ ) = span{}. (c) FU (τ , U ∗ )  ∈ Now, given d and di as defined in (44) and (45), respectively, we can state the following theorems. Theorem 14 Suppose j is a positive integer such that λ j < f 0 and d j = dk for any  integer k = j. Then, d j , U ∗ is a bifurcation point of the equation F (d, U ) = 0 with respect to curve (d, U ∗ ) for d > 0. There exists a one–parameter family of non–trivial solutions  (s) = (d (s) , u (s) , v (s)) of (89) for |s| sufficiently small, where d (s) , u (s) , v (s) are continuous functions and ⎧ d (0) = d j ⎪ ⎪ ⎨ u (s) = u ∗ + sφ j + o (s) , (110) v (s) = v ∗ + sb j φ j + o (s) , ⎪ ⎪ ⎩ λ −f b = ( j 0 ) > 0. j

f1

∗ The zero set of F consists of two curves (d, U ) and  (s) in a neighborhood of  ∗ the bifurcation point d j , U .

Theorem 15 Under the same assumption 13, the projection of the bifur of Theorem  cation curve  j on the d-axis contains d j , ∞ . If d > d and d = dk for any integer k > 0, then the problem (89) possesses at least one constant positive solution. The one–dimensional case was studied further in [43], where they considered the case

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⎧ ⎨ du = dt ⎩ dv dt

  4uv + a − u − 1+u 2 ,   " 2 4uv = σ c ∂∂2 vx + b u − 1+u , 2 ∂2u ∂ 2!x

(111)

over the interval (0, π) with the no–flux boundary conditions u x (0, t) = u x (π, t) = vx (0, t) = vx (π, t) = 0. The findings of [43] are summarized in the following theorem. Theorem 16 Subject to (H1), system (111) undergoes a Hopf–bifurcation at (u ∗ , v ∗ ) 3α2 −5 when b = ( σα ) . Then, (1) If (H2) is satisfied, the Hopf–bifurcation is supercritical and the bifurcating periodic solutions are unstable. (2) If (H4) is satisfied, the Hopf–bifurcation is subcritical and the bifurcating periodic solutions are unstable. (3) If either (H4) or (H5) is satisfied, the Hopf–bifurcation is subcritical and the bifurcating periodic solutions are orbitally asymptotically stable. This concludes our discussion of the Lengyel–Epstein system’s Hopf–bifurcation. For simplicity, we have decided to omit the case of the two–dimensional system studied in [41] and are content with simply referring interested readers to the literature.

5 Modified Lengyel–Epstein Systems Over the years since the first realization of the CIMA reaction and its mathematical model, which we studied in this chapter, numerous modifications have been made to the original model. These new models were either based on modifications made to the reaction itself, such as the intensity of light applied to the reactants, or based on theoretical conceptualization. In this section, our aim is to simply list the most interesting of these models and refer the reader to the relevant literature. 1. In [31], the authors observed the existence of oscillating spots in a hexagonal lattice using the photosensitive chemical reaction made up of chlorine dioxide, iodine, and malonic acid (CDIMA). The CDIMA reaction was first realized and modeled in [19] and studied briefly in [36]. The findings of [31] confirmed previous theoretical propositions concerning the existence of Hopf–Turing mixed modes. The CDIMA reaction–diffusion model is of the form ⎧ ∂u 4uv 2 ⎪ = a − cu − 1+u ⎨ 2 − φ + ∇ u, ∂t ⎪ ⎩ ∂v = σ(cu − uv + φ + d∇ 2 v), 1+u 2 ∂t

(112)

The Lengyel–Epstein Reaction Diffusion System

337

where the parameter φ denotes the intensity of illumination applied to the reaction. The value of φ was found to control the type of patterns in the reaction with its minimum value producing spots and its maximum value producing stripes. The same model was studied in [38], where the authors used computer simulations to obtain the 3D patterns produced by the reaction under specific constraints. 2. In [37], the author studied the effects of additive white Gaussian noise (AWGN) on pattern formation in the Lengyel–Epstein model following the work of [27] concerning the Brusselator reaction–diffusion system. The main motivation for this consideration is that Turing–type systems are supposed to be the basis for morphogenesis, which must be stable against noise. The resulting noisy model is of the form ⎧ 4uv ∂u ⎪ ⎨ = ∇2u + a − u − + ζu ξu (r, t) , ∂t 1 + u 2   (113) uv ∂v ⎪ ⎩ + ζv ξv (r, t) , = σ c∇ 2 v + b u − 2 ∂t 1+u where ξu (r, t) is a Gaussian noise generator and ζu controls the intensity of the noise. The author found that the patterns can sustain a certain level of noise before losing their structure. He also found that the sustainable level is higher with inverted hexagonal patterns compared to other types. 3. The photosensitive CDIMA model (112) assumed a constant illumination intensity. The authors of [10] took it a step further and considered an oscillatory illumination φ (t) leading to the modified model: ⎧ uv ∂u ⎪ ⎨ = Δu + a − cu − 4 − φ (t) , ∂t 1 + u2   uv ∂v ⎪ ⎩ . = dΔv + σ cu − + φ (t) ∂t 1 + u2

(114)

This oscillatory illumination gave rise to three different types of periodic patterns with very interesting properties including chaos–like features. This same model was studies again in [17]. The authors considered solutions with wavelengths that are exactly half of the forcing illumination wavelength and showed that the width of the resonant response is a non–monotonic function of the illumination amplitude and that it decays to zero for high amplitudes. 4. In [46], the authors considered the modified Lengyel–Epstein model ⎧ ∂u 4uv ⎪ ⎨ = Δu − cu + + a − ψ + χω υ, ∂t 1 + u2 θuv ∂v ⎪ ⎩ = dΔv + bθu − + θψ + χω w, ∂t 1 + u2

(115)

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in Q ∞ :=  × (0, +∞). Here, ω is an open subset of  with the associated characteristic function χω , θ is another dimensionless parameter, ψ (x, t) is a given function, and υ, w ∈ L 2 (Q). 5. Another interesting study is that of Gambino et al. [15], where the authors considered a Lengyel–Epstein model with density dependent diffusion parameters. The considered model is

m

⎧ u ∂ ∂u ∂u 4uv ⎪ ⎪ , = d + γ a − u − u ⎨ ∂t ∂x u0 ∂x 1 + u2 (116)

n

⎪ v uv ∂ ∂v ∂v ⎪ ⎩ , = cdv + γcb u − ∂t ∂x v0 ∂x 1 + u2 where γ represents the relative strength of the reaction terms. Obviously, for u v m, n > 0, the ratios and help scale up or down the rates of diffusion du u0 v0 and cdv for concentrations u and v, respectively. The main finding of this study is that the nonlinear density dependent diffusion increases the tendency of pattern formation in the Lengyel–Epstein system. 6. In [45], Zheng built on the work in [46] and considered the optimal control problem for the Lengyel–Epstein system with obstacles

T [g (t, u (t)) + h (w (t))] dt,

min L (u, v, w) = 0

subject to ⎧ ∂u 4uv ⎪ ⎨ = Δu − cu + − κ∂ I[σ∗ ,σ∗ ] (u) + a − φ, ∂t 1 + u2 θuv ∂v ⎪ ⎩ = δΔv + bθu − + θφ + Bw, ∂t 1 + u2 in Q :=  × (0, T ), and

(117)

F (u) ⊂ s.

In this model, κ > 0, σ∗ , σ ∗ ∈ R are specific constants and ∂ I[σ∗ ,σ∗ ] (u) denotes an obstacle that is subdifferential of the indicator function I[σ∗ ,σ∗ ] (u) in the interval [σ∗ , σ ∗ ]. The state constraint F (u) ⊂ s represents the physical background of the CDIMA reaction model. Finally, Bw is the control term. 7. The author of [18] considered a Lengyel–Epstein system with a single diffusion in v, i.e. ⎧ vu ∂u ⎪ ⎪ =a−u+4 , ⎨ ∂t 1 + u2 (118) vu ∂v ⎪ ⎪ ⎩ = dv Δv + bu − b , ∂t 1 + u2

The Lengyel–Epstein Reaction Diffusion System

339

and studied the uniform boundedness of solutions as well as the existence and stability of steady states. 8. One of the more recent studies concerning the Lengyel–Epstein model was carried out in [29] where the authors study the formation of Turing patterns in the superdiffusion model ⎧ ∂u 4uv ⎪ ⎨ , = ∇γu + a − u − ∂t  1 + u 2  uv ∂v ⎪ ⎩ , = σ c∇ γ v + b u − ∂t 1 + u2

(119)

where ∇ γ denotes the Riesz fractional operator with 1 < γ < 2, which is defined as ∇γu = − 2 cos − 2 cos with γ R L D−∞,x u

=

and γ

R L D x,+∞ =

  1 γ γ  πγ  R L D−∞,x u + R L Dx,+∞ u 2   1 γ γ  πγ  R L D−∞,y u + R L D y,+∞ u , 2

∂2 1  (n − γ) ∂ 2 x

∂2 1  (n − γ) ∂ 2 x

x −∞

+∞

(x − s)1−γ u (s, y, t) ds,

(s − x)1−γ u (s, y, t) ds.

x

The authors were able to derive sufficient conditions for the diffusion–driven instability and consequently the existence of Turing patterns. The authors, then, proceed to study the dynamics and nature of these patterns and derive the amplitude equations. 9. In [16, 42], the authors consider the time feedback control of the Lengyel– Epstein system. They introduced a delayed feedback in the external illumination of the form φ (t) = φ0 − P [v (t − τ ) − v (t)] , where P is the feedback intensity, φ0 is the reference light intensity, and τ is the time delay. With some manipulation, the resulting system becomes ⎧ ∂u 4uv ∂v ⎪ ⎨ = Δu + a − u − − φ0 + τ P , 2 ∂t 1 + u

∂t ∂v ∂v uv ⎪ ⎩ + φ0 − τ σb P . = σcΔv + σb u − 2 ∂t 1+u ∂t

(120)

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The authors of [42] derive conditions for Turing instability and show the effect of the delay on the stability and how it leads to a Hopf–bifurcation. 10. In [35], the authors consider a slight variation to the photosensitive CDIMA reaction model (112) where the spatial distribution of the illumination intensity is nonhomogeneous, i.e. ⎧ ∂u 4uv ⎪ ⎨ − φ (x, y) , = Δu + a − u − ∂t

  1 + u2 uv ∂v ⎪ ⎩ = σ cΔv + b u − + φ (x, y) . ∂t 1 + u2

(121)

The authors establish the boundedness of solutions and produce conditions for the linear and nonlinear stability of the system.

6 Generalizations of the Lengyel–Epstein Model Although the Lengyel–Epstein system is not the only one that falls under the Turing– type umbrella, it is the most well studied. In order to generalize the work that has been carried out on this model to encompass other similar systems, a generalization is in order. Perhaps the earliest generalization is that of [14], where the author considered the general system described by ⎧ cuv ∂u ⎪ ⎪ =a−u− , ⎨ ∂t 1 + uγ

uv ∂v ⎪ ⎪ ⎩ , =b u− ∂t 1 + uγ

(122)

for some positive integer γ. The author established the existence of Turing patterns subject to a  1. c+1 Another interesting generalization was recently published in [1, 3], where the authors considered a system of the form ⎧ ∂u ⎪ ⎨ − d1 Δu = a − μu − λϕ (u) v, ∂t ∂v ⎪ ⎩ − d2 Δv = σ (u − ϕ (u) v) , ∂t

(123)

where the constants d1 , d2 , a, λ, μ, and σ are assumed to be positive, and the function ϕ is assumed to be non–negative and continuously differentiable on R+ such that

The Lengyel–Epstein Reaction Diffusion System

⎧ ⎪ ⎪ ⎨

341

ϕ (0) = 0,

  for u ∈ 0, μa ,   ⎪ ⎪ ⎩ ϕ (u) ≥ ϕ (u) u. for u ∈ 0, a . μ ϕ (u) > 0

u Clearly, with d1 = 1, ϕ (u) = 1+u 2 , λ = 4, μ = 1, d2 = σc, and σ → σb, (123) reduces to the original Lengyel–Epstein system. By means of this generalized system, the authors not only allowed for other similar models but also achieved the exact same conditions for the local and global asymptotic stability of the Lengyel–Epstein system itself. Furthermore, in [3] sufficient conditions were derived for the nonexistence of Turing pattern, which again coincide with previous results for the Lengyel–Epstein system. A further generalization of the form

 ∂u ∂t ∂v ∂t

− d1 Δu = ( f (u) − λv) ϕ (u) , − d2 Δv = σ (g (u) − v) ϕ (u) ,

(124)

was proposed in [2], where ⎧ ϕ (0) = 0, f (δ) = 0 ⎪ ⎪ ⎪ ⎪ ⎨ g(u), f (u) , ϕ (u) > 0 g  (u) ≥ 0 ⎪ ⎪ ⎪ ∃α ∈ (0, δ) ⎪ ⎩ (α − u) [ f (u) − λg (u)] > 0

for some for for such that for

δ ∈ R+ , u ∈ (0, δ) , u ∈ (0, δ) , λg (α) = f (α) , u ∈ (0, α) ∪ (α, δ) .

The authors showed that under specific parameters, the proposed generalized form encompasses some other well known models such as the FitzHugh–Nagumo system [13, 33], which represents an excitation system such as a neuron in Human physiology. Again, the derived conditions for the stability of the system coincide with previous results pertaining to the special cases considered, namely the Lengyel– Epstein system and the FitzHugh–Nagumo system.

7 Numerical Results Although we have now reviewed most of the literature related to the Lengyel–Epstein system, it does not seem fit to close the chapter without showing some numerical simulation results that confirm the findings. Let us consider the parameter sets listed in Table 2. The first four examples are solved numerically in the ODE case, i.e. system (25). As can be seen in Figs. 2, 3, 4 and 5, the first two examples are stable in the ODE sense with a center at (3, 10), whereas the remaining two have periodic solutions (a limit circle). This confirms the ODE stability condition (29) derived earlier in the chapter.

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Table 2 Simulation parameters for the Lengyel-Epstein model (19) of the CIMA reaction. Note that in the 2D diffusion case, a Gaussian disturbance w (x, y) is added to u 0 and v0 to introduce spatial non–homogeneity ODE case 2D Diffusion case Parameter Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 a b c σ u0 v0

15 5 0.5 2 3.5 10.5

15 5 9 2 3.5 10.5

Fig. 2 Solutions of the Lengyel–Epstein ODE system (25) with parameter set 1 from Table 2

20 3 1 2 3.5 10.5

15 11 3

1 2 3.5 10.5

15 1.2 2 8 3.5 10.5

15 1.2 3 8 3.5 10.5

15 2 4.5 8 3.5 10.5

The Lengyel–Epstein Reaction Diffusion System

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Fig. 3 Solutions of the Lengyel–Epstein ODE system (25) with parameter set 2 from Table 2

Figures 2, 3, 4 and 5 show the development of Turing patterns as dots and stripes appear due to the diffusion driven instability. The simulations assume zero Dirichlet boundaries and spatially non–homogeneous initial conditions by means of an additive Gaussian disturbance. Clearly, the patterns vary with parameters, especially c, which is proportional to the diffusion constant of substance v (x, t) (Figs. 6, 7 and 8).

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Fig. 4 Solutions of the Lengyel–Epstein ODE system (25) with parameter set 3 from Table 2

8 Open Problems Although the literature on this subject is substantial, numerous aspects remain unexplored and may be investigated in the future. For the sake of brevity, we are going to confine our attention to four main problems. The following problems can be examined either for the original system, or the generalized system (123): Case 1 A variety of reaction–diffusion models have been studied subject to fractional differentiation. The main idea behind this shift is that fractional calculus allows for a more general modeling strategy as opposed to the conventional one relying on the decomposition of physical phenomena into their constituents. For this reason, it would be interesting to examine the dynamics of the fractional continuous–time

The Lengyel–Epstein Reaction Diffusion System

345

Fig. 5 Solutions of the Lengyel–Epstein ODE system (25) with parameter set 4 from Table 2

Lengyel–Epstein model, which in the case of the generalization (123) may be formulated as  α Dt u = a − μu − λϕ (u) v, (125) β Dt v = σ (u − ϕ (u) v) . A good question to ask would be: is the fractional model stable under the same conditions and what kind of patterns do we obtain? Also, it is interesting to study the global asymptotic stability for α = β . Case 2 Another interesting problem is the dynamics of the generalized Lengyel– Epstein system when the Laplacian operator has a fractional exponent, i.e.

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Fig. 6 Solutions of the Lengyel–Epstein system (19) in the 2D case with parameter set 5 from Table2

Fig. 7 Solutions of the Lengyel–Epstein system (19) in the 2D case with parameter set 6 from Table 2

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Fig. 8 Solutions of the Lengyel–Epstein system (19) in the 2D case with parameter set 7 from Table 2



α

u t + d1 (−Δ) 2 u = a − μu − λϕ (u) v, β vt + d2 (−Δ) 2 v = σ (u − ϕ (u) v) ,

(126)

for 0 < α, β < 2. The main question here is: under which conditions do solutions exist globally in time for this type of system? The reader is refered to [29], where the author considered part of this problem. Case 3 The current Lengyel–Epstein system assumes a diagonal diffusion matrix implying that the diffusion of one substance is solely due to its own concentration. However, this may not be true for the CIMA reaction in general as cross–diffusion terms are more likely to appear and may not be negligible. Hence, it is desirable to study the model subject to a non–diagonal diffusion matrix, which for (123) is of the form  u t − d11 Δu − d12 Δv = a − μu − λϕ (u) v, (127) vt − d21 Δu − d22 Δv = σ (u − ϕ (u) v) . It is interesting to see whether Turing patterns still exist in this scenario and under what conditions. In other words, one should examine the current conditions for Turing instability and whether they still apply when cross diffusion is present. Case 4 The Lengyel–Epstein system we have seen in this chapter contains diagonal differentiation with respect to time. Investigating the non–diagonal case may give more insight into the dynamics of the CIMA reaction. For the generalized case (123), this amounts to studying the dynamics of

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τ11 u t + τ12 vt − d1 Δu = a − μu − λϕ (u) v, τ21 u t + τ22 vt − d2 Δv = σ (u − ϕ (u) v) .

(128)

This system can also be considered a generalization of Luikov’s system of equations with reaction terms. Case 5 To the best of our knowledge, no studies have been dedicated to the dynamics of the Lengyel–Epstein system in evolving spatial domains. One may consider, for instance, Q as an unbounded open subset of R+ × Rn+1 defined by Q = {(t, x) ; t ∈ R+ , x ∈ (−t0 − t, t0 + t) × ω} , where n is a positive number, ω is a bounded open subset of Rn , and t0 > 0. Also, for t > 0, one may define #0  # t 0 t Qt  t

= (−t0 , t0 ) , = (−t0 − t, t0 + t) , = {0} × (−t0 , t0 ) × ω, = {t} × (−t0 − t, t0 + t) × ω, = {(s, x) ∈ Q; s ≤ t} , = ∂ Q\0 , =  ∩ Qt ,

where ∂ Q is the boundary of Q and Q t is the closure of Q t . The generalized system (123) can, therefore, be modified to 

with the initial data

u t − d1 Δu = a − μu − λϕ (u) v, in Q t , vt − d2 Δv = σ (u − ϕ (u) v) , in Q t , 

(129)

u (0, x) ≥ 0, in 0 , v (0, x) ≥ 0,

and homogeneous Neumann boundaries ∂v ∂u = = 0 in t . ∂ν ∂ν Subject to this non–cylindrical domain, it would be interesting to study the large time and space size behavior as well as the characteristics of Turing’s instability. Case 6 In (123), ϕ (u) was assumed to be sublinear meaning that the mapping is non–increasing. This is equivalent to the condition (0, ∞)  s → ϕ(s) s ϕ (u) ≥ ϕ (u) u

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assumed in [1, 3]. One important question that comes to mind is: what happens if ϕ (u) is non sublinear but rather superlinear corresponding to the mapping (0, ∞)  being non–decreasing? Another interesting problem is: if the invariant s → ϕ(s) s regions of (123) as considered in [1, 3] are changed, what other reaction–diffusion systems will fall under the general model? Case 7 Finally, any combination of the above mentioned problems can and should be investigated. This would give more insight into the realistic dynamics of the chemical reaction. Acknowledgements The authors would like to acknowledge Dr. Ibrahim Suleiman from the University of Balqa’ in Jordan for his assistance in completing this chapter. We are also deeply indebted to Prof. Irving Epstein from Brandeis University, USA, and Prof. Gaetana Gambino from the University of Palermo, Italy, for their useful insights and directions.

References 1. Abdelmalek, S., Bendoukha, S.: On the global asymptotic stability of solutions to a generalized Lengyel-Epstein system. Nonlinear Anal. R. World Appl. 35, 397–413 (2017) 2. Abdelmalek, S., Bendoukha, S., Kirane, M.: The global existence and asymptotic stability of solutions for a reaction-diffusion system, to appear 3. Abdelmalek, S., Bendoukha, S., Rebiai, B.: On the stability and nonexistence of Turing patterns for the generalised Lengyel-Epstein model. Math. Method Appl. Sci. 1–11 (2017). https://doi. org/10.1002/mma.4457 4. Ahmad, S., Ambrosetti, A.: A Textbook on Ordinary Differential Equations. Springer International Publishing, Switzerland (2014) 5. Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Cambridge (1985) 6. Casten, R.G., Holland, C.J.: Stability properties of solutions to systems of reaction–diffusion equations. SIAM J. Appl. Math. 33(2), 353 (1977) 7. Castets, V.V., Dulos, E., Boissonade, J., DeKepper, P.: Experimental evidence of a sustained standing Turing–type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953–2956 (1990) 8. Conway, E., Hoff, D., Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J. Appl. Math. 35(1), 1–16 (1978) 9. Crandall, M., Rabinowitz, P.: Bifurcation from simple Eigen values. J. Funct. Anal. 8, 321–340 (1971) 10. Cuias-Vazquez, D., Carballido-Landeira, J., Pé rez-Villar, V., Munuzuri1, A.P.: Chaotic behaviour induced by modulated illumination in the Lengyel-Epstein model under Turing considerations. Chaotic Model. Simul. (CMSIM) 1, 45–51 (2012) 11. De Mottoni, P., Rothe, F.: Convergence to homogeneous equilibrium state for generalized Volterra–Lotka systems with diffusion. SIAM J. Appl. Math. 37(3), 648–663 (1979) 12. DeKepper, P., Epstein, I.R., Orban, M., Kustin, K.: Batch oscillations and spatial wave patterns in chlorite oscillating systems. J. Phys. Chem. 86, 170–171 (1982) 13. Doelman, A., van Heijster, P., Kaper, T.J.: Pulse dynamics in a three component system: existence analysis. J. Dyn. Diff. Equ. 21(5), 73–115 (2009) 14. Engelhardt, R.: Modelling Pattern Formation in Reaction-Diffusion Systems: An Investigation of Turing’s Theory of Morphogenesis with Special Reference to Highly Non Linear and Bistable Models. University of Copenhagen, Denmark (1994) 15. Gambino, G., Lombardo, M.C., Sammartino, M.: Turing instability and pattern formation for the Lengyel-Epstein system with nonlinear diffusion. Acta. Appl. Math. 132, 283–294 (2014)

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16. Ghosh, P.: Control of the Hopf-turing transition by time-delayed global feedback in a reactiondiffusion system. Phys. Rev. E 84(1 Pt 2), 016222 (2011) 17. Haim, L., Hagberg, A., Meron, E.: Non-monotonic resonance in a spatially forced LengyelEpstein model. Chaos 25, 064307 (2015) 18. Harting, S.: Reaction–diffusion–ODE systems: de–novo formation of irregular patterns and model reduction, PhD dissertation, Ruprecht-Karls-Universitat, Heidelberg, Germany (2016) 19. Horvath, A.K., Dolnik, M., Zhabotinsky, A.M., Epstein, I.R.: Kinetics of photoresponse of the chlorine dioxide-iodine-malonic acid reaction. J. Phys. Chem. A 104(24), 5766 (2000) 20. Jang, J., Ni, W., Tang, M.: Global bifurcation and structure of Turing patterns in the 1D Lengyel– Epstein model. J. Dyn. Diff. Equ. 16(2), 297–320 (2004) 21. Keshet, L.E.: Mathematical Methods in Biology, vol. 46. Classics in Applied Mathematics. SIAM, Philadelphia (2005) 22. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, Berlin (1998) 23. LaSalle, J.P.: Invariance principle in the theory of stability. In: Hale, J.K., LaSalle, J.P. (eds.) International Symposium on Differential Equations and Dynamical Systems. Academic Press, New York (1967) 24. Lengyel, I., Epstein, I.R.: Modeling of turing structures in the chlorite-iodide-malonic acidstarch reaction system. Science 251, 650–652 (1991) 25. Lengyel, I., Epstein, I.R.: A chemical approach to designing Turing patterns in reactiondiffusion system. Proc. Natl. Acad. Sci. USA 89, 3977–3979 (1992) 26. Lengyel, I., Rabai, G., Epstein, I.R.: Batch oscillation in the reaction of chlorine dioxide with iodine and malonic acid. J. Am. Chem. Soc. 112, 4606–4607 (1990) 27. Leppanen, T., Karttunen, M., Barrio, R.A., Kaski, K.: The effect of noise on Turing patterns. Prog. Theor. Phys. Suppl. 150, 367–370 (2003) 28. Lisena, B.: On the global dynamics of the Lengyel-Epstein system. Appl. Math. Comput. 249, 67–75 (2014) 29. Liu, B., Wu, R., Iqbal, N.: Turing patterns in the Lengyel–Epstein system with superdiffusion. Int. J. Bifurc. Chaos 27(8), 1730026 (2017) 30. Marsden, J.E., McCracken, M.: The Hopf–Bifurcation and Its Applications. Applied Mathematical Sciences, vol. 19. Springer, Berlin (1976) 31. Miguez, D.G., Alonso, S., Munuzuri, A.P., Sagues, F.: Experimental evidence of localized oscillations in the photosensitive chlorine dioxide-iodine-malonic acid reaction. Phys. Rev. Lett. 97, 178301 (2006) 32. Ni, W.M., Tang, M.: Turing patterns in the Lengyel-Epstein system for the CIMA reaction. Trans. Am. Math. Soc. 357, 3953–3969 (2005) 33. Oshita, Y., Ohnishi, I.: Standing pulse solutions for the FitzHugh-Nagumo equations. Jpn. J. Indust. Appl. Math. 20, 101–115 (2003) 34. Othmer, H.G., Scriven, L.E.: Interactions of reaction and diffusion in open systems. Ind. Eng. Chem. Fundam. 8, 302–313 (1969) 35. Rionero, S., Vitiello, M..: On the dynamics of the Lengyel–Epstein model with forcing intensity. Ric. mat. 1–16(2017) 36. Rudiger, S., Miguez, D.G., Munuzuri, A.P., Sagué s, F., Casademunt, J.: Dynamics of Turing patterns under spatiotemporal forcing. Phys. Rev. Lett. 90, 128301 (2003) 37. Scholz, C.: Morphology of Experimental and Simulated Turing Patterns. Friedrich-AlexanderUniversity, Germany (2009) 38. Shoji, H., Ohta, T.: Computer simulations of three dimensional Turing patterns in the LengyelEpstein model. Phys. Rev. Lett. 91, 032913 (2015) 39. Silva, F.A.D.S., Viana, R.L., Lopes, S.R.: Pattern formation and Turing instability in an activator–inhibitor system with power–law coupling. Physica A 419, 487–497 (2015) 40. Turing, A.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. 237(641), 37–72 (1952) 41. Wang, L., Zhao, H.: Hopf bifurcation and Turing instability of 2-D Lengyel-Epstein system with reaction-diffusion terms. Appl. Math. Comput. 219, 9229–9244 (2013)

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Prediction and Control of Buckling: The Inverse Bifurcation Problems for von Karman Equations Natalia I. Obodan, Victor J. Adlucky and Vasilii A. Gromov

Abstract The chapter presents novel approaches to predict and control buckling of thin-walled structures; mathematically, these approaches are formalized as the first and second inverse bifurcation problems for von Karman equations. Both approaches are based upon the method employed to solve the direct bifurcation problem for the equations in question. The approach considered was applied to several difficult problems of actual practice, viz., for the first inverse problem, to the problems of optimal thickness distribution and optimal external pressure distribution for a cylindrical shell, optimal curvature for a cylindrical panel as well; for the second inverse problem, to the problem to predict buckling of a cylindrical shell under an external pressure.




Keywords Non-linear boundary problem for von Karman equations The inverse bifurcation problem Identification of parameters Identification of pre-bifurcation state Control of bifurcations Buckling of thin-walled structures Early-warning signs
















Natalia I. Obodan–Deceased 15 July 2018. N. I. Obodan (Deceased)
V. J. Adlucky School of Applied Mathematics, Oles Honchar Dnepropetrovsk National University, Gagarina Avenue, 72, Dnepropetrovsk, Ukraine e-mail: [email protected] V. J. Adlucky e-mail: [email protected] V. A. Gromov (&) School of Data Analysis and Artificial Intelligence, National Research University Higher School of Economics, Kochnovskii Passage, 3, Moscow, Russian Federation e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_11

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1 Introduction Nonlinear systems usually depend upon both internal and external parameters; as these parameters vary, the system can transform itself drastically—it can bifurcate. When nonlinear systems are used in actual practice, challenging task is to establish correspondence between desirable values of external parameters and those of internal parameters in such a way that the system is optimal with respect to some quality functional, while retaining its full functionality. Mention in passing, that these demands seem to be natural but may be contradictory for real-world systems —‘optimization leads to catastrophe’ [1]. On the other hand, nonlinear behaviour necessitates an approach to identify pre-bifurcation states of the system subject to external contingency impact—such approach makes it possible to predict bifurcations and thereby catastrophic changes of system state. Mathematically, the problems in question are stated as the inverse bifurcation problems. The term ‘inverse bifurcation problem’ is conventionally used in two distinct senses. The first statement suggests that, for a given vector of external parameters that defines bifurcation points, one determines internal parameters values such that they meet a specific property. This problem may be described as the bifurcation control problem. The second statement of the inverse bifurcation problem implies that one attempts to identify pre-bifurcation states and those that do not precede bifurcation, provided a sequence of system states is observed. This problem may be described as the bifurcation prediction problem. The present chapter is concerned with novel approaches to solve both inverse bifurcation problems for the nonlinear elliptic equations of the von Karman-type and their applications. Systems modeled by the von Karman equations [2] are used extensively in aerospace, biomedical, and nanotechnology applications, in shipbuilding, in oil and gas industry, and so on. This can be attributed to the fact that the equations govern stress-strain state of thin-walled systems, structures that ensure the maximum buckling load for a given mass of the structure. Guarracino [3] explores the limits of applicability of von Karman equations in real-world shell structures; the paper [4] discusses the same limits for carbon nanotubes. In the context of nonlinear behaviour of thin-walled shells, the first of the above bifurcation problems is embodied in the problem of the worst initial imperfection or the infavourable load [5] (that is the imperfection\load corresponding to the lowest possible buckling load). The second problem is also a subject of much current interest as far as it manifests itself in actual practice as the problem of rapid sustainability assessment of a damaged thin-walled structure. On the other hand, robust design [6], which is growing more popular in engineering, implies that one is able to identify every possible buckling state rapidly. Obodan et al. [7] studies nonlinear boundary problems of von Karman equations and presents their primary, secondary, and tertiary bifurcation paths as well as bifurcation properties in function of parameters. By way illustration, Fig. 1a shows branching structure for von Karman equations defined on closed cylindrical domain

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Fig. 1 a The branching structure of the von Karman equations [2] (defined on closed cylindrical domain) for a constant right-hand term F ð xÞ ¼ const ¼ q and for homogeneous boundary problems. On the x-axis, typical value of u1 (divided over the parameter “shell thickness”) is plotted; on the y-axis, an amplitude of a right-hand member k, (divided over a minimum bifurcation value of the respective linear problem). The central cross-sections of the solutions (function u1 ) are placed near the corresponding sections of bifurcation paths. Circles denote solution singular points. The figure displays a pre-bifurcation path (it coincides with y-axis); primary bifurcation paths O  A  B  C (it corresponds to regular solution 1), M  N (regular solution 3), that with a minimum point H (regular solution 3); secondary bifurcation paths A  D  G  E  F  H (localized solutions 4, 5, and 6) and A  K  L  M (localized solutions 7 and 8). b A sketch of a loaded cylindrical shell: the system is governed by the von Karman equations; the solutions exhibited near post-bifurcation paths in Fig. 1a correspond to its post-buckling shapes

for a constant right-hand member and homogeneous boundary conditions—this corresponds to a uniformly loaded cylindrical shell, the structure of paramaunt inportance in aerospace engineering (Fig. 1b). Circles symbolize singular points of solution. The branching structure presented suggests that a set of singular points (as opposed to the analogous sequence of the respective linear eigenvalue problem) are neither descending nor ascending sequence; furthermore, the minimum bifurcation values are located not on the pre-bifurcation equilibrium path but on secondary bifurcation paths. That fact makes the inverse bifurcation problems difficult since some perturbations can ‘throw’ the system to secondary and tertiary bifurcation path at values of external parameters significantly lesser than desirable. The remainder of the chapter is organized as follows. The next section discusses related works concerned with both inverse bifurcation problems; the third presents the von Karman equations and formally states both problems. The third on deals with the method to solve direct bifurcation problem; the fourth discusses the existence of a solution of the first inverse bifurcation problem. The sixth and

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seventh sections outline methods to solve the first and second inverse problems, respectively. The eighth and nineth sections provide results for a series of applications. Finally, the last section presents conclusions and directions for future research.

2 Related Works Let us start with papers concerned with the first inverse bifurcation problem for von Karman equations; such studies are partially inspired by new technologies to manufacture composite thin-walled structures and the respective concept of topological optimization [8]. The concept suggests that bifurcation set of von Karman equations can be optimized by the right choice of coefficients of the equations, assumed to be functions of multiple variables—the optimal functions appear to be far from uniform one. The vital problem here, the problem that many researchers attempt to gain a better insight into, is to estimate minimum buckling loads for such systems [9, 10]. For example, Henrichsen et al. [11] propose to choose parameters of von Karman equations (with initial imperfections) out of finite set of predefined alternatives to determine the optimal structure of the bifurcation set—to this end authors propose to use iteration process such that each iteration comprises determination of the worst imperfection with respect to buckling load. A class of possible imperfections used consists of linear combinations of symmetric solutions of linear eigenvalue problem—the combinations, however, are markedly asymmetric functions. The present chapter is concerned with an approach based upon methods to solve both inverse (not bifurcation!) problems for PDEs and direct bifurcation problems for them. So the remainder of this section outlines available approaches to these two problems. The coefficient inverse problem for nonlinear boundary problem of PDEs (particularly, of von Karman equations) is conventionally solved with the employment of various optimization methods like the Newton, the Gauss-Newton, the gradient descent and others [12]. This approach implies that some regularization technique is applied to iterative process: Engl and Kügler [13] presented a brilliant review of various inverse problems and regularization techniques with a particular emphasis on nonlinear ones. Alternatively, one can employ a neural network approximation for the inverse operator that maps traces of the direct (forward) problem solution onto unknown functions of the inverse one. To this end, these solutions are discretized and a learning sample for a neural network is selected in such a way that the values of the inverse problem function corresponding to its vectors form a compact set. Such approach guarantees that the inverse problem is regularized provided the direct problem does not possess singular points on its definition domain and that the traces of the direct problem solution corresponding to the inverse problem solution are close enough to the specified ones. Both formulations require that the Fréchet

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differentiability of the inverse operator be proved [14]. The paper [15] considers the inverse problem for non-linear time-dependent Schrödinger equation; the problem is formulated using variational approach. The author investigates the existence, uniqueness, and Fréchet differentiability of its solution. The paper [16] deals with inverse problems in elasticity with application to cancer identification. Reviewing the papers that deal with the problem to predict bifurcation (the second inverse bifurcation problem), one should note that by and large all available approaches can be broadly classified into two groups. For the method of the first group, one identifies values of parameters of the observed system (those of initial imperfection, the external load, flaws or irregularities of shell material) and then performs non-linear bifurcation analysis for the system with the identified parameters to estimate bifurcation values [17]. The second group suggests that one utilizes a salient pre-bifurcation feature to predict bifurcation; such features are collectively called ‘flags’, ‘fingerprints’, or ‘precursors’ of bifurcation; recently, terms ‘earlywarning signs’ and ‘tipping points’ are growing more popular [18–20. Sometimes methods of these groups are named model and modelless, respectively [21]. The method proposed by Stull et al. [22] exemplifies the first approach as applied to theory of thin-walled structures; it implies that the stochastic inverse problem is solved (in the framework of Bayesian statistics) to identify initial imperfections and thereby to estimate the first bifurcation point value (the buckling load). Vibration Correlation Technique, experimental method proposed by Abramovich et al. exemplifies the second approach [23]; the authors propose to use characteristic oscillations preceding to buckling as precursors of bifurcation and formulate operational guidelines to prevent buckling. For the overwhelming majority of methods used to identify a pre-bifurcation (pre-buckling) state, it is necessary to perform a very large number of arithmetic operations. Since speed of buckling processes associated with thin-walled shells is very high and the buckling can be due to off-design contingencies, the use of such methods for rapid sustainability assessment of thin-walled shells is computationally prohibitive. This makes necessary to develop a method able to divide all operations required to solve the problem under study into two non-equal parts. The substantially greater part should be performed before the system starts functioning, (off-line); the significantly lesser—directly used to identify pre-buckling states—is to be carried out when the system is in operation (on-line) [24]; such separation is extremely important to make a decision in-time for the system buckling due to off-design contingencies. The necessary precondition to apply the method is to trace all bifurcation path of the respective non-linear boundary problem. Thus the complete branching structure with all existing bifurcation paths (that is solution of the direct bifurcation problem) is a necessary precondition to solve both inverse bifurcation problems in the framework of the approaches discussed below in this chapter. The seminal papers concerned with bifurcation paths of the von Karman equations (for example, Thompson and Hunt [25]) deal mainly with the primary bifurcation paths. However, the non-linear boundary problem features the spectrum crowding and the secondary and tertiary bifurcation paths that makes it

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necessary to use a radically new method. To the best of authors’ knowledge, the paper [26] presented the secondary bifurcation paths associated with a cylindrical shell subjected to uniform external pressure; the solutions of the secondary paths correspond to localized buckling shapes, frequently encountered in actual practice (see also Obodan et al. [27], Obodan and Gromov [7]). The paper [28] considers the nonlinear boundary problem under investigation for cylindrical panel subjected to the lumped force; the authors employ the finite element method combined with arc-length technique to trace primary bifurcation paths. The papers [29, 30] are concerned with ‘jump’ technique to switch to bifurcation paths. The authors succeeded in tracing some secondary bifurcation paths for an axially-compressed cylindrical shell (the compression is uniform). Hu and Burgueño [31] explore, both numerically and experimentally, bifurcation solutions corresponding to non-uniform compression; the compression function is a linear combination of linear buckling modes. Zhao et al. [32] combine linear and nonlinear techniques to prevent localized buckling (associated with tertiary bifurcation paths [7]) by means of optimally placed grids-stiffeners. The monograph [7] presents the bifurcation structure including various primary (for cylindrical panel (see also Obodan and Gromov [33])); primary and secondary (for cylindrical shell subjected to a uniform pressure); primary, secondary, and tertiary (for cylindrical shell subjected to a uniform axial compression) bifurcation paths; papers [34, 35] provides results for spherical shell. One should emphasize that usual nonlinear computation can be carried out using any conventional finite elements package, while the problem to construct bifurcation structure makes it necessary to develop and implement specific methods. These methods may lean upon the finite element method [5, 28] or may require development new methods to solve nonlinear boundary problems for PDEs [7, 36]—we favour the second alternative as it avoids lots of computational difficulties of the bifurcation theory. By way of illustration, one may point to the generalized Kantorovich method [7, 27, 33, 37] or similar method of variational iteration [36]; the later in conjunction with iterative linearization procedure and reduction of the Karman equation into Germain-Lagrange equations makes it possible to find solution of the linear static von Karman equation for a plate [36]. The article mentioned also furnishes a proof of convergence for the combined technique [36]. Sometimes, it is a matter of importance to ascertain dynamics of buckling, or to put it differently, of transition to nonlinear solutions associated with bifurcation paths. Localized solutions, both theoretically interesting and practically significant, (starting from early work [38–40]) are associated with ‘bifurcation structure in which a sequence of saddle-node bifurcations leads to the growth of localized states by subsequent addition of structures at the fronts of a localized state is known as homoclinic snaking’ [41, 42] and considered to be edge states that are fully nonlinear equilibrium states on the basin boundary of the unbuckled state’s basin of attraction. Their existence is attributed to a phenomenon of spatial localization [41], observed in a broad class of dynamical partial differential equations. It is worth stressing that the nonlinear structures such as beams, plates, and shells exhibit a wide range of various dynamic regimes: harmonic, subharmonic,

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quasi-periodic, and chaotic [36, 43–47]. Furthermore, a noise, inherent in any real-world system, tends to make dynamical regimes more involved [44], up to the point that new modifications of conventional scenarios of transition to chaos emerge [44]. Another dynamical regime of fundamental interest is standing waves (soliton-like solutions) found by Awrejcewicz et al. [46]; this type of behaviour should be distinguished in a proper way from localised buckling modes in the second bifurcation problem. Actually, the above-mentioned complex behaviour of dynamical von Karman equations is the raison d’etre to explore branching structure of static von Karman equations (that is to solve the direct bifurcation problem) in order to solve both inverse bifurcation problems in question.

3 Problem Statement The direct bifurcation problem. von Karman equations
Nonlinear max  are assumed to be min max  x  x ; x  x  x defined on the domain X : xmin ; x ¼ fx1 ; x2 g (with 1 2 1 1 2 2 piecewise-differentiable boundary C ¼ @ X) [2]: L ¼ 0;

ð1Þ

L ¼ fL‘1 ; L2 gT

  ik jl L1 ¼ rij Aijkl 1 ðH Þðrkl u1 Þ  1 1 rkl u2 Bij þ rij u1 ¼ kF ðx1 ; x2 Þ   ik jl L2 ¼ rij Aijkl 2 ðH Þðrkl u2 Þ  1 1 rkl u1 Bij þ rij u1 ¼ 0

ð2Þ

Possible boundary conditions, defined on a part C, are as follows: ui ¼ ui ; ri ui ¼ hi ; i; j ¼ 1; 2;

ð3aÞ

rii u2 ¼ Tii ; rij u2 ¼ Tij i; j ¼ 1; 2:

ð3bÞ

or

Here, u ¼ fu1 ; u2 gT denotes unknown functions of the problem in question; 2 ri ¼ @x@ i ; rkp ¼ @x@i @xj ; i; j ¼ 1; 2; H ð xÞ stands for a vector-function that describes properties of the system, k parameter. Asterisk designates predefined values of the respective functions on the boundary. Hereinafter the summation agreement over repeating indexes is applied. In what follows we use the following notations: L, R, and h are the length, radius and thickness of the shell. E and l denote the Young’s modulus and Poisson’s ratio of shell material, respectively. The functions H ð xÞ ¼ fHk ð xÞg represent geometrical, mechanical or other ð2Þ properties of the system. If H ð xÞ and a right-hand member F ð xÞ belong to W2 ðXÞ and solution u belong to specific functional space V ðXÞ ¼ V1 ðXÞ  V2 ðXÞ; namely,

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those constructed from elements of W2 ðXÞ with the employment of closure operation in ‘energy’ norms 

us; vs



R ¼ Aijkl s ðH Þrij us rkl vs dX; R X ijkl ¼ As ðH Þrij us rkl us dX; s ¼ 1; 2: ;

Vs ðXÞ

kus kV s ð X Þ

ð4Þ

X

ðu; vÞV ðXÞ ¼ ðu1 ; v1 ÞV1 ðXÞ þ ðu2 ; v2 ÞV2 ðXÞ ; then solution of the problem (1), (3a, 3b) exists [2]. The spaces are defined in such a way that their elements satisfy particular boundary conditions (3a, 3b). In the present chapter, unless otherwise stated, we assume that the vector-function u belongs to the space VðXÞ; corresponding to particular conditions (3a, 3b). Provided the operator gradLu ðu; H; kFÞ exists and is invertible at all functions ð2Þ FðxÞ; HðxÞ 2 W2 ðXÞ; according to the implicit function theorem, a continuous solution uðH; kFÞ exists and is unique. If, vice versa, Lu ðu; H; kFÞ ¼ 0; that is the operator kernel is nontrivial, or put it differently, several different solutions or families solutions exist; these solutions depends upon parameter and k functions FðxÞ; HðxÞ as well. The pair ðkcr ; ucr Þ such that Lu ðucr ; kcr ; H0 ; F 0 Þ ¼ 0;

ð5Þ

for some functions FðxÞ ¼ F0 ðxÞ and HðxÞ ¼ H0 ðxÞ is a singular point of the problem (1), (3a, 3b). Singular points are classified into bifurcation and limit, depending on whether a solution of the problem exists for k [ kcr . A bifurcation set comprises all parameters values that correspond to singular points of the problem in question. The first inverse bifurcation problem. Actual practice dictates statements of the first inverse bifurcation problems such that either F(x) or H(x) is specified, while another function is determined. The present chapter discuss an approach to solve the first inverse bifurcation problems for the case of a specified F(x) (an approach to solve the problem for a specified H(x) is similar); Sect. 8 presents solutions of the problem under discussion for both cases. A quasisolution of the first inverse bifurcation problem [48] is defined as H  ¼ arg min JðHÞ ~ H2H

ð6Þ

with a functional Z J¼

C 2 ðH ÞdX þ u1 ðk  kcr ðH ÞÞ:

ð7Þ

X

Hereinafter C(H) is a specified functional, which is assumed to be continuous with respect to H; u 2 U; where U symbolizes a compact set of the space VðXÞ; u1 ~ is defined as follows: denotes a Lagrange multiplier; the set H

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8 9 < Aijkl ; H 2 W ð2Þ ðXÞ; A  Aijkl  A ; H  H  H ; c  dH  c ; = min max min max min max p p 2 ~ ¼ dk H : ; signcmin ¼ signcmax

ð8Þ The condition (5) determines the bifurcation value kcr ¼ kcr ðHÞ; which is amplitude of the right-hand member corresponding to the first singular point located on a pre-bifurcation equilibrium path (the path starts at the origin of coordinates); k is a desirable value of the parameter k. The second inverse bifurcation problem. Solving the second inverse bifurcation problem, one assumes to have a sequence of monitored deformed shapes (possibly affected by off-design impact) that constitutes the primary input data for the problem under consideration. In this context, the solution of the problem is a set of typical sequences that precede bifurcation (precursors of bifurcation). Mathematically, the problem to identify pre-bifurcation state is stated as Cbif ¼ arg min qC ðC; C  Þ;

ð9Þ

C2=

where qC ðC; C Þ is the Euclidean distance between a subsequence of observed solutions C  and the sequence of possible solutions (determined by the respective bifurcation precursor) C; = is the
set of all available precursors.  The observed sequence C ¼ u0 ; u1 ; . . .; uM reflects system behaviour using n o ðiÞ vectors of observations ui ¼ u1 ðXk Þ ; each vector comprises the values of the i-th observation of the function u1 calculated at fixed points Xk 2 X; k ¼ 1; n; . . .; K; of the definition domain (observation points). Conventionally, the neighbouring vectors are observed at regular intervals. A nbifurcation precursor is defined as a typical sequence of solution vectors o C ¼ uC1 ; uC2 ; . . .; uCMC ; for it, the neighbouring vectors may be associated with

unequal intervals. These intervals are characteristic of the particular precursor. In these terms, a solution of the problem to identify pre-bifurcation state is the functional mapping F : C ! = that maps an observed sequence of states C onto a precursor belonging to =. This statement allows interpretation in the context of the ‘reality model’ concept: a reality model is a model of the object under examination that is in agreement with its observed state. Respectively, if one chooses the bifurcation precursor closest to the observed sequence, one thereby chooses a reality model. One should emphasize that the observed sequence C  may be either a sequence without temporal dimension or a sequence unfolding in time. For the former case, both the observed sequence and sequences of solution functions that constitute bifurcation precursors include points of the same set, namely, that of possible solutions of the nonlinear boundary problem (1), (3a, 3b). The later case implies that the observed sequence includes solutions of the dynamical von Karman

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equations whereas bifurcation precursors are associated with solutions of the static von Karman equations as before. The method employed (see below, Sect. 7) is presumed that sections of the first sequence are close enough to bifurcation precursors. Respectively, it is worth stressing that if one compares bifurcation solutions (nonlinear buckling modes) of the static (see, for example, Obodan et al. [7], Fujii and Noguchi [29], Fujii et al. [30], Arbelo et al. [49], Hao et al. [50–52] and experimental studies by Krasovsky et al. [53]) and dynamic (see, for example, Kreilos and Schneider [41, 42] and references therein and experimental studies by Yamaki [54], Virot [55]) von Karman equations one may conclude that they are similar enough. This justifies the application of the approach to the equations under consideration. It means that the approach is a kind of a method to generate bifurcation precursors for the dynamic von Karman equations, exploring equations that govern a structure of their attractors that is the static von Karman equations. Generally, the inverse bifurcation problems (of both types) for PDEs can be classified as: 1. The problems to predict and control bifurcations for a non-constant (perturbed) right-hand member. 2. The problems to predict and control bifurcations for a non-constant parameters of the direct problem. 3. The problems to predict and control bifurcations for unknown boundary conditions provided several types of them are possible. The methods proposed in this chapter to solve the first and second inverse bifurcation problems, make it necessary to construct a solution of the respective direct bifurcation problems. This, in turn, includes solving nonlinear boundary problems for PDEs, tracing pre-bifurcation and bifurcation paths, and locating singular points on them. The outcome of this process is the complete bifurcation structure for the nonlinear boundary problem of PDEs.

4 Method to Solve the Direct Bifurcation Problem The generalized solution of the non-linear boundary problem in question is a pair of function u 2 VðXÞ satisfying the integral equations Z

   1ik 1jl Bij v1  ri v1 rj u1 rkl u2 dX;

  Z 1 ¼  1ik 1jk Bij u1  ri u1 rj u1 rkl v2 dX; 2

ðu1 ; v1 ÞV1 ðXÞ ¼ ðu2 ; v2 ÞV2 ðXÞ

ð10Þ

for arbitrary functions v ¼ ðv1 ; v2 Þ 2 VðXÞ: To construct solutions of the non-linear boundary problem (1), (3a, 3b), given by its generalized solution (10), approximate sequence is generated with the employment of the representation the vector of unknown functions

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363 ðnÞ

uðsnÞ ðx1 ; x2 Þ ¼ hðsnÞ ðx1 ÞgðsnÞ ðx2 Þ þ hs ðx1 ÞgðsnÞ ðx2 Þ; s ¼ 1; 2; n ! 1; uðsnÞ ðx1 ; x2 Þ 2 Vs ðXÞ

ð11Þ ðnÞ

on the iterations of the method. Each iteration comprises calculation both hs ðx1 Þ and gðsnÞ ðx2 Þ. The functions with overbars are known from the previous iteration. The representation (11) implies that a solution of the non-linear boundary problem (1), (3a, 3b) is sought as the sequence of solutions for ordinary differential equations     dh max ¼ f x1 hðx1 Þ; hðx1 Þ; ax2 ; q ; x1 2 xmin 1 ; x1 dx1

ð12Þ

  dg max : ¼ f x2 ðgðx2 Þ; gðx2 Þ; ax1 ; qÞ; x1 2 xmin 2 ; x2 dx2

ð13Þ

For the sake of brevity, the superscripts corresponding to iteration in n s are omitted o d h1 ðx1 Þ d s h2 ðx1 Þ (12), (13). The vectors hðx1 Þ and gðx2 Þ are hðx1 Þ ¼ , dxs1 ; dxs1 ns o d g1 ðx2 Þ d s g2 ðx2 Þ gð x 1 Þ ¼ , s ¼ 0::3; hðx1 Þ ¼ hðn1Þ ðx1 Þ, gðx2 Þ ¼ gðn1Þ ðx2 Þ. dxs ; dxs 2

2

Elements of the vectors ax1 ; ax2 are definite integrals of components of hðx1 Þ; hðx1 Þ and gðx2 Þ; gðx2 Þ; respectively. If the components of gðx2 Þ are calculated first then components of hðx1 Þ are replaced by the appropriate components of hðx1 Þ while the vector ax1 is formed. It is worth stressing that an order of the Eqs. (12), (13) does not depend on the way the initial PDEs (1) are approximated, but does depend only on their order. Ordinary differential Eqs. (12), (13) must be completed by point-wise boundary conditions derived from boundary conditions (3a, 3b). The iterative generalized Kantorovich method (IGKM) refers to an iterative process such that the systems (12), (13) are resolved separately and successively; therefore solution of the non-linear boundary problem for PDEs is found using solutions for the sequence of boundary problems for ODEs. The process is completed, if the norms of solutions differences for several consequent iterations are small. To solve ODEs of the sequence mentioned above, one employs the reduction of nonlinear boundary problem (for ODE) to the equivalent Cauchy problem; the method is based upon the Newton iteration formula. To obtain good initial approximations for the IGKM (that is of fundamental importance for the convergence of the Newton method), one employs parameter continuation. It is worth noting that the Newton method used to solve the problems (12) and (13) implies that one checks numerically the condition (5). Since the parameter is continued from zero k  0, the algorithm locates all singular points of a pre-bifurcation equilibrium path. Furthermore, branching equations solved for singular points allow tracing bifurcation paths and, eventually, constructing the complete bifurcation structure of PDEs under study. Obodan et al. [7] describes the IGKM in greater detail, Ref. (Gromov) examines its convergence, Obodan and

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Lebedev [26], Obodan et al. [7], Obodan and Gromov [27] and Obodan and Gromov [33] presents solutions of the direct bifurcation problem for a number of thin-walled structures used in actual practice.

5 The Existence of a Solution to the First Inverse Bifurcation Problem The necessary prerequisite to solving the optimization problem (6) is to prove that the relation kcr ðHÞ determined by the condition (5) is continuous. To this end, represent u as u ¼ u0 þ ~ u and then express conditions (5) explicitly as:     ~1 ¼ rij Aijkl u1  1ik 1jl rkl ~ u2 Bij ð xÞ þ rij u01 þ rkl u02 rij ~u1 ¼ 0; L 1 rkl ~    ~2 ¼ rij Aijkl L u2  1ik 1jl rkl ~ u1 Bij ð xÞ þ rij u01 ¼ 0; 2 rkl ~

ð14Þ ð15Þ

with ~ ui jC ¼ 0; ri ~ ui jC ¼ 0; i; j ¼ 1; 2 or rii ~ u2 ¼ 0; rij ~u2 ¼ 0; i; j ¼ 1; 2. Thus the relations for the generalized solution are reworked in the following way: Z u1 ; v1 ÞV1 ðXÞ ¼ ð~ ð~ u2 ; v2 ÞV2 ðXÞ ¼

Z

   1ik 1jl Bij v1  ri v1 rj u01 rkl ~ u2  rkl u02 ri v1 rj ~u1 dX;   1ik 1jk Bij ~ u1  ri u01 rj ~ u1 rkl v2 dX; ~u 6¼ 0;

ð16Þ

In the expression (16), the relation of generalized solution (10) is assumed to be satisfied for u0 ; the terms of the second order of smallness with respect to ~u are neglected. Now, to establish the continuity of the relationship kcr ðHÞ we need rearrange the linearized generalized solution (16) as aðH Þ  bðH Þ ¼ 0; aðH Þ ¼ fa1 ðH Þ; a2 ðH ÞgbðH Þ ¼ fb1 ðH Þ; b2 ðH Þg; where

Z u2 dX; 1ik 1jl Bij v1 rkl ~

a1H ¼ ð~ u1 ; v1 ÞV1 ðXÞ þ Z b1H ¼

X

  u2 þ rkl u02 rj ~ u1 ri v1 dX; 1ik 1jl rj u01 rkl ~

X

Z u1 rkl v2 dX; 1ik 1jl Bij ~

u2 ; v2 ÞV2 ðXÞ þ a2H ¼ ð~ Z b2H ¼

X

  u1 rkl v2 dX; ~ 1ik 1jl ri u01 rj ~ u 6¼ 0;

X

and investigate the functionals a(H), b(H).

ð17Þ

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~; u0 Þ is met the condition (16), if DH is a change in Provided the point P1 ¼ ðH; u the function H such that kDH k  e, we need estimate differences aH þ DH ðH þ DH Þ aH ðH Þ; bH þ DH ðH þ DH Þ  bH ðH Þ for the point P2 ¼ ðH þ DH; ~u; u0 þ Du0 Þ. In these expressions, the functions that appear in the subscripts determine the coefficients Aijkl p ; Bij ; on the other hand, the functions that appear in the arguments influences the functional indirectly, through the function u0 . If DAijkl p ðH; DH Þ  ~ following e; kDu0 ðH; DH Þk  dðeÞ; then, taking into account properties of the set H, Vorovich [2], it is straightforward to prove that dðe) ! 0 with e!0. ð1Þ

ð2Þ

V ðXÞ and V ðXÞ symbolize functional spaces corresponding to points P1 and P2, respectively; it is worth emphasizing that the distances in these spaces are defined with the employment of the functions H and H þ DH; respectively. With the equivalence of these spaces [2], expressions (4) allows one to introduce     Dp  up ðH Þ; vp ð2Þ  up ðH Þ; vp ð1Þ Vp ðXÞ Vp ðXÞ Z ¼ 1ik 1jl DAijkl p ðH; DH Þrij up rkl vp dX;

p ¼ 1; 2;

X

and then to estimate the quantity, using the Sobolev embedding theorem [2], as follows: Dp  me up

ð 1Þ

Vp ðXÞ

vp

ð 1Þ

Vp ðXÞ

; m ¼ const:

ð18Þ

The operators Dp defined by the following expressions   D p up ; v p

ð2Þ

Vp ðXÞ

  ¼ up ; v p

ð 1Þ

Vp ðXÞ

; p ¼ 1; 2;

ð19Þ

permit, taking into account the inequality (19), the following estimates 1  me  Dp  1 þ me; p ¼ 1; 2:

ð20Þ

Finally, if one estimates all terms of aH, bH (17) with the employment of the embedding theorems and the previous inequality, one would obtain uk jaH þ DH ðH þ DH Þ  aH ðH Þj  m1 ek~

ð 1Þ

V ðXÞ

jbH þ DH ðH þ DH Þ  bH ðH Þj  m2 dku0 k

ð1Þ

kvkV1X ;

ð 1Þ

V ðXÞ

ð1Þ

ð1Þ

k~ukV1X kvkV ðXÞ :

This proves the continuity of the functionals aH, bH.

ð21Þ

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Furthermore, Litvinov proves that, if conditions ensure that the functionals aH, ~ then the mapping KðH Þ ¼ kcr ðH Þ  inf kcr ðH Þ (K : H ~ ! l1;0 , where bH, H 2 H, l1;0 is the normalized space of sequences n ¼ fn1 ; n2 ; . . .; nn ; . . .g that converge to 0; the norm is defined as knkl1;0 ¼ supjnn j) is also continuous [56]. The constants n

~ are supposedly such that the (nonlinear) spectrum of sinthat define the space H gular points persists to be infinite, similar to that of the conventional von Karman equations corresponding to the constant H. The infinite spectrum along with nonmonotonous dependences of its elements on ~ result in that the element of the spectrum such that kcr ðH Þ ¼ functions H 2 H inf kcr ðH Þ varies with H. Consequently, it is necessary to minimize the functional ~ and spectrum elements KðH Þ in order to (7) with respect to both functions H 2 H determine the function H  that met the condition (6): kmin ¼

inf

inf kðH Þ

kcr ðH Þ2KðH Þ H2H

ð22Þ

A necessary condition for the existence of the solution kðH  Þ, apart from continuity of kðH Þ with respect to H; proved above, is its Frechet differentiability. According to the Theorem 1.9.2 [56], the Frechet derivative exists and is given by the following expression:

1 @bH @bH k ðH0 Þ ¼  ðH0 ; k0 Þ
ðH0 ; k0 Þ: @k @H 0

ð23Þ

This fact completes the proof of the existence of a solution to the first inverse bifurcation problem.

6 Method to Solve the First Inverse Bifurcation Problem ~ (8), The functional (7), subject to constraints that define the admissible functions H may be modified as follows: Z  2   T2 ðH  Hmin Þ þ u  T3 ðHmax  H Þ þ dX þ u1 ðkmin ðH Þ  k Þ; J¼ C ðH Þ þ u ð24Þ  2; u  3 are Lagrange multipliers determined by where u1 ; u

Prediction and Control of Buckling …

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u2 ; H  Hmin ; 0; otherwise

u3 ; H Hmax 3 ¼ u : 0; otherwise 2 ¼ u

ð25Þ

The condition signcmin ¼ signcmax is assumed to be satisfied beforehand. The unknown function H ð xÞ is approximated by finite elements: H ¼ fHi g; i ¼ 1; N , where Hi denotes values in the i-th node of finite element mesh. Likewise,  2; u  3 are approximated using the same mesh; their nodal values the functions u1 ; u  2i ; u  3i . are denoted as u1i ; u With the finite-element approximation, the condition (6) may be rewritten as: Mi ¼ 2

dC dkmin C þ u2i  u3i þ u1 ¼ 0; dHi dHi

i ¼ 1; N:

ð26Þ

 2i ; u  3i and Hi , we propose to To calculate the unknown quantities u1i ; u employ the Newton-Raphson method; the following expression shows its iteration formula: aðkÞ ¼ aðk1Þ 

1 @D D: @a

ð27Þ


 Here, a ¼ aj denotes a vector to be found; k is the iteration number; D ¼ fDi g n o @Di is a discrepancy used to determine a , @D @a ¼ @aj ; i; j ¼ 1; N is a numerical analogue to the Frechet matrix (see, for example, Obodan et al. [7]). Namely, to determine the quantity u1 we used the discrepancy kmin  k ;

ð28aÞ

 2 the discrepancies the quantities u D1 ¼ fD1i g ¼ fHi  Hmin g;

ð28bÞ

 3 the discrepancies the quantities u D2 ¼ fD2i g ¼ fHmax  Hi g

ð28cÞ

and, finally, the vector H is determined with the employment of M ¼ fMi g:

ð28dÞ

In these terms, the algorithm to solve the first inverse bifurcation problem is as follows:

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Algorithm 1 0. Initialize Hi0 , k , u1 ¼ u2 ¼ u3 ¼ 0, ‘ ¼ 0. While H ðlÞ  H ðl1Þ e 1. Determine kð‘Þ cr by solving the direct bifurcation problem and taking into account the condition (5). ðl Þ

cr 2. Calculate d ð‘Þ ¼ dk dH .

3. If for any component d ð‘Þ holds true that sign

ðl Þ

di

ðl1Þ di

¼ 1, then interrupt the cycle

and restart the algorithm with new initial values.  1; u  2; u  3 with the employment of the Newton method (27), using 4. Calculate u discrepancies (28a, 28b, 28c). 5. Calculate H with the employment of the Newton method (27), using discrepancies (28d). 6. ‘ ¼ ‘ þ 1 Endwhile

7 Method to Solve the Second Bifurcation Problem Large-scale simulation reveals (Fig. 2) that sequences of von-Karman-equations solutions can be broadly classified into two types. For the first type, one observes the capricious alternations of various solutions (Fig. 3), while the second type shows steady development of a concrete solution (Fig. 4). The simulation also reveals that sequences of the second type precede bifurcation, whereas those of the first type do not lead to bifurcation. This kind of sequences is obviously related to chaotic sequences of von-Karman-equations solutions revealed by Awrejcewicz and Krysko [45]. It is worth emphasizing that separate solutions that belong to the sequences of the first and second types may be the same, but the sequences themselves are qualitatively distinct. Of principal importance is that the sequences of the first type appear to be similar to those of bifurcation paths of von Karman equations, the sequences of the second type do not bear resemblance to the post-bifurcation sequences. We have taken this fact of importance as a starting point for novel method to identify pre-bifurcation states: the typical sequences of the second type serves as precursors of bifurcation. Such precursors belong among the topological precursors [57, 58]. This approach to construct precursors of bifurcation implies that one is fully aware of all post-bifurcation solutions of the respective nonlinear boundary problem; in turn, it means that the direct bifurcation problem of the nonlinear boundary problem has already been solved (all its bifurcation paths have been traced)—see, Sect. 4. Then, the problem to extract typical sequences (and thereby to construct precursors of bifurcation) from a set of all sequences of post-bifurcation solutions observed along the bifurcation paths at hand is that of clustering.

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Fig. 2 The observed sequences of solution for various k. The equations are perturbed by a localized perturbation during s ¼ 0:002 s

Fig. 3 A sequence of observed solutions that does not lead to bifurcation: a, b show the solution at the specified point of definition domain; c, d exhibit sequences of cross-sections x1 ¼   0:5 xmax þ xmin of the function u1 ; subfigures b, d correspond to smaller step between 1 1 observations. R=h ¼ 500, k ¼ 0:92; A ¼ 50:0; s ¼ 0:002

Clustering of the sequences of solutions. The algorithm consists of two parts. The first one is used to cluster the sequences of post-bifurcation solutions and then reveal precursors of bifurcation (the centres of the clusters). The second part is employed to identify pre-bifurcation states using these precursors.

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Fig. 4 A sequence of observed solutions that precedes bifurcation (the limit solution corresponds to the secondary bifurcation path with the localized solution with a single local dent): a, b show the solution at the specified point of definition domain; (c, d) exhibit sequences of cross-sections  þ xmin x1 ¼ 0:5 xmax of the function u1 ; subfigures b, d correspond to smaller step between 1 1 observations. R=h ¼ 500, k ¼ 0:52; A ¼ 200:0 ; s ¼ 0:002

The sample for the first part is generated with the employment of the vectors zi concatenated from post-bifurcation solutions in compliance with a certain pattern. A pattern is defined as a preset sequence of distances between positions of solutions such that these (non-successive) solutions are to be placed on the successive positions in the sample vector to be generated—a step along a bifurcation path is assumed to be constant. Thus each pattern is a S  1-dimension integer vector ðp1 ; . . .; pS1 Þ, pj 2 f1; . . .; Pmax g; j ¼ 1; . . .; S  1 and each sample vector is a ðS  1ÞK-dimension vector. The parameter Pmax dictates the maximum distance between positions of solutions that become successive in the vector to be generated. @ðS; Pmax Þ denotes a set of all possible patterns of the specified length S. Interestingly, samples selected from the vectors of concatenated successive solutions prove less efficient than those based on the vectors concatenated according to various patterns. Before the clustering technique is applied, each solution is normalized to its maximum value u1max (we use the function u1 only); this allows one to cluster typical solutions profiles (rather than their amplitudes). If the typical sequences are compared with the observed sequence of solutions (the second part of the algorithm), solutions of the later are normalized to their maxima likewise. The distance throughout the algorithm is the Euclidean.

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The centres of the clusters constitute a set of precursors of bifurcation = used to solve the inverse bifurcation problem in order to identify pre-bifurcation states for nonlinear boundary problem of von Karman equations. It is worth noting that each centre is an average representation of system behaviour (the system is approaching to bifurcation) in the neighbourhood of the respective section of the bifurcation path. The representation both deteriorates the identification quality due to averaging and improves it due to cancelling random components (inevitably present in real-world measurements) of opposite signs. The clustering technique used strike a compromise between these factors. When the algorithm attempts to identify the sequence observed (its second part), the sequence is compared with the centres of all clusters computed for all patterns. Namely, for current position (algorithm tries to accomplish identification from the very first observations) vectors from previous observations are concatenated in compliance with each possible pattern in such a way that the last vector position coincides with the current position of the observed sequence and then the distances between such vectors and the respective centres of clusters are calculated. If the minimum distance does not exceed a threshold value q, then the observed sequence is judged to be identified and such that precedes to bifurcation and takes place in the neighbourhood of that section of that bifurcation path the selected cluster is associated with. Otherwise, it is judged to be unidentified and such that does not lead to bifurcation and takes place in the neighbourhood of pre-bifurcation equilibrium pass. The clustering employed is the modified [59]. It was applied to samples corresponding to all possible patterns of the length three @ð3; 10Þ; each pattern generated its own sample. To test the algorithm efficiency, we used sequences different from those that had been employed to generate learning sample.

8 Solutions of the First Inverse Bifurcation Problem The approach discussed in Sect. 6 was applied to solve a number of problems of practical importance. Simulation was carried out for E = 2  104 MPa (Young’s modulus), m ¼ 0:3 (Poisson’s ratio). Conditions (3a, 3b) defined on the boundaries C of the definition domain are those of simple support. We assume, unless otherwise stated, that the right-hand member F ðx1 ; x2 Þ is a constant function (that is a shell is subjected to a uniform external pressure) and the boundary conditions are that of simple support r22 u2 ¼ u1 ¼ r11 u1 ¼ r22 u1 ¼ 0:

ð29Þ

Optimal thickness function for a cylindrical shell. For this problem, the function H ð xÞ, the solution of the first inverse bifurcation problem, is the thickness function (distribution of thicknesses as a function of coordinates of the shell middle surface) while the function C ðH Þ (integrand of the functional (7)) is a specific weight of the ~ (8) results in constraints. shell; a definition of the set H

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We consider two different models for the shell: 1. The axisymmetric model. This implies that a solution of the problem (1), (3a, 3b) does not depend on the circumferential coordinate x2 . 2. The non-axisymmetric model. This implies that the solution depends on both longitudinal and circumferential coordinates. The algorithm discussed above was employed to perform a simulation for a cylindrical shell with a radius R ¼ 0:06 m, a thickness h ¼ 0:0003 m (R=h ¼ 200), ~ is and lengths: 1 – L ¼ 0:12 m (L=R ¼ 2); 2 – L ¼ 0:24 m (L=R ¼ 4). The set H determined using minimum and maximum thicknesses of Hmin ¼ 0:00015 m and Hmax ¼ 0:0004 m, respectively. Figure 5 illustrates iterations of the algorithm considered for the shell of the first length; dashed line stands for initial approximation, thick solid line for the last iteration (a shell with such thickness distribution features a bifurcation load k ); dotted line for symbolizes the thickness of the shell of constant thickness featuring the same bifurcation load k . Figure 6 shows the same plots for the second shell. These figures allow one to draw the conclusion that optimal thickness functions are localized; by way of illustration, the first shell is essentially  max  thicker in the neighmin bourhood of its central cross-section x1 ¼ 0:5 x1 þ x1 ; the second shell in the  neighbourhood of the central cross-section and the cross-section x1 ¼ 0:375 xmax 1 þ xmin 1 Þ. The (non-constant) optimal thickness functions results in a decreased shell weight; the first shell is 17% lighter than its counterpart of a constant thickness with the same bifurcation load, all other things being equal. For the second shell, the decreasing rate is 12%. For the shells with the (axisymmetric) optimal thickness distributions, the complete bifurcation structure was constructed using the method outlined in the Sect. 4. The structure is the destruction of the complete bifurcation structure associated with a shell of a constant thickness (Fig. 1). For that case, the first singular point located on the equilibrium path starting from the origin of 4.50E-04 H 3.80E-04 3.10E-04 2.40E-04 1.70E-04 1.00E-04 0

L/4

L/2

Fig. 5 Iterations of the algorithm considered for the shell of the first length; dashed line stands for initial approximation, thick solid line for the last iteration; dotted line for symbolizes the thickness of the shell of constant thickness featuring the same bifurcation load k

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5.00E-04 H

3.50E-04

2.00E-04 0

L/4

L/2

Fig. 6 Iterations of the algorithm considered for the shell of the second length; dashed line stands for initial approximation, thick solid line for the last iteration; dotted line for symbolizes the thickness of the shell of constant thickness featuring the same bifurcation load k

coordinates is a limit one; the respective solution is regular in circumferential direction. Furthermore, the path of localized solutions bifurcate near this limit point, and the first limit point of this bifurcation path correspond to the parameter value lesser than k but greater than the similar value for the shell of constant thickness H ð xÞ ¼ const. ~ For non-axisymmetric shells, we apply the same algorithm (with the same set H) under constraint that the shell thickness is constant in circumferential direction. In particular, for non-axisymmetric thickness distribution H ðx1 ; x2 Þ ¼ H0 ½ð1  dÞ þ d cos mxR 2 , it appears that bifurcation loads varies with the parameter m, being greater for m equal or multiple to eigenwavenumber for the shell of such geometry and lesser for others. Figure 7 exhibits bifurcation loads for non-axisymmetrical external loads F ðx1 ; x2 Þ ¼ ð1  dÞ þ d cos mxR 2 for various wavenumbers of the thickness function H ðx1 ; x2 Þ. Optimal curvature of cylindrical panel. The problem of optimal curvature, in context of inverse problems, is the first inverse bifurcation problem with unknown function H ð xÞ ¼ B22 ¼ const for B11 ; B12 ¼ 0 and constant Aijkl . For that case, C ðH Þ  0, and the functional (7) acquires the form J ðH Þ ¼ ðk  kcr Þ2 ; the defi~ (8) is naturally reduced to nition of the set H

 dH ð2Þ ~ H ¼ H 2 W2 ðXÞ; Hmin  H  Hmax ; cmin   cmax ; signcmin ¼ signcmax : dkcr ð30Þ

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cr 1.2 1.1 1. 0.9 0.8

0

2

4

6

8

m

Fig. 7 Bifurcation loads against wavenumbers of the thickness function H ðx1 ; x2 Þ, for non-axisymmetrical external loads F ðx1 ; x2 Þ ¼ ð1  dÞ þ d cos mxR 2

With these constraints, the functional J ðH Þ takes the form  2 ðH  Hmin Þ þ u  3 ðHmax  H Þ J ðH Þ ¼ ðk  kcr Þ2 þ u

ð31Þ

The remaining constraint signcmin ¼ signcmax is satisfied approximately, in the framework of the Algorithm 1. The boundaries like x1 ¼ const (curvilinear edges of a panel) are simply supported (29), as usual, whereas those like x2 ¼ const (straight edges of a panel) are free: r11 u1 ¼ r22 u1 ¼ r11 u2 ¼ r12 u2 ¼ 0:

ð32Þ

Figure 8 presents dependence H ðkcr Þ for rectangular in plan cylindrical panel (L  2a) with length L, chord 2a (L=a ¼ 8), and thickness h ¼ 0:01a under a aH

1.1 1.0 0.9 0.8 0.7 0.6 0.5

0.4 0.0

0.2

0.4

Fig. 8 Dependence H ðkcr Þ for a cylindrical panel

0.6

0.8

λ cr

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Fig. 9 Nonlinear bifurcation modes associated with different aH (shown in the centre of a subfigure)

aH =0.6

aH =0.7

aH =1.0

aH =0.95

uniform normal load. Bifurcation loads are, as usual, normalized by bifurcation value for the respective closed cylindrical shell. The figure shows it is possible that ~ (30) exist; the fact is attributable to a highly nonmonotonous several compact sets H dependence H ðkcr Þ [7, 33], which, in turn, is explained by different nonlinear buckling modes (bifurcation solutions) of the problem (1), (3a, 3b) associated with different H (Fig. 9). Optimal load distribution. We consider a cylindrical shell with parameters L=R ¼ 4; R=h ¼ 200, subjected to non-uniform with respect to the coordinate x2 external pressure qðx2 Þ ¼ k0 F ðx2 Þ. Then functional (7) takes the form J ðH Þ ¼ ðk  kcr Þ2 :

ð33Þ

With finite-element approximation of the function F ðx2 Þ, one obtains governing relations similar to (26), and then optimal load distribution. The results are tabulated in the Table 1 for various k . The second lines (for some k ) in the table show that it is possible that two (or more) different optimal load distibutions simultaneously exist for that value of Table 1 The calculated optimal load distributions x2 =ðpRÞ k

0.000

0.125

0.250

0.375

0.500

0.625

0.750

0.875

1.000

1.10

1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.768 0.981 0.824 0.962 0.856 0.933 0.890

0.341 0.924 0.453 0.854 0.531 0.752 0.622

0.081 0.831 0.158 0.691 0.228 0.515 0.330

0.009 0.707 0.031 0.500 0.062 0.287 0.125

0.000 0.556 0.003 0.309 0.009 0.121 0.029

0.000 0.383 0.000 0.146 0.000 0.031 0.003

0.000 0.195 0.000 0.038 0.000 0.003 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.15 1.20 1.25

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k . Convergence of the iterative process to one or another solution depends on an initial approximation. The multiplicity of the solutions for a given k may be attributed to the multiple solutions of the directed bifurcation problem that exist for the critical load [7].

9 Solutions of the Second Inverse Bifurcation Problem To simulate observed sequences (cf. Figs. 3 and 4), we employed sequences of finite-element solutions of von Karman equations with the constant right-hand member, perturbed by localized function with an amplitude A and duration time s. The results of that Section correspond to von  Karman equations defined on close min cylindrical domain; cross-sections x1 ¼ 0:5 xmax of the function u1 are used þ x 1 1 to present solutions in the figures of the present section. Figure 10 exemplifies solving of the problem to identify pre-bifurcation state (the second inverse bifurcation problem); the input data is the observed sequence preceding to bifurcation (the middle row) whereas a tool to solve the problem is a precursor of bifurcation (the lower row). For comparison, the upper row shows the sequence that does not lead to bifurcation. Despite the separate solutions are similar for the two types of sequences, the algorithm does not take the upper row as the sequence preceding to bifurcation (the algorithm has found no precursors close enough to the observed sequence). If one considers the resultant clusters (their centres constitute the bifurcation precursors used to predict bifurcations), one is able to ascertain correspondence between them and sections of bifurcation paths of the nonlinear boundary problem for von Karman equations. A single section between two neighbouring extrema of a path is associated, usually, with one trough three clusters. In average, a part of a path that correspond to a single cluster exhibits 5–7% variation in parameter k. To summarize, since the number of clusters associated with a single path is rather small and, on the other hand, the number of bifurcation path corresponding the practically important region of parameter space is limited [7], the total number of bifurcation

Fig. 10 Identification of the observed sequence preceding to bifurcation (the middle row) with the employment of a precursor of bifurcation (the lower row). For comparison, the upper row shows the sequence that does not lead to bifurcation (s ¼ 0:002 s)

Prediction and Control of Buckling …

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Fig. 11 The observed sequences (the middle rows) and the bifurcation precursors that the algorithm puts into correspondence to them (the lower rows); for bifurcations with regular (a, b) and localized (c) bifurcation solutions

precursors is limited too. This fact provides the basis for an efficient system able to identify a pre-bifurcation state rapidly. Figure 11 shows other typical observed sequences that precede bifurcations with regular and localized bifurcation solutions and the bifurcation precursors that the algorithm puts into correspondence to them. In particular, the observed sequence of Fig. 11 a correspond to the starting bifurcation process with final regular solution with typical variability (number of waves) in circumferential direction equal to 5 (the path O  A  B  C, Fig. 1), Fig. 11b to the starting bifurcation process with final regular solution with typical variability equal to 4 (the path with the lower limit point H, Fig. 1), and, finally, the last subfigure exhibits a sequence preceding to the bifurcation with a localized bifurcation mode (the section A  D of the bifurcation path A  D  E  G  F  H, Fig. 1).

10

Conclusions and Directions of Future Research

The chapter discusses novel approaches to predict and control buckling of thin-walled systems. Mathematically, these approaches are formalized as the first and second inverse bifurcation problems for the static von Karman equations—the

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problem to determine the optimal properties of the model (to control bifurcation) and the one to identify pre-bifurcation state (to predict bifurcation), respectively. Both approaches are based upon the method employed to solve the direct bifurcation problem for the equations in question; this makes necessary to find solutions of the nonlinear boundary problem and to trace its bifurcation paths and, in turn, to develop respective numerical methods (like that discussed briefly in the Sect. 4 of the present chapter). For both inverse problems, the necessary precondition to utilize the methods discussed in this chapter is to prove the existence of a solution to the problem. It is worth emphasizing that these approaches are equally applicable to a differential statement of a nonlinear boundary problem for PDEs, its variational statement, and the statement based upon a generalized solution. The approach considered was applied to several difficult problems of actual practice, viz., (for the first inverse problem) to the problems of optimal thickness distribution (for a cylindrical shell), optimal curvature (for a cylindrical panel), and optimal external pressure distribution (for a cylindrical shell); (for the second inverse problem) to the problem to predict buckling of a cylindrical shell under an external pressure. The approaches proposed in the present chapter to solve the direct, the first and second inverse bifurcation problems can be extended to apply to a broad class of static nonlinear boundary problems for partial differential equations such that they admit only a finite number of bifurcation paths corresponding to a restricted region of parameter space. Particularly, equations featuring spatial localization appear to be a focus of attention. On the other hand, a wide-ranging simulation showed that typical sequences of post-bifurcation solutions of the static problem can be used as kind of bifurcation precursors for the dynamic problem.

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9. Lindgaard, E., Dahl, J.: On compliance and buckling objective functions in topology optimization of snap-through problems. Struct. Multidisc. Optim. 47, 409–421 (2013) 10. Lindgaard, E., Lund, E., Rasmussen, K.: Nonlinear buckling optimization of composite structures considering “worst” shape imperfections. Int. J. Solids Struct. 47, 3186–3202 (2010) 11. Henrichsen, S.R., Lindgaard, E., Lund, E.: Buckling optimization of composite structures using a discrete material parametrization considering worst shape imperfections. In: E. Onãte, X. Oliver, A. Huerta (eds.) Proceedings of 11th World Congress on Computational Mechanics, N.-Y. (2014) 12. Smołka, M.: Differentiability of the objective in a class of coefficient inverse problems. Comput. Math Appl. 73, 2375–2387 (2017) 13. Engl, H.W., Kügler, P.P.: Nonlinear inverse problems: theoretical aspects and some industrial applications. In: Capasso and Periaux (eds.), Multidisciplinary Methods for Analysis, Optimization and Control of Complex Systems, Springer Heidelberg, Series Mathematics in Industry, pp. 3–48 (2005) 14. Dierkes, T., Dorn, O., Natterer, F., Palamodov, V., Sielschott, H.: Fréchet derivatives for some bilinear inverse problems. SIAM J. Appl. Math. 62(6), 2092–2113 (2002) 15. Yildirim Aksoy, N.: Variational method for the solution of an inverse problem. J. Comput. Appl. Math. 312, 82–93 (2017) 16. Babaniyi, O.A., Oberai, A.A., Barbone, P.E.: Direct error in constitutive equation formulation for plane stress inverse elasticity problem. Comput. Methods Appl. Mech. Eng. 314, 3–18 (2017) 17. Obodan, N.I., Guk, N.A:. The Inverse Problems of Thin-Walled Shell Theory. Berlin: Lambert academic publishing; (2012) (in Russian) 18. Gowda, K., Kuehn, C.: Early-warning signs for pattern-formation in stochastic partial differential equations. Commun. Nonlin. Sci. Numer. Simulat. 22, 55–69 (2015) 19. Wiesenfeld, K.: Virtual Hopf phenomenon: a new precursor of period-doubling bifurcations. Phys. Rev. A 32(3), 1744–1751 (1985) 20. Zulkuparov, M-G.M., Malinetskii, G.G., Podlazov, A.V.: The inverse bifurcation problem for noisy dynamic systems. Preprint of Keldysh Institute of Applied Mathematics (Russian Academy of Science) (in Russian) (2005). http://keldysh.ru/papers/2005/prep39/prep2005_ 39.pdf 21. Lim, J., Epureanu, B.I.: Forecasting a class of bifurcations: theory and experiment. Phys. Rev. E 83(1), 162–165 (2011) 22. Stull, C.J., Nichols, J.M., Earls, C.J.: Stochastic inverse identification of geometric imperfections in shell structures. Comput. Methods Appl. Mech. Eng. 200, 2256–2267 (2011) 23. Abramovich, H., Govich, D., Grunwald, A.: Damping measurements of laminated composite materials and aluminum using the hysteresis loop method. Prog. Aerosp. Sci. 78, 8–18 (2015) 24. Obodan, N.I., Adlucky, V.J., Gromov, V.A.: Rapid identification of pre-buckling states: a case of cylindrical shell. Thin-Walled Struct. 124, 449–457 (2018) 25. Thompson, J.M.T., Hunt, G.W.: A General Theory of Elastic Stability. Wiley, London (1978) 26. Obodan, N.I., Lebedev, A.G..: The Secondary Branching in the Nonlinear Theory of Thin-Walled Shells. Reports of the Academy of Science of Ukranian SSR. Series 2, vol. 12, pp. 38–41 (1980) (in Russian) 27. Obodan, N.I., Gromov, V.A.: Numerical analysis of the branching of solutions to nonlinear equations for cylindrical shells. Int. Appl. Mech. 42(1), 90–97 (2006) 28. Zhou, Y., Stanciulescu, I., Eason, T., Spottswood, M.: Nonlinear elastic buckling and postbuckling analysis of cylindrical panels. Finite Elem. Anal. Des. 96, 41–50 (2015) 29. Fujii, F., Noguchi, H.: Symmetry-breaking bifurcation and post-buckling strength of a compressed circular cylinder. In: Solid Mechanics and Fluid Mechanics: Computational Mechanics for the Next Millennium, pp. 563–568. Pergamon: Amsterdam 30. Fujii, F., Noguchi, H., Ramm, E.: Static path jumping to attain postbuckling equilibria of a compressed circular cylinder. Comp. Mech. 26, 259–266 (2000)

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Numerical Solution with Special Layer Adapted Meshes for Singularly Perturbed Boundary Value Problems Deepti Kaur and Vivek Kumar

Abstract The objective of this chapter is to present a comparative study of simple upwind finite difference method on various non-uniform meshes existing in the literature for resolving the boundary layer of two-point singularly perturbed problems. Our exposition begins with the Bakhvalov mesh and its modification using Padé approximation, then continues with the piecewise uniform Shishkin-type meshes and to the most recent W-grid using Lambert W-function. A new kind of mesh of Shishkin type that incorporates an idea by Roos et al. ( Roos, Teofanov and Uzelac, Appl. Math. Lett. 31, 7–11 (2014) [1]) using Lambert W-function has also been proposed and using this mesh, the method gives better results as compared to the results using the standard Shishkin mesh. For various meshes, the computed solution is uniformly convergent with respect to the small perturbation parameter. Numerical results on a test problem are presented which validate the theoretical considerations. Keywords Singularly perturbed · Layer-adapted mesh · Finite difference method · Bakhvalov mesh · Shishkin mesh · Uniform convergence

1 Introduction In this chapter, we carry out a survey of a few of the most important ideas from the literature for constructing layer-adapted meshes of Bakhvalov and Shishkin type for finding the numerical solution of singularly perturbed boundary value problems. Such problems involve a small parameter,  multiplied with the highest derivative. These singularly perturbed problems occur in numerous fields of science and engineering, D. Kaur University of Delhi, Department of Mathematics, Delhi 110007, India e-mail: [email protected] V. Kumar (B) Delhi Technological University, Department of Applied Mathematics, Delhi 110042, India e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_12

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such as fluid mechanics, quantum mechanics, hydrodynamics, elasticity etc. The major concern with these problems is the sudden change in the dependent variable within one or more thin “layer region(s)” thus exhibiting multiscale phenomena. That is, for such problems the solution varies rapidly in a thin layer while away from this layer the solution varies slowly and smoothly. It is well known that to approximate the solution of such problems, classical numerical methods on uniform meshes are inadequate as unreasonably large number of mesh points are required to produce satisfactory accurate solution [2–5]. Therefore, classical numerical treatment of these problems on uniform mesh gives major computational difficulties as the error becomes unbounded for arbitrary small values of the perturbation parameter . So, construction of non-uniform meshes which can resolve the layer structure of these problems has been an active field of research over the last few decades. As the construction of a layer-adapted mesh plays a very crucial role to find approximate solution of singularly perturbed problems, therefore developing a new kind of mesh adapted to layer structure of these problems has always been a challenging task. After carefully studying the construction of piecewise uniform Shishkin mesh, a new kind of mesh of Shishkin type has been proposed and numerical results have been compared with the other existing meshes. The chapter is organized as follows: in Sect. 2 we describe the construction of layer-adapted meshes of Bakhvalov and Shishkin type. In Sect. 3, numerical results on a test problem are presented. Sect. 4 is then devoted to the discussion and comparison of the different layer-adapted meshes for the simple upwind finite difference method.

2 Construction of Layer Adapted Meshes We consider the one-dimensional singularly perturbed boundary value problem given as − u  (x) − b(x)u  (x) + c(x)u(x) = f (x),

for x ∈ (0, 1), u(0) = (1) = 0, (1) where  is a small positive parameter and b(x) ≥ β > 0 for x ∈ [0, 1]. The above problem has a unique solution having an exponential boundary layer of width O() at x = 0. It was proved by Kellogg and Tsan [6] that u and its derivatives up to an arbitrary prescribed order q (depending on the smoothness of the data) can be bounded by |u (k) (x)| ≤ C{1 + −k e−βx/ }, for k = 0, 1, . . . , q and x ∈ [0, 1], where here and throughout the chapter, C denotes a generic positive constant independent of  and the number of mesh points used. For problem (1), the solution is smooth in most of the solution domain apart from a small region near x = 0 where the solution changes rapidly.

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Definition 1 (Uniform Convergence) Let u  be the unique solution of singularly perturbed problem (1) and UN be a numerical approximation of u  obtained by a numerical method where N is the discretization parameter. The numerical method is said to be uniformly convergent with respect to the perturbation parameter  if there exist a positive integer N0 and a function v that are both independent of  such that u  − UN  ≤ v(N ) for N ≥ N0 , with lim v(N ) = 0.

N →∞

where . is the discrete maximum norm. One of the successful approach to approximate the solution of (1) by a numerical method which is uniformly convergent with respect to  known as fitted mesh method is to construct a mesh which allows resolution of the structure of the layer. We consider simple upwind finite difference method applied on nonuniform layeradapted mesh  : 0 = x0 < x1 < . . . < x N = 1 having N subintervals on [0, 1]. The local mesh sizes are denoted by h i = xi − xi−1 , for i = 1, . . . , N and h = max h i i=1,...,N

denotes the maximal step size to obtain approximate solutions: −δ 2 UN (xi ) − b(xi )D + UN (xi ) + c(xi )UN (xi ) = f (xi ), for i = 1, . . . , N − 1, UN (0) = UN (N ) = 0

(2)

where D + , D − are the forward and backward finite difference operators, respectively for the first-order derivative defined by D + UN (xi ) =

UN (xi+1 )−UN (xi ) xi+1 −xi

D − UN (xi ) =

UN (xi )−UN (xi−1 ) xi −xi−1

while δ 2 is the second order centered difference operator defined by δ 2 UN (xi ) =

2(D + UN (xi ) − D − UN (xi )) . xi+1 − xi−1

2.1 Bakhvalov Mesh We recall a basic concept for describing layer-adapted meshes. Definition 2 (Mesh generating function) A strictly monotone function φ : [0, 1] → [0, 1] which maps a uniform mesh ti = i/N , i = 0, 1, . . . , N onto a layer-adapted mesh in x by xi = φ(ti ), i = 0, 1, . . . , N is called a mesh generating function.

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Bakhvalov [7] was perhaps the first person to design a special graded mesh for capturing the boundary layer of a singularly perturbed problem. His essential idea was to use a uniform mesh in t near x = 0, then to transform this mesh back to the x-axis by inverting the exponential boundary layer function, so that the meshpoints xi near x = 0 are given by  −βxi  i q 1 − e σ = ti = N which is equivalent to xi =

  −σ i , ln 1 − β qN

where q ∈ (0, 1) is the approximate fraction of the meshpoints taken to resolve the layer and the parameter σ > 0 affects the density of the mesh within the layer. The above definition of xi is changed as one moves away from the boundary layer by using a uniform mesh with the transition point τ chosen in a manner that the underlying mesh generating function is C 1 [0, 1]. The Bakhvalov mesh generating function is  φ(t) =

λ(t) :=

−σ β

ln(1 − qt )

for t ∈ [0, τ ],



π(t) := λ(τ ) + λ (τ )(t − τ ) for t ∈ [τ , 1]

(3)

where τ is defined implicitly by λ(τ ) + λ (τ )(1 − τ ) = 1

(4)

so that the function π(t) is a tangent line to λ(t) at the contact point (τ , λ(τ )) which passes through the point (1, 1). The function λ(t) generates mesh points in the boundary layer in the neighbourhood of x = 0 while the function π(t) generates mesh points outside the boundary layer. A major drawback of Bakhvalov’s original mesh is that the nonlinear equation (4) cannot be solved explicitly for τ . However, the iteration τ 0 = 0,

τ i+1 = q −

1+

σ (1 − τ i ) β σ ln(1 − τ i /q) β

for i = 0, 1, . . .

(5)

is proved to converge to τ in [7] with 0 ≤ τ i < τ i+1 < τ for all i. As it is difficult to determine the exact value of τ explicitly, various authors have devised ways to modify the above construction to yield meshes of Bakhvalov-type , (B-type meshes). Kopteva [8] uses the original Bakhvalov mesh with τ = q − σ β which is the first iteration value in (5). With this choice of τ , a refined grid having reasonable approximation properties is obtained but the mesh generating function is no longer C 1 [0, 1]. The generated mesh will be called Bakhvalov mesh for carrying out numerical experiments (Fig. 1).

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Mesh generating functions for B−type meshes

Fig. 1 Mesh generating functions for B-type meshes using (3) and (6)

1

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Vulanovic [9] devised to approximate the exponential in the original Bakhvalov mesh by its (0, 1) Padé approximation, so the resulting mesh generating function is  φ(t) =

μ(t) :=

σ t β (q−t) 

for t ∈ [0, τ ],

μ(τ ) + μ (τ )(t − τ ) for t ∈ [τ , 1].

(6)

Then, instead of nonlinear equation (4), a quadratic equation is obtained from which τ can be computed easily,

τ=

q−



σq (1 β

1+

−q +

σ ) β

σ β

.

The mesh generated is called B-mesh. Linß [10] introduced a quantity for characterizing the uniform convergence results for various finite difference methods on an arbitrary nonuniform mesh defined by v [ p] () := max

i=1,...,N



xi

(1 + −1 e−βs/ p )ds.

xi−1

It is established in Chap. 4 of [10] that the error estimate of a simple upwind finite difference method (2) satisfies u  − UN  ≤ Cv [1] (). For simple upwind finite difference method (2) on B-type meshes, the following error estimate holds true:

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u  − UN  ≤ C N −1 i.e, first-order uniform convergence is obtained in the discrete maximum norm.

2.2 Shishkin Mesh Another alternative class of special layer-adapted mesh whose construction is much simpler is the piecewise uniform Shishkin mesh [11, 12] having transition point α given by

σ ln N . α = min q, β The mesh generating function for the standard Shishkin mesh is  φ(t) =

αt q

for t ∈ [0, q],

1 − (1 −

1−t α) 1−q

for t ∈ [q, 1].

The parameter q ∈ (0, 1) denotes the amount of the meshpoints taken to resolve the layer and the choice of the transition point α has been made so that on [α, 1], the −βx term e  which corresponds to the layer is smaller than N −σ . In practice, q ≥ α as otherwise a uniform mesh is adequate. The numerical schemes using Shishkin meshes are simpler to analyze because of their construction than the schemes using B-type meshes but the -uniform error estimate on Shishkin mesh deteriorates, u  − UN  ≤ C N −1 ln N , for simple upwind scheme (2). Since the error estimate on Shishkin mesh is deteriorated by the logarithmic factor, so this initiated the construction of Shishkin-type meshes for improving its performance while retaining its simplicity. They are the meshes having the same transition point α as given by Shishkin and using an equidistant submesh on [α, 1]. A classification of this class of meshes can be found in [13]. The mesh generating function for such meshes is  φ(t) =

σ ˜ φ(t) β

1 − (1 −

σ β

ln

1−t N ) 1−q

for t ∈ [0, q], for t ∈ [q, 1]

˜ ˜ where the function φ˜ is monotonically increasing on [0, q] with φ(0) = 0 and φ(q) = ln N . Then, the mesh characterizing function is defined as ˜

ψ(t) = e−φ(t)

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which is monotonically decreasing with ψ(0) = 1 and ψ(q) = 1/N . The function ψ is used to give the characterization of -uniform error estimates of a numerical method. It is proved in [10] that for a Shishkin-type mesh  with σ ≥ 1, under the assumption that there exists a constant C such that ˜ ≤ C N, max φ(t)

t∈[0,q]

the error of the simple upwind scheme (2) satisfies v [1] () ≤ C(h + max |ψ  (t)|N −1 ). t∈[0,q]

−(ln N )t

For standard Shishkin mesh, the mesh characterizing function is ψ(t) = e q with max |ψ  | = lnqN . Linß [14] modified standard Shishkin mesh by incorporating an idea by Bakhvalov [7]. In order to avoid the solution of a nonlinear equation which gives the transition point in the original Bakhvalov mesh, Linß [14] chooses the same transition point α as in the standard Shishkin mesh for constructing such a mesh. The interval [α, 1] is uniformly divided into (1 − q)N subintervals whereas [0, α] is partitioned into q N −βxi subintervals by inverting the boundary layer function e σ . The meshpoints xi near x = 0 are given by e

−βxi σ

= Ai + B, for i = 0, 1, . . . , q N −βxi

where the constants A and B are chosen such that x0 = 0 and xq N = α so that e σ is a linear function in i. Such a mesh is called Bakhvalov-Shishkin mesh. The resulting mesh generating function is  φ(t) =

−σ β

 ln 1 +

1 − (1 −

σ β

(1−N )t qN

ln



1−t N ) 1−q

for t ∈ [0, q], for t ∈ [q, 1]

which unlike B-type meshes is not C 1 [0, 1] but only C 0 [0, 1] and depends upon the discretization parameter N . For this mesh, the mesh characterizing function is ψ(t) = 1 − (1 − 1/N ) with max |ψ  | =

t q

1 1 (1 − 1/N ) ≤ . q q

The above mesh satisfies the following error estimate for simple upwind scheme (2): u  − UN  ≤ C( + N −1 )

390

D. Kaur and V. Kumar

It is noted that the convergence is destroyed if  ≥ N −1 but generally  ≤ N −1 in practice, so u  − UN  ≤ C N −1 . Hence, in contrast to the standard Shishkin mesh, the convergence is not spoiled by a logarithmic factor. Roos and Linß [13] gave a modification of the Bakhvalov-Shishkin mesh by using (0, 1) Padé approximation for the exponential term in the sense of [9]. For the generated mesh, called the modified Bakhvalov-Shishkin mesh, ˜ = φ(t)

t ln N q + (q − t) ln N

so its mesh characterizing function is ψ(t) = e−t ln N /(q+(q−t) ln N ) . For this mesh max |ψ  | ≤

4 , q

h ≤ C(1 +  ln2 N )N −1 so

v [ p] () ≤ C(1 +  ln2 N )N −1

and the error for finite difference method (2) satisfies u  − UN  ≤ C N −1 . A polynomial S-mesh was proposed by Roos and Linß [13] by choosing ˜ = ln N (t/q)m , m ≥ 1. φ(t) For this mesh,

with

ψ(t) = e− ln N ( q )

t m

max |ψ  (t)| ≤ C(ln N ) m 1

so the following error estimate holds true: u  − UN  ≤ C N −1 (ln N ) m . 1

Numerical Solution with Special Layer Adapted Meshes for Singularly …

391

For m > 1, the accuracy of simple upwind finite difference method (2) is improved and for m = 1, we recover the standard Shishkin mesh. Roos and Linß [13] proposed a mesh with rational characterizing function ψ by taking ˜ = ln(1 + (N − 1)(t/q)m ), for m > 1. φ(t) The mesh characterizing function for this mesh is ψ(t) =

1 . 1 + (N − 1)(t/q)m

For this mesh, the error estimate is u  − UN  ≤ C N −1+1/m . Vulanovic [15] gave generalized S-mesh denoted by S(m) where the positive integer m indicates that the mesh is divided into m + 1 subintervals by introducing additional transition points defined by τ1 =

σ σ σ σ ln ln . . . ln N, τ2 = ln ln . . . ln N, . . . . . . , τm−1 = ln ln N , τm = ln N . β  β  β β m times

m−1 times

By taking, τ0 = 0 and τm+1 = 1, [0, 1] =

m+1 

In ,

In = [τn−1 , τn ].

n=1

Then each interval In is dissected into Nn subintervals uniformly such that N1 + N2 + · · · + Nm+1 = N by taking qn = NNn , n = 1, 2, . . . , m + 1 so that q1 + q2 + · · · qm+1 = 1. Thus the refined part of the mesh is first dissected into m subintervals In and the coarse mesh is taken in Im+1 . For this mesh, the accuracy is improved as . . ln N . u  − UN  ≤ C N −1 ln ln . m times

From Fig. 2, it is evident that mesh points in standard Shishkin mesh are not as close to x = 0 as in Bakhvlov-Shishkin mesh are. Also, standard Shishkin mesh is not as dense close to x = 0 as the polynomial S-mesh, S(2) and S(3) are but more denser than the Shishkin mesh with rational ψ. Among various Shiskin-type meshes, S(3) mesh has the highest density (Table 1).

392

D. Kaur and V. Kumar

Fig. 2 Mesh generating function for Shishkin-type meshes: —- Standard Shishkin mesh, —Bakhvalov-Shishkin mesh, * Modified Bakhvalov-Shishkin mesh, + Polynomial S-mesh (m = 2), + Shishkin mesh with rational ψ, —- S(2) with q1 = 14 , q2 = 41 , - - S(2) 9 3 with q1 = 16 , q2 = 16 , –*– S(3) with 7 1 q1 = 16 , q2 = 14 , q3 = 16

Mesh generating function for Shishkin−type meshes

1

0.8

t

0.6

0.4

0.2

0

0

0.2

0.4

x

0.6

0.8

1

Table 1 Mesh characterizing function ψ(t) and error estimates for various Shishkin-type meshes Mesh Mesh characterizing Error estimates function ψ(t) Standard Shishkin mesh Bakhvalov-Shishkin mesh

−(ln N )t q

e 1 − (1 − 1/N ) qt

Modified Bakhvalov-Shishkin mesh e−t ln N /(q+(q−t) ln N ) Polynomial S-mesh Shishkin mesh with rational ψ

e

− ln N ( qt )m

1 1+(N −1)(t/q)m

S(m)

u  − UN  ≤ C N −1 ln N u  − UN  ≤ C N −1 u  − UN  ≤ C N −1 1

u  − UN  ≤ C N −1 (ln N ) m u  − UN  ≤ C N −1+1/m u  − UN  ≤ C N −1 ln . . ln N  ln . m times

Recently, Roos et al. [1] introduced a modified Bakhvalov mesh generated using the Lambert W-function [16] which is an essential tool for solving equations. As the solution of xe x = a is x = W (a), so the general approach for solving equations using Lambert W-function is to manipulate all occurrences of the unknown variable x into the form g(x)eg(x) for some function g. As the meshpoints in Bakhvalov mesh near x = 0 are given by   −βxi 2i 1 − e σ = ti = N for q = 1/2, so in [1], a mesh is generated by taking xi = ξ(i/N ), i = 0, 1, . . . , N − 1, x N = 1, where ξ is given implicitly by the solution of the equation

Numerical Solution with Special Layer Adapted Meshes for Singularly …

393

ξ(t) − e−βξ(t)/(σ) + 1 − 2t = 0. Since the solution of the equation a x = x + b is x = −b − W (−a −b ln a), so using the W-function, the meshpoints are given explicitly by   σ β β(1− 2iN ) 2i σ , i = 0, 1, . . . , N − 1, x N = 1. −1+ W e xi = N β σ The above mesh is called modified Bakhvalov mesh that can be generated using the lambert W function in MATLAB or Product Log function in Mathematica and has the advantage of not using the transition point as in the Bakhvalov mesh. It is established in [1] that for simple upwind difference method on modified Bakhvalov mesh, the following error estimate holds true: u  − UN  ≤ C N −1 . Now, we propose a new modification of piecewise uniform Shishkin mesh by redefining the transition point so as to improve its performance. We choose the transition point in the Shishkin mesh as

σ α1 = min q, W (N ) , β by making use of the Lambert W-function. The mesh generated by the above choice of the transition point is called Shishkin W -grid. The idea for the above choice of transition point comes from the construction of the modified Bakhvalov mesh [1] as W (N ) satisfy ln(ln N ) ≤ W (N ) ≤ ln N and W (N ) solves e−W (N ) = W (N )N −1 . Asymptotically, W (N ) behaves like ln N but with this choice of the transition point a better layer resolving mesh is obtained, so more accurate numerical results are expected. For Shishkin W-grid, the mesh generating function is  φ(t) =

σ ˜ φ(t) β

1 − (1 −

σ 1−t W (N ) 1−q β

for t ∈ [0, q], for t ∈ [q, 1]

˜ ˜ = W (N )t is monotonically increasing on [0, q] with φ(0) =0 where the function φ(t) q ˜ and φ(q) = W (N ). As shown in Fig. 3, Shishkin W-grid mesh is denser than the standard Shishkin mesh. Then, the mesh characterizing function ψ(t) for this mesh is defined as

394

D. Kaur and V. Kumar

Fig. 3 Mesh generating function

1

0.8

t

0.6

0.4 Standard Shishkin Mesh Shishkin W−grid

0.2

0

0

0.2

0.6

0.4

0.8

1

x

Fig. 4 Mesh characterizing function

1 Standard Shishkin Mesh

0.8

Shishkin W−grid

t

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

x

˜ = W (ψ(t)−1 ), φ(t) ˜ i.e., ψ(t)−1 = W −1 (φ(t)). W (N )t 1 q e− q So, ψ(t) = = −1 ˜ t W (N ) W (φ(t)) which is monotonically decreasing with ψ(q) = 1/N . It is evident from Fig. 4 that the mesh characterizing function for Shishkin W-grid has the same behaviour as the mesh characterizing function for the standard Shishkin mesh. So, for simple upwind finite difference method (2) on Shishkin W-grid, the error estimate cannot be worst than the standard Shishkin mesh as supported by numerical results. The -uniform error estimate on Shishkin W-grid also satisfies

Numerical Solution with Special Layer Adapted Meshes for Singularly …

395

u  − UN  ≤ C N −1 ln N , for simple upwind scheme (2).

3 Numerical Results In this section, we test and compare the performance of the simple upwind finite difference method (2) an various layer-adapted meshes discussed before. Our test problem is from [10]: − u  − u  + 2u = e x−1 ,

for x ∈ (0, 1), u(0) = (1) = 0.

(7)

The exact solution is u  (x) =

e x−1 em 1 x − em 2 x − em 1 x+m 2 −1 + em 2 x+m 1 −1 + , ( − 1)(em 1 − em 2 ) 1−

where m 1 and m 2 are the roots of the characteristic equation m 2 + m − 2 = 0, given by m1 = m2 =

√ −1+ 1+8 , 2 √ −1− 1+8 . 2

(8)

This problem has regular boundary layer of width O() at x = 0. In Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 we present the errors E(N ) in the discrete maximum norm E(N ) = u  − UN  where UN is the numerical solution of the problem (7) obtained when simple upwind finite difference method (2) is applied on various layer-adapted meshes discussed for several values of  and different number of mesh intervals N . We also present the numerical order of convergence (Fig. 5) Or d(N ) =

ln E(N ) − ln E(2N ) . ln 2

4 Discussion and Conclusion It is evident from Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 that the simple upwind difference method (2) is uniform with respect to the perturbation parameter  for the various layer-adapted meshes discussed above as the errors change only slightly

396

D. Kaur and V. Kumar

Table 2 E(N ) and Or d(N ) for Bakhvalov mesh, q = 0.5 N 

16

2−10

4.0499(−2) 2.1481(−2) 1.0942(−2) 5.4838(−3) 2.7378(−3) 1.3667(−3) 6.8281(−4) 0.9148

2−12

1.0011

0.9778

1.0007

1.0053

1.0048

1.0031

0.9804

1.0025

1.0066

1.0059

1.0040

0.9822

1.0037

1.0073

1.0063

1.0043

0.9835

1.0045

1.0079

1.0066

1.0045

0.9846

1.0053

1.0082

1.0069

1.0046

4.2580(−2) 2.2400(−2) 1.1313(−2) 5.6341(−3) 2.8004(−3) 1.3934(−3) 6.9439(−4) 0.9267

2−24

1024

4.2484(−2) 2.2366(−2) 1.1303(−2) 5.6309(−3) 2.7995(−3) 1.3931(−3) 6.9431(−4) 0.9285

2−22

1.0023

512

4.2363(−2) 2.2322(−2) 1.1289(−2) 5.6268(−3) 2.7981(−3) 1.3927(−3) 6.9416(−4) 0.9243

2−20

1.0022

256

4.2201(−2) 2.2261(−2) 1.1269(−2) 5.6201(−3) 2.7958(−3) 1.3918(−3) 6.9382(−4) 0.9228

2−18

0.9966

128

4.1957(−2) 2.2163(−2) 1.1233(−2) 5.6066(−3) 2.7905(−3) 1.3896(−3) 6.9290(−4) 0.9208

2−16

0.9732

64

4.1508(−2) 2.1964(−2) 1.1152(−2) 5.5733(−3) 2.7764(−3) 1.3836(−3) 6.9030(−4) 0.9182

2−14

32

0.9855

1.0057

1.0086

1.0070

1.0048

4.2660(−2) 2.2428(−2) 1.1322(−2) 5.6365(−3) 2.8010(−3) 1.3936(−3) 6.9445(−4) 0.9276

0.9862

1.0063

1.0089

1.0071

1.0049

256

512

Table 3 E(N ) and Or d(N ) for B-mesh, q = 0.5 N 

16

2−10

4.3359(−2) 2.3346(−2) 1.2132(−2) 6.1832(−3) 3.1212(−3) 1.5681(−3) 7.8589(−4) 0.8932

2−12

0.9438

0.9723

0.9863

0.9933

0.9967

0.9443

0.9714

0.9865

0.9933

0.9966

0.9447

0.9718

0.9862

0.9934

0.9966

0.9448

0.9722

0.9863

0.9932

0.9966

0.9448

0.9724

0.9865

0.9933

0.9965

4.5442(−2) 2.4155(−2) 1.2549(−2) 6.3956(−3) 3.2276(−3) 1.6213(−3) 8.1256(−4) 0.9117

2−24

0.9966

4.5404(−2) 2.4140(−2) 1.2541(−2) 6.3918(−3) 3.2259(−3) 1.6205(−3) 8.1219(−4) 0.9114

2−22

0.9931

4.5331(−2) 2.4111(−2) 1.2526(−2) 6.3848(−3) 3.2228(−3) 1.6190(−3) 8.1140(−4) 0.9108

2−20

0.9863

1024

4.5187(−2) 2.4056(−2) 1.2498(−2) 6.3723(−3) 3.2168(−3) 1.6158(−3) 8.0982(−4) 0.9095

2−18

0.9724

128

4.4909(−2) 2.3951(−2) 1.2447(−2) 6.3481(−3) 3.2039(−3) 1.6094(−3) 8.0659(−4) 0.9069

2−16

0.9444

64

4.4381(−2) 2.3756(−2) 1.2350(−2) 6.2949(−3) 3.1775(−3) 1.5962(−3) 7.9994(−4) 0.9017

2−14

32

0.9447

0.9724

0.9866

0.9933

0.9966

4.5460(−2) 2.4163(−2) 1.2553(−2) 6.3975(−3) 3.2286(−3) 1.6217(−3) 8.1276(−4) 0.9118

0.9448

0.9725

0.9866

0.9934

0.9966

Numerical Solution with Special Layer Adapted Meshes for Singularly …

397

Table 4 E(N ) and Or d(N ) for standard Shishkin mesh, q = 0.5 N 

16

2−10

3.6002(−2) 2.2037(−2) 1.3110(−2) 7.6196(−3) 4.3312(−3) 2.4185(−3) 1.3319(−3) 0.7081

2−12

0.8606

0.7491

0.7833

0.8152

0.8407

0.8607

0.7492

1.7833

0.8153

0.8407

0.8606

0.7492

0.7833

0.8153

0.8407

0.8606

0.7492

1.7833

0.8153

0.8409

0.8603

0.7492

1.7833

0.8153

0.8407

0.8606

3.6245(−2) 2.2189(−2) 1.3201(−2) 7.6704(−3) 4.3589(−3) 2.4339(−3) 1.3404(−3) 0.7079

2−24

1024

3.6244(−2) 2.2189(−2) 1.3201(−2) 7.6704(−3) 4.3589(−3) 2.4339(−3) 1.3404(−3) 0.7079

2−22

0.8407

512

3.6244(−2) 2.2189(−2) 1.3201(−2) 7.6703(−3) 4.3589(−3) 2.4334(−3) 1.3404(−3) 0.7079

2−20

0.8149

256

3.6241(−2) 2.2187(−2) 1.3200(−2) 7.6697(−3) 4.3586(−3) 2.4337(−3) 1.3403(−3) 0.7079

2−18

0.7829

128

3.6229(−2) 2.2180(−2) 1.3196(−2) 7.6672(−3) 4.3572(−3) 2.4329(−3) 1.3399(−3) 0.7079

2−16

0.7493

64

3.6184(−2) 2.2151(−2) 1.3179(−2) 7.6576(−3) 4.3519(−3) 2.4300(−3) 1.3382(−3) 0.7079

2−14

32

0.7492

0.7833

0.8153

0.8407

0.8606

3.6245(−2) 2.2189(−2) 1.3201(−2) 7.6705(−3) 4.3589(−3) 2.4339(−3) 1.3404(−3) 0.7079

0.7492

0.7833

0.8154

0.8407

0.8606

Table 5 E(N ) and Or d(N ) for Bakhvalov-Shishkin mesh, q = 0.5 N 

16

2−10

3.7681(−2) 2.0908(−2) 1.0828(−2) 5.4617(−3) 2.7332(−3) 1.3656(−3) 6.8245(−4) 0.8498

2−12

1.0007

0.9479

0.9867

0.9998

1.0027

1.0024

0.9466

1.9863

0.9996

1.0029

1.0027

0.9463

0.9862

0.9996

1.0027

1.0027

0.9462

0.9862

0.9996

1.0027

1.0027

0.9464

0.9861

0.9996

1.0027

1.0027

3.8143(−2) 2.1251(−2) 1.1029(−2) 5.5674(−3) 2.7845(−3) 1.3896(−3) 6.9349(−4) 0.8439

2−24

1024

3.8142(−2) 2.1251(−2) 1.1028(−2) 5.5673(−3) 2.7844(−3) 1.3896(−3) 6.9348(−4) 0.8438

2−22

1.0011

512

3.8141(−2) 2.1249(−2) 1.1028(−2) 5.5669(−3) 2.7843(−3) 1.3895(−3) 6.9343(−4) 0.8442

2−20

0.9988

256

3.8136(−2) 2.1245(−2) 1.1025(−2) 5.5656(−3) 2.7835(−3) 1.3891(−3) 6.9324(−4) 0.8440

2−18

0.9873

128

3.8114(−2) 2.1229(−2) 1.1015(−2) 5.5601(−3) 2.7807(−3) 1.3876(−3) 6.9251(−4) 0.8443

2−16

0.9493

64

3.8026(−2) 2.1163(−2) 1.0976(−2) 5.5387(−3) 2.7698(−3) 1.3823(−3) 6.9000(−4) 0.8454

2−14

32

0.9462

0.9862

0.9996

1.0027

1.0027

3.8143(−2) 2.1251(−2) 1.1029(−2) 5.5674(−3) 2.7845(−3) 1.3896(−3) 6.9349(−4) 0.8439

0.94622

0.9862

0.9996

1.0027

1.0027

398

D. Kaur and V. Kumar

Table 6 E(N ) and Or d(N ) for modified Bakhvalov-Shishkin mesh, q = 0.5 N 

16

2−10

3.8332(−2) 2.1138(−2) 1.0988(−2) 5.6235(−3) 2.8608(−3) 1.4497(−3) 7.3240(−4) 0.8587

2−12

0.9851

0.9447

0.9672

0.9754

0.9806

0.9847

0.9448

0.9674

0.9756

0.9805

0.9847

0.9449

0.9675

0.9756

0.9805

0.9846

0.9449

0.9675

0.9756

0.9805

0.9846

0.9449

0.9675

0.9756

0.9806

0.9845

3.8772(−2) 2.1425(−2) 1.1129(−2) 5.6914(−3) 2.8943(−3) 1.4668(−3) 7.4125(−4) 0.8557

2−24

1024

3.8772(−2) 2.1424(−2) 1.1129(−2) 5.6914(−3) 2.8942(−3) 1.4667(−3) 7.4125(−4) 0.8558

2−22

0.9807

512

3.8771(−2) 2.1424(−2) 1.1129(−2) 5.6912(−3) 2.8941(−3) 1.4667(−3) 7.4122(−4) 0.8558

2−20

0.9750

256

3.8766(−2) 2.1420(−2) 1.1127(−2) 5.6903(−3) 2.8973(−3) 1.4665(−3) 7.4111(−4) 0.8558

2−18

0.9664

128

3.8745(−2) 2.1406(−2) 1.1120(−2) 5.6870(−3) 2.8921(−3) 1.4657(−3) 7.4068(−4) 0.8559

2−16

0.9439

64

3.8661(−2) 2.1352(−2) 1.1093(−2) 5.6739(−3) 2.8857(−3) 1.4624(−3) 7.3898(−4) 0.8565

2−14

32

0.9449

0.9675

0.9756

0.9805

0.9846

3.8773(−2) 2.1425(−2) 1.1129(−2) 5.6914(−3) 2.8943(−3) 1.4668(−3) 7.4125(−4) 0.8557

0.9449

0.9675

0.9756

0.9805

0.9846

Table 7 E(N ) and Or d(N ) for Polynomial S-mesh (m = 2), q = 0.5 N 

16

2−10

3.7712(−2) 2.1687(−2) 1.1910(−2) 6.4283(−3) 3.4182(−3) 1.7992(−3) 9.3989(−4) 0.7982

2−12

0.9368

0.8651

0.8900

0.9112

0.9259

0.9365

0.8653

0.8901

0.9112

0.9258

0.9364

0.8654

0.8901

0.9112

0.9258

0.9364

0.8652

0.8901

0.9112

0.9258

0.9364

0.8653

0.8901

0.9112

0.9258

0.9364

3.8082(−2) 2.1899(−2) 1.2021(−2) 6.4861(−3) 3.4489(−3) 1.8155(−3) 9.4864(−4) 0.7983

2−24

1024

3.8082(−2) 2.1899(−2) 1.2021(−2) 6.4861(−3) 3.4489(−3) 1.8154(−3) 9.4863(−4) 0.7983

2−22

0.9259

512

3.8081(−2) 2.1898(−2) 1.2021(−2) 6.4859(−3) 3.4489(−3) 1.8154(−3) 9.4860(−4) 0.7983

2−20

0.9112

256

3.8077(−2) 2.1896(−2) 1.2019(−2) 6.4852(−3) 3.4485(−3) 1.8152(−3) 9.4849(−4) 0.7989

2−18

0.8897

128

3.8059(−2) 2.1886(−2) 1.2014(−2) 6.4825(−3) 3.4470(−3) 1.8144(−3) 9.4808(−4) 0.7982

2−16

0.8647

64

3.7989(−2) 2.1845(−2) 1.1993(−2) 6.4715(−3) 3.4412(−3) 1.8113(−3) 9.4642(−4) 0.7983

2−14

32

0.8653

0.8901

0.9112

0.9258

0.9364

3.8082(−2) 2.1899(−2) 1.2021(−2) 6.4862(−3) 3.4489(−3) 1.8155(−3) 9.4864(−4) 0.7983

0.8653

0.8901

0.9112

0.9258

0.9364

Numerical Solution with Special Layer Adapted Meshes for Singularly …

399

Table 8 E(N ) and Or d(N ) for Shishkin mesh with rational ψ, q = 0.5 N 

16

2−10

3.8787(−2) 2.6753(−2) 1.8732(−2) 1.3328(−2) 9.4622(−3) 6.7097(−3) 4.7611(−3) 0.5359

2−12

0.4949

0.5147

0.4919

0.4947

0.4962

0.4952

0.5149

0.4922

0.4946

0.4963

0.4953

0.5149

0.4922

0.4948

0.4963

0.4952

0.5150

0.4921

0.4948

0.4963

0.4953

0.5150

0.4921

0.4948

0.4963

0.4953

3.9022(−2) 2.6922(−2) 1.8839(−2) 1.3394(−2) 9.5055(−3) 6.7388(−3) 4.7805(−3) 0.5355

2−24

0.4959

1024

3.9022(−2) 2.6922(−2) 1.8839(−2) 1.3394(−2) 9.5054(−3) 6.7388(−3) 4.7805(−3) 0.5355

2−22

0.4942

512

3.9021(−2) 2.6921(−2) 1.8839(−2) 1.3394(−2) 9.5053(−3) 6.7387(−3) 4.7805(−3) 0.5355

2−20

256

3.9018(−2) 2.6919(−2) 1.8838(−2) 1.3393(−2) 9.5048(−3) 6.7384(−3) 4.7802(−3) 0.5359

2−18

0.4910

128

3.9007(−2) 2.6911(−2) 1.8833(−2) 1.3389(−2) 9.5027(−3) 6.7369(−3) 4.7793(−3) 0.5355

2−16

0.5142

64

3.8963(−2) 2.6879(−2) 1.8813(−2) 1.3378(−2) 9.4946(−3) 6.7315(−3) 4.7756(−3) 0.5356

2−14

32

0.5150

0.4921

0.4948

0.4963

0.4953

3.9022(−2) 2.6922(−2) 1.8839(−2) 1.3394(−2) 9.5055(−3) 6.7388(−3) 4.7805(−3) 0.5355

0.5150

0.4921

0.4948

Table 9 E(N ) and Or d(N ) for S(2) with q1 = 41 , q2 =

0.4963

1 4

0.4953

as discussed in [13]

N 

16

2−10

3.4606(−2) 1.9825(−2) 1.1004(−2) 5.9778(−3) 3.1876(−3) 1.6777(−3) 8.7468(−4) 0.8037

2−12 2−14

1024

0.9397

0.9257

0.8495

0.8806

0.9071

0.9395

0.8494

0.8807

0.9072

0.9257

0.9394

3.4882(−2) 1.9997(−2) 1.1098(−2) 6.0275(−3) 3.2141(−3) 1.6920(−3) 8.8231(−4) 0.8496

0.8807

0.9071

0.9257

0.9394

3.4886(−2) 1.9999(−2) 1.1099(−2) 6.0281(−3) 3.2144(−3) 1.6922(−3) 8.8239(−4) 0.8495

0.8807

0.9072

0.9256

0.9394

3.4886(−2) 2.0000(−2) 1.1099(−2) 6.0282(−3) 3.2145(−3) 1.6923(−3) 8.8242(−4) 0.8496

0.8806

0.9071

0.9256

0.9394

3.4887(−2) 2.0000(−2) 1.1099(−2) 6.0282(−3) 3.2145(−3) 1.6923(−3) 8.8243(−4) 0.8027

2−24

0.9259

512

3.4869(−2) 1.9989(−2) 1.1094(−2) 6.0251(−3) 3.2128(−3) 1.6913(−3) 8.8193(−4)

0.8026 2−22

0.9071

256

0.8029

0.8027 2−20

0.8803

128

3.2077(−3) 1.6886(−3) 8.8046(−4)

0.8027 2−18

0.8493

64

3.4816(−2) 1.9956(−2) 1.1075(−2) 6.0155−3)

0.8027 2−16

32

0.8495

0.8806

0.9071

0.9256

0.9394

3.4887(−2) 2.0000(−2) 1.1099(−2) 6.0282(−3) 3.2145(−3) 1.6923(−3) 8.8243(−4) 0.8027

0.8495

0.8806

0.9071

0.9256

0.9394

400

D. Kaur and V. Kumar

Table 10 E(N ) and Or d(N ) for S(2) with q1 =

9 16 ,

q2 =

3 16

as discussed in [15]

N 

16

2−10

5.0558(−2) 2.7673(−2) 1.4559(−2) 7.4788(−3) 3.7888(−3) 1.9054(−3) 9.5481(−4) 0.8695

2−12

0.9968

0.9231

0.9598

0.9791

0.9899

0.9952

0.9223

0.9594

0.9788

0.9895

0.9948

0.9221

0.9593

0.9787

0.9894

0.9947

0.9221

0.9592

0.9787

0.9894

0.9947

0.9220

0.9592

0.9787

0.9894

0.9947

5.1029(−2) 2.7988(−2) 1.4771(−2) 7.5974(−3) 3.8552(−3) 1.9419(−3) 9.7451(−4) 0.8665

2−24

0.9916

1024

5.1029(−2) 2.7988(−2) 1.4771(−2) 7.5973(−3) 3.8552(−3) 1.9418(−3) 9.7449(−4) 0.8665

2−22

0.9811

512

5.1028(−2) 2.7987(−2) 1.4770(−2) 7.5969(−3) 3.8549(−3) 1.9417(−3) 9.7443(−4) 0.8665

2−20

256

5.1022(−2) 2.7983(−2) 1.4768(−2) 7.5955(−3) 3.8542(−3) 1.9413(−3) 9.7420(−4) 0.8673

2−18

0.9610

128

5.0999(−2) 2.7968(−2) 1.4758(−2) 7.5898(−3) 3.8511(−3) 1.9396(−3) 9.7327(−4) 0.8667

2−16

0.9266

64

5.0912(−2) 2.7909(−2) 1.4718(−2) 7.5668(−3) 3.8386(−3) 1.9327(−3) 9.6955(−4) 0.8673

2−14

32

0.9220

0.9592

0.9787

0.9893

0.9947

5.1029(−2) 2.7988(−2) 1.4771(−2) 7.5974(−3) 3.8552(−3) 1.9419(−3) 9.7451(−4) 0.8665

0.9220

0.9592

0.9787

Table 11 E(N ) and Or d(N ) for S(3) with q1 =

7 16 ,

0.9893

q2 = 14 , q3 =

0.9947

1 16

as discussed in [15]

N 

16

2−10

5.8836(−2) 3.2095(−2) 1.6458(−2) 8.3279(−3) 4.1898(−3) 2.0941(−3) 1.0406(−3) 0.8744

2−12

1.0089

0.9629

0.9826

0.9906

0.9997

1.0079

0.9628

0.9825

0.9904

0.9994

1.0075

0.9627

0.9825

0.9904

0.9994

1.0074

0.9627

0.9825

0.9903

0.9994

1.0074

0.9627

0.9826

0.9904

0.9994

1.0073

5.9483(−2) 3.2537(−2) 1.6695(−2) 8.4490(−3) 4.2527(−3) 2.1273(−3) 1.0582(−3) 0.8704

2−24

1024

5.9482(−2) 3.2537(−2) 1.6695(−2) 8.4489(−3) 4.2527(−3) 2.1272(−3) 1.0582(−3) 0.8704

2−22

1.0006

512

5.9480(−2) 3.2535(−2) 1.6694(−2) 8.4486(−3) 4.2525(−3) 2.1271(−3) 1.0581(−3) 0.8704

2−20

0.9911

256

5.9473(−2) 3.2530(−2) 1.6691(−2) 8.4471(−3) 4.2517(−3) 2.1267(−3) 1.0579(−3) 0.8705

2−18

0.9828

128

5.9442(−2) 3.2509(−2) 1.6679(−2) 8.4413(−3) 4.2487(−3) 2.1252(−3) 1.0571(−3) 0.8706

2−16

0.9636

64

5.9319(−2) 3.2425(−2) 1.6635(−2) 8.4184(−3) 4.2368(−3) 2.1189(−3) 1.0537(−3) 0.8714

2−14

32

0.9627

0.9826

0.9904

0.9994

1.0074

5.9483(−2) 3.2537(−2) 1.6695(−2) 8.4490(−3) 4.2527(−3) 2.1273(−3) 1.0582(−3) 0.8704

0.9627

0.9826

0.9904

0.9994

1.0074

Numerical Solution with Special Layer Adapted Meshes for Singularly …

401

Table 12 E(N ) and Or d(N ) for Modified Bakhvalov mesh N 

16

2−10

3.9487(−2) 2.1100(−2) 1.0812(−2) 5.4331(−3) 2.7147(−3) 1.3562(−3) 6.7777(−4) 0.9041

2−12

1.0007

0.9699

0.9953

1.0021

1.0037

1.0026

0.9740

0.9983

1.0039

1.0042

1.0030

0.9771

1.0004

1.0053

1.0050

1.0035

0.9794

1.0020

1.0062

1.0057

1.0039

0.9812

1.0031

1.0069

1.0061

1.0042

4.2265(−2) 2.2288(−2) 1.1279(−2) 5.6240(−3) 2.7974(−3) 1.3925(−3) 6.9415(−4) 0.9232

2−24

1024

4.2106(−2) 2.2230(−2) 1.1261(−2) 5.6185(−3) 2.7958(−3) 1.3920(−3) 6.9399(−4) 0.9215

2−22

1.0012

512

4.1903(−2) 2.2155(−2) 1.1237(−2) 5.6109(−3) 2.7934(−3) 1.3912(−3) 6.9374(−4) 0.9194

2−20

1.0010

256

4.1631(−2) 2.2051(−2) 1.1202(−2) 5.5994(−3) 2.7895(−3) 1.3899(−3) 6.9326(−4) 0.9168

2−18

0.9928

128

4.1241(−2) 2.1894(−2) 1.1146(−2) 5.5796(−3) 2.7822(−3) 1.3871(−3) 6.9213(−4) 0.9135

2−16

0.9646

64

4.0619(−2) 2.1628(−2) 1.1042(−2) 5.5391(−3) 2.7656(−3) 1.3793(−3) 6.8840(−4) 0.9093

2−14

32

0.9826

1.0040

1.0075

1.0064

1.0044

4.2393(−2) 2.2334(−2) 1.1293(−2) 5.6282(−3) 2.7987(−3) 1.3929(−3) 6.9426(−4) 0.9246

0.9838

1.0047

1.0079

1.0067

1.0045

Table 13 E(N ) and Or d(N ) for Shishkin W-grid, q = 0.5 N 

16

2−10

3.3379(−2) 1.9468(−2) 1.1107−2)

6.2530(−3) 3.4926(−3) 1.9362(−3) 1.0653(−3)

0.7778

0.8402

2−12

0.8619

0.8096

0.8291

0.8408

0.8516

0.8629

0.8095

0.8292

0.8408

0.8516

0.8629

0.8095

0.8292

0.8408

0.8516

0.8629

0.8096

0.8292

0.8408

0.8516

0.8629

0.8095

0.8292

0.8408

0.8516

0.8629

3.3644(−2) 1.9662(−2) 1.1219(−2) 6.3145(−3) 3.5256(−3) 1.9538(−3) 1.0743(−3) 0.7749

2−24

1024

3.3644(−2) 1.9662(−2) 1.1219(−2) 6.3145(−3) 3.5256(−3) 1.9538(−3) 1.0743(−3) 0.7749

2−22

0.8512

512

3.3643(−2) 1.9662(−2) 1.1218(−2) 6.3143(−3) 3.5255(−3) 1.9537(−3) 1.0742(−3) 0.7749

2−20

256

3.3639(−2) 1.9659(−2) 1.1217(−2) 6.3135(−3) 3.5251(−3) 1.9535(−3) 1.0741(−3) 0.7749

2−18

0.8288

128

3.3627(−2) 1.9650(−2) 1.1212(−2) 6.3105(−3) 3.5233(−3) 1.9525(−3) 1.0736(−3) 0.7751

2−16

0.8096

64

3.3577(−2) 1.9613(−2) 1.1190(−2) 6.2985(−3) 3.5167(−3) 1.9489(−3) 1.0716(−3) 0.7757

2−14

32

0.8095

0.8292

0.8408

0.8516

0.8629

3.3644(−2) 1.9662(−2) 1.1219(−2) 6.3146(−3) 3.5256(−3) 1.9538(−3) 1.0743(−3) 0.7749

0.8095

0.8292

0.8408

0.8516

0.8629

402 Fig. 5 Exact solution and Numerical solution of problem (7) for  = 2−10 generated by Shishkin W-grid for N = 32

D. Kaur and V. Kumar 0.25

0.2

0.15

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1

when  → 0. The result in Table 2 is given for the Bakhvalov mesh for q = 0.5. As Table 3 shows, the error becomes slightly worse when logarithm in Bakhvalov mesh is replaced by its (0, 1) Padé approximation in B-mesh. It is apparent from Table 4 that the performance of standard Shishkin mesh is inferior to that of B-type meshes which has prompted efforts to improve it. The error becomes better for the Bakhvalov-Shishkin mesh which blends the ideas of Bakhvalov and Shishkin in contrast to the standard Shishkin mesh, as noted in Table 5. However, as observed before the error becomes slightly worse in Table 6 when Bakhvalov-Shishkin mesh is modified by using (0, 1) Padé approximation for the exponential term in the modified Bakhvalov-Shishkin mesh. We got smaller errors for polynomial S-mesh (m = 2) in Table 7 as opposed to the standard Shishkin mesh but worse than the BakhvalovShishkin mesh. It follows from Table 8 that the results for the Shishkin mesh with rational ψ are even worse than the standard Shishkin mesh. As Tables 9, 10 and 11 shows, S(2) and S(3) are an improvement over the standard Shishkin mesh. The mesh S(2) in Table 9 contains 50% of mesh points in the interval (0, τ2 ] and the numerical results shows that it reaches the accuracy of B-type meshes but it is not as dense close to x = 0 as the B-type meshes are. However, the S(2) mesh in Table 10 and S(3) mesh in Table 11 preserves 75% of mesh points in the interval (0, τm ] for m = 2 and m = 3, respectively. The numerical results for the Bakhvalov mesh in Table 2 and its modification using Lambert-W function in Table 12 differ only a bit but the modified Bakhvalov mesh has the advantage of not using the transition point as in the Bakhvalov mesh whose exact value cannot be determined explicitly. From Table 13, it is evident that for our test problem, the new proposed Shishkin W-grid gives better results as compared to the results using the standard Shishkin mesh in Table 4. In order to emphasize the differences between various meshes, we present the percentage of the number of mesh points in the interval [0, ] for  = 2−10 in Table 14. From Table 14, it is observed that B-type meshes keep a high percentage of

Numerical Solution with Special Layer Adapted Meshes for Singularly …

403

Table 14 Percentage of mesh points in [0, ] Mesh

N 16

64

512

Bakhvalov mesh B-mesh Standard Shishkin mesh Bakhvalov-Shishkin mesh Modified Bakhvalov-Shishkin mesh Polynomial S-mesh (m = 2) Shishkin mesh with rational ψ S(2) with q1 = 1/4, q2 = 1/4 S(2) with q1 = 9/16, q2 = 3/16 S(3) with q1 = 7/16, q2 = 1/4, q3 = 1/16 Modified Bakhvalov mesh Shishkin W-grid

23.53 17.64 11.76 23.52 23.52 23.52 11.76 11.76 29.41 52.94 23.53 11.76

20 16.92 6.15 20 21.54 18.46 6.15 9.23 20 47.69 20 9.23

19.30 16.76 4.09 19.49 19.49 14.23 1.95 6.82 15.40 36.26 19.69 5.46

mesh points in the interval [0, ] while Shishkin-type meshes are not as dense close to x = 0 as the B-type meshes are. Among Shishkin-type meshes, S(3) with q1 = 7/16, q2 = 1/4, q3 = 1/16 in Table 11 contains a high percentage of mesh points in [0, ] but numerical results for this mesh shows that a large number of mesh points in the boundary layer do not guarantee smaller error. All numerical experiments on the test problem confirm the theoretically obtained results of order of convergence. The numerical results shows that it is extremely useful to use graded (B-type) meshes as the error on these meshes is considerably smaller in contrast to the piecewise equidistant meshes of Shishkin-type which have attracted much attention because of their simple structure [17, 18].

References 1. Roos, H.-G., Teofanov, L., Uzelac, Z.: A modified Bakhvalov mesh. Appl. Math. Lett. 31, 7–11 (2014) 2. Farrell, P.A., Hegarty, A., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman and Hall (2000) 3. Kumar, V., Srinivasan, B.: A novel adaptive mesh strategy for singularly perturbed parabolic convection diffusion problems. Differ. Equ. Dyn. Syst. https://doi.org/10.1007/s12591-0170394-2 4. Kumar, V., Srinivasan, B.: An adaptive mesh strategy for singularly perturbed convection diffusion problems. Appl. Math. Model. 39(7), 2081–2091 (2015) 5. Srinivasan, B., Kumar, V.: The versatility of an entropy inequality for the robust computation of convection dominated problems. Procedia Comput. Sci. 108C, 887–896 (2017) 6. Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32, 1025–1039 (1978)

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7. Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of boundary layer. Zh. Vychisl. Mater. Fiz. 9, 841–859 (1969). (In Russian) 8. Kopteva, N.V.: On the convergence, uniform with respect to the small parameter, of a scheme with central difference on refined grids. Comput. Math. Math. Phys. 39(10), 1594–1610 (1999) 9. Vulanovic, R.: On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 13, 187–201 (1983) 10. Linß, T.: Layer-adapted Meshes for Reaction-Convection-Diffusion problems. Lecture Notes in Mathematics. Springer, Heidelberg (1985) 11. Miller, J.J.H., O’Riordan, E., Shishkin, I.G.: Fitted Numerical Methods for Singular Perturbation Problems. Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific, Singapore (1996) 12. Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations. Second doctorial thesis, Keldysh Institute, Moscow (1990). (In Russian) 13. Roos, H.-G., Linß, T.: Sufficient conditions for uniform convergence on layer-adapted grids. Computing 63(1), 27–45 (1999) 14. Linß, T.: An upwind difference scheme on a novel Shishkin-type mesh for a linear convectiondiffusion problem. J. Comput. Appl. Math. 110(1), 93–104 (1999) 15. Vulanovic, R.: A priori meshes for singularly perturbed quasilinear two-point boundary value problems. IMA J. Numer. Anal 21, 349–366 (2001) 16. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math 5, 329–359 (1996) 17. Cakir, M., Amirali, I., Kudu, M., Amiraliyev, G.M.: Convergence analysis of the numerical method for a singularly perturbed periodical boundary value problem. J. Math. Comput. Sci. 16, 248–255 (2016) 18. Kudu, M., Amirali, I., Amiraliyev, G.M.: A layer analysis of parameterized singularly perturbed boundary value problems. IJAM 29(4), 439–449 (2016)

Use of Galerkin Technique in Some Water Wave Scattering Problems Involving Plane Vertical Barriers B. N. Mandal and Soumen De

Abstract The explicit solutions exist for normal incidence of the surface wave train or a single thin plane vertical barrier partially immersed or completely submerged in deep water. However, for oblique incidence of the wave train and/or for finite depth water, no such explicit solution is possible to obtain. Some approximate mathematical techniques are generally employed to solve them approximately in the sense that quantities of physical interest associated with each problem, namely the reflection and transmission coefficients, can be obtained approximately either analytically or numerically. The method of Galerkin approximations has been widely used to investigate such water wave scattering problems involving thin vertical barriers. Use of Galerkin method with basis functions involving somewhat complicated functions in solving these problems has been carried out in the literature. Choice of basis functions as simple polynomials multiplied by appropriate weights dictated by the edge conditions at the submerged end points of the barrier providing fairly good numerical estimates for the reflection and transmission coefficients have been demonstrated in this article. Keywords Oblique scattering · Linear theory · First kind integral equations · Galerkin approximations · Reflection and transmission coefficients

Introduction Wave interaction by structures is a very interesting topic and has been studied extensively in the fields of research concerning water waves, acoustics waves, electromagnetic waves, etc. employing a wide range of mathematical techniques. Particularly, B. N. Mandal (B) Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 108, India e-mail: [email protected] S. De Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700 009, India e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_13

405

406

B. N. Mandal and S. De

the problems of water wave interaction with obstacles of various geometrical shapes have been studied in the literature somewhat extensively after the Second world War due to rapid growth in ocean related activities like construction of off-shore structures, breakwaters, off-shore drilling for oil extraction from oil fields below the ocean floor, construction of very large floating structures (VLFS) etc. While handling water wave problems, some simplified assumptions are made in order to formulate the problem. The linearsed theory is developed from the basic assumptions that the amplitude of the wave motion is small compared to the wave length, and the velocity components, the free surface elevation together with their partial derivatives are small quantities so that their products and powers can be neglected in the equations of fluid mechanics. Also, water is considered to be a homogeneous, inviscid, incompressible liquid. A substantial amount of research work on water waves is based on the linearised theory. Water wave scattering problems involving floating and submerged bodies of various geometrical shapes form an important area of research. There may be several types of configurations of the bodies, these may be thin or thick, may be straight or curved. There exist various mathematical techniques to tackle these problems, some being dependent on the geometry of the body. Floating and submerged bodies in the form of thin plane vertical barriers are the first type of bodies investigated the water wave literature due to their simplicity in engineering design of models for breakwaters and also in the mathematical analysis of the related wave scattering problems. These have received much attention in the water wave literature because of the ability to solve some two-dimensional water wave scattering problems involving the vertical barriers explicitly (cf. [1–3, 8, 9, 21, 29, 34, 35]). The explicit solutions exist for normal incidence of the surface wave train on a single thin plane vertical barrier partially immersed or completely submerged in deep water. However, for oblique incidence of surface water waves, these problems do not possess explicit solutions. Some approximate mathematical techniques have generally been employed to solve these problems approximately in the sense that the reflection and transmission coefficients for each problem could be obtained approximately either analytically or numerically. Oblique scattering problems involving a partially immersed or completely submerged thin vertical barrier have been studied by [13, 14, 16] employing the Wiener–Hopf technique of complex variable theory. They obtained the asymptotic forms of the reflection and transmission coefficients for large wavenumber. The partially immersed vertical barrier problem for oblique incidence has also been investigated by [10] who formulated the problem in terms of two first kind integral equations, one involving the horizontal component of velocity across the gap as in [34] and the other involving the difference of potential across the barrier as in [19]. These integral equations cannot be solved explicitly. For the case of normal incidence these reduce to the forms which, however, can be solved explicitly. Reference [10] used single-term Galerkin approximations involving the explicit solutions of the corresponding integral equations for the case of normal incidence given by [34] for the horizontal component of velocity across the gap below the barrier and by [19] for the difference of potential across the barrier. Both these solutions are generally referred to as Ursell’s solutions. Reference [10]

Use of Galerkin Technique in Some Water Wave Scattering Problems …

407

then obtained very close upper and lower bounds for the reflection and transmission coefficients for all wavenumbers and angles of incidence for the partially immersed barrier problem so that averages of upper and lower bounds provide fairly accurate numerical estimates for these. However, they noted that use of Ursell’s exact solutions for the submerged barrier problem does not produce accurate results. This fact and to obtain more accurate numerical results, led us to think that perhaps multi-term Galerkin approximations need to be employed. Then the question arises about what should be the basis functions of the multi-term approximations. Ursell’s solutions are somewhat complicated and as such the single-term approximations of the integral equations arising in the oblique scattering problem appears to be complicated. It is also not apparent about what should be the other terms of the set of basis functions of Galerkin approximation if the first term is taken as Ursell’s solution. Although for the case of uniform finite depth, multi-term approximations involving Chebyshev polynomials have been used in the literature (cf. [21, 30]), there is practically no work so far on the choice of basis functions for the case of infinitely deep water. This has motivated us to employ multi-term expansion involving simple polynomials multiplied by suitable weights for the deep water problem. The forms of the weights can be found from a knowledge of the behavior of the unknown functions of the integral equations at the end points. Very accurate upper and lower bounds for the reflection and transmission coefficients are obtained for all angles of incidence and all wave lengths using multi-term Galerkin approximations involving simple polynomials multiplied by suitable weight functions for the solutions of integral equations arising in the problems of oblique scattering by thin vertical barriers. It may be noted that [20] employed an eigenfunction expansion to analyze oblique scattering by thin vertical barriers of different configurations present in finite depth water. The same method has been employed earlier by [18] to study oblique scattering by a rectangular trench and by [4] to study the same by porous structures. Scattering of water waves by a pair of thin vertical barriers of various geometrical configurations produces another important class of water wave problems, which have been investigated in the literatures by [7, 11, 12, 15, 19, 26–28, 30] and others. Closed form expressions for the reflection and transmission coefficients are found only when the pair of thin barriers, either partially immersed up to the same depth below the mean free surface (cf. [19]) or submerged completely from the same depth and extending infinitely downwards (cf. [15]), are present in deep water for normally incident surface waves. Highly accurate approximate solutions of the twobarrier problems could be, however, obtained for several other geometries of the barriers and also for oblique incidence of the wave train by employing a variety of mathematical techniques like Galerkins approximations, matching of eigenfunctions etc. (cf. [11, 12, 17, 26]). Though the study of water wave scattering by identical thin vertical barriers is quite extensive in the water wave literature, only a handful of works are available in the literature involving non-identical vertical barriers. Reference [27] first tackled a scattering problem involving two non-identical vertical barriers, one of which is partially immersed and the other is fully submerged and extends infinitely downwards, by adapting a variational approach. She established the existence of total transmission and total reflection at an infinite sequence of wavenumbers only

408

B. N. Mandal and S. De

for overlapping barriers while for non-overlapping barriers she found at most a finite number of zeros of each of reflection and transmission coefficients. Reference [26] employed a matching of eigenfunction technique to study water wave scattering by two thin vertical barriers partially immersed upto two different depths below the mean free surface. He observed non-existence of zeros of reflection coefficient when the lengths of immersion of the barriers are unequal. Reference [12] extended [27] deep water problem to water of finite depth and obtained very accurate numerical estimates for the reflection and transmission coefficients by employing multi-term Galerkin approximations. The problem of scattering of an obliquely incident wave train by two non-identical thin vertical barriers, partially immersed in deep water, has been investigated by employing Havelocks expansion of water wave potential. Then following Havelocks inversion formulae, the problem is ultimately reduced to solving coupled integral equations of first kind (cf. [31–33]). In fact, two complementary formulations are obtained in the mathematical analysis, one is in terms of horizontal component of velocity and the other involves difference of potentials. On application of Galerkin’s single-term approximations, both these two formulations produce fairly accurate numerical estimates for the reflection and transmission coefficients. The estimates for the reflection coefficient obtained using any of these formulations are depicted graphically against the wavenumber. The graphs of the reflection coefficient exhibit total transmission i.e. zeros of reflection coefficient at discrete frequencies only for two identical barriers. Almost similar behaviour of the reflection coefficient was observed by [26] for water wave scattering by two unequal partially immersed barriers in uniform finite depth water. When the separation length between the barriers is very small, the results of [34] for a single partially immersed barrier are recovered. This indicates that for very small barrier spacing, the two partially immersed barriers become nearly a single partially immersed barrier whose length of immersion is equal to the length of immersion of the longer one. These problems have applications in the construction of offshore structures in the form of breakwaters required to protect the seashore from the impact of rough sea during storms.

1 Oblique Scattering by a Thin Vertical Barrier in Deep Water This section is concerned with scattering of obliquely incident surface waves by a thin vertical barrier. Four basic configurations of the thin vertical barrier are generally considered which are (i) partially immersed barrier, (ii) submerged bottom standing barrier (for finite depth water) or completely submerged barrier extending infinitely downwards (for deep water), (iii) submerged plate, and (iv) wall with submerged gap. There is no explicit solutions for the problems of oblique scattering of water waves by a thin vertical barrier of various geometrical configurations present in deep water. However, there exists some approximate methods to solve these problems approximately in the sense that the reflection and transmission coefficients could be

Use of Galerkin Technique in Some Water Wave Scattering Problems …

409

obtained numerically. Oblique scattering problems involving a partially immersed or completely submersed thin vertical barrier was studied by [10, 13, 14, 16, 23–25] by using various methods. The scattering of obliquely incident surface waves by a thin vertical barrier which may be either partially immersed or completely submerged extending infinitely downwards in deep water was considered by [5]. Very accurate upper and lower bounds for the reflection and transmission coefficients are obtained for all angles of incidence and all wavelengths using two-term Galerkin approximation involving simple polynomials multiplied by suitable weight functions to the solutions of integral equations arising in the problems of oblique scattering by thin vertical barriers partially immersed or completely submerged in deep water. This is in contrast with the use of [34] exact solutions in the Galerkin approximations as has been done by Evans and Morris (1972) for the partially immersed barrier problem. As mentioned above, Evans and Morris (1972) noted that use of [34] exact solution for the submerged barrier problem does not produce accurate results. Three and fourterm Galerkin approximations have also been considered and it is observed that the lower and upper bounds improve. However, as the improvement is in the sixth decimal place, we feel that two-term approximations provide fairly accurate numerical results. The problem of oblique scattering by fixed thin vertical plate submerged in deep water was considered, by [6], by employing single-term Galerkin approximation involving constant as basis multiplied by appropriate weight function after reducing it to solving a pair of first kind integral equations. Upper and lower bounds of reflection and transmission coefficients, when evaluated numerically are seen to be very close so that their averages produce fairly accurate numerical estimates for these coefficients. Numerical estimates for the reflection coefficient are depicted graphically against the wave number for different values of various parameters. The numerical results obtained by the present method are found to be in an excellent agreement with the known results.

1.1 Mathematical Formulation of the Problem A train of surface water waves represented by the velocity potential Re{χ0 (x, y, z) e−iσt } with (1) χ0 (x, y, z) = e−K y+iμx+iνz , where μ = K cos α, ν = K sin α, K = σg , g is the acceleration due to gravity, σ is the circular frequency, traveling from the direction of negative infinity is incident at an angle α to the normal of the vertical barrier occupying the position x = 0, y ∈ L, −∞ < z < ∞, the y−axis being taken vertically downwards into the fluid region. Here L ≡ L 1 = (0, a) for a partially immersed barrier and L ≡ L 2 = (b, ∞) for a barrier submerged below the mean free surface from a depth b and extending infinitely downwards. If Re{χ(x, y, z)e−iσt } represents the velocity potential of the 2

410

B. N. Mandal and S. De

resulting motion, then χ(x, y, z) satisfies the Laplace equation in the fluid region and on the free surface it satisfies the linearized condition ∂χ + K χ = 0 on y = 0, ∂y

(2)

while on the barrier it satisfies ∂χ = 0, x = 0, y ∈ L . ∂x

(3)

In view of the geometrical symmetry of the problem, the z−dependence of χ(x, y, z) throughout can be assumed to be eiνz so that we can write χ(x, y, z) = φ(x, y)eiνz ,

(4)

χ0 = φ0 (x, y)eiνz ,

(5)

φ0 (x, y) = e−K y+iμx .

(6)

(∇ 2 − ν 2 )φ = 0, y ≥ 0,

(7)

∂φ = 0, y = 0, ∂y

(8)

∂φ = 0, x = 0, y ∈ L , ∂x

(9)

r 2 ∇φ is bounded as r −→ 0

(10)

and also

where Thus φ(x, y) satisfies

Kφ +

1

where r is the distance from a submerged edge of the barrier,



and φ(x, y) →

∇φ −→ 0 as y −→ ∞

(11)

T φ0 (x, y) as x → ∞, φ0 (x, y) + Rφ0 (−x, y) as x → −∞

(12)

where T and R are the transmission and reflection coefficients respectively and are to be determined. Here the notation φ0 (x, y) is used for φinc (x, y).

Use of Galerkin Technique in Some Water Wave Scattering Problems …

411

1.2 Method of Solution A solution for φ(x, y) satisfying the Eq. (7) and the conditions (8), (11) and (12) is given by φ(x, y) =

⎧ ∞ ⎨ T φ0 (x, y) + 0 A(k)S(k, y)e−k1 x dk, x > 0, ⎩

φ0 (x, y) + Rφ0 (−x, y) +

∞ 0

(13) B(k)S(k, y)ek1 x dk, x < 0

1  where k1 = k 2 + ν 2 2 with k1 = k when ν = 0 and S(k, y) = k cos(ky) − K sin(ky). Let f (y) =

∂φ (0, y), 0 < y < ∞, ∂x

(14)

(15)

and g(y) = φ(x + 0) − φ(x − 0), 0 < y < ∞,

(16)

f (y) = 0 for y ∈ L

(17)

g(y) = 0 for y ∈ L = (0, ∞) − L .

(18)

then

and

The unknown constants R, T and the unknown functions A(k) and B(k) are related to f (y) and g(y) as given by 2i K T =1− R =− μ



f (y)e−K y dy,

(19)

L

2 A(k) = −B(k) = − f (y)S(k, y)dy, πk1 (k 2 + K 2 ) L g(y)e−K y dy, R = −K

(20)

(21)

L

A(k) =

1 π(k 2 + K 2 )

g(y)S(k, y)dy.

(22)

L

In deriving relations (19) and (20) the condition (17) and in deriving the relations (21) and (22) the condition (18) have been utilized in the appropriate Havelock inversion formula. For the sake of completeness, we state the Havelock’s integral expansion of a function F(y) defined on (0, ∞) and satisfying Dirichlet condition, as given by

412

B. N. Mandal and S. De



F(y) = F0 e−K y +

∞

F(k)S(k, y)dy, y > 0

0

where

∞

F0 = 2K

F(y)e−K y dy,

0

F(k) =

1 2 π(k + K 2 )



∞

F(y)S(k, y)dy. 0

The expressions for F0 and F(k) together are known as the Havelock’s inversion formula. Use of the condition (9) in the form ∂φ (±0, y) = 0, y ∈ L ∂x in the representation (13) for φ(x, y) produces an integral equation for g(y) as given by g(u)M(y, u)du = πiμ(1 − R)e−K y , y ∈ L

(23)

L



where

∞

M(y, u) = lim

→+0 0

k1 S(k, y)S(k, u) − k e dk, k2 + K 2

(24)

the exponential term being introduced to ensure convergence of the integral. From the relation (24) we note that M(y, u) is real since each term in the integral is real and is a symmetric function of y and u, since the integral involves S(k, y) and S(k, u). Again, as φ(x, y) is continuous across the gap, use of the representation (13) along with the relations (20) produces an integral equation for f (y) as given by L

π f (u)N (y, u) = − Re−K y , y ∈ L 2

where

∞

N (y, u) = 0

S(k, y)S(k, u) dk, k1 (k 2 + K 2 )

(25)

(26)

so that N (y, u) is also real since each term in the integral is real and symmetric function of y and u, since the integral involves S(k, y) and S(k, u). Let us write 2 f (y), y ∈ L, (27) F(y) = − πR

Use of Galerkin Technique in Some Water Wave Scattering Problems …

G(y) = −

1 g(y), y ∈ L , πiμ(1 − R)

413

(28)

then G(y) and F(y) satisfy the integral equations

G(u)M(y, u)du = e−K y , y ∈ L ,

(29)

F(u)N (y, u)du = e−K y , y ∈ L.

(30)

L

L

It may be noted that the functions G(y) and F(y) in (29) and (30) respectively must be real, since the kernels and forcing functions are real. The relations (19) and (21) are now recast as F(y)e−K y dy = C (31) L

and



G(y)e−K y dy = L

where C=

1 π2 K 2 C

1− R cos α. iπ R

(32)

(33)

It is important to note that C is real, since F(y) and G(y) are real valued functions. It is interesting to note that the real part of (1 − R) i.e, T is zero. It may be mentioned here that bounds of |R| and |T |(= |1 − R|) have been computed numerically by a method described in Sect. 1.3. R and T are in general complex and their real and imaginary parts cannot be computed numerically by this method.

1.3 Upper and Lower Bounds for C Following Evans and Morris (1972), we define an inner product < f, g >=

f (y)g(y)dy.

(34)

L

Then obviously < f (y), g(y) > is symmetric and linear. Also the operator M defined by (Mg)(y) =< M(y, u), g(u) > (35)

414

B. N. Mandal and S. De

is linear, self-adjoint and positive semi-definite. For the solution of (29) we choose multi-term Galerkin approximation as G(y) ≈

N

αn gn (y), y ∈ L

(36)

n=0

where gn (y)(n = 0, 1, 2, . . . , N ) are suitable basis functions. To find the unknown constants αn (n = 0, 1, 2, . . . , N ) appearing in relation (36), we substitute (36) into the integral Eq. (29) and multiply by gm (y) and integrate over L to obtain the linear system N

αn L mn = G m0 , m = 0, 1, . . . , N

(37)

n=0

where L mn =< (Mgn )(y), gm (y) >,

(38)

G m0 =< e−K y , gm (y) > .

(39)

Thus the unknown constants αn (n = 0, 1, 2, . . . , N ) are obtained by solving the linear system (37). Using the same argument as in [21, 30], it can be shown that, after using the relation (32) C ≤ A, (40) where A=

π2 K 2

1 N n=0

αn G n0

.

(41)

Thus A can be regarded as an upper bound of the unknown constant C. Again, if we define another inner product by f (y)g(y)dy

>=

(42)

L

and another operator N by (N f )(y) =>,

(43)

then it is obvious that > is linear, symmetric and also the operator N is linear, self-adjoint and positive semi-definite. For the solution of the integral Eq. (30), we choose multi-term Galerkin approximation as

Use of Galerkin Technique in Some Water Wave Scattering Problems …

F(y) ≈

N

βn f n (y), y ∈ L,

415

(44)

n=0

where f n (y)(n = 0, 1, 2, . . . , N ) are suitable basis functions. To find the unknown constants βn (n = 0, 1, 2, . . . , N ) appearing in relation (44), we substitute (44) into integral Eq. (30), multiply by f m (y) and integrate over L to obtain the linear system N

βn K mn = Fm0 , m = 0, 1, 2, . . . , N

(45)

n=0

where K mn =>,

(46)

Fm0 => .

(47)

Thus the unknown constants βn (n = 0, 1, . . . , N ) are obtained by solving the linear system (45). Again, it can be shown that, after using the relation (31), B≤C where B=

N

βn Fn0 .

(48)

(49)

n=0

Hence for the unknown real constant C, we find B≤C ≤ A

(50)

where A and B are given by (41) and (49) respectively. Thus upper and lower bounds for |R| and |T | are obtained as

where

R1 ≤ |R| ≤ R2 , T1 ≤ |T | ≤ T2

(51)

1 1 R1 =   21 , R2 =  1 , 2 2 2 2 1 + π A sec α 1 + π B 2 sec2 α 2

(52)

π B sec α π A sec α T1 =   21 , T2 =  1 . 1 + π 2 A2 sec2 α 1 + π 2 B 2 sec2 α 2

(53)

416

B. N. Mandal and S. De

Fig. 1 Partially immersed barrier [34]

In the next sections two different configurations of the barrier are considered, upper and lower bounds for the reflection and transmission coefficients are evaluated in each case for the various values of the different parameters involved.

1.4 Partially Immersed Vertical Barrier In this case L = (0, a) so that L = (a, ∞) (See Fig. 1). The corresponding problem was considered by Evans and Morris (1972) by utilizing one-term Galerkin approximation using [34] explicit solution as the one-term approximation. However, here we use multi-term Galerkin approximation involving simple polynomials multiplied by an approprite weight function. For G(y) we choose N

y n

y 2  21

G(y) ≈ 1 − αn ,0 < y < a a a n=0

(54)

where αn (n = 0, 1, 2, . . . , N ) are unknown constants and here

y 2  21 y n gn (y) = 1 − , (n = 0, 1, 2, . . . , N ), 0 < y < a. a a

(55a)

Here we give analytical results only for N = 1 i.e., two-term approximation. After substituting g0 (y) and g1 (y) in the expressions (37), (38), (39), and (41), A is obtained as 1 (55b) A= 2 2 π K {α0 M(K a) + α1 N (K a)} where

Use of Galerkin Technique in Some Water Wave Scattering Problems …



a

M(K a) = 0

N (K a) =

a



y 2  21 e−K y 1 − dy, a

e−K y

0

with P(K a) =

y a

1−

y 2  21 a

dy,

417

(55c)

(55d)

α0 =

N (K a)P(K a) − M(K a)S(K a) , R(K a)P(K a) − S 2 (K a)

(55e)

α1 =

R(K a)M(K a) − N (K a)S(K a) , R(K a)P(K a) − S 2 (K a)

(55f)

π2 4



∞ 0

R(K a) =

∞

0

k1 {K H1 (ka) − k J1 (ka)}2 dk, k 2 (k 2 + K 2 )

(55g)

k1 {kU (ka) − K V (ka)}2 dk, (k 2 + K 2 )

(55h)



a

U (ka) = 0

a

1−

y 2  21

cos(ky)dy,

(55i)

y 2  21

y 1− sin(ky)dy, a a

(55j)

a

0

V (ka) =

y

a

and π 2



∞

k1 {K H1 (ka) − k J1 (ka)} {K V (ka) − kU (ka)} dk, 2 + K 2) k(k 0 (55k) H1 (ka) being the Struve function and J1 (ka) being the Bessel function. Again, we choose multi-term Galerkin approximation for F(y)(a < y < ∞) as given by  − 21

N

y n y 2 −K y −1 βn ,a < y < ∞ (56) F(y) ≈ e a a n=0 S(K a) =

where βn (n = 0, 1, . . . , N ) are unknown constants and here f n (y) = e−K y

 − 21  y 2 y n −1 , (n = 0, 1, . . . , N ), a < y < ∞. a a

(57a)

Here also we give analytical results only for N = 1 i.e., two-term approximation.

418

B. N. Mandal and S. De

After substituting f 0 (y) and f 1 (y) in the expressions (45), (46), (47) and (49), B is obtained as (57b) B = β0 0 (K a) + β1 1 (K a) with 0 (K a) = a K 0 (2K a)

(57c)

1 (K a) = a K 1 (2K a)

(57d)

where K 0 (2K a), K 1 (2K a) are modified Bessel functions and β0 =

E(K a)0 (K a) − 1 (K a)D(K a) , C(K a)E(K a) − D 2 (K a)

(57e)

β1 =

1 (K a)C(K a) − 0 (K a)D(K a) , C(K a)E(K a) − D 2 (K a)

(57f)



∞

C(K a) = 0



∞

W (ka) =

e−K y

 y 2 a

a

− 21 {k cos(ky) − K sin(ky)} dy, −1

∞

E(K a) = 0



∞

X (ka) =

e

−K y

y  y 2 a

a



∞

D(K a) = 0

(k 2

(W (ka))2 dk, k1 (k 2 + K 2 )

a

(X (ka))2 dk, k1 (k 2 + K 2 )

− 21 {k cos(ky) − K sin(ky)} dy, −1

k1 {kY (ka) − K Z (ka)} {k I (ka) − K L(ka)} dk, + K 2)

∞

Y (ka) =

e

−K y

 y 2 a

a



∞

Z (ka) =

e

−K y

a

a

I (ka) =

∞

e a

−K y

 y 2

a

(57h)

(57i)

(57j)

(57k)

− 21 −1 cos(ky)dy,

(57l)

− 21 −1 sin(ky)dy,

(57m)

y  y 2 a

(57g)

− 21 −1 cos(ky)dy,

(57n)

Use of Galerkin Technique in Some Water Wave Scattering Problems …

419

Table 1 Lower and upper bounds for the reflection coefficient |R| for various values of the parameters K a, α and N = 1 α = 0◦

α = 30◦

α = 60◦

α = 75◦

Ka

R1

R2

|R| (Ursell)

R1

R2

R1

R2

R1

R2

0.2

0.066941

0.066973

0.066965

0.056831

0.056914

0.032538

0.032610

0.016813

0.016833

0.6

0.602115

0.603552

0.603287

0.539135

0.539160

0.336211

0.336260

0.179751

0.179876

1.0

0.944813

0.947251

0.947059

0.927107

0.927120

0.792101

0.792137

0.590347

0.590399

1.4

0.992869

0.993722

0.993400

0.991520

0.993981

0.962755

0.964392

0.872246

0.877391

1.8

0.996852

0.999116

0.999028

0.998413

0.998873

0.991418

0.997192

0.971160

0.988032

Table 2 Lower and upper bounds for the reflection coefficient |R| for various values of the parameters K a, α and N = 2, 3 N=2 Ka 0.2 0.6 1.0 1.4 1.8 0.2 0.6 1.0 1.4 1.8

α = 0◦ R1 0.066947 0.602117 0.944818 0.992872 0.996864 N=3 0.066952 0.602125 0.944822 0.992878 0.996879

R2 0.066970 0.603549 0.947243 0.993720 0.999107

α = 30◦ R1 0.056839 0.539142 0.927109 0.991525 0.998418

0.066967 0.603544 0.947240 0.993701 0.999035

0.056844 0.539150 0.927109 0.991533 0.998423

L(ka) =

∞

e a

−K y

R2 0.056902 0.539155 0.927116 0.993981 0.998860

α = 60◦ R1 0.032543 0.336235 0.792114 0.962759 0.991422

0.056871 0.539166 0.927113 0.993970 0.998851

0.032551 0.336239 0.792118 0.962779 0.991429

y  y 2 a

a

R2 0.032601 0.336244 0.792129 0.964380 0.997178

α = 75◦ R1 0.016815 0.179755 0.590347 0.872252 0.971163

R2 0.016829 0.179870 0.590391 0.877384 0.988028

0.032590 0.336241 0.792123 0.964372 0.997171

0.016819 0.179760 0.590352 0.872278 0.971165

0.016823 0.179866 0.590387 0.877293 0.988022

− 21 −1 sin(ky)dy.

(57o)

R1 and R2 are now evaluated for various values of the parameters K a, α and are displayed in Table 1. For α = 0, |R| is known exactly in terms of modified Bessel functions (cf. Ursell (1947)). However, here we put α = 0 in the expressions for R1 and R2 for obtaining numerical estimates for |R| for the case of normal incidence and the bounds are also compared with exact values given in Table 1. It is observed from the Table 1 that R1 and R2 in most cases coincide within 3 to 4 decimal places and hence their average provides a very accurate estimate for the reflection coefficients. Similar computations have been carried out for T1 , T2 and it is found that T1 and T2 in most cases coincide within 3 to 4 decimal places and hence their averages also provide very accurate estimates for |T |. However, these results are not displayed here.

420

B. N. Mandal and S. De 1 Ka=1.8 Ka=1.4

0.9 0.8 Ka=1.0

0.7

|R|

0.6 0.5 Ka=0.6

0.4 0.3 0.2 0.1 0

Ka=0.2 0

10

20

40

30

50

60

70

80

90

α (in degree)

Fig. 2 |R| versus α for different values of K a

By taking more terms in the expansions of G(y) and F(y), the bounds can be improved. For N = 2 (three-term expansion) and N = 3 (four-term expansion), Table 2 displays the bounds of reflection coefficients for different values of α and wavenumber K a. The details of analytical calculations, however, are not given here. From this Table 2 it is observed that by choosing N = 2, 3 more accurate bounds for |R| are obtained, although the results obtained for N = 1 are fairly accurate. It is also seen that estimates for |R| and |T | obtained by the present method satisfy the energy identity |R|2 + |T |2 = 1. This provides some check on the correctness of the method. Also, in view of the energy identity |R|2 + |T |2 = 1, henceforth we confine our attention on |R| only. |R| calculated in this manner is depicted graphically against α for different values of K a in Fig. 2 and the curves coincide with the corresponding curves given by Evans and Morris (1972). Again Fig. 3 depicts |R| against K a for different values of α. It may be noted that for α = 0◦ , the curve of |R| coincides with that given in [34].

1.5 Submerged Barrier Extending Infinitely Downwards In this case L = (b, ∞) so that L = (0, b) (see Fig. 4). As mentioned earlier, Evans and Morris (1972) reported that use of one-term Galerkin approximation involving the corresponding solution for the case of normal incidence did not provide results as good as surface barrier problems. It may be noted that the single-term approximation for the integral equation satisfied by the difference of velocity potential function

Use of Galerkin Technique in Some Water Wave Scattering Problems …

421

1

α=00

0.9 0.8

α=300 0 α=60

0.7

|R|

0.6

α=750

0.5

α=850

0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

Ka

Fig. 3 |R| versus K a for different values of α Fig. 4 Submerged barrier extending infinitely downwards (Ursell (1947))

across the submerged barrier in terms of Ursell’s solution is given by g0 (y) = e

−K y



y

√ b

eK v v 2 − b2

dv, y > b.

As when y becomes large, g0 (y) ≈ O( by ), and because of this Evans and Morris (1972) perhaps could not obtain good results. This has apparently led us to employ multi-term Galerkin approximation involving simple polynomials multiplied by suitable weight functions. We choose multi-term Galerkin approximation for G(y) as

422

B. N. Mandal and S. De

 y 2

G(y) ≈ e−K y

b

 21

N

y n −1 αn ,b < y < ∞ b n=0

(58)

where αn (n = 0, 1, 2, . . . , N ) are unknown constants and here gn (y) = e

−K y

 y 2 b

 21  y n −1 , (n = 0, 1, 2, . . . , N ), b < y < ∞. b

(59a)

We give analytical results only for N = 1 i.e., two-term approximation. After substituting g0 (y) and g1 (y) in the expressions (37), (38), (39), and (4.8), A is obtained as 1 (59b) A= 2 2 π K {α0 G(K b) + α1 H (K b)} where

G(K b) =

∞

e−2K y

b



∞

H (K b) =

e−2K y

b

(59d)

α1 =

P(K b)H (K b) − R(K b)G(K b) , P(K b)Q(K b) − R 2 (K b)

(59f)

∞ 0



∞

Q(K b) = 0

0

b

(59e)



∞

b

 21 − 1 dy,

G(K b)Q(K b) − R(K b)H (K b) , P(K b)Q(K b) − R 2 (K b)

P(K b) =



y  y 2

(59c)

α0 =

with

R(K b) =

  21 y 2 − 1 dy, b

k1 {kC(kb) − K D(kb)}2 dk, (k 2 + K 2 )

(59g)

k1 {k E(kb) − K F(kb)}2 dk, (k 2 + K 2 )

(59h)

k1 {K D(kb) − kC(kb)} {K F(kb) − k E(kb)} dk, (k 2 + K 2 )

∞

C(kb) =

e−K y

b



∞

D(kb) = b

e−K y

  21 y 2 − 1 cos(ky)dy, b  y 2 b

 21 − 1 sin(ky)dy,

(59i)

(59j)

(59k)

Use of Galerkin Technique in Some Water Wave Scattering Problems …



∞

E(kb) =

e−K y

y  y 2 b

b



∞

F(kb) =

b

y  y 2

e−K y

b

b

b

423

 21 − 1 cos(ky)dy,

(59l)

 21 − 1 sin(ky)dy.

(59m)

Again, we choose multi-term Galerkin approximation for F(y)(0 < y < b) as given by  1 N y 2 − 2 y n βn ,0 < y < b (60) F(y) ≈ 1 − b b n=0 where βn (n = 0, 1, . . . , N ) are unknown constants and here f n (y) =

 1 y 2 − 2 y n 1− , (n = 0, 1, . . . , N ), 0 < y < b. b b

(61a)

Here also we give analytical results only for N = 1 i.e., two-term approximation. After substituting f 0 (y) and f 1 (y) in the expressions (45), (46), (47) and (49), B is obtained as (61b) B = β0 M(K b) + β1 N (K b) where M(K b) = N (K b) = b with

2

 {L1 (K b) − I1 (K b)} + 1

(61d)

M(K b)Q(K b) − R(K b)N (K b) , P(K b)Q(K b) − R 2 (K b)

(61e)

β1 =

N (K b)P(K b) − R(K b)M(K b) , P(K b)Q(K b) − R 2 (K b)

(61f)

b2 π 2 Q(K b) = 4

π

(61c)

β0 =

b2 π 2 P(K b) = 4

b2 π 2 ∞ R(K b) = 4 0

bπ {I0 (K b) + L0 (K b)} , 2

0

∞

0 ∞



2 π − H1 (kb)



{k J0 (kb) − K H0 (kb)}2 dk, k1 (k 2 + K 2 )

 2k π

2 − kH1 (kb) − K J1 (kb) dk, k1 (k 2 + K 2 )

(61g)

(61h)

 k 2 J0 (kb) − k K H0 (kb) − K J1 (kb) {k J0 (kb) − K H0 (kb)} k1 (k 2 + K 2 )

dk

(61i)

424

B. N. Mandal and S. De

Table 3 Lower and upper bounds for the reflection coefficient |R| for various values of the parameters K b, α and N = 1 α = 0◦ Kb

R1

α = 30◦ R2

|R| Dean [2]

R1

α = 60◦ R2

R1

α = 75◦ R2

R1

R2

0.01

0.832513

0.832525

0.832524

0.784316

0.784346

0.568783

0.578798

0.342531

0.342558

0.2

0.473397

0.474880

0.473508

0.406818

0.406835

0.232156

0.232272

0.120077

0.120227

0.4

0.331251

0.331941

0.331742

0.264610

0.264658

0.142210

0.142351

0.071853

0.071879

0.6

0.240181

0.247209

0.241027

0.193619

0.193828

0.103137

0.103504

0.049812

0.049915

where I0 (K b), I1 (K b), J0 (kb), J1 (kb) are the Bessel functions and L0 (K b), L1 (K b), H0 (kb), H1 (kb) are the Struve functions. R1 and R2 are now evaluated for various values of the wavenumber K b and the angle of incidence α and are displayed in Table 3. For α = 0◦ , |R| is known explicitly in terms of modified Bessel functions (cf. [8]). For the case of normal incidence, we put α = 0◦ in the expressions for the bounds and obtain their values numerically and compare with exact values given in Table 3. From the Table 3 it is observed that R1 and R2 in most cases coincide upto 3 to 4 decimal places. This is in contrast to the observation by Evans and Morris (1972) who however employed the deep water solutions of integral equations for normal incidence of the incident wave train in the one-term Galerkin approximation for the solution of the corresponding integral equations arising in the mathematical analysis for the case of obliquely incident wave train. Thus the average of R1 and R2 obtained here provides a very accurate estimate for the reflection coefficient in this case. The numerical estimates of |R| obtained in this manner are depicted graphically in Fig. 5 against α for different values of K b and in Fig. 6 against K b for different values of α. It is seen that the curve of |R| corresponding to α = 0◦ coincide with the curve of |R| given by [8]. Choice of more terms in the expansions of F(y)(0 < y < b) and G(y)(b < y < ∞) produces finer bounds as is evident from Table 4 in which results are tabulated for N = 2 (three-term expansion) and N = 3 (four-term expansion). As in the case of partially immersed barrier, here also, the details of analytical calculations are not given. It is observed from the Table 4 that although more accurate bounds are obtained for N = 2 and 3, the bounds obtained for N = 1 are also fairly accurate. It may be noted that the factor e−K y in the expansion of G(y)(b < y < ∞) plays an important role for obtaining fairly accurate bounds.

Use of Galerkin Technique in Some Water Wave Scattering Problems …

425

0.5 0.45 0.4

Kb=0.2

0.35

|R|

0.3 0.25 0.2

Kb=0.8

0.15 0.1

Kb=1.0 Kb=1.4 Kb=1.8

0.05 0

0

10

20

30

40

50

60

70

80

90

α (in degree) Fig. 5 |R| versus α for different values of K b 1 0.9 0.8 0.7

|R|

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0

α=0 0 α=30 0 α=60 0 α=75 0 α=85 0.5

1

Kb Fig. 6 |R| versus K b for different values of α

1.5

2

426

B. N. Mandal and S. De

Table 4 Lower and upper bounds for the reflection coefficient |R| for various values of the parameters K b, α and N = 2, 3 N=2 Ka 0.01 0.2 0.4 0.6 0.01 0.2 0.4 0.6

α = 0◦ R1 0.832515 0.473423 0.331309 0.240693 N=3 0.832518 0.473431 0.331325 0.240718

R2 0.832525 0.474878 0.331919 0.247152

α = 30◦ R1 0.784320 0.406822 0.264617 0.193629

0.832525 0.474870 0.331914 0.247110

0.784324 0.406822 0.264619 0.193641

R2 0.784333 0.406834 0.264655 0.193808

α = 60◦ R1 0.568825 0.232167 0.142211 0.103149

0.784327 0.406832 0.264649 0.193800

0.568872 0.232171 0.142217 0.103157

R2 0.578708 0.232270 0.142347 0.103500

α = 75◦ R1 0.342536 0.120080 0.071864 0.049823

R2 0.342550 0.120222 0.071875 0.049908

0.578674 0.232256 0.142336 0.103492

0.342542 0.120103 0.071867 0.049827

0.342545 0.120216 0.071869 0.049864

Fig. 7 Submerged plate

1.6 Oblique Scattering by a Thin Vertical Plate Here L = (a, b) so that L = (0, a) + (b, ∞) (See Fig. 7). The function g(y) in (36) is chosen as 1 (y − a)(b − y), a < y < b. (62) g(y) = b After substituting g(y) in (41), A is obtained as 1 A= 2 2 π K

∞ 0

k1 k 2 +K 2

where p(a, b, k) =

1 b

a

b

[kp(a, b, k) − K q(a, b, k)]2 dk (r (a, b, K ))2  (y − a)(b − y) cos(ky)dy,

(63)

Use of Galerkin Technique in Some Water Wave Scattering Problems …

1 q(a, b, k) = b r (a, b, K ) =





b

427

(y − a)(b − y) sin(ky)dy,

a



1 b

 e−K y (y − a)(b − y)dy.

b

a

Again, we choose f (y) in (44) as f (y) =

⎧ ⎨

a ,0 a−y

⎩ e−K y



< y < a,

b ,b y−b

< y < ∞.

(64)

After substituting this f (y) in the expressions (49), B is obtained as B = ∞ 0

where

[M(K a) + N (K b)]2 1 [kU (a, k) + K V (a, k) + kW (b, k, K ) − K X (b, k, K )]2 dk k1 (k 2 +K 2 ) (65)  a a dy, e−K y M(K a) = a−y 0



∞

N (K b) =

e

−2K y

b

U (a, k) =

a



a



0

V (a, k) = 0



∞

W (b, k, K ) =

a cos(ky)dy, a−y a sin(ky)dy, a−y  b cos(ky)dy, y−b

e−K y

b





∞

X (b, k, K ) =

e b

b dy, y−b

−K y

b sin(ky)dy. y−b

The integrals appearing in (63) and (65) are simple to evaluate numerically. Numerical estimation for the upper(R2 or T2 ) and lower (R1 or T1 ) bounds of the reflection and transmission coefficients |R| and |T | are obtained for different values of the various parameters. In Table 5 the lower and upper bounds of |R| for different values of various parameters are presented. From this table it is seen that the two bounds of |R| coincide upto 3 to 4 decimal places. Similar results are found

428

B. N. Mandal and S. De

Table 5 Lower and upper bounds of the reflection coefficient |R| for various values of the parameters K b, α and μ(= ab ) = 0.5 α = 15◦ Kb 0.05 0.4 0.8 1.6 2.4 3.0

R1 0.000442 0.017022 0.037450 0.045591 0.032430 0.021992

α = 45◦ R2 0.000442 0.017051 0.037481 0.045599 0.032445 0.021999

R1 0.000346 0.012437 0.027287 0.032631 0.022473 0.014773

α = 75◦ R2 0.000372 0.012473 0.027352 0.032669 0.022680 0.014796

R1 0.000118 0.004553 0.009962 0.001170 0.007881 0.005074

α = 85◦ R2 0.000119 0.004563 0.009981 0.011733 0.00794 0.005290

R1 0.000043 0.001508 0.003352 0.003920 0.002663 0.001603

R2 0.000046 0.001552 0.003377 0.003991 0.002699 0.001692

1 0.9 0.8

μ=0

0.7

μ=0.01

|R|

0.6 0.5

μ=0.05

0.4 0.3 0.2

μ=0.25

0.1 0

0

0.5

1

1.5

2

2.5

3

Kb

Fig. 8 Graph of |R| versus K b for different values of μ(= a/b)

for the two bounds for |T | which are not given here. The average of an upper and lower bound of |R|(|T |) thus produces fairly good numerical estimate for |R|(|T |). The numerical results obtained by the present method satisfy the energy identity |R|2 + |T |2 = 1. This provides a check on the correctness of the results obtained here. Because of the energy identity, we confine our attention on |R| only. In Fig. 8 2 |R| is depicted against the wave number K b(= σgb ) for different values of μ(= ab ) and for normal incidence (α = 0). From the Fig. 8 it is seen that the curve of |R| corresponding to normal incidence almost coincides with the curve of |R| in Fig. 9 of [9]. This provides another check on the correctness of the results obtained using the present method. Geometrical significance of the limiting case μ = 0 is that the plate intersects the free surface. This indicates that submerged plate behaves like

Use of Galerkin Technique in Some Water Wave Scattering Problems …

429

0.6 0

α=15

0.5

α=450

|R|

0.4

0.3

0.2

0

α=75

0.1

0

α=850

0

0.5

1

1.5

2

2.5

3

Kb

Fig. 9 Graph of |R| versus K b for different values of α and μ = 0.05

a partially immersed barrier in deep water. For each finite μ, |R| first increases to maximum as K b increases and then decreases to zero for further increases of K b. Thus for each finite value of μ, |R| → 0 as K b → ∞. In Figs. 9 and 10, |R| is plotted against the wavenumber K b for different incident angles and for μ = 0.05 and 0.1 respectively. From these figures it is seen that the curve of |R| almost coincides with the curve of |R| in Figs. 6 and 7 of [22]. Further it is also observed that for most of the cases the results displayed in Table 5 coincide upto 3 to 4 decimal places with the results in Table 1 of [22]. These provide another check on the correctness of the results obtained using the present method. Reflection coefficient |R| first increases as K b increases and then decreases for further increases of K b in the Figs. 9 and 10. Also in Fig. 11 the curve for |R| is depicted against α for different K b and for μ = 0.5. From Fig. 8 and from Table 5 it is seen that for fixed μ and K b, |R| decreases as α increases from 0◦ to 90◦ . This is obvious since the incident wave then almost grazes along the plate. Here we have used Havelock’s expansion of water wave potential for the problem of water wave scattering by submerged plate to reduced the problem to the solution of pair of integral equations involving the difference of potentials and the horizontal component of velocity across the barriers. These integral equations are solved by using single-term Galerkin technique involving a constant as basis. Numerical values of the upper and lower bounds for the reflection coefficient are seen to be very close. Their averages give actual values of reflection coefficients for all practical purposes. The present method produces numerical results which are in good agreement with the earlier results obtained by [22].

430

B. N. Mandal and S. De 0.6

0.5

|R|

0.4

0

α=15

0.3

0

α=45

0.2 0

α=75

0.1

α=850 0

0

0.5

1

1.5

2

2.5

3

Kb

Fig. 10 Graph of |R| versus K b for different values of α and μ = 0.1 0.05 0.045 0.04

Kb=3.0 Kb=2.4

0.035

Kb=1.6

|R|

0.03 0.025

Kb=0.8

0.02

Kb=0.4

0.015 0.01 0.005 0

Kb=0.05

0

10

20

30

40

α

50

60

70

Fig. 11 Graph of |R| versus α for different values of K b and μ = 0.5

80

90

Use of Galerkin Technique in Some Water Wave Scattering Problems …

431

References 1. Banerjea, S.: Scattering of water waves by a vertical wall with gaps. Aust. Math. Soc. Ser. B 37, 512–529 (1996) 2. Banerjea, S., MandaI, B.N.: Scattering of water waves a submerged thin vertical wall with a gap. Aust. Math. Soc. Ser. B 39, 318–331 (1998) 3. Chakrabarti, A.: Solution of two singular integral equations arising in water wave problems. ZAMM 69, 457–459 (1989) 4. Dalrymple, R.A., Losada, M.A., Martin, P.A.: Reflection and transmission from porous structures under oblique wave attack. J. Fluid Mech. 224, 625–644 (1991) 5. Das, B.C., De, Soumen, Mandal, B.N.: Oblique scattering by thin vertical barriers: solution by multi-term Galerkin technique using simple polynomials as basis. J. Mar. Sci. Technol. (2018). https://doi.org/10.1007/s00773-017-0520-4 6. Das, B.C., De, Soumen, and Mandal, B.N.: The problem of oblique scattering by a thin vertical submerged plate in deep water-Revisited, Springer Proceedings in Mathematics and Statistics (2018) (Accepted) 7. Das, Pulak, Dolai, D.P., Mandal, B.N.: Oblique water wave diffraction by two parallel thin barriers with gaps. J. Eng. Math. 123, 163–171 (1997) 8. Dean, W.R.: On the reflection of surface waves by a submerged plane barrier. Proc. Camb. Philos. 41, 231–238 (1945) 9. Evans, D.V.: Diffraction of water waves by a submerged vertical plate. J. Fluid Mech. 40, 433–451 (1970) 10. Evans, D.V., Morris, A.C.N.: The effect of a fixed vertical barrier on oblique incident surface waves in deep water. J. Inst. Math. Appl. 9, 198–204 (1972a) 11. Evans, D.V., Morris, C.A.N.: Complementary approximations to the solution of a problem in water waves. J. Inst. Math. Appl. 10, 1–9 (1972b) 12. Evans, D. V. and Porter, R.: Complementary methods for scattering by thin barriers, Chapter 1, Mathematical Techniques for water waves. In: Mandal, B.N. (ed.) pp. 1–43. Computational Mechanics Publications, Southampton and Boston (1997) 13. Faulkner, T.R.: The diffraction of an obliquely incident surface wave by a submerged plane barrier. ZAMP 17, 699–707 (1965) 14. Faulkner, T.R.: The diffraction of an obliquely incident surface wave by a vertical barrier of finite depth. Proc. Camb. Philos. Soc. 62, 829–838 (1966) 15. Jarvis, R.J.: The scattering of surface waves by two vertical plane barriers. Inst. Math. Appl. 7, 207–215 (1971) 16. Jarvis, R.J., Taylor, B.S.: The scattering of surface waves by a vertical plane barrier. Proc. Camb. Philos. Soc. 66, 417–422 (1969) 17. Kanoria, M., Mandal, B.N.: Oblique wave diffraction by two parallel vertical barriers with submerged gaps in water of uniform finite depth. J. Tech. Phys. 37, 187–204 (1996) 18. Kirby, T.J., Dalrymple, R.A.: Propagation of obliquely incident water waves over a trench. J. Fluid Mech. 133, 47–63 (1983) 19. Levine, H., Rodemich, E.: Scattering of surface waves on an ideal fluid. In: Mathematics and Statistics Laboratory, Technical Report No 78, Stanford University, USA (1958) 20. Losada, I.J., Losada, M.A., Roldan, A.J.: Propagation of oblique incident waves past rigid vertical thin barriers. Appl. Ocean Res. 14, 191–199 (1992) 21. MandaI, B.N., Chakrabarti, A.: Water Wave Scattering by Barriers. WIT Press, Southampton (2000) 22. Mandal, B.N., Das, P.: Oblique diffraction of surface waves by a submerged vertical plate. J. Eng. Math. 30, 459–470 (1996) 23. Mandal, B.N., Goswami, S.K.: A note on the diffraction of an obliquely incident surface wave by a partially immersed fixed vertical barrier. Appl. Sci. Res. 40, 345–353 (1983) 24. Mandal, B.N., Goswami, S.K.: A note on the scattering of surface wave obliquely incident on a submerged fixed vertical barrier. J. Phys. Soc. Jpn. 53(9), 2980–2987 (1984a)

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25. Mandal, B.N., Goswami, S.K.: The scattering of an obliquely incident surface wave by a submerged fixed vertical plate. J. Math. Phys. 25, 1780–1783 (1984b) 26. McIver, P.: Scattering of water waves by two surface-piercing vertical barriers, IMA. J. Appl. Math. 35, 339–355 (1985) 27. Morris, C.A.N.: A variational approach to an unsymmetric water wave scattering problem. J. Eng. Math. 9, 291–300 (1975) 28. Newman, J.N.: Interaction of water waves with two closely spaced vertical obstacles. J. Fluid Mech. 66, 97–106 (1974) 29. Porter, D.: The transmission of surface waves through a gap in a vertical barrier. Proc. Camb. Philos. Soc. 71, 411–421 (1972) 30. Porter, R., Evans, D.V.: Complementary approximations to waves scattering by vertical barriers. J. Fluid Mech. 294, 155–180 (1995) 31. Roy, R., Basu, U., Mandal, B.N.: Oblique water wave scattering by two unequal vertical barriers. J. Eng. Math. 97, 119–133 (2016a) 32. Roy, R., Basu, U., Mandal, B.N.: Water wave scattering by two submerged thin vertical unequal plates. Arch. Appl. Mech. 86, 1681–1692 (2016b) 33. Roy, R., Basu, U., Mandal, B.N.: Water wave scattering by a pair of thin vertical barriers with submerged gaps. J. Eng. Math. 105, 85–97 (2017) 34. Ursell, F.: The effect of a fixed barrier on surface wave in deep water. Proc. Camb. Soc. 43, 374–382 (1947) 35. Williams, W.E.: Note on the scattering of water wavesby a vertical barrier. Proc. Camb. Philos. Soc. 62, 507–509 (1966)

Dynamics of a Class of Leslie–Gower Predation Models with a Non-Differentiable Functional Response Viviana Rivera-Estay, Eduardo González-Olivares, Alejandro Rojas-Palma and Karina Vilches-Ponce

Abstract The main peculiarity of the Leslie–Gower type models is the predator growth equation is the logistic type, in which the environmental carrying capacity is proportional to the prey population size. This assumption implies the predators are specialists. Considering that the predator is generalist, the environmental carrying capacity is modified adding a positive constant. In this work, the two simple classes of Leslie–Gower type predator-prey models are analyzed, considering a non-usual functional response, called Rosenzweig or power functional responses, being its main feature that is non-differentiable over the vertical axis. Just as Volterra predatorprey model, when the Rosenzweig functional response is incorporated, the systems describing the models have distinctive properties from the original one; moreover, differences between them are established. One of the main properties proved is the existence of a wide set of parameter values for which a separatrix curve, dividing the phase plane in two complementary sectors. Trajectories with initial conditions upper this curve have the origin or a point over the vertical axis as their ω-limit. Meanwhile those trajectories with initial conditions under this curve can have a positive equilibrium point, or a limit cycle or a heteroclinic curve as their ω-limit. The marked differences between the two cases studied shows as a little change in the mathematical expressions to describe the models can produce rich dynamics. In other words, little perturbations over the functions representing predator interactions

V. Rivera-Estay · E. González-Olivares Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile e-mail: [email protected] E. González-Olivares (B) Instituto de Filosofía y Ciencias de la Complejidad, Santiago, Chile e-mail: [email protected] A. Rojas-Palma · K. Vilches-Ponce Departamento de Matemática Física y Estadística, Universidad Católica del Maule, Talca, Chile e-mail: [email protected] K. Vilches-Ponce e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_14

433

434

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have significant consequences on the behavior of the solutions, without change the general structure in the classical systems. Keywords Bifurcation · Stability · Limit cycle · Predator-prey model · Functional response AMS Subject Classifications 92D25 · 34C23 · 58F14 · 58F21

1 Introduction This paper concerns with a special class of bi-dimensional autonomous ordinary differential equation systems (ODE), describing the predation interaction. The action of the predator consuming the prey, called functional response or consumption function rate [5, 31], is represented by a less-canonical increasing functional response, which was suggested in a seminal paper by the American ecologist Rosenzweig [28]. This functional response is described by the function h (x) = q x α , with 0 < α < 1, [3, 8, 27, 30] and called Rosenzweig or αth power functional responses [28], being its main feature that is non-differentiable when x = 0 [30]. This functional response appears also proposed in the bio-economic literature [10] and denominated as compensatory power functional response; it is a particular case of a more general function called Cobb-Douglas type production function [10], described by h (x) = q x α y β , with 0 < α, β < 1 [10, 14]. The function h (x) = q x α , with 0 < α < 1, has a strong implication on the dynamics of the system such as happens in the Volterra model [31], a compartmentalized Gause-type model [12, 33, 34], which is a system non-Lipschitzian [8, 30] in the y − axis. Thus, for each point in the vertical axis pass two trajectories, i.e., the solutions do not fulfill the assumptions of the Picard-Lindelof theorem or the Existence and Uniqueness Theorem which not applies there. Another interesting property in the Gause-type model with Rosenzweig functional response is the existence of a separatrix curve  determined by the stable manifold of the non-hyperbolic equilibrium (0, 0). Trajectories with initial conditions over the curve  attain the vertical axis on finite time. This function is therefore unsuitable for modeling a interaction where the predator is approaching satiety [23]. It also has some other problems [23], since when it is incorporated into the basic Volterra model [31], after Vito Volterra Italian mathematician (1860–1940), these do not satisfy the conditions of the Kolmogorov Theorem (see [20]). In that model could produce a situation where there is neither non-zero stable populations nor stable oscillations but where one or both species become extinct [8, 16, 23, 30].

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A special case is given the function by h (x) = q x 2 , considering the value α = 21 , proposed by the Russian ecologist Gause [13], called root square functional response [21, 32] and used to describe the anti-predator behavior (APB) named as prey herd behavior [6, 33]. It is said that a prey species exhibits herd behavior when the individuals have collective social conduct to avoid predation, realizing each a typical reaction equal to that effectuated by the majority of the other members of the gaggle [6, 21]. This APB has received increasing attention of the modellers after to the work of Ajraldi et al. [1], analyzing its consequences in Gause-type predator-prey models [34], and it has compared with other APB called group defence formation [32]. On the other hand, many types of modeling have been proposed for this important interaction, from the seminal Lotka–Volterra model [31], a model obeying massaction principle [7]. One of this type of model was presented by the English ecologist Leslie [18], which was analyzed partially in [19] incorporating explicitly the dependence of this rate on the quotient between the population sizes of prey and predators [25]. Leslie assumed that the equation representing predators evolution is a function of logistic-type growth, in which the conventional environmental carrying capacity of predators K y is proportional to the prey population size x = x (t), that is K y = K (x) = nx [20, 31]. An interesting case of this type model is the so called May–Holling–Tanner model [29], in which the predator consumption rate is hyperbolic, a Holling type II functional response [31]. The model proposed by Leslie does not fit to the Lotka–Volterra scheme and it is not defined in x = 0. For this fact, it has been strongly criticized, since it presents anomalies in their predictions, because it permits that even in very low prey population density, when the consumption rate per predator is almost zero, predator population might increase, yet if the predator/prey ratio is very small [31]. It is well-known that for the Leslie–Gower model [15] there exists a wide set of parameter values for which the unique positive equilibrium point is globally asymptotically stable [9, 11]; this property is proved constructing a suitable Lyapunov function [3, 17]. Nonetheless, in the case of severe scarcity, some predator species can switch over to other available food, i.e., the predators are generalists. This situation can be modeled by adding a positive constant c in the carrying capacity K y (x), being described now by K (x) = nx + c, being a function of the prey population size and the other available food [20, 31]. If x = 0, then K (0) = c, concluding that the predator is generalist since it searches an alternative food. Therefore, we will analyze the two different cases of the Leslie–Gower scheme or modified Leslie–Gower model [2, 4, 17], making an adequate comparison between the two models and also with the original Leslie–Gower model. Conditions on the parameter space are established, for which there exists at least a limit cycle for c = 0, i.e., there exist oscillations of the population sizes. This property distinguishes the class of models here studied with the original model proposed by Leslie [18, 19], in which do not exist periodic trajectories.

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Moreover, it also exists a remarkable difference between the models considering c = 0 or c > 0, as we will show in the present work. The rest of the paper is organized as follow: In the next section, the modified Leslie–Gower model is presented; in Sect. 3, the main properties of the model are proved; in Sect. 4 some numerical simulations are shown to reinforced the analytical results, and in the last section we discuss the obtained results, giving some ecological meanings and interpretations.

2 The Model The class of modified Leslie–Gower models to be analyzed is described by following bi-dimensional autonomous ordinary differential equations system:  dx X μ (x, y) :

dt dy dt

 =r 1−  = s 1−



x − qxα y y  y nx+c x K

(1)

where x = x (t) and y = y (t), express the population sizes of prey and predators, respectively for t ≥ 0, (measured as number of individuals, biomass or density per unit area or volume). The parameters are all positive, that is, μ = (r, K , q, s, n, α, c) ∈ R5+ × ]0, 1[ × (R+ ∪ {0}) and they have the following ecological meanings [20, 31] (Table 1). In system (1) is assumed that the carrying capacity of the population of predators is variable and dependent on the quantity of prey available at each time t ≥ 0 [20]. System (1) is defined in   0 = (x, y) ∈ R2 /0 < x, 0 ≤ y , when c = 0, or

   = (x, y) ∈ R2 /0 ≤ x, 0 ≤ y ,

when c > 0. Table 1 Parameter meanings in system (1) Parameters Meanings r K q s n c α

Intrinsic prey growth rate Prey environmental carrying capacity Consuming rate per capita of the predators Intrinsic predator growth rate Measure of the quality of food Amount of alternative food available for the generalist predators Shape parameter that represents a kind of aggregation efficiency [32]

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The equilibrium points of system (1) or singularities of vector field X μ (x, y) are (K ,0), (0, c) when c > 0, and (xe , ye ) over the null isoclines y = nx + c and y = qr 1 − Kx x 1−α . As the functional response h (x) = q x α , with 0 < α < 1, and system (1) are nondifferentiable in x = 0, it is required a non-usual analysis to established all properties of proposed model. To simplify the calculation, we make a change of coordinates and a time rescaling, + 2 described by the diffeomorphism  : R2 × R+ 0 → R × R0 , such that  u+  ku, nkv, r

c Kn

 τ

= (x, y, t),

⎞ K 0 0 The Jacobian matrix of  is D(u, v, τ ) = ⎝ 0 n K 0 ⎠. 1 0 ur r and 2 det D(u, v, τ ) = n Kr u > 0. i.e, ϕ is a diffeomorphism preserving the time orientation. Furthermore,   α c qk n s , , α, = (Q, S, α, C) , −1 (r, k, q, s, n, α, c) = r r Kn ⎛

It has the following result: Proposition 1 Topologically equivalent systems The vector field X μ (x, y) or system (1) is topologically equivalent to the system  du Yη (u, v) :

dτ dv dτ

= ((1 − u) u − Qu α v) (u + C) = S (u + C − v) v,

with η = (Q, S, α, C) ∈ R2 × ]0, 1[ × (R+ ∪ {0}). Proof Let x = K u and y = n K v; replacing it has  Uμ :

K du = r (1 − u) K u − q (K u)α n K v dt   v n K v. n K dv = s 1 − n Kn Ku+c dt

Simplifying and factoring it obtains  Uμ :

du dt dv dt

= r (1 − u) u − = s(1 −

v u+ Kcn

)v.

q K αn α u v r



(2)

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By means of a time rescaling given by τ =

r u+ Kcn

t, and applying the Chain Rule

we have = = , with z = x or z = y. Replace and simplifying it is obtained dz dt

dz dτ dτ dt

dz r dτ u+ Kcn

 Vμ :

du dτ dv dτ

=

(1 − u) u −

=

s r

(u +

c Kn

Defining the new parameter by Q = (2) is obtained.

q K αn α u v r

 u+

c Kn



− v)v. q K αn , r

C=

c Kn

and S = rs , the new system 

System (2) or vector field Yη (u, v) is a topologically equivalent to a continuous extension of system (1); it is defined in the first quadrant i.e., in:   ¯ = (u, v) ∈ R2 /0 ≤ u, 0 ≤ v ,  Remark 2 We note that system (2) is of Kolmogorov type [12], when C = 0, i.e., the axis are invariant set. That is not true when C > 0. The equilibrium points of system (2) when C > 0 are (0, 0), (1, 0), (0, C) and (H, H + C) determined by the intersection of the isoclines v=

1 (1 − u)u 1−α and v = u + C. Q

When C = 0, the equilibrium points of system (2) are (0, 0), (1, 0) and (H, H ), since (0, C) collapse with (0, 0); the abscissas of that equilibrium point, denoted by H , satisfies the transcendental equation: 1 − u − Qu α = 0

(3)

We note that Eq. (3) has a unique positive real root in the open interval ]0, 1[. Graphically, the decreasing straight line g1 (u) = 1 − u and the increasing power function g2 (u) = Qu α , with 0 < α < 1, have a unique intersection in the interval ]0, 1[. When C > 0, it has Qu1 α (1 − u)u − (u + C) = 0 obtaining the transcendental equation: (4) (1 − u) u − (u + C) Qu α = 0 and the increasing power The graphic of the rational function g3 (u) = (1−u)u u+C function g2 (u) = Qu α , with 0 < α < 1, can have one or two intersections, when u lies in the interval ]0, 1[ or none. Thus, different case must be analyzed according to if G satisfies the Eq. (4) (see Figs. 1 and 2).

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1 0.9 0.8 0.7 g (C=0.2) 3

v

0.6

g (α=0.2) 2

0.5

g (α=0.3)

0.4

g (α=0.6)

2 2

g (α=0.9) 2

0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u

Fig. 1 Different cases are obtained for the intersection among the graphics of the curve g3 (u) = (1−u)u α u+C and the curve g2 (u) = Qu for distinct values of α = 0.2, α = 0.3, α = 0.6 and α = 0.9, when C = 0.2 and Q = 1. Then, there exist two positive equilibrium points, one or none, when u lies in the interval ]0, 1[ 1 0.9 g (C=0.1)

0.8

3

g (Q=1.5) 2

0.7

g2 (Q=1) g (Q=0.5)

v

0.6

2

0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u α Fig. 2 Intersection among the graphics of the curve g3 (u) = (1−u)u u+C and the curve g2 (u) = Qu for distinct values of Q = 0.5, Q = 1 and Q = 1.5, when C = 0.1 and α = 0.4. Thus, there also exist two positive equilibrium points, one or none, when u lies in the interval ]0, 1[

3 Main Results For system (2) or vector field Yη (u, v), we have the following properties: Lemma 3 Existence of an invariant region   ¯ : 0 ≤ u ≤ 1, v ≥ 0 is a positively invariant region. The set ¯ = (u, v) ∈ 

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Proof By setting u = 1, we obtain  du Yη (1, v) :

dτ dv dτ

= −Qv = S(1 − v)v,

Hence, the orbits with initial condition outside of the set will get on the set and the orbits inside of the set do not get out.  Remark 4 In system (1) the invariant region is given by  = {(x, y) ∈  : 0 ≤ x ≤ K , y ≥ 0} .

3.1 Case 1: Kolmogorov Type System First we will consider the case C = 0. We recall that in system (2) the point (0, C) collapses with (0, 0). Lemma 5 Boundedness of solutions The solutions are uniformly bounded. Proof We consider W (τ ) = u + 1S v, W (τ ) +

dW (τ ) 1 = u 2 − u 3 − Qu α+1 v + uv − v2 + u + v dτ S 1 ≤ u 2 + uv − v2 + u + v S 1 ≤ 12 + v − v2 + 1 + v s   2   1 + 1/S 1 + 1/S 2 ≤ 2+ v− − 2 2 2   1 + 1/S ≤ 2+ v− = R, 2

then dW (τ ) ≤ eτ R dτ (W (τ )eτ ) ≤ eτ R

eτ W (τ ) + eτ

integrating it has,

W (τ )eτ ≤ Reτ + L.

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When τ = 0 W (0) ≤ R + L W (0) − R ≤ L There is n ∈ N such that C ≤ n (W (0) − R), then W (τ )eτ ≤ eτ R + n (W (0) − R) W (τ ) ≤ R + e−τ n (W (0) − R) , Clearly, when τ → ∞ then W (τ ) ≤ R.



The Jacobian matrix of system (2) for C = 0 is  DYη (u, v) =

3.1.1

 2u − 3u 2 − (α + 1) Qu α v −Qu α+1 . Sv S (u − 2v)

Nature of Equilibria Over the Axis

Lemma 6 Nature of equilibrium free of predators The equilibrium (1, 0) is a saddle equilibrium point for all parameter values. Proof The Jacobian matrix DYη (u, v) evaluated in (1, 0) is:  DYη (1, 0) =

−1 −Q 0 S

 ,

whose eigenvalues are λ1 = −1 < 0 , λ2 = S > 0. The result follows from the linearization Theorem of Hartman and Grobman [9]  since detDYη (1, 0) = −S < 0. Theorem 7 Existence of a a hyperbolic and a parabolic sector The non-hyperbolic equilibrium point (0, 0) has a hyperbolic and a parabolic ¯ sector [24] determined by the stable manifold W s (0, 0) = . Proof Clearly, system (1) is not differentiable at point (0, 0), but the system (2), topologically equivalent to system (1), is differentiable at the origin. Nonetheless, the matrix of the linearization of vector field Yη (u, v) in the the origin is the null matrix. Then, to study this anomaly we will use the horizontal blowing-up method [11, 30], defining two new variables r and p and making the change of variables given by u = r , v = r s. dv dr dr dv dr ds = dτ , dτ = s dτ + r dτ . Thus, dp = r1 dτ − rp dτ Then, du dτ dτ

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Replacing and factoring we have  du dτ dv dτ

Yη (r, p) :

= (1 − r − Qr α p) r 2 = S(1 − p)r 2 p,

Therefore,  dr

= (1 − r − Qr α p) r 2

dτ dp dτ

Z η (r, p) :

= S (1 − p) r p − (1 − r − Qr α p) r p,

Rescaling the time by T = r τ , we have  Z¯ η (r, p) :

dr dT dp dT

dr dτ

=

dr dT dT dτ



dp dτ

=

dp dT dT dτ

and becomes

= (1 − r − Qr α p) r = (S (1 − p) − (1 − r − Qr α p)) p,

 The equilibrium points of Z¯ η are (0, 0) and 0, The Jacobian matrix of Z¯ η is D Z¯ η (r, p) =

and

S−1 S

 .

 α Qpr α α − 2r −Qr α+1 1 − Qpr  −α−1 . p Qpr α + 1 S + r − 2Sp + 2Qpr α − 1

Not defined for r = 0. dp dr Let us r = X α , p = Y . So, dT = α1 X α −1 ddTX and dT = The new system after of to replace and to factor is 1

1

dY dT

.

⎧

 ⎨ d X = α 1 − X α1 − Q X Y X dT

 Z˜ η (X, Y ) : dY 1 ⎩ α − QXY = S − Y − 1 − X Y. (1 ) dT  The equilibrium points of Z˜ η are (0, 0) and 0, Z˜ η is  D Z˜ η (X, Y ) =

1



and the Jacobian matrix of

1

−Q X 2 α

X α −1 + QY α

S + X α − 2SY + 2Q X Y − 1

α − X α α − X α − 2QX Y α Y α

S−1 S

1

 Evaluating, in (0, 0) and 0,

S−1 S



1

we obtain

D Z˜ η (0, 0) =



 α 0 , 0 S−1

 .

Dynamics of a Class of Leslie–Gower Predation Models …

and

   S−1 D Z˜ η 0, = S

443

 α 0   . S−1 Q S−1 − (S − 1) αS S

Thus, assuming S > 1 in Z˜ we have (a) the point (0,  0) is a hyperbolic repeller, and is a hyperbolic saddle. (b) the point 0, S−1 S Assuming S < 1 in Z˜ we have (c) the point (0, 0) is a hyperbolic saddle, and is a hyperbolic repeller, but it lies in the fourth quadrant. (d) the point 0, S−1 S Then, in Z¯ η we have a non-hyperbolic  and saddle.  repeller in Yη it has, if S > 1, the saddle By the blowing-down, (0, 0) and 0, S−1 S   ¯ determined by the stable manidetermines the separatrix curve , point 0, S−1 S fold W s (0, 0).  Remark 8 By the diffeomorphism ϕ, in system (1) there is a separatrix curve  that born in the neighborhood of the point (0, 0), although system (1) is not defined there, dividing the trajectories in the phase plane. We define the set

= {(x, y) ∈ /0 ≤ x, 0 ≤ y ≤ y , such that (x, y ) ∈ } , and the invariant set  is divided in the set and S =  − . In the following figure we show the sectors determined by the curve separatrix  in system (1). Remark 9 Let us W u (1, 0), the unstable manifold of the hyperbolic saddle point ¯ = W s (0, 0) , the stable manifold of the non-hyperbolic saddle point (1, 0) and  (0, 0) , trajectories in system (2). The relative position of both manifold determines ¯ = φ (See Fig. 3). a heteroclinic curve, when W u (1, 0) ∩  Notice that the unique positive equilibrium (H, H ) is in the region   ¯ = (u, v) ∈ /0 ¯ ≤ u, 0 ≤ v ≤ v , such that (u, v ) ∈  ¯ .

Its nature depends of the relation among vu and vs (Fig. 4). Remark 10 In system there exists a separatrix curve of the point (0, 0), which is not an equilibrium point in this system.

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Separatrix curve Σ

Line x = K

S y

Λ

0 0

x

K

Fig. 3 This figure schematically shows the sets and S =  − , determined by the separatrix curve , associated to the point (0, 0) in the phase plane of the system (1)

v

v

vs

vu

W s(0,0)

W u (1,0)

u

W (1,0)

vu

s

v

W (0,0)

s

(ue ,ve )

(u ,v ) e

(0,0)

e

u*

(1,0)

u

(0,0)

u*

(1,0)

u

¯ = W s (0, 0) and the unstable manifold W u (1, 0), Fig. 4 Relative position of the stable manifold  ¯ in the region of system (2)

3.1.2

Nature of the Positive Equilibrium Point

For study of the equilibrium (H, H ), let us make a change of parameters in the system (2), given by: 1− H , Q= Hα

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with H ∈ ]0, 1[. Getting the system  du Zν =

dτ dv dτ

 = (1 − u) u −

1−H α u v Hα



u,

(5)

= S (u − v) v,

¯ = R2 × ]0, 1[. The vector field X ν or system (5) has an with ν = (S, H, α) ∈  ¯ equilibrium (H, H ) and it is topologically equivalent to the system (2) in . Theorem 11 Nature of the positive equilibrium Let   ¯ K = (u, v) ∈ /0 ¯ ≤ u ≤ 1, 0 ≤ v ≤ v , such that (u, v ) ∈  ¯ ⊂ . ¯

¯ 1 . Then, the positive equilibrium (a) Assuming vu < vs , for (u ∗ , vu ) and (u ∗ , vu ) ∈ (H, H ) is (i) an attractor, if and only if, S > 1 − α − 2H + α H . (ii) a repeller, if and only if, S < 1 − α − 2H + α H . (iii) a weak (fine) focus, if and only if, S = 1 − α − 2H + α H . ¯ K . Then, the positive equilibrium (b) Assuming vu > vs , for (u ∗ , vu ) and (u ∗ , vu ) ∈ (H, H ) is a repeller (focus or node), and (0, 0) is an almost global stable point [26]. Proof The Jacobian matrix evaluated the equilibrium (H, H ) is  D Z ν (H, H ) =

2H − 3H 2 − (α + 1) (1 − H ) H − (1 − H ) H SH −S H

 .

Then, det D Z ν (H, H ) = S H 2 (α + H (1 − α)) > 0, and tr D Z ν (H, H ) = H (1 − α − 2H + α H − S) . Hence, the nature of the equilibrium (H, H ), depends on the sign of trace, which at once depends on the sign of the factor ρ(ν) = 1 − α − 2H + α H − S. (a) Assuming vu < vs . (i) If S > 1 − α − 2H + α H , the equilibrium (H, H ) is an attractor of system ¯ (4) for all trajectories with initial conditions in .

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(ii) If S < 1 − α − 2H + α H , the equilibrium (H, H ) is a repeller. ¯ K is a compact region; then the Poincaré–Bendixson Theorem Clearly, ¯ K and there exist at least a limit cycle in system (5). applies in ¯ have a this limit cycle Therefore, all trajectories with initial conditions in as their ω-limit. (iii) If S = 1 − α − 2H + α H , the equilibrium (H, H ) is a weak focus. (b) Assuming vu > vs , (H, H ) is a repeller (focus or node). By the Existence and Uniqueness Theorem, the trajectories born in that vicinities of (0, 0) not cross the unstable manifold W u (1, 0) of equilibrium (1, 0), and they have the point (0, 0) as their ω − limit. From the cases above, the result is proved.



Remark 12 The set

K = {(x, y) ∈ /0 ≤ x ≤ K , 0 ≤ y ≤ y , such that (x, y ) ∈ } , is a compact region in system (1). Corollary 13 Condition of transversality If S = 1 − α − 2H + α H , there exists at least limit cycle generated by Hopf bifurcation surrounding the positive equilibrium point (H, H ). Proof Deriving tr D X ν (H, H ) = H (1 − α − 2H + α H − S), respect to the parameter S it has, ∂ T r D X ν (H, H ) = −H =, ∂S and the results follows applying the transversality Theorem [24].



Conjecture 14 Uniqueness of a limit cycle The limit cycle generated by Hopf bifurcation surrounding the positive equilibrium point (H, H ) is unique. In a ongoing work, we will prove the last result using method of Lyapunov quantities to determine the weakness of the equilibrium point (H, H ).

3.2 Case 2: Non-Kolmogorov Type System Now, we will analyze the case C > 0.

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As we say above, in this case, system (2) has four equilibrium points. The Jacobian matrix of system (2) for C > 0 is  DYη (u, v) =

 −Qu α+1 2u − 3u 2 − (α + 1) Qu α v . Sv S (u + C − 2v)

Lemma 15 Boundedness of solutions The solutions of system (2) are uniformly bounded. Proof Similar to the case C = 0.

3.2.1



Nature of Equilibria Over the Axis

Lemma 16 Nature of (1, 0) The equilibrium (1, 0) is a saddle for all parameter values. Proof The Jacobian matrix DYη (u, v) evaluated in (1, 0) is:  DYη (1, 0) =

−1 −Q 0 SC

 ,

whose eigenvalues are λ1 = −1 < 0 , λ2 = SC > 0. The statement follows since det DYη (1, 0) = −SC < 0.



Lemma 17 Nature of (0, C) The equilibrium point (0, C) is a nonhyperbolic attractor. Proof The Jacobian matrix DYη (u, v) evaluated in (0, C) is:  DYη (0, C) =

0 0 SC −SC

 .

As the Jacobian matrix has an eigenvalue equal to zero, therefore we apply the Central Manifold Theorem. Let us u = h (v) = a2 v2 + a3 v3 + a4 v4 + · · · + an vn . Then, its derivative is du = dv

du dτ dv dτ

.

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Separatrix curve

v

(0,C)

Λ

(0,0)

u

(1,0)

¯ is determined by the stable manifold  ¯ = W s (0, C) in the phase plane Fig. 5 The subregión of the system (2) v

vu

s

vs

W s(0,C)

u

W (1,0)

v

v

v

W s(0,C)

u

(0,C)

(0,0)

W u(1,0) (G,G+C)

(0,C)

(G,G+C)

u*

(1,0)

(0,0)

u*

(1,0)

Fig. 6 Relative position of the stable manifold of (0, C) and the unstable manifold of (1.0)

After an algebraic work we obtain a2 = a3 = a4 = a5 · · · = 0. This means that the tangent curve in v = C is approximately the axis− v. Then we have that the axis v is invariant. Therefore, we have that (0, C) is a non-hyperbolic attractor point.  Remark 18 We define the set   ¯ = (u, v) ∈ /0 ¯ ≤ u , 0 ≤ v ≤ v , such that (u, v ) ∈  ¯ ,

¯ is divided in the set ¯ and S =  ¯ − . ¯ and the phase plane  (See Fig. 3). In the following figure we show the subregions determined by the curve separatrix ¯ in system (2) (Fig. 5). 

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Remark 19 In system (2), we take account newly W u (1, 0), the unstable manifold ¯ = W s (0, C), the stable of the hyperbolic saddle point (1, 0), but now we consider  manifold of the non-hyperbolic attractor point (0, C). Then, the relative position ¯ = φ (See of both manifold determines a heteroclinic curve, when W u (1, 0) ∩  Fig. 6). We note that the unique positive equilibrium (G, G + C) is in the region   ¯ = (u, v) ∈ /0 ¯ ≤ u , 0 ≤ v ≤ v , such that (u, v ) ∈  ¯ ,

and its nature depends of the relation among vu and vs (see Fig. 6). Lemma 20 Nature of the origin The equilibrium point (0, 0) is a nonhyperbolic repeller Proof The Jacobian matrix evaluated in the equilibrium (0, 0) is  DYη (0, 0) =

0 0 0 SC



Newly, we will use the Central Manifold Theorem in similar form to above Lemma 

3.2.2

Nature of the Positive Equilibria

Here we assume the existence of a unique positive equilibrium (G, G + C). It has a similar nature of the point (H, H ) when C = 0, according a la relative position of ¯ = W s (0, C) and the the ordinates vu and vs of the points over the stable manifold  u unstable manifold W (1, 0), respectively. Theorem 21 Nature of a unique positive equilibrium Let   ¯ 1 = (u, v) ∈ /0 ¯ ≤ u ≤ 1, 0 ≤ v ≤ v , such that (u, v ) ∈  ¯ ⊂ . ¯

¯ 1 . Then, the positive equilibrium (a) Assuming vu < vs , for (u ∗ , vu ) and (u ∗ , vu ) ∈ (G, G + C) is (i) an attractor, if and only if, S > G(2G−α+Gα+1) . C+G G(2G−α+Gα+1) . (ii) a repeller, if and only if, S < C+G . (iii) a weak (fine) focus, if and only if, S = G(2G−α+Gα+1) C+G

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¯ K . Then, the positive equilibrium (b) Assuming vu > vs , for (u ∗ , vu ) and (u ∗ , vu ) ∈ (G, G + C) is a repeller (focus or node), and (0, 0) is an almost global stable point [22, 26]. Proof The Jacobian matrix is  DYη (G, G + C) =

2G − 3G 2 − (α + 1) QG α (G + C) −QG α+1 S (G + C) − (C + G) S

 ,

Then, det DYη (G, G + C) = S (C + G) T, where T = −2G + 3G 2 + G α Q (C + 2G + Cα + Gα) , and tr DYη (G, G + C) = 2G − 3G 2 − (α + 1) QG α (G + C) − (C + G) S. From Eq. (4) it has Q = depends on the sign of

(1−G)G ; (G+C)G α

then, replacing it has det DYη (G, G + C)

T = (1 − α) G 2 + (2C + (1 − C) α) G − C (1 − α) , and tr DYη (G, G + C) = 2G − 3G 2 − (α + 1) (1 − G) G − (C + G) S = G (−2G − α + Gα + 1) − (C + G) Assuming T > 0, it has the nature of the equilibrium depends on the sign of trDYη (G, G + C). Then, the thesis is held according to the different relations existent for the parameter S.  Remark 22 Other dynamical situations for the non-Kolmogorov case will be analyzed in a prospective work; as an example, when two positive equilibrium coexist, one of them is always a saddle point. Only we will present some graphical simulations in the following section, when there exists a unique positive equilibrium point.

4 Numerical Simulations Some simulations are shown to comprobe the diverse dynamics of the class of Leslie–Gower type models studied. Case 1. Kolmogorov type system Here we present simulations for the case C = 0.

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1. The origin is an almost global attractor [22, 26] (see Fig. 7). 2. Existence of an elliptic sector [24] (see Fig. 8)

v

1

0.5 (ue,ve)

0

(0,0)

0

(1,0)

0.5

1

u

Fig. 7 For C = 0, α = 0.15, Q = 0.75 and S = 0.125, the positive equilibrium point (u e , ve ) is a repeller focus and (0, 0) is a almost global attractor [22, 26]

v

1

0.5

0

(1,0)

(0,0)

0

0.5

1

u

Fig. 8 For C = 0, α = 0.15, Q = 1.9 an S = 0.75, there exists an elliptic sector, generated by the stable and unstable manifolds of the non-hyperbolic equilibrium (0, 0)

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3. Existence of a stable limit cycle (see Fig. 9) Case 2. Non-Kolmogorov type system Now we show simulations for the case C > 0. 4. Existence of bistability phenomenon (see Fig. 10)

v

1

0.5

Stable limit cycle (ue,ve)

0

(1,0)

(0,0)

0

0.5

1

u

Fig. 9 For C = 0, α = 0.15, Q = 0.75 an S = 0.175, there exists an stable limit cycle surrounding the positive equilibrium point (u e , ve )

v

1

0.5

(0,C)

0

(G,G+C)

(1,0)

(0,0)

0

0.5

u

1

Fig. 10 For C = 0.1995, α = 0.6005, Q = 1.2 an S = 0.75, the point (G, G + C) is an attractor node and (0, C) is also attractor

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5. Existence of a like-cusp equilibrium point (see Fig. 11) 6. Positive equilibrium point is a almost global attractor [22, 26] (see Fig. 12)

v

1

0.5

(G,G+C) (0,C)

0

(1,0)

(0,0)

0

0.5

1

u

Fig. 11 For C = 0.1995, α = 0.6, Q = 1.2 an S = 0.75, the point (G, G + C) is a like-cusp point and (0, C) is an almost global attractor [22, 26]

v

1

0,5

(G,G+C)

0

(0,C) (0,0)

0

(1,0)

0.5

1

u

Fig. 12 For C = 0.001, α = 0.425, Q = 1.25 an S = 0.3, the point (G, G + C) is a focus almost global attractor [22, 26]

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7. Existence of heteroclinic curve (see Fig. 13)

v

1

0,5

Heteroclinic curve

0

(0,C)

(G,G+C) (1,0)

(0,0)

0

0.5

1

u

Fig. 13 For C = 0.00001, α = 0.19, Q = 1.25 an S = 0.2725, the point (G, G + C) is a repeller focus and there exists an heteroclinic curve

5 Conclusions In this work we have studied the dynamics of two modified Leslie–Gower type predation models [18, 20, 31], considering a non-differentiable functional response proposed by Rosenzweig [28] described by h (x) = q x α , with 0 < α < 1; furthermore, we had analyzed two cases assuming the predators are specialist or generalist, i.e., c = 0 or c > 0 in the environmental carrying capacity of predators. According to our knowledge, this is the first work in that the Rosenzweig functional response is assumed in a Leslie–Gower model. In order to simplify the calculations, a reparameterization and a time rescaling were made, obtaining the topologically equivalent system (2). When c = 0, it was established that the system (2) can have up to 3 singularities, two of them are hyperbolic and the origin is non-hyperbolic, being the collapses of the points (0, c) and (0, 0). The nature of this equilibrium was difficult to analyze since it has a complicated behavior; using the directional blowing-up method to determine the nature of this ¯ determined by the stable point, we verify that there exists at least a separatrix curve  manifold of the point (0, 0) (by the diffeomorphism, a separatrix curve  exists in the original system (1)). Then, a slightest deviation in the initial population sizes, respect to the curve , it can signify the coexistence of populations or the extinction of both. This situation can occur in many marine fisheries due to overfishing, often resulting in extreme stock depletion, if not actual extinction [10].

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For this case, we also show that the unique positive equilibrium point (H, H ) is locally asymptotically stable for certain parameter conditions; for other conditions in the parameter space, there is at least one limit cycle, a fact that establishes a clear difference with the dynamic of the Leslie–Gower model proposed in 1948 by Leslie [18]. ¯ of the trajectories in the Although in the system there exists a separatrix curve  phase plane, there are no trajectories that attain at the vertical axis on finite time [3], as it happens with the Volterra model [31] (a Gause-type model) using the same functional response [3, 30]. We reiterate that the particular case α = 21 has been used to represent the called prey herd behavior; we estimate that any value α ∈ ]0, 1[ can be employed to describe this behavior, since the properties here shown are valid for that special case. When c > 0, it is assumed that predators have an alternative food, that is, the variable environmental carrying capacity, depending on the size of the prey population is expressed by K y = K (x) = nx + c, where c represents the available alternative food. In this case, the system (2) has four equilibrium points, being two of them nonhyperbolic; (0, 0) is a repeller and (0, C) is an attractor, which implies the prey population can go to extinction meanwhile the predator population persists. It also exists constraint in the parameter for which the prey population attains the maximum size (environmental carrying capacity) and the predator population goes to extinction. Moreover, both populations can remain in the neighborhood of the a coexistence point (local stable positive equilibrium point) or they persist having oscillatory population sizes (existence of a limit cycle) around that point. According to the numerical simulations already carried out, it can verify that this new modified Leslie–Gower model has a strong difference with the model in which c = 0, since there exist trajectories that on finite time reach the vertical axis; furthermore, possibly the system is non-Lipschitzian on that axis, i.e., there exist two solutions passing for each point of the vertical with ordinate v > C. This last property will be established in a prospective work. We emphasize that a small change in the mathematical expression for the linear functional response, it produces an interesting dynamics in the new model, meaningfully distinct respect to the original Leslie–Gower model. Moreover, by considering specialist or generalist predators (modifying the variable environmental carrying capacity of predators) are obtained two nearly models, but with significantly different mathematical properties. We warn that it must have great care with the use of a more complicated functional response, since non-usual mathematical properties can emerge, as the nonuniqueness of the solutions; these results could have not an adequate ecological interpretations in specific interactions of the real world. However, complex ecological relations on the predation interactions implies the use of more sophisticated mathematical tools. Acknowledgements This work has been sponsorship by Mathematical Modeling and Pattern Recognition (GMMRP), Chile (www.biomatematica.cl). The second author was partially financed

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by the DIEA-PUCV 124.730/2012 project. The fourth author was supported by Conicyt PAIAcademia 79150021 project.

References 1. Ajraldi, V., Pittavino, M., Venturino, E.: Modeling herd behavior in population systems. Nonlinear Anal.: R. World Appl. 12, 2319–2338 (2011) 2. Arancibia-Ibarra, C., González-Olivares, E.: A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey. In: Mondaini, R. (ed.) BIOMAT 2010 International Symposium on Mathematical and Computational Biology, pp. 146–162. World Scientific Co. Pte. Ltd., Singapore 3. Ardito, A., Ricciardi, P.: Lyapunov functions for a generalized Gause-type model. J. Math. Biol. 33, 816–828 (1995) 4. Aziz-Alaoui, M.A., Daher Okiye, M.: Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math. Lett. 16, 1069– 1075 (2003) 5. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific Publishing Co. Pte. Ltd., Singapore (1998) 6. Bera, S.P., Maiti, A., Samanta, G.P.: Modelling herd behavior of prey: analysis of a preypredator model. World J. Model. Simul. 11, 3–14 (2015) 7. Berryman, A.A., Gutierrez, A.P., Arditi, R.: Credible, parsimonious and useful predator-prey models - a reply to Abrams, Gleeson, and Sarnelle. Ecology 76, 1980–1985 (1995) 8. Bravo, J.L., Fernández, M., Gámez, M., Granados, B., Tineo, A.: Existence of a polycycle in non-Lipschitz Gause-type predator-prey models. J. Math. Anal. Appl. 373, 512–520 (2011) 9. Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, 2nd edn. Springer, Berlin (2006) 10. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York (1990) 11. Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Springer, Berlin (2006) 12. Freedman, H.I.: Deterministic Mathematical Model in Population Ecology. Marcel Dekker, New York (1980) 13. Gause, G.F.: The Struggle for Existence. Dover, New York (1934) 14. González-Olivares, E., Sáez, E., Stange, E., Szantó, I.: Topological description of a nondifferentiable bio-economics model. Rocky Mt. J. Math. 35(4), 1133–1155 (2005) 15. González-Olivares, E., Mena-Lorca, J., Rojas-Palma, A., Flores, J.D.: Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey. Appl. Math. Model. 35, 366–381 (2011) 16. Hesaaraki, M., Moghadas, S.M.: Existence of limit cycles for predator-prey systems with a class of functional responses. Ecol. Model. 142, 1–9 (2001) 17. Korobeinikov, A.: A Lyapunov function for Leslie-Gower predator-prey models. Appl. Math. Lett. 14, 697–699 (2001) 18. Leslie, P.H.: Some further notes on the use of matrices in population mathematics. Biometrica 35, 213–245 (1948) 19. Leslie, P.H., Gower, J.C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometricka 47, 219–234 (1960) 20. May, R.M.: Stability and Complexity in Model Ecosystems, 2nd edn. Princeton University Press, Princeton (2001) 21. Melchionda, D., Pastacaldi, E., Perri, C., Banerjee, M., Venturino, E.: Social behavior-induced multistability in minimal competitive ecosystems. J. Theor. Biol. 439, 24–38 (2018) 22. Monzón, P.: Almost global attraction in planar systems. Syst. Control Lett. 54, 753–758 (2005)

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23. Myerscough, M.R., Darwen, M.J., Hogarth, W.L.: Stability, persistence and structural stability in a classical predator-prey model. Ecol. Model. 89, 31–42 (1996) 24. Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, Berlin (2001) 25. Ramos-Jiliberto, R., González-Olivares, E.: Regulaci de la tasa intrínseca de crecimiento poblacional de los depredadores: modificación auna clase de modelos de depredación. Rev. Chil. Hist. Nat. 69, 271–280 (1996). (in spanish) 26. Rantzer, A.: A dual to Lyapunov’s stability theorem. Syst. Control Lett. 42(3), 161–168 (2001) 27. Rivera-Estay, V.: Un modelo de Leslie-Gower con respuesta funcional no diferenciable (A Leslie-Gower type model with non-differentiable functional respose). Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaí so, Licenciate final work (2013). in spanish 28. Rosenzweig, M.L.: Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science 171, 385–387 (1971) 29. Sáez, E., González-Olivares, E.: Dynamics on a predator-prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999) 30. Sáez, E., Szántó, I.: A polycycle and limit cycles in a non-differentiable predator-prey model. Proc. Indian Acad. Sci. Math. Sci. 117, 219–231 (2007) 31. Turchin, P.: Complex Population Dynamics: A Theoretical/Empirical Synthesis. Mongraphs in Population Biology. Princeton University Press, Princeton (2003) 32. Venturino, E., Petrovskii, S.: Spatiotemporal behavior of a prey-predator system with a group defense for prey. Ecol. Complex. 14, 37–47 (2013) 33. Vilches-Ponce, K., Dinámicas de un modelo de depredaci ón del tipo Gause con respuesta funcional no diferenciable (Dynamics of a Gause type predator-prey model with non-differentiable functional response) Master thesis, Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaíso (2009), in spanish 34. Vilches, K., González-Olivares, E., Rojas-Palma, A.: Prey herd behavior modeled by a generic non-differential functional response. Math. Model. Nat. Phenom. 13(3), 26 (2018)

Entire Solutions of a Nonlinear Diffusion System Drago¸s-P˘atru Covei

Abstract In this chapter, a diffusion model system of equations is analyzed, and entire solutions are established under some conditions on its nonlinearity. The starting point of this work is raised by the following question: can one establish new results related to the existence and asymptotic behaviour of solutions for such systems as the one considered? We believe that this question deserves investigation, which can be structured in several scientific research objectives. The results achieved in the chapter, generated by the above question, are of high interest in the academic society and industry and want to convey a great variety of applications.

1 Context and the Diffusion Model Stochastic control problems arise in Finance, Economics, Management Science, Operations Research, Physics and Population Dynamics; for more on this we refer the reader to Alvarez [1], Arnold [2], Bensoussan, Sethi, Vickson and Derzko [3], Ghosh, Arapostathis and Marcus [7], Lasry and Lions [19] and Smooke [28]. The value function for stochastic control problems in diffusion processes framework is characterized by the Hamilton Jacobi Bellman (HJB) equation. This in turn leads to a non linear and possibly degenerate system of partial differential equations of parabolic or elliptic type. Let us present the setting. Consider W a N −dimensional Brownian motion on a filtered probability space (1) (Ω, {Ft }0≤t≤T , F , P), where {Ft }0≤t≤T is the completed filtration generated by W. The state process is a diffusion whose dynamics is given by the following stochastic differential equation

D.-P. Covei (B) Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st district, 010374 22, Bucharest, Romania e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_15

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D.-P. Covei

d X (t) = g(X (t), U (t), t)dt + σ (X (t), U (t), t)dW (t).

(2)

Here X (t) in R N is the state of the system at time t, and U (t) in R N is the control at time t. When the benchmark time horizon is infinite, we take g, σ independent of time and this makes the process X (t) stationary. Then, one defines the discounted cost functional as   ∞ −λt e [ f (X (t)) + C(X (t), U (t), t)] dt|X (0) = x . (3) J (x, U ) = E 0

Let A(u) be the infinitesimal generator of diffusion X , second order differential operator defined by A(u)v =

 i, j

i j (x, u)

 ∂ 2v ∂v + g j (x, u) , ∂ xi ∂ x j ∂x j j

(4)

where  = 21 σ σ . As we posit ourselves in the infinite horizon case, the value function is defined as (5) v(x) = inf J (x, U ). U

The value function is characterized by the following HJB equation − λv + inf [A(U )v + C(U ) + f ] = 0, x ∈ R N , U

(6)

and a transversality condition (due to the infinite horizon). Regime switching refers to the situation when the characteristics of the state process are affected by several regimes (e.g. in finance bull and bear market with higher volatility in the bear market). We allow for regime switching in our model; regime switching refers to the situation when the characteristics of the state process are affected by several regimes (e.g. in finance bull and bear market with higher volatility in the bear market). The regime switching is captured by a continuous time homogeneous Markov chain ε(t) adapted to Ft with two regimes good and bad, i.e., for every t ∈ [0, ∞) and ε(t) ∈ {1, 2}. In a specific application, ε(t) = 1 could represent a regime of economic growth while ε(t) = 2 could represent a regime of economic recession. In another application, ε(t) = 1 could represent a regime in which consumer demand is high while ε(t) = 2 could represent a regime in which consumer demand is low. The Markov chain’s rate matrix is   −a1 a1 , (7) A= a2 −a2

Entire Solutions of a Nonlinear Diffusion System

461

for some a1 > 0, a2 > 0. Diagonal elements Aii are defined such that Aii = −  Ai j , j=i

(8)

where A11 = −a1 , A12 = a1 , A21 = a2 , A22 = −a2 . In this case, if pt = E[ε(t)] ∈ R2 , then dε(t) = Aε(t). dt 

Moreover

t

ε(t) = ε(0) +

Aε(u) du + Mt ,

(9)

(10)

0

where M(t) is a martingale with respect to Ft . Let us consider a Markov modulated controlled diffusion with controls in feedback form (11) d X (t) = cε(t) (X (t))dt + kε(t) dW (t), for some constants k1 > 0, k2 > 0, and X (0) = x ∈ R N. Here, at every time t, the demand rate cε(t) and the volatility kε(t) depend on the regime ε(t). We allow the demand to take on negative values, which represent items return (due to spoilage). We consider the class of admissible controls, A , which are the feedback controls for which the above SDE has a unique strong solution. Next, for each c ∈ A the cost functional is defined by 

∞

J (x, c, i) = E

e 0

−λt

 1 2 [a(|X (t)|) + |c|ε(t) (X (t))] dt|ε(0) = i . 2

(12)

Here | · | stands for the Euclidean norm in R N and a(|x|) is a positive function. Our objective is to minimize the functional J , i.e. determine the value function vi (x) = inf J (x, c, i),

(13)

and the optimal control. Here the infimum is taken over all admissible controls c ∈ A . In order, to obtain the HJB equation (6), we apply the martingale/supermartingale principle; search for a function u(x, i) such that the stochastic process M c (t) defined below  t 1 e−λu [a(|X (u)|) + |c|2ε(u) (X (u))] du, (14) M c (t) = e−λt u(X (t), ε(t)) − 2 0

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is supermartingale and martingale for the optimal control. If this is achieved and the following transversality condition holds true lim E[e−λt u(X (t), ε(t))] = 0,

(15)

vi (x) = −u(x, i) = inf J (x, c, i).

(16)

t→∞

then c∈A

The supermartingale/martingale requirement leads to the following HJB equation   |c|2 ki Δu(x, i) + sup ∇u(x, i)c − = a(|x|) + (λ + ai )u(x, i) − ai u(x, j), 2 2 c∈A (17) for i, j ∈ {1, 2}. First order condition yields the candidate optimal control cˆi (x) = ∇u(x, i) = −∇vi (x),

(18)

and this leads to the system |∇u(x, i)|2 ki Δu(x, i) + = a(|x|) + (λ + ai )u(x, i) − ai u(x, j), 2 2

(19)

for i, j ∈ {1, 2}. Alternatively this system can be written in terms of vi (x), (i = 1, 2) to get  2 − k21 Δv1 (x) + |∇v12(x)| = a(|x|) − (λ + a1 )v1 (x) + a1 v2 (x), (20) 2 − k22 Δv2 (x) + |∇v22(x)| = a(|x|) − (λ + a2 )v2 (x) + a2 v1 (x). The change of variable v1 (x) = k1 w1 and v2 (x) = k2 w2 transform the system (20) into 

k2

− 21 Δw1 (x) + k2 − 22 Δw2 (x) +

k12 |∇w1 (x)|2 2 k22 |∇w2 (x)|2 2

= a (|x|) − (λ + a1 ) k1 w1 + a1 k2 w2 , = a (|x|) − (λ + a2 ) k1 w2 + a2 k1 w1 ,

(21)

[a (|x|) − (λ + a1 ) k1 w1 + a1 k2 w2 ] , [a (|x|) − (λ + a2 ) k2 w2 + a2 k1 w1 ] .

(22)

or, equivalently 

−Δw1 (x) + |∇w1 (x)|2 = −Δw2 (x) + |∇w2 (x)|2 =

The change of variable

2 k12 2 k22

Entire Solutions of a Nonlinear Diffusion System

463

u 1 (x) = e−w1 (x) and u 2 (x) = e−w2 (x) transform the system (22) into 

Δu 1 = u 1 (x) [ k22 (a (|x|) + (λ + a1 ) k1 ln u 1 − a1 k2 ln u 2 )], 1 Δu 2 = u 2 (x) [ k22 (a (|x|) + (λ + a2 ) k2 ln u 2 − a2 k1 ln u 1 )],

(23)

2

since

Δu 1 (x) = e−w1 (x) (−Δw1 (x) + |∇w1 (x)|2 ), Δu 2 (x) = e−w2 (x) (−Δw2 (x) + |∇w2 (x)|2 ).

(24)

This motivates the study of more generally class of problems

Δu 1 = H1 (x, u 1 , u 2 ) for x ∈ R N (N ≥ 1), Δu 2 = H2 (x, u 1 , u 2 ) for x ∈ R N (N ≥ 1),

(25)

where H1 and H2 are given functions. Since, at this time there are no mathematical results on system (22) or the general system (25), we restrict our discussion to a system of nonlinear partial differential equations for which the existence of solutions are established essential in the study of (22) or to others. For example, the solutions for the considered system can serve as a sub-solutions or a super- solutions for the mathematical model (22) or others models of the real world as time independent Schrödinger system type, see Grosse and Martin [9].

2 Radial Symmetry of Solutions to Diffusion Systems and Historical Notes Our goal is to solve a nonlinear system of the following type

Δφ1 u := H1 (|x| , u, v), x ∈ R N (N ≥ 3), Δφ2 v := H2 (|x| , u, v), x ∈ R N (N ≥ 3),

(26)

where H1 , H2 : [0, ∞) × [0, ∞) × [0, ∞) → [0, ∞) , are continuous and increasing functions in (u, v) and Δφi u (i = 1, 2) stands for the φi -Laplacian operator defined as Δφi ◦ := div(φi (|∇ ◦ |)∇◦). The mathematical motivation for the study of the system (26) stems from [17, 18, 30]. The paper [17] has considered entire large radial solutions for the elliptic system

Δu = a1 (|x|) vα , Δv = a2 (|x|) u β , x ∈ R N (N ≥ 3),

(27)

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where 0 < α ≤ 1, 0 < β ≤ 1, a1 and a2 are nonnegative continuous functions on R N, and they established that a necessary and sufficient condition for this system to have a nonnegative entire large radial solution (i.e., a nonnegative spherically symmetric solution (u, v) on R N that satisfies lim u (x) = lim v (x) = ∞), is |x|→∞



∞

α   t ta1 (t) t 2−N s N −3 Q (s) ds dt = ∞,

0



∞

|x|→∞

0

 ta2 (t) t



t

2−N

0

s

N −3

(28)

β dt = ∞,

P (s) ds

(29)

0



where P (r ) =

r



r

τ a1 (τ ) dτ and Q (r ) =

0

τ a2 (τ ) dτ.

0

It is well known, see Yang [30], that if a : [0, ∞) → [0, ∞) is a spherically symmetric continuous function and the nonlinearity f : [0, ∞) → [0, ∞) is a continuous, increasing function with f (0) ≥ 0 and f (s) > 0 for all s > 0 which satisfies 

∞ 1

1 dt = ∞, f (t)

(30)

then the single equation 

Δu = a (|x|) f (u) for x ∈ R N (N ≥ 3), lim u (|x|) = ∞,

(31)

|x|→∞

has a nonnegative radial solution if and only if a satisfies  lim Aa (t) = ∞, Aa (t) :=

t→∞



t

s

s 1−N 0

z N −1 a(z)dzds.

(32)

0

A direct computation shows that 1 lim Aa (t) = t→∞ N −2



∞

ra (r ) dr.

(33)

0

However, there is no counterpart result for systems (26), where H1 , H2 satisfy a condition of the form (30), excepting certain cases from [6]. One purpose of this chapter is to fill this gap. The paper [18] extended the result of [17] to a more general case by requiring αβ ≤ 1. They showed that if αβ > 1, then (27) has an entire large solution if either (28) and (29) fails to hold, i.e., a1 and a2 must satisfy (at least) one of the conditions

Entire Solutions of a Nonlinear Diffusion System



∞

0



465

 α  t ta1 (t) t 2−N s N −3 Q (s) ds dt < ∞,

(34)

0

∞

 β  t ta2 (t) t 2−N s N −3 P (s) ds dt < ∞.

0

(35)

0

To summarize, when αβ > 1 a sufficient condition to ensure the existence of a positive entire large solution for the system (27) is that a1 and a2 satisfy (34) or (35). Therefore, it remains unknown whether or not this is a necessary condition. However, we know from the reference [30] that this is not true for the single equation (31). The second purpose of this chapter is to prove that this does not happen in the case of the systems either. Let us point out that if a1 and a2 satisfy

∞ (1) 0 ra1 (r ) dr = ∞, ∞ (2) 0 ra2 (r ) dr = ∞,

(36)

then they also satisfy (28) and (29). Moreover, if they satisfy

∞ (3) 0 ra1 (r ) dr < ∞, ∞ (4) 0 ra2 (r ) dr < ∞,

(37)

then they also satisfy (34) and (35). In both cases, however, the converse is not true. For further results on this, see [4–6, 22, 24–26] and the references therein. In the present chapter, we are interested in the more general class of nonlinear systems of the form (26). This, actually, is the third goal of our chapter motivated by the φi −Laplacian operator applications in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. Several results related to our goals were obtained by Gregorio [8], Franchi– Lanconelli and Serrin [10], Hamydy–Massar–Tsouli [12], Keller [13], Kon’kov [15], Jaroˆs–Takaˆsi [16], Lieberman [20], Losev–Mazepa [22], Li–Zhang–Zhang [21], Luthey [23], Mazepa [24], Naito–Usami [25, 26], Osserman [27], Smooke [28], Zhang–Zhou [31] and Zhang [32]. We expect that our work, while focussing on a very specific problem, will also lead to general insights and new methods with potential applications to a much wider class of problems of the form (22)/(25).

3 Statement of the Problem The system considered is (26). Next we state our work hypothesis. Throughout the chapter we let α, β ∈ (0, ∞) be arbitrary parameters and we assume φi and Hi (i = 1, 2) satisfy the following conditions: (O1) φi ∈ C 1 ((0, ∞) , (0, ∞)), lim tφi (t) = 0 and lim tφi (t) = ∞; t→0

t→∞

466

D.-P. Covei

(O2) tφi (t) > 0 is strictly increasing for t > 0; (O3) there exist positive constants k i , k i , the continuous and increasing functions θ i , θ i : [0, ∞) → [0, ∞) and the continuous functions ψ i , ψ i : [0, ∞) → [0, ∞) such that k i θ i (s1 )ψ i (s2 ) ≤ h i−1 (s1 s2 ) ≤ k i θ i (s1 )ψ i (s2 ) for all s1 , s2 > 0,

(38)

where h i−1 is the inverse function of h i (t) = tφi (t) for t > 0. (H) there exist a 1 , a 2 , a 1 , a 2 : [0, ∞) → [0, ∞) and f 1 , f 2 : [0, ∞) → [0, ∞) such that a 1 (|x|) f 1 (v) ≤ H1 (|x| , u, v) ≤ a 1 (|x|) f 1 (v) , ∀u, v ≥ 0 and x ∈ R N , (39) a 2 (|x|) f 2 (u) ≤ H2 (|x| , u, v) ≤ a 2 (|x|) f 2 (u) , ∀u, v ≥ 0 and x ∈ R N , (40) where the functions a 1 , a 2 , a 1 , a 2 , f 1 and f 2 satisfy some appropriate conditions that will be described below: (A) a 1 , a 2 , a 1 , a 2 : [0, ∞) → [0, ∞) are spherically symmetric continuous functions (i.e., a i (x) = a i (|x|), a i (x) = a i (|x|) for i = 1, 2); (C1) f 1 , f 2 : [0, ∞) → [0, ∞) are continuous, increasing and f 1 (s1 ) · f 2 (s2 ) > 0 for all s1 , s2 > 0; (C2) there exist continuous and increasing functions g1 , g2 : [0, ∞) → [0, ∞) and the continuous functions ξ 1 , ξ 2 : [0, ∞) → [0, ∞) such that f 1 (t1 · w1 ) ≤ g1 (t1 ) · ξ 1 (w1 ) , ∀w1 ≥ 1 and ∀ t1 ≥ M1 · θ 2 ( f 2 (α)), f 2 (t2 · w2 ) ≤ g2 (t2 ) · ξ 2 (w2 ) , ∀ w2 ≥ 1 and ∀ t2 ≥ M2 · θ 1 ( f 1 (β)), where

M1 ≥ max 1,

β θ 2 ( f 2 (α))



and M2 ≥ max 1,

α θ 1 ( f 1 (β))

(41) (42)

 ;

(C3) there are continuous functions ξ 1 , ξ 2 : [0, ∞) → [0, ∞) such that f 1 (m 1 w1 ) ≥ ξ 1 (w1 ) , ∀ w1 ≥ 1, f 2 (m 2 w2 ) ≥ ξ 2 (w2 ) , ∀ w2 ≥ 1,

(43) (44)

where



 m 1 ∈ 0, min β, θ 2 ( f 2 (α)) and m 2 ∈ 0, min α, θ 1 ( f 1 (β)) . Remark 1 Assumptions (O3), (C2) and (C3) are further discussed in Krasnosel’skii and Rutickii [14] (see also Soria [29]). The class of nonlinearities considered by Lair [17, 18] are covered by our results.

Entire Solutions of a Nonlinear Diffusion System

467

Remark 2 (see [11, Lemma 2.1]) Suppose φi (i = 1, 2) satisfies (O1), (O2) and (O4) there exist li , m i > 1 such that Φi (t) · t ≤ m i for any t > 0, Φi (t)

li ≤



where

t

Φi (t) =

φi (s) sds, t > 0;

(45)

(46)

0

(O5) there exist a0i , a1i > 0 such that a0i ≤

Φi (t) · t ≤ a1i for any t > 0, Φi (t)

(47)

then assumption (38) holds true with ψ i = ψ i = h i−1 , k i = k i = 1  

θ i (t) = min t 1/m i , t 1/li , θ i (t) = max t 1/m i , t 1/li .

(48) (49)

Remark 3 (see [11] for more information) The function Φi appears in physical applications, such as nonlinear elasticity, plasticity, newtonian fluids, and plasma physics: Nonlinear Elasticity: p Φi (t) = 1 + t 2 − 1,  p−1 φi (t) = 2 p 1 + t 2 ,

(50) (51)

where t > 0 and p > 21 ; Plasticity: Φi (t) = t p (ln (1 + t))q ,   lnq−1 (t + 1)  p−1 pt + qt q−2 ln (t + 1) + qt p−1 , φi (t) = t +1

(52) (53)

where t > 0, p > 1 and q > 0; Generalized Newtonian fluids:  Φi (t) = φi (t) = t

t

0 −p

where t > 0, 0 ≤ p ≤ 1 and q > 0;

q s 1− p sinh−1 s ds,

(54)

ar csinh q t,

(55)

468

D.-P. Covei

Plasma Physics: tp tq + , p q p−2 + t q−2 , φi (t) = t

Φi (t) =

(56) (57)

where t > 0 and 1 < p < q.

4 The Mathematical Results As stated we start with the formulation of our results, which represents an improvement of [6]. We introduce some notations needed in the sequel. The reader may just as well glance through this subchapter and return to it when necessary  s k i ψ i (s 1−N z N −1 a i (z)dz)ds , i = 1, 2, 0 0   y  r    P 1,2 (r ) = ψ 1 y 1−N t N −1 a 1 (t)ξ 1 1 + A a2 (t) dt dy, 0 0   y  r    P 2,1 (r ) = ψ 2 y 1−N t N −1 a 2 (t)ξ 2 1 + A a1 (t) dt dy, 

A ai (t) =

t

0

0

P 1,2 (∞) = lim P 1,2 (r ) , P 2,1 (∞) = lim P 2,1 (r ) , r →∞ r →∞  s  t k i ψ i (s 1−N z N −1 a i (z)dz)ds, i = 1, 2, A ai (t) = 0 0    y  r  −1 1−N N −1 y h1 t a 1 (t)ξ 1 1 + A a2 (t) dt dy, P 1,2 (r ) = 0 0    y  r  −1 1−N N −1 y h2 t a 2 (t)ξ 2 1 + A a1 (t) dt dy, P 2,1 (r ) = 0

0

P 1,2 (∞) = lim P 1,2 (r ) , P 2,1 (∞) = lim P 2,1 (r ) , r →∞ r →∞  r 1  dt, H1,2 (∞) = lim H1,2 (s) , H1,2 (r ) = s→∞ α θ 1 (g1 M1 θ 2 ( f 2 (t) )  r 1  dt, H2,1 (∞) = lim H2,1 (s) . H2,1 (r ) = s→∞ β θ 2 (g2 M2 θ 1 ( f 1 (t) ) Let us point that  (r ) = H1,2

and



1

 > 0 for r > α, θ 1 (g1 M1 θ 2 ( f 2 (r ) )

(58)

Entire Solutions of a Nonlinear Diffusion System  H2,1 (r ) =



1

 > 0 for r > β. θ 2 (g2 M2 θ 1 ( f 1 (r ) )

469

(59)

−1 on [0, H1,2 (∞)) and H2,1 has the inverse Then H1,2 has the inverse function H1,2 −1 function H2,1 on [0, H2,1 (∞)). We are ready at this point to state the following first result:

Theorem 1 Assume that H1,2 (∞) = H2,1 (∞) = ∞ and (A), hold. Furthermore, if f 1 and f 2 satisfy the hypotheses (C1) and (C2) then the system (26) has one positive radial solution (u, v) ∈ C 1 ([0, ∞)) × C 1 ([0, ∞)) with (u (0) , v (0)) = (α, β) .

(60)

If in addition, f 1 and f 2 satisfy the hypothesis (C3), P 1,2 (∞) = ∞ and P 2,1 (∞) = ∞ then limr →∞ u (r ) = ∞ and limr →∞ v (r ) = ∞. Conversely, if ξ i = ξ i (i = 1, 2), h i−1 = ψ i (i = 1, 2) and (C1), (C2), (C3) hold true, and (u, v) is a positive entire large solution of (26) such that (u (0) , v (0)) = (α, β), then a1 and a2 satisfy P 1,2 (∞) = P 1,2 (∞) = ∞ and P 2,1 (∞) = P 2,1 (∞) = ∞. Theorem 1 covers all known results about the large solutions for (26), and thus it achieves our first goal. Next, we are interested in the existence of entire bounded radial solutions for the system (26). This is the gist of our next result: Theorem 2 Suppose that H1,2 (∞) = H2,1 (∞) = ∞ and (A), holds true. Furthermore, if f 1 and f 2 satisfies (C1) and (C2) then the system (26) has one positive radial solution (u, v) ∈ C 1 ([0, ∞)) × C 1 ([0, ∞)) with (u (0) , v (0)) = (α, β) .

(61)

Moreover, if P 1,2 (∞) < ∞ and P 2,1 (∞) < ∞ then limr →∞ u (r ) < ∞ and limr →∞ v (r ) < ∞. We now turn to a more refined result concerning the solutions of system (26). This is presented in the next Theorem 2. Theorem 3 Assume that (A) holds true. If (C1), (C2), P 1,2 (∞) < H1,2 (∞) < ∞ and P 2,1 (∞) < H2,1 (∞) < ∞ are satisfied, then the system (26) has one positive bounded radial solution (u, v) ∈ C 1 ([0, ∞)) × C 1 ([0, ∞)) with (u (0) , v (0)) = (α, β) .

(62)

 −1 k 1 P 1,2 (r ) , α + P 1,2 (r ) ≤ u (r ) ≤ H1,2 −1 k 2 P 2,1 (r ) . β + P 2,1 (r ) ≤ v (r ) ≤ H2,1

(63)

such that



The next result presents the case when one of the components is bounded while the other is large.

470

D.-P. Covei

Theorem 4 Assume that H1,2 (∞) = H2,1 (∞) = ∞ and (A), holds true. Furthermore, if f 1 and f 2 satisfies the hypotheses (C1) and (C2) then the system (26) has one positive radial solution (u, v) ∈ C 1 ([0, ∞)) × C 1 ([0, ∞)) with (u (0) , v (0)) = (α, β) .

(64)

Moreover, the following holds true: (1) If in addition, f 2 satisfy the condition (44), P 1,2 (∞) < ∞ and P 2,1 (∞) = ∞ then limr →∞ u (r ) < ∞ and limr →∞ v (r ) = ∞. (2) If in addition, f 1 satisfy the condition (43), P 1,2 (∞) = ∞ and P 2,1 (∞) < ∞ then limr →∞ u (r ) = ∞ and limr →∞ v (r ) < ∞. Theorem 5 Assume that (A) holds true. If (C1), (C2), (43), H1,2 (∞) = ∞, P 1,2 (∞) = ∞ and P 2,1 (∞) < H2,1 (∞) < ∞ are satisfied, then the system (26) has one positive radial solution (u, v) ∈ C 1 ([0, ∞)) × C 1 ([0, ∞)) with (u (0) , v (0)) = (α, β) ,

(65)

such that limr →∞ u (r ) = ∞ and limr →∞ v (r ) < ∞. Theorem 6 Assume that the hypothesis (A) holds. If (C1), (C2), (44), P 2,1 (∞) = ∞, H2,1 (∞) = ∞ and P 1,2 (∞) < H1,2 (∞) < ∞ are satisfied, then the system (26) has one positive radial solution (u, v) ∈ C 1 ([0, ∞)) × C 1 ([0, ∞)) with (u (0) , v (0)) = (α, β) ,

(66)

such that limr →∞ u (r ) < ∞ and limr →∞ v (r ) = ∞. Remark 4 In the special case of M1+ =

∞



∞

= 0

s

k i ψ 2 (s 1−N

0



and M2+



z N −1 a 2 (z)dz)ds

(67)

z N −1 a 1 (z)dz)ds,

(68)

0

 k 1 ψ 1 (s

s

1−N 0

we get improved results: (a) If M1+ ∈ (0, ∞) then the condition (41) is not necessary but H1,2 (r ) must be replaced by   r  1 β  dt, M1 ≥ max 1, H1,2 (r ) = · 1 + M1+ , θ 2 ( f 2 (α)) a θ 1 ( f 1 M1 θ 2 ( f 2 (t) ) (69)

y

r  and therefore P 1,2 (r ) = 0 ψ 2 y 1−N 0 t N −1 a 1 (t)dt dy. (b) If M2+ ∈ (0, ∞) then the condition (42) is not necessary but H2,1 (r ) must be replaced by

Entire Solutions of a Nonlinear Diffusion System

471

  α  dt, M2 ≥ max 1, · 1 + M2+ . θ 1 ( f 1 (β)) b θ 2 ( f 2 M2 θ 1 ( f 1 (t) ) (70)

y

r  and therefore P 2,1 (r ) = 0 ψ 1 y 1−N 0 t N −1 a 2 (t)dt dy. (c) If M1+ ∈ (0, ∞) and M2+ ∈ (0, ∞) then the conditions (41) and (42) are not necessary but H1,2 (r ) and H2,1 (r ) must be replaced by (69) and (70). Here P 1,2 (r ) and P 2,1 (r ) are defined as in a), b). (d) If m 1 ≥ 1 then ξ 1 = f 1 . (e) If m 2 ≥ 1 then ξ 2 = f 2 . (f) If m 1 ≥ 1 and m 2 ≥ 1 then ξ 1 = f 1 and ξ 2 = f 2 . 

H2,1 (r ) =

r

1



Remark 5 (see [31]) When 

1 0



1

 dt = ∞, θ 1 (g1 M1 θ 2 ( f 2 (t) )

(71)

one can see that there is α > 0 sufficiently small such that P 1,2 (∞) < H1,2 (∞) < ∞,

(72)

P 1,2 (∞) < ∞ and H1,2 (∞) < ∞,

(73)

holds, provided where α is given as in (C2).

5 The Arzelà–Ascoli Theorem The main tool in our proof of the theorems is a variant of the Arzelà –Ascoli Theorem which are going to present. Let r1 , r2 ∈ R with r1 ≤ r2 and (K = [r1 , r2 ] , d K (x, y)) ,

(74)

be a compact metric space, with the metric d K (x, y) = |x − y| , and let C ([r1 , r2 ]) = {g : [r1 , r2 ] → R |g is continuous on [r1 , r2 ] } ,

(75)

denote the space of real valued continuous functions on [r1 , r2 ] and for any g ∈ C ([r1 , r2 ]), let g ∞ = max |g (x)| , (76) x∈[r1 ,r2 ]

472

D.-P. Covei

be the maximum norm on C ([r1 , r2 ]). Remark 6 Let us point out that in the case of g 1 , g 2 ∈ C ([r1 , r2 ]) and    d g 1 , g 2 = g 1 − g 2 ∞

(77)

(C ([r1 , r2 ]) , d) is a complete metric space. Next we define the notions of boundedness, equicontinuity, and uniform convergence for sequences of functions. Definition 1 We say that the sequence {gn }n∈N from C ([r1 , r2 ]) is bounded if there exists a positive constant C such that gn (x) ∞ ≤ C for each x ∈ [r1 , r2 ]. (Equivalently: |gn (x)| ≤ C for each x ∈ [r1 , r2 ] and n ∈ N∗ ). Definition 2 We say that the sequence {gn }n∈N from C ([r1 , r2 ]) is equicontinuous if for any given ε > 0, there exists a number δ > 0 (which depends only on ε) such that |gn (x) − gn (y)| < ε for all n ∈ N (78) whenever d K (x, y) < δ for every x, y ∈ [r1 , r2 ]. Definition 3 Let {gn }n∈N be a family of functions defined on [r1 , r2 ]. The sequence {gn }n∈N converges uniformly to g (x) if for every ε > 0 there is an N (which depends only on ε) such that |gn (x) − g (x)| < ε for all n > N and x ∈ [r1 , r2 ] .

(79)

At this point we are ready to state Arzelà–Ascoli Theorem. Theorem 7 (Arzelà–Ascoli theorem) Let {gn }n∈N be a sequence in C  ([r1 , r2 ]), bounded and equicontinuous. There exists a subsequence denoted gn k k∈N which converges uniformly to g on C ([r1 , r2 ]).

6 The Proofs of Mathematical Results 6.1 The Solvability of the System with no Boundary Conditions Radially symmetric solutions of the problem (26) correspond to solutions of the ordinary differential equations system N −1      φ u (r ) u (r ) = r N −1 H1 (r, u (r ) , v (r ))) on [0, ∞) , r N −1 1      φ2 v (r ) v (r ) = r N −1 H2 (r, u (r ) , v (r ))) on [0, ∞) , r

(80)

Entire Solutions of a Nonlinear Diffusion System

473

subject to the initial conditions  (u (0) , v (0)) = (α, β) and u  (0) , v (0) = (0, 0) , since (u (r ) , v (r )) is a radially symmetric positive entire solution of the system (26). Integrating (80) from 0 to r , we obtain

  φ1 (u  (r ))u  (r ) = φ2 (v (r ))v (r ) =

1 r N −1 1 r N −1

r N −1

0r s N −1 H1 (s, u (s) , v (s))) ds, on [0, ∞) , H2 (s, u (s) , v (s))) ds, on [0, ∞) . 0 s

(81)

Taking into account the equations (81), it is easy to see that u (r ) is an increasing function on [0, ∞) of the radial variable r , and so it is v (r ). Thus, the radial solutions of the system (80) solve the system of integral equations



r 1−N t N −1 H1 (s, u (s) , v (s))) ds)dt, r ≥ 0, u(r ) = α + 0 h −1 1 (t 0 s

r 1−N t N −1 (t s H v(r ) = β + 0 h −1 2 (s, u (s) , v (s))) ds)dt, r ≥ 0. 2 0

(82)

We define inductively {u m }m≥0 and {vm }m≥0 on [0, ∞) as follows ⎧ ⎨ u 0 (r ) = α, v0 (r

r) = β,1−N t N −1 u m (r ) = α + 0 h −1 H1 (s, u m−1 (s), vm−1 (s))) ds)dt, r ≥ 0, 1 (t

0 s ⎩ r 1−N t N −1 vm (r ) = β + 0 h −1 (t s H 2 (s, u m−1 (s), vm (s))) ds)dt, r ≥ 0. 2 0 (83) Obviously, for all r ≥ 0 and m ∈ N it holds that u m (r ) ≥ α, vm (r ) ≥ β and v0 ≤ v1 . Our assumptions yield u 1 (r ) ≤ u 2 (r ), for all r ≥ 0, so v1 (r ) ≤ v2 (r ), for all r ≥ 0. Continuing on this line of reasoning, we obtain that the sequences {u m }m and {vm }m are increasing on [0, ∞). We next establish bounds for the non-decreasing sequences {u m }m and {vm }m . From (83) we obtain the following inequalities 

r

vm (r ) = β + 

0 r

1−N h −1 2 (t 1−N h −1 2 (t



t

s N −1 H2 (s, u m−1 (s), vm (s))) ds)dt

0



t

s N −1 a 2 (s) f 2 (u m (s))ds)dt 0 0  t  r −1 h 2 ( f 2 (u m (t))t 1−N z N −1 a 2 (z)dz)dt ≤β+ 0 0  t  r k 2 θ 2 ( f 2 (u m (t)) ψ 2 (t 1−N z N −1 a 2 (z)dz)dt ≤β+ 0 0  r  t 1−N k 2 ψ 2 (t z N −1 a 2 (z)dz)dt ≤ β + θ 2 ( f 2 (u m (r )))

≤β+

0

0

(84)

474

D.-P. Covei

β

+ A a2 (r )) θ 2 ( f 2 (u m (r )) β ≤ θ 2 ( f 2 (u m (r )) ( + A a2 (r )) θ 2 ( f 2 (α)) ≤ θ 2 ( f 2 (u m (r )) (

≤ M1 θ 2 ( f 2 (u m (r )) (1 + A a2 (r )) and, in the same vein  u m (r ) = α +

r

0

 ≤α+

r

0

 ≤α+

0

r

1−N h −1 1 (t 1−N h −1 1 (t 1−N h −1 1 (t



t

s N −1 H1 (s, u m−1 (s), vm−1 (s))) ds)dt

0



t

s N −1 a 1 (s) f 1 (vm−1 (s))ds)dt

0



t

s N −1 a 1 (s) f 1 (vm (s))ds)dt

(85)

0

≤ M2 θ 1 ( f 1 (vm (r )) (1 + A a1 (r )). Moreover, using (84), by an elementary computation it follows that    r 1−N N −1 u m (r ) ≤ h −1 s a (s) f (v (s))ds r 1 1 m 1 0    r  −1 1−N N −1 ≤ h1 r s a 1 (s) f 1 M1 θ 2 ( f 2 (u m (s)) (1 + A a2 (s)) ds 0    r   1−N N −1 ≤ h −1 ds M 1 + s a (s)g θ ( f (s)) ξ A (86) r (u (s) 1 1 1 2 2 m a2 1 1 0      1−N r N −1 g ≤ h −1 θ ( f (r )) s a (s)ξ A M 1 + ds r (u (s) 1 1 2 2 m 1 a 1 2 1 0    r   1−N N −1 s a 1 (s)ξ 1 1 + A a2 (s) ds . ≤ k 1 θ 1 (g1 M1 θ 2 ( f 2 (u m (r )) )ψ 1 r 0

Arguing as above, and making use of the second inequality (85), one can show that    r  1−N N −1 u , v ds vm (r ) = h −1 s H r (s, (s) (s))) 2 m−1 m 2 0    r 1−N N −1 ≤ h −1 r s a (s) f (u (s))ds (87) 2 1 m−1 2 0    r   1−N N −1 ≤ k 2 θ 2 (g2 M2 θ 1 ( f 1 (vm (r )) )ψ 2 r s a 2 (s)ξ 2 1 + A a1 (s) ds . 0

Combining the previous relations (86) and (87), we further obtain

Entire Solutions of a Nonlinear Diffusion System

475

  r    (u m (r ))  ≤ k 1 ψ 1 r 1−N s N −1 a 1 (s)ξ 1 1 + A a2 (s) ds , θ 1 (g1 M1 θ 2 ( f 2 (u m (r )) ) 0 (88)   r    (vm (r ))  ≤ k 2 ψ 2 r 1−N s N −1 a 2 (s)ξ 2 1 + A a1 (s) ds . θ 2 (g2 M2 θ 1 ( f 1 (vm (r )) ) 0 (89)

Integrating the inequalities (88) and (89) from 0 to r , yields that  u m (r ) α

 vm (r ) −1 k  dt ≤ P 1,2 (r ) ,  dt ≤ P 2,1 (r ) . 2 θ 1 (g1 M1 θ 2 ( f 2 (t) ) θ 2 (g2 M2 θ 1 ( f 1 (t) ) β −1



k1

(90)

Also, going back to the setting of H1,2 and H2,1 we rewrite (90) as H1,2 (u m (r )) ≤ k 1 P 1,2 (r ) and H2,1 (vm (r )) ≤ k 2 P 2,1 (r ) ,

(91)

which plays a basic role in the proof of our main results. Since H1,2 (resp. H2,1 ) −1 −1 (resp. H2,1 ) strictly increasing on is a bijection with the inverse function H1,2     0, H2,1 (∞) (resp. 0, H1,2 (∞) ), the inequalities (91) can be reformulated as   −1 −1 u m (r ) ≤ H1,2 k 1 P 1,2 (r ) and vm (r ) ≤ H2,1 k 2 P 2,1 (r ) .

(92)

Then, we have found that upper bounds for {u m (r )}m≥0 and {vm (r )}m≥0 , which are dependent of r . We point to the reader that the corresponding estimates (92) are sometimes essential. Next we prove that the sequences {u m (r )}m≥0 and {vm (r )}m≥0 , are bounded and equicontinuous on [0, c0 ] for arbitrary c0 > 0. To do this, we take   −1 −1 k 1 P 1,2 (c0 ) and C2 = H2,1 k 2 P 2,1 (c0 ) , C1 = H1,2 and since



(93)



(u m (r )) ≥ 0 and (vm (r )) ≥ 0, it follows that u m (r ) ≤ u m (c0 ) ≤ C1 and vm (r ) ≤ vm (c0 ) ≤ C2 . We have proved that

(94)

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D.-P. Covei

{u m (r )}m≥0 and {vm (r )}m≥0 , are bounded on [0, c0 ] for arbitrary c0 > 0. Using this fact in (86) and (87) we show that the same is true for (u m (r )) and (vm (r )) . By construction we verify that u m (r )

= ≤ ≤ ≤ ≤ ≤

   r 1−N N −1 r s H1 (s, u m−1 (s), vm−1 (s))) ds 0    r 1−N N −1 r s a (s) f (v (s))ds h −1 1 1 m−1 1 0    r 1−N N −1 r s a (s) f (v (s))ds h −1 1 1 m 1 0  r  a (s) f (v (s))ds h −1 1 1 m 1  0   r a f (v (s))ds h −1 1 ∞ 1 m 1 0   r  a f (C ) ds h −1 1 ∞ 1 2 1

h −1 1

(95)

0

≤ h −1 1 ( a 1 ∞ f 1 (C 2 )c0 ) on [0, c0 ] . We follow the argument used in (95) to obtain (vm (r )) ≤ h −1 2 ( a 2 ∞ f 2 (C 1 )c0 ) on [0, c0 ] .

(96)

To summarize, we have found that (u m (r )) ≤ h −1 1 ( a 1 ∞ f 1 (C 2 )c0 ) on [0, c0 ] ,  (vm (r )) ≤ h −1 2 ( a 2 ∞ f 2 (C 1 )c0 ) on [0, c0 ] .

(97) (98)

Finally, it remains to prove that {u m (r )}m≥0 and {vm (r )}m≥0 , are equicontinuous on [0, c0 ] for arbitrary c0 > 0. Let ε1 , ε2 > 0 be arbitrary. To verify equicontinuity on [0, c0 ] observe that the mean value theorem yields   |u m (x) − u m (y)| = (u m (ζ1 ))  |x − y| ≤ h −1 1 ( a 1 ∞ f 1 (C 2 )c0 ) |x − y| ,   −1  |vm (x) − vm (y)| = (vm (ζ2 ))  |x − y| ≤ h 2 ( a 2 ∞ f 2 (C1 )c0 ) |x − y| ,

for all n ∈ N and all x, y ∈ [0, c0 ] and for some ζ1 , ζ2 . Then it suffices to take

(99) (100)

Entire Solutions of a Nonlinear Diffusion System

δ1 =

ε1 h −1 1

( a 1 ∞ f 1 (C2 )c0 )

and δ2 =

477

ε2 h −1 2

( a 2 ∞ f 2 (C1 )c0 )

(101)

to see that {u m (r )}m≥0 and {vm (r )}m≥0 , are equicontinuous on [0, c0 ]. Since {u m (r )}m≥0 and {vm (r )}m≥0 , theorem are bounded and equicontinuous on [0, c0 ] we can apply the Arzelà–Ascoli 

with [r1 , r2 ] = [0, c0 ]. Thus, there exists a subsequence, denoted u m 1 (r ), vm 1 (r ) that converges uniformly on [0, 1] × [0, 1]. Let  (m 1 ,m 1 )→∞ u m 1 (r ) , vm 1 (r ) → (u 1 (r ) , v1 (r )) uniformly on [0, 1] .

(102)



Likewise, the subsequence u m 1 (r ) , vm 1 (r ) is bounded and equicontinuous on the interval [0, 2]. Hence, it must contain a convergent subsequence 

u m 2 (r ) , vm 2 (r ) ,

(103)

that converges uniformly on [0, 2] × [0, 2]. Let  (m 2 ,m 2 )→∞ → u m 2 (r ) , vm 2 (r ) (u 2 (r ) , v2 (r )) uniformly on [0, 2] × [0, 2] . (104) Note that

and

{u m 2 (r )} ⊆ {u m 1 (r )} ⊆ {u m (r )}m≥2 ,

(105)

 

vm 2 (r ) ⊆ vm 1 (r ) ⊆ {vm (r )}m≥2 .

(106)

These relationships imply that u 2 (r ) = u 1 (r ) and v2 (r ) = v1 (r ) on [0, 1] .

(107)

Proceeding in this fashion we obtain a countable collection of subsequences such that {u m n } ⊆ .... ⊆ {u m 1 (r )} ⊆ {u m (r )}m≥n (108) and

 {vm n } ⊆ .... ⊆ vm 1 (r ) ⊆ {vm (r )}m≥n

and a sequence {(u n (r ) , vn (r ))} such that

(109)

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(u n (r ) , vn (r )) ∈ C [0, n] × C [0, n] for for (u n (r ) , vn (r )) = (u 1 (r ) , v1 (r )) for (u n (r ) , vn (r )) = (u 2 (r ) , v2 (r )) ... ... (u n (r ) , vn (r )) = (u n−1 (r ) , vn−1 (r )) for

n = 1, 2, 3, ... r ∈ [0, 1] r ∈ [0, 2] ... r ∈ [0, n − 1] .

(110)

These relationships show that there exists a sequence {(u n (r ) , vn (r ))} that converges to (u (r ) , v (r )) ,

(111)

on [0, ∞) satisfying (u n (r ) , vn (r )) = (u (r ) , v (r )) if 0 ≤ r ≤ n.

(112)

This convergence is uniformly on bounded intervals, implying (u (r ) , v (r )) ∈ C [0, ∞) × C [0, ∞) ,

(113)

and the family {(u n (r ) , vn (r ))} is also equicontinuous. Moreover, since {u m }m and {vm }m are increasing on [0, ∞), we see that {(u m , vm )}m≥0 itself converges to (u, v) . The solution (u (r ) , v (r )) constructed in this way has radially symmetry. Going back to the system (80), the radial solutions of (26) are the solutions of the ordinary differential equations system (80). We conclude that radial solutions of (26) with u (0) = α, v (0) = β satisfy: 

r

u(r ) = α + 0



r

v(r ) = β + 0

1−N h −1 1 (t 1−N h −1 2 (t



t

s N −1 H1 (s, u (s) , v (s))) ds)dt, r ≥ 0, (114)

0



t

s N −1 H2 (s, u (s) , v (s))) ds)dt, r ≥ 0. (115)

0

We are now ready to give a complete proof of the Theorems 1–4.

Entire Solutions of a Nonlinear Diffusion System

6.1.1

479

Existence of Entire Large Solutions

Proof of Theorem 1 From (115) we obtain the following inequalities 

r

v(r ) = β + 0



r



1−N h −1 2 (t



1−N h −1 2 (t

t

s N −1 H2 (s, u (s) , v (s))) ds)dt

0 t

s N −1 a 2 (s) f 2 (u(s))ds)dt  r  z 1−N ≥β+ h −1 ( f (α)z s N −1 a 2 (s)ds)dz 2 2 ≥β+

0

0

0

≥ β + θ 2 ( f 2 (α))A a2 (r ) ≥ m 1 (1 + A a2 (r )),

(116)

0

and, in the same vein  u(r ) = α +

r

0

 ≥α+

0

r

1−N h −1 1 (t 1−N h −1 1 (t

≥ m 2 (1 + A a1 (r )).

 

t

s N −1 H1 (s, u (s) , v (s))) ds)dt

0 t 0

s N −1 a 1 (s) f 1 (v(s))ds)dt

(117)

If P 1,2 (∞) = P 2,1 (∞) = ∞, we observe that  u (r ) = α +

r

0

1−N h −1 1 (t



t

s N −1 H1 (s, u (s) , v(s))ds)dt

0

 t 1−N h −1 (t s N −1 a 1 (s) f 1 (v(s))ds)dt 1 0 0    y  r  −1 1−N N −1 y h1 t a 1 (t) f 1 m 1 (1 + A a2 (t) dt dy ≥α+ 0 0    y  r  −1 1−N N −1 y h1 t a 1 (t)ξ 1 1 + A a2 (t) dt dy (118) ≥α+ 0 0    y  r  y 1−N h −1 t N −1 a 1 (t)ξ 1 1 + A a2 (t) dt dy ≥α+ 1 

≥α+

r

0

0

= α + P 1,2 (r ) . Analogously, we refine the strategy above to prove:

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y

r   y 1−N 0 t N −1 a 2 (t)ξ 2 1 + A a1 (t) dt dy v (r ) ≥ β + 0 h −1 2 = β + P 2,1 (r ) ,

(119)

and passing to the limit as r → ∞ in (118) and in the above inequality we conclude that (120) lim u (r ) = lim v (r ) = ∞, r →∞

r →∞

which yields the result. In order to prove the converse let (u, v) be an entire large radial solution of (26) such that (u, v) = (α, β). Then, (u, v) satisfy

r 1−N t N −1 H1 (s, u (s) , v(s))ds)dt, r ≥ 0, u(r ) = α + 0 h −1 1 (t

0 s r 1−N t N −1 v(r ) = β + 0 h −1 (t s H2 (s, u (s) , v(s))ds)dt, r ≥ 0, 2 0

(121)

and, so H1,2 (u (r )) ≤ k 1 P 1,2 (r ) and H2,1 (v (r )) ≤ k 2 P 2,1 (r ) .

(122)

By passing to the limit as r → ∞ in (122) we find that a1 and a2 satisfy P 1,2 (∞) = P 2,1 (∞) = ∞, since (u, v) is large and H1,2 (∞) = H2,1 (∞) = ∞. This completes the proof. We next consider:

6.1.2

Existence of Entire Bounded Solutions

Proof of Theorem 2 If P 1,2 (∞) < ∞ and P 2,1 (∞) < ∞, then using the same arguments as in (114) and (115) we can see that

and

 −1 k 1 P 1,2 (∞) < ∞, u (r ) ≤ H1,2

(123)

 −1 k 2 P 2,1 (∞) < ∞, v (r ) ≤ H2,1

(124)

for all r ≥ 0. Hence (u, v) is bounded and this completes the proof. Proof of Theorem 3 We deduce from (91) and the conditions of the theorem that

Entire Solutions of a Nonlinear Diffusion System

481

H1,2 (u m (r )) ≤ k 1 P 1,2 (∞) < k 1 H1,2 (∞) < ∞, H2,1 (vm (r )) ≤ k 2 P 2,1 (∞) < k 2 H2,1 (∞) < ∞.

(125)

−1 −1 and H2,1 are strictly increasing on [0, ∞), we find out On the other hand, since H1,2 that  −1 k 1 P 1,2 (∞) < ∞, u m (r ) ≤ H1,2 (126)

and

 −1 k 2 P 2,1 (∞) < ∞. vm (r ) ≤ H2,1

(127)

Then, the non-decreasing sequences {u m (r )}m≥0 and {vm (r )}m≥0 , are bounded above for all r ≥ 0 and all m. Putting these two facts together yields (u m (r ) , vm (r )) → (u (r ) , v (r )) as m → ∞

(128)

and the limit functions u and v are positive entire bounded radial solutions of system (26).

6.1.3

Existence of Entire Semi-Bounded Solutions

Proof of Theorem 4 We split it into two cases. Case (1): By an analysis similar to the Theorems 1 and 4 above, we have that  −1 k 1 P 1,2 (∞) < ∞ and v (r ) ≥ b + k 2 P 2,1 (r ) . u (r ) ≤ H1,2

(129)

P 1,2 (∞) < ∞ and P 2,1 (∞) = ∞

(130)

lim u (r ) < ∞ and lim v (r ) = ∞.

(131)

So, if

then r →∞

r →∞

In order, to complete the proofs it remains to proceed to the Case (2): In this case, we invoke the proof of Theorem 2. An easy computation yields that  −1 k 2 P 2,1 (r ) . u (r ) ≥ α + k 1 P 1,2 (r ) and v (r ) ≤ H2,1 (132) Our conclusion follows by letting r → ∞ in (132). Proof of Theorem 5 and 6 completed: It is a straightforward adaptation of the above proofs.

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7 Learning from an Example Let us illustrate one of the above theorems with an example. For the rest of the theorems, relevant examples can be constructed in a similar way. The simplest case occurs when  1 σ1 , x ∈ R N (N ≥ 3), Δu := 1+|x| 2v (133) |x| σ2 Δv := 1+|x|2 u , x ∈ R N (N ≥ 3), where 0 < σ1 ≤ 1, 0 < σ2 ≤ 1. In this case, for |x| 1 , a 2 (|x|) = a 2 (|x|) = , 2 1 + |x| 1 + |x|2 f 1 (v) = g1 (v) = ξ 1 (v) = vσ1 , f 2 (u) = g2 (u) = ξ 2 (u) = u σ2 ,

a 1 (|x|) = a 1 (|x|) =

(134) (135)

we notice that H1,2 (∞) = H2,1 (∞) = ∞,

(136)

P 1,2 (∞) = P 2,1 (∞) = ∞,

(137)

and thus by Theorem 1 we get that the system (133) has positively radial solution (u, v) such that lim u (|x|) = ∞ and lim v (|x|) = ∞. |x|→∞

|x|→∞

Moreover, the solution (u(|x|), v(|x|)) for the problem (133) is the limit of the sequence {(u m (|x|), vm (|x|))}m≥0 constructed inductively as follows ⎧ ⎪ = β, ⎨ u 0 (|x|) = α, v0 (|x|)

|x| 1−N t N −1 1 σ1 u m (|x|) = α + 0 t v (s)dsdt, r ≥ 0, 0 s 1+s 2 m−1 ⎪ ⎩ v (|x|) = β + |x| t 1−N t s N −1 s u σ2 (s)dsdt, r ≥ 0. m 0 0 1+s 2 m−1

(138)

8 Open Problem There are some interesting problems related to this chapter that one can investigate. Here, we point to one open problem. We consider the system of partial differential equations

Entire Solutions of a Nonlinear Diffusion System

⎧ N ⎪ ⎪ −Δu = λ1 g1 (v) for x ∈ R N , ⎨ −Δv = λ2 g2 (u) for x ∈ R , 0 < u (x) ≤ l1 for x ∈ R N , ⎪ ⎪ ⎩ 0 < v (x) ≤ l2 for x ∈ R N ,

483

(139)

where λ1 , λ2 are suitable parameters, the nonlinearities g1 , g2 : (0, ∞) → R are continuous and there exists s0 ∈ (0, ∞) such that: (I1) g1 (s), g2 (s) are non-decreasing for all s ∈ (s0 , ∞); (I2) lims→0+ gi (s) = gi (s0 ) = 0 for i = 1, 2; (I3) for i = 1, 2 we assume gi (s) < 0∀s ∈ (0, s0 ) ,

(140)

gi (s) > 0∀s ∈ (s0 , ∞) .

(141)

and We conjecture that there exists l1 , l2 ∈ [0, s0 ] such that the system (139) has a positive solution 2,α N  2,α N  R × Cloc R for some α ∈ (0, 1) , (142) (u, v) ∈ Cloc with (u (x) , v (x)) → (l1 , l2 ). Moreover, such a solution gives the existence of a sub/super-solution solution for (23) which combined with our above proofs and the sub- and super-solution method guarantees the existence of a solution for (23). The existence of a solution for (23) will be such that x → (∇ ln u(x), ∇ ln v(x)) , (143) is locally Lipschitz. Moreover it can be shown that |∇ ln u(x)| ≤ A1 (1 + |x|) and |∇ ln v(x)| ≤ A2 (1 + |x|).

(144)

Next, we can prove the transversality condition (15). These conditions in turn guarantee existence of a unique strong solution for the (11) which makes the candidate  control cˆ1 (x), cˆ2 (x) optimal. Acknowledgements The author would like to thank Professor Traian A. Pirvu for valuable comments and suggestions which further improved this chapter.

References 1. Alvarez, O.: A quasilinear elliptic equation in R N . In: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 126, pp. 911–921 (1996) 2. Arnold, L.: Stochastic Differential Equations. Wiley, New York (1974)

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3. Bensoussan, A., Sethi, S.P., Vickson, R., Derzko, N.: Stochastic production planning with production constraints. SIAM J. Control Optim. 22, 920–935 (1984) 4. Covei, D.-P.: On the radial solutions of a system with weights under the Keller-Osserman condition. J. Math. Anal. Appl. 447, 167–180 (2017) 5. Covei, D.-P.: Boundedness and blow-up of solutions for a nonlinear elliptic system. Int. J. Math. 25(9), 1–12 (2014) 6. Covei, D.-P.: Existence Theorems for a Class of Systems Involving Two Quasilinear Operators, to be published in vol. 83. Izvestiya: Mathematics (2019). https://doi.org/10.1070/IM8731 7. Ghosh, M.K., Arapostathis, A., Marcus, S.I.: Optimal control of switching diffusions with application to flexible manufacturing systems. SIAM J. Control. Optim. 31(5), 1183–1204 (1993) 8. Gregorio, D.: A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller-Osserman condition. Mathematische Annalen 353(1), 145–159 (2012) 9. Grosse, H., Martin, A.: Particle Physics and the Schrodinger Equation. Cambridge Monographs on Particle Physic’s, Nuclear Physics and Cosmology (1997) 10. Franchi, B., Lanconelli, E., Serrin, J.: Existence and uniqueness of nonnegative solutions of quasilinear equations in R N . Adv. Math. 118, 177–243 (1996) 11. Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Annali di Matematica 186(3), 539–564 (2007) 12. Hamydy, A., Massar, M., Tsouli, N.: Existence of blow-up solutions for a non-linear equation with gradient term in R N . J. Math. Anal. Appl. 377, 161–169 (2011) 13. Keller, J.B.: On solution of Δu = f (u). Commun. Pure Appl. Math. 10, 503–510 (1957) 14. Krasnosel’ski˘ı, M.A., Ruticki˘ı, Y.B.: Convex Functions and Orlicz Spaces, Translated from the first Russian edition by Boron, L.F., Noordhoff, P., LTD. - Groningen - the Netherlands (1961) 15. Kon’kov, A.A.: On properties of solutions of quasilinear second-order elliptic inequalities. Nonlinear Anal.: Theory, Methods Appl. 123–124, 89–114 (2015) 16. Jaros, J., Takasi, K.: On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations. In: Proceedings of the Royal Society of Edinburgh, vol. 145A, pp. 1007–1028 (2015) 17. Lair, A.V.: A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems. J. Math. Anal. Appl. 365(1), 103–108 (2010) 18. Lair, A.V.: Entire large solutions to semilinear elliptic systems. J. Math. Anal. Appl. 382, 324–333 (2011) 19. Lasry, J.M., Lions, P.L.: Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with State Constraints. Mathematische Annalen, vol. 283, pp. 583–630 (1989) 20. Lieberman, G.M.: Asymptotic behavior and uniqueness of blow-up solutions of quasilinear elliptic equations. J. d’Analyse Math ématique 115, 213–249 (2011) 21. Li, H., Zhang, P., Zhang, Z.: A remark on the existence of entire positive solutions for a class of semilinear elliptic systems. J. Math. Anal. Appl. 365, 338–341 (2010) 22. Losev, A.G., Mazepa, E.A.: On asymptotic behavior of positive solutions of some quasilinear inequalities on model Riemannian manifolds. Ufa Math. J. 5, 83–89 (2013) 23. Luthey, Z.A.: Piecewise Analytical Solutions Method for the Radial Schrodinger Equation, Ph.D. Thesis in Applied Mathematics, Harvard University, Cambridge (1974) 24. Mazepa, E.A.: The Positive Solutions to Quasilinear Elliptic Inequalities on Model Riemannian Manifolds. Russian Mathematics, vol. 59, 18–25 (2015) 25. Naito, Y., Usami, H.: Entire Solutions of the Inequality div(A(|Du|)Du) ≥ f (u). Mathematische Zeitschrift, vol. 225, pp. 167–175 (1997) 26. Naito, Y., Usami, H.: Nonexistence results of positive entire solutions for quasilinear elliptic inequalities. Can. Math. Bull. 40, 244–253 (1997) 27. Osserman, R.: On the inequalityΔu ≥ f (u). Pac. J. Math. 7, 1641–1647 (1957) 28. Smooke, M.D.: Error estimates for piecewise perturbation series solutions of the radial Schrödinger equation. SIAM J. Numer. Anal. 20, 279–295 (1983)

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Goal Programming Models for Managerial Strategic Decision Making Cinzia Colapinto, Raja Jayaraman and Davide La Torre

Abstract The Goal Programming (GP) model is an important Multiple Objective Programming (MOP) technique that has been widely utilized for strategic decision making in presence of competing and conflicting objectives. The GP model aggregates multiple objectives and allows obtaining satisfying solutions where the deviations between achievement and the aspirations levels of the attributes are to be minimized. The GP model is easy to understand and to apply: it is based on mathematical programming techniques and can be easily solved using software packages such as LINGO, MATLAB, and AMPL. The GP describes the spectrum of the Decision Maker’s preferences through a user-friendly and learning decision-making process. This chapter aims to present the state-of-the-art of GP models and highlight its applications to strategic decision making in portfolio investments, marketing decisions and media campaign.

C. Colapinto Department of Management, Ca’ Foscari University of Venice, Venice, Italy e-mail: [email protected] C. Colapinto Graduate School of Business, Nazarbayev University, Astana, Kazakhstan R. Jayaraman (&) Department of Industrial & Systems Engineering, Khalifa University of Science & Technology, Abu Dhabi, UAE e-mail: [email protected] D. La Torre Dubai Business School, University of Dubai, Dubai, UAE e-mail: [email protected] D. La Torre Department of Economics, Management, and Quantitative Methods, University of Milan, Milan, Italy © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_16

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1 Introduction Success in value investing mainly depends on the Decision Maker (DM)’s understanding of the business, however there are some basic frameworks the DM can use to analyze any business decisions. Indeed, quantitative techniques in portfolio selection and management can be valuable to managers, as they can better understand interrelationships among assets and the marketplace, and use this knowledge to their advantage. Nowadays the portfolio approach is useful in a wide range of fields. For instance, it could be applied to optimize paid search marketing, as diversification of a search-marketing portfolio can ameliorate the advertiser’s bottom line. Portfolio Management can also be used to select a portfolio of new product development projects: the DM manages the product pipeline and makes decisions about the product portfolio trying to achieve different goals such as maximization of the profitability of the portfolio, balance and support of the strategy of the firm. Indeed, rapidly changing technologies, shorter product life cycles, and intense global competition makes portfolio management for product innovation crucial for a company’s survival and success. Obviously, Research and Development (R&D) investments are often treated like financial investments in the stock market. R&D managers strive to maximize the value of the portfolio, leveraging return on R&D spending by appropriately designing a balanced portfolio, and a portfolio investment strategy that is aligned with the company’s overall business strategy. Goal Programming (GP) formulations are a particular class of optimization models, which are well-suited for portfolio construction under multiple competing objectives and investment goals. This chapter is organized as follows. Section 2 provides a brief background of Multiple Objective programming (MOP) and Pareto optimality. Section 3 provides an overview of popular goal programming techniques. Section 4 presents three different applications of GP formulations. We close the chapter with a discussion about the advantages and possible applications and extensions of GP model in decision making.

2 Background: Multiple Objective Programming Multiple Objective Programming is a discipline that considers decision-making situations involving multiple and conflicting criteria. Some examples of conflicting criteria that have been considered in literature includes cost or price, quality, satisfaction, risk, and others. For instance, in portfolio management the DM is interested in getting high returns but at the same time reducing the risks: the stocks that have high return typically are also associated with a high risk of losing value. In a service industry, customer satisfaction and the cost of providing service are two conflicting criteria that very often need to be considered simultaneously.

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Considering multiple criteria explicitly leads to more informed and better decisions. However, typically a unique optimal solution does not exist and it is necessary to use DM’s preferences to differentiate between available solutions. Many important advances have been made in this field since the start of the modern MOP discipline in the early 1960s, including new approaches, innovative methods, hybrid techniques, and sophisticated computational algorithms. The general formulation of a MOP model can be formulated as follows [33]:
Given a set of p criteria f1 ; f2 ; . . .; fp optimize the vector f1 ð xÞ; f2 ð xÞ; . . .; fp ð xÞ under the condition that x 2 DRn where D designates the set of feasible solutions.
 Let us define a vector-valued function f ðxÞ :¼ f1 ð xÞ; f2 ð xÞ; . . .; fp ð xÞ ; according to this, a classical MOP problem can be formulated as (assuming all objectives have to be minimized): Min f ðxÞ x2D

We say that a point ^x 2 D is a global Pareto optimal solution or global Pareto efficient solution if f ðxÞ 2 f ð^xÞ þ ðRpþ nf0gÞc for all x 2 D, this definition of optimal solution is based on the notion of Pareto ordering induced by the cone. Practically speaking, a Pareto optimal solution describes a state in which the input variables are distributed in such a way that it is not possible to improve a single criterion without also causing at least one other criterion to become worse off than before the change. In other words, a state is not Pareto efficient if there exists a certain change in allocation of input variables that may result in some criteria being in a better position with no criterion being in a worse position than before the change. If a point x 2 D is not Pareto efficient, there is potential for a Pareto improvement and an increase in Pareto efficiency. We refer the readers to recent advances and various mathematical techniques of MOP in Zopounidis and Pardalos [42].

3 Goal Programming Models: Some of the Existing Variants The GP model is based on mathematical programming commonly solved using powerful mathematical programming software such as AMPL, Lindo, GAMS and CPLEX. The GP satisfies the spectrum of the DM’s preferences where some trade-offs can be made through a user-friendly and learning decision-making process. It is important to point out the investment decisions are actually taken by the DM and the mathematical model is to assist and not substitute the DM. The central idea of GP is the determination of the goal levels gi; i ¼ 1; . . .; pfor the objective function and the minimization of any (positive or negative) deviation

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from these levels. Charnes et al. [13], and Charnes and Cooper [12] first introduced GP and over the decades GP Models have been applied in several fields and it is still the most popular technique within the field of MOP. If we assume to optimize simultaneously p different conflicting criteria f1 ; f2 ; . . .; fp , the GP model is an aggregating methodology that allows to obtain a solution representing the best compromise that can be achieved by the DM. We present the mathematical formulations of three popular and commonly used models namely, Weighted, Stochastic and Fuzzy GP.

3.1

Weighted Goal Programming

The DM can show different appreciation of the positive and negative deviations based on their relative importance in the objective. The Weighted GP (WGP) model can express this different appreciation through corresponding weights wiþ and w i respectively. The mathematical formulation of the WGP, as applied to the portfolio selection problem is as follows: Min Z =

p X

 wiþ diþ þ w i di

i¼1

Subject to: fi ðxÞ  diþ þ d i ¼ gi ; n X xj ¼ 1

i ¼ 1; . . .; p

j¼1

x2D diþ ; d i  0;

i ¼ 1; . . .; p

Among many applications we can cite Callahan [10] and Kvanli [22] illustrate a WGP investment planning model. Blancas et al. [8] propose a synthetic sustainability indicator based on a WGP approach to support the decision making process in the field of tourism. Jha et al. [20] include the practical aspect of segmentation and develop a model which deals with optimal allocation of advertising budget (through a WGP model) for multiple products which is advertised through different media in a segmented market.

3.2

Stochastic Goal Programming

In many financial contexts, the DM has to take decisions under uncertainty. Hence the objective functions and the corresponding goals are, in general, random variables. The Stochastic GP (SGP) model deals with the uncertainty related to the

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491

decision making situation as we assume that the goal values are stochastic and follow a specific probability distribution. The general formulation of the SGP is as follows: Min Z =

p X





diþ þ d i

i¼1

Subject to: 





g; fi ðxÞ  diþ þ d i ¼ i

n X

i ¼ 1; . . .; p

xj ¼ 1

j¼1

x2D 



diþ ; d i  0;

i ¼ 1; . . .; p

where ~ gi 2 Nðli ; ri Þ. Martel and Aouni [29] develop the concept of satisfaction function in order to incorporate explicitly the DM’s preferences. A satisfaction function F is taking values in [0,1]. Therefore, it has a value of 1 when the DM is totally satisfied; otherwise it is monotonically decreasing and can take values between 0 and 1 (see Fig. 1). We can identify three different thresholds, namely: (a) the indifference threshold (aid): total satisfaction when the deviations are within the interval, (b) the nil satisfaction threshold (aio): there is no satisfaction when the deviations reach this threshold but the solution is not rejected, (c) the veto threshold (aiv): rejecting any solution that lead to deviations larger than this threshold. Considering veto thresholds leads to a partially non-compensatory model in the sense that a bad performance on one objective cannot be compensated by a good

Fig. 1 The general form of satisfaction function

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one on another objective. The SGP with satisfaction function [3], which incorporates explicitly the DM’s preferences, is as follows: Max Z ¼

p X

wiþ F

  diþ þ w F d i i

i¼1

Subject to: 





g; fi ðxÞ  diþ þ d i ¼ i

n X

i ¼ 1; . . .; p

xj ¼ 1

j¼1

x2D 

 0  d i  aiv ;

0

 diþ

 aivþ ;

i ¼ 1; . . .; p i ¼ 1; . . .; p

Aouni et al. [3] integrate the DM’s preferences in a decision situation where the DM wants to invest a certain amount of capital in the Tunisian stock exchange market where stocks returns are not known with certainty. An alternative way to include randomness is to consider the so-called scenariobased models [2, 4, 19]. If we assume that the space of all possible events or scenarios X ¼ fx1 ; x2 ; . . .; xN g with associated probabilities pðxs Þ ¼ ps is finite and the objective functions and the corresponding goals are depending on the scenario xs , the above SGP model with satisfaction function can be readily extended to: Max Z =

p X

   wiþ F diþ þ w i F di

i¼1

Subject to: fi ðx; xs Þ  diþ þ d i ¼ gi ðxs Þ; n X xj ¼ 1 j¼1

x2D  0  d i  aiv ðxs Þ; 0  diþ

where xs 2 X is fixed.

 aivþ ðxs Þ;

i ¼ 1; . . .; p i ¼ 1; . . .; p

i ¼ 1; . . .; p

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3.3

493

Fuzzy Goal Programming

The Fuzzy GP (FGP) model was developed to deal with some decisional situations where the DM can only give vague and imprecise goal values; in other words aspiration levels are not known precisely. The FGP is based on the fuzzy sets theory developed by Zadeh [37] and the concept of membership functions introduced by Zimmerman [39]. Narasimhan [30] and Hannan’s [18] FGP formulations also use the concept of membership functions to deal with the fuzziness of the goal values using triangular membership functions. The general formulation of the membership function requires two acceptability degrees (lower and upper) [41] and the functions are assumed to be linear. Dhingra et al. [15], Rao [31] and Zimmerman [38, 40] have developed an approximation procedure for the non-linear membership functions. The general formulation of FGP model as developed by Hannan [18] is as follows: Max Z ¼ k Subject to: fi ðxÞ gi  diþ þ d ; i ¼ Di Di n X xj ¼ 1

i ¼ 1; . . .; p

j¼1

x2D k þ diþ þ d i  1; k; diþ ; d i

 0;

i ¼ 1; . . .; p

i ¼ 1; . . .; p

where Di is the constant of deviation of the aspiration levels gi . This constant is pre-specified by the DM. Arenas-Parra et al. [7] have utilized FGP approach for the portfolio selection problem.

3.4

Other GP Variants

Finally, we introduce other GP variants used to study multi-criteria problems, highlighting their current use of combined/integrated models. The Lexicographic or pre-emptive GP (LGP) is based on the optimization of the objectives according to their relative importance: the most important objectives will be at the highest levels of priority and they will be optimized first and so on. Thus the objectives at the lowest levels of priority will have a marginal impact on the final decision. Wang et al. [35] use a combined analytical hierarchy process - LGP approach for supplier selection problem in supply chains.

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In the case the returns of assets are not normally distributed, higher moments (such as skewness and kurtosis) have to be considered: the DM copes with a trade-off between competing and conflicting objectives, i.e., maximizing expected return and skewness, while minimizing variance and kurtosis, simultaneously. Lai [23] used the Polynomial GP (PGP) in order to explore incorporation of investor’s preferences in the construction of a portfolio with skewness. The Min-Max GP model [32] seeks the minimization of the maximum deviation from any single goal in portfolio selection: it uses essentially the same concepts as the WGP, except instead of minimizing the sum of deviations this model seeks the solution that minimizes the worst unwanted deviation from any single goal. More recently, the interactive multiple GP model (IMPG)’s incorporates all the advantages of “traditional” GP, while circumventing the unnecessary burden of obtaining a “complete” picture of the DM’s preference pattern [17]. Indeed, an interactive procedure progresses by seeking this information from the investor, removing the need to make the preference structure more explicit. Lee and Shim [26] present an interactive GP model starting on the original work by Lee et al. [24] in strategic management. For a more extensive literature review on GP models we refer the readers to Aouni et al. [6] and Colapinto et al. [14].

4 Applications We present examples that illustrate different applications of GP formulations. In details, we are going to describe three GP models applied to: (a) portfolio management, (b) strategic marketing, and (c) media planning.

4.1

A SGP Model with Satisfaction Function for Portfolio Management

In portfolio selection problems, the Financial DM (FDM) considers simultaneously several factors such as: return, risk, liquidity, gross book value per share, capitalization ratio, and stock market value of each firm. These objectives are usually incommensurable and conflicting and the best portfolio requires some compromises among various criteria by the FDM. The trade-offs are based on the FDM’s structure of preferences. The first bi-criteria portfolio selection model was proposed by Markowitz [28]. The main objective of the classical model is to obtain the best portfolio that may maximize the FDM’s return while simultaneously minimize the risk of financial losses. Given a space of events X ¼ fx1 ; x2 ; . . .; xN g with associated probabilities pðxs Þ ¼ ps , we assume that the FDM takes his/her decisions on a stochastic linear multi-criteria optimization model formulated as:

Goal Programming Models for Managerial Strategic Decision Making

Max Z1 ¼ Min Z2 ¼ Subject to:

Xn

x i¼1 i

Xn j¼1

Xn

495

lj ðxs Þxj

r ðxs Þxj j¼1 j

¼1

x2D where: (a) (b) (c) (d) (e) (f)

Z1 is the stochastic return of the portfolio Z2 is the stochastic risk of the portfolio xi is the proportion of the budget to be invested in security j li is the stochastic return of security j ri is the stochastic risk of security j D is the set of feasible solutions.

In the above stochastic model x describes possible different scenarios. We propose a GP model to deal with the above stochastic context, using the notion of deterministic equivalent formulation introduced by Caballero et al. [9]. The notion of deterministic equivalent consists of replacing the initial stochastic objective functions, which are difficult to be analyzed, with the expected value of all objective functions. In this manner the stochastic problem is reduced to a deterministic model. Of course a lot of information is lost by reducing the stochastic model to a deterministic one and this can lead to different definitions of the concept of efficient solution and efficient sets that can be obtained for the same stochastic problem. Since they deal with different characteristics of the initial problem they are non-comparable. Caballero et al. [9] assert that the concepts of efficient solution under certain conditions for a stochastic multi-objective programming are closely related. Given any particular problem from the established relationships the concept of efficiency is the one that best fits the preferences of the FDM. The GP model with satisfaction function is illustrated using simulated scenarios (see Tables 1 and 2) obtained from the data of the Tunisian stock exchange market available in Ben Abdelaziz et al. [1]. The choice of the ten stocks is supposed to be independent. The deterministic equivalent formulation of the above stochastic model reads as: Max Z1 ¼ Min Z2 ¼ Subject to:

Xn

x j¼1 j

x2D

Xn

 E lj xj  E r j xj j¼1

j¼1 Xn

¼1

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Table 1 Rate of return of stocks from stock exchange market Events/securities

S1

S2

S3

S4

S5

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Mean Events/securities

−1.6407 1.1965 –1.2962 0.6120 –0.5761 –0.6970 –1.0075 0.2642 2.2613 0.6736 –0.02099 S6

0.0322 −0.6556 0.1922 0.7931 0.6148 1.3447 0.3509 2.1381 0.7589 –0.2285 0.53408 S7

–0.2907 0.9379 –0.3401 –0.1693 0.8296 0.5318 1.1727 –0.3094 0.6688 1.9818 0.50131 S8

–0.7878 0.3610 –0.2735 1.7931 0.7547 –0.0932 –2.0650 –0.1729 1.1859 0.2622 0.09645 S9

–0.6686 0.9944 1.0520 0.1135 2.6507 –0.5764 –0.2296 0.4338 0.3012 1.3112 0.53822 S10

1.1357 0.5725 1.3342 2.4851 0.1180 2.3149 –0.4660 1.0750 1.1440 2.5944 1.23078

–1.3194 –0.4361 –0.1004 0.0215 0.0941 0.0337 –0.1055 –0.4435 –0.7442 –0.3541 –0.51393

1.2922 –0.6783 –1.0707 0.6340 –0.1958 1.4385 –0.8466 0.9723 –1.6181 0.7902 0.07177

1.0438 0.2358 –0.0925 0.4969 –0.1010 0.7004 –1.5535 0.0004 0.0938 0.3037 0.11278

1.2933 –0.9456 1.7890 –1.1960 0.0107 0.3077 0.5256 0.6331 0.8684 1.0807 0.43669

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Mean

Table 2 Risks of stocks from the stock exchange market Events/securities

S1

S2

S3

S4

S5

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Mean Events/securities

1.1666 2.2283 2.1728 1.1592 6.1608 3.9946 1.6560 0.6371 2.7000 1.9834 2.3859 S6

2.4065 0.3635 2.5525 1.3802 0.4548 0.0734 1.5056 2.1861 1.1961 0.0559 1.21746 S7

3.3958 1.4174 0.0085 0.2343 0.3859 1.0999 0.2071 0.4652 0.0000 5.4431 1.26572 S8

2.7400 0.0719 0.4493 0.3811 0.4605 0.0645 0.1649 2.6861 1.8632 0.0388 0.89203 S9

6.1057 0.0381 0.1024 0.0083 4.3228 2.6128 0.7742 0.0819 1.2887 1.4534 1.67883 S10

x1 x2 x3

0.0153 0.1253 0.0004

0.0990 0.0007 0.0574

0.2566 0.9951 0.0393

0.0001 0.2327 0.1631

0.652 0.9584 0.2175 (continued)

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Table 2 (continued) Events/securities

S6

S7

S8

S9

S10

x4 x5 x6 x7 x8 x9 x10 Mean

0.1943 2.1875 0.4980 1.9964 2.0239 0.3023 5.2098 1.25532

0.0012 0.0260 0.2152 0.3535 0.0541 0.0424 0.0000 0.08495

0.0314 1.0185 0.0076 3.1890 0.0475 0.4626 0.0006 0.60482

2.3941 0.0772 0.0082 0.0955 0.4885 0.0350 0.4227 0.39171

1.0640 1.5360 0.0296 0.2576 0.9609 0.1495 0.2212 0.60471

Table 3 Budget values

Events

Budget values

Events

Budget values

x1 x2 x3 x4 x5 Mean

1,001 1,003 1,005 997 996

x6 x7 x8 x9 x10 1000

990 1,001 995 1,002 1,010

where (a) (b) (c) (d) (e) (f)

Z1 is the expected return of the portfolio Z2 is the expected risk of the portfolio xj is the proportion of the budget to be invested in security j  E lj : the expected return of security j  E rj : the expected risk of security j D is the set of feasible solutions and it takes into account the portfolio diversification.

Let us suppose that the budget is a random variable whose distribution values are listed in Table 3 according to the events’ occurrence. On the other hand, let us suppose that the following additional financial constraints are to be satisfied: • S1 þ S2  600 • S6  400 • S2  100 Let g1 and g2 be the two random aspiration levels for the objective functions Z1 and Z2 whose values are listed in the following Table 4. As satisfaction function, let us consider the following expression:

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Table 4 Aspiration levels

Events

g1

g2

x1 x2 x3 x4 x5 Mean

850.61 830.15 820.11 810.55 800.43

910.01 x6 910.04 x7 910.02 x8 910.10 x9 910.02 x10 g1 = 830.780

Fðx; aÞ ¼

Events

g1

g2

832.13 910.01 820.15 910.04 817.83 910.06 840.74 910.20 885.10 910.20 g2 = 910.07

1 1 þ a2 x 2

where a is a parameter. This function exhibits the behavior to be considered as a satisfaction function and it is easy to verify the following properties: (a) (b) (c) (d) (e) (f)

Fð0Þ ¼ 1 Fð þ 1Þ ¼ 0 F 00 ðxÞ ¼ 0 , x ¼ 1=2a 0:9  FðxÞ  1 , 0  x  1=3a 0  FðxÞ  0:1 , x  3=a 0  FðxÞ  0:01 , x  3=a

Natural candidates for the indifference threshold and the dissatisfaction threshold are, respectively, aid ¼ 1=3a and aio ¼ 3=a. Let us assume the veto threshold aiv ¼ 2  aio ¼ 6=a. In the following model let us choose a ¼ 0:1. The GP Model þ  with satisfaction function and weights w1þ ¼ 0:5; w 1 ¼ 0:5 w2 ¼ 0:1; w2 ¼ 0:1 can be written as: þ  Max Z ¼ 0:5Fðd1þ Þ þ 0:5Fðd 1 Þ þ 0:1Fðd2 Þ þ 0:1Fðd2 Þ

Subject to:  0:02099  S1 þ 0:53408  S2 þ 0:50131  S3 þ 0:09645  S4 þ 0:53822  S5 þ 1:23078  S6  0:51393  S7 þ 0:07177  S8 þ 0:11278  S9 þ 0:43669  S10  d1þ þ d 1 ¼ 830:780 2:38588  S1 þ 1:21746  S2 þ 1:26572  S3 þ 0:89203  S4 þ 0:39171  S9 þ 0:60471  S10  d2þ þ d 2 ¼ 910:07 S1 þ S2 þ S3 þ S4 þ S5 þ S6 þ S7 þ S8 þ S9 þ S10 ¼ 1000 S1 þ S2  600 S6  400 S2  100 S1; S2; S3; S4; S5; S6; S7; S8; S9; S10  0 þ  d1þ ; d 1 ; d2 ; d2  0

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The solution provided by LINGO [27] is a portfolio with investments only in S2 (100 units), S6 (400 units), and S10 (500 units).

4.2

Strategic Marketing Decisions Using GP with Satisfaction Function

New information technologies are facilitating more complex interactions that are organized by networks. According to Castells [11], the network enterprise is a new form of organization characteristic of economic activity, but gradually extending its logic to other domains and organizations. Moreover, most firms are complex organizations that market many different products in many different business areas, thus a portfolio approach is well suited. The process of evaluating and implementing strategic marketing decisions is characterized by high levels of uncertainty, potential synergies between different options, and long term consequences. Moreover, each decision can affect all the business functions, thus contemplated marketing actions must be evaluated and analyzed. Strategies and their implementation plan must be developed and executed at the corporate, business and products levels. Besides Product marketing plan, a company needs a strategic marketing plan. According to Wiersema [36], the strategic marketing perspective is defined as having the dual task of providing a marketplace perspective on the process of determining corporate direction, and guidelines for the development and execution of marketing programs that assist in attaining the corporate objectives. The application of GP models allows a more efficient analysis for decision making in marketing. In a company there exist potential conflicts between different areas (such as the production and marketing areas, see Taylor III and Anderson, [34], thus GP model can deal with the complex trade-off decisions involved in this kind of situations, which require greater coordination and integration between different business functions. We extend the case study presented in Lee and Nicely [25] concerning Raynebo, Inc., a company managed by Mr. Rayne. The company is a lessor of color television receivers in a large metropolitan area. In the same area, there are three other competitors and Raynebo, Inc. is holding 25% of the market share. Mr. Rayne has defined three broad goals for his company, in order of priority as follows: 1. Achieve the minimum ROI of 15% and strive for the target ROI of 20%. 2. Maintenance or improvement of market volume which is now 1250 leased sets. 3. Retention of present workforce by minimizing the turnover amongst the workers. The company is expected to have two new clients and delivery for each client 50 sets of leased TVs. One client will be served in March and the other in June. In order to guarantee that the two new clients buy the products from the company, the management decided to give 2 months free service for them.

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Moreover, the company’s main expenses can be categorized as salaries and promotional activities. The company dedicates $12,500 yearly for promotional activities and is targeting to increase them to $14,500. A sum of $118,004 of the expensive goes for salaries to all of its employees and also the management is intending on increasing the salaries by 5%. Mr. Rayne has set several goals for the next business year. The goals and their equations are presented below. For the modelling purposes the following are the names and meanings of the variables: • • • • • •

X1 X2 X3 X4 X5 X6

denotes denotes denotes denotes denotes denotes

to to to to to to

number of leased sets from present investment base. number of leased sets to result for normal growth. number of leased sets required by new client first of March. salaries of all employees. promotional activities. number of leased sets required by new client first of June.

The model has several constraints and they are presented below. The number of television sets to be leased from the original investment base must be limited to sets in stocks, that is (P1) X1 d1þ ¼ 1250 In addition to that, the management of Raynebo, Inc. has set the following goals in ordinal ranking of priorities: (P2) Achieve a ROI of at least 15% þ 192 X1 þ 150 X2 þ 81:5 X3 þ 44 X6 X4 X5 þ d 2  d2 ¼ 85; 996

(P3) Maintain at least 1200 leased sets þ X 1 þ d 3 d3 ¼ 1; 200

(P4) Retain the current level of employment in terms of wages and salaries X4 d4þ ¼ 118; 004 (P5) Maintaining the promotional expenditure at their current level X5 d þ 5 ¼ 12; 500 (P6) Achieve normal market growth of 2% or 25 additional leased sets above the present level of 1250. þ X1 þ X2 þ d 6 d6 ¼ 1; 275

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(P7) Achieve the target ROI of 20% þ 192 X1 þ 136 X2 þ 68:67 X3 þ 34:67 X6 X4 X5 þ d 7  d7 ¼ 11; 099

(P8) Grant a pay increase of at least 5% to employees X4 d8þ ¼ 123; 904:2 (P9) Increase the promotional expenditure by at least $2000 X5 d9þ ¼ 12; 500 (P10) Expand volume of business by at least 100 sets for the two new clients each 50 sets. X3 þ d 10 ¼ 50 X6 þ d 11 ¼ 50 As in the previous example, let us suppose that the satisfaction function assumes the following form: 1 Fðx; aÞ ¼ 1 þ a2 x 2 We analyze the proposed model with a 2 f1; 0:1; 0:01g. The GP model with the satisfaction function is shown below:     þ   þ  Max Z ¼ F d2þ þ F d2þ þ F d þ F d þ F d5þ 2 þ F d3 3 þ F d4    þ F dþ þ þ F d6    þ     þ F d7þ þ F d þ F d9þ þ F d 7 þ F d8 10 þ F d11 Subject to: X1 d1þ ¼ 1250 192 X1 þ 150 X2 þ 81:5 X3 þ 44 X6 X4 X5 þ d  2  d þ 2 ¼ 85996 þ X1 þ d 3 d3 ¼ 1200 X4 d4þ ¼ 118004 X5d5þ ¼ 12500 X1 þ X2  d6þ þ d 6 ¼ 1275 192 X1 þ 136 X2 þ 68:67 X3 þ 34:67 X6 X4 X5  d7þ þ d 7 ¼ 110996 þ X4d8 ¼ 123904:2 X5d9þ ¼ 12500 X3 þ d 10 ¼ 50 X6 þ Z 11 ¼ 50 þ þ þ  þ þ þ  þ  þ þ   d1 ; d2 ; d 2 ; d3 ; d3 ; d4 ; d5 ; d þ ; d þ ; d7 ; d7 ; d8 ; d9 ; d10 ; d11 [ ¼ 0 X1 ; X2 ; X3 ; X4 ; X5 ; X6 [ ¼ 0

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Table 5 Output of the GP model with satisfaction function Variable/parameter a

a=1

a = 0.1

a = 0.01

X1 X2 X3 X4 X5 X6

1250.002 31.12276 50.00209 123904.2 14500 50.00105

1251.257 28.30444 51.66706 123904.2 14500 50.80718

1250 29.64506 52.3378 123904.2 14500 51.17972

Table 6 Priorities Achieved by the GP Model with satisfaction function Output

GP with satisfaction function formulation

Lee & Nicely formulation

Achieve a ROI of at least 15% Maintain at least 1200 leased sets Retain the current level of employment in terms of wages and salaries Maintaining the promotional expenditure at their current level Achieve normal market growth of 2% Achieve the target ROI of 20% Grant a pay increase of at least 5% to employees Increase the promotional expenditure by at least $2000 Expand volume of business by at least 100 sets for the two new clients each 50 sets

Achieved Achieved Achieved

Achieved Achieved Achieved

Achieved

Achieved

Achieved Achieved Achieved Achieved

Achieved Achieved Achieved Not Achieved

Achieved

Achieved

Using LINGO [27] with three different values for a produced the following results. From Table 5, it is obvious that the values are more or less close to each other for different values of a. The results (Table 6) show that all the predefined goals have been achieved using the proposed model with satisfaction functions. Clearly, the company now can serve its two new clients in March and June since the under achievement for d−10 = 0, and d−11 = 0 for a values. It also achieves the growth of 2% in the market d−6 = 0 for all a values. Moreover, the ROI has reached 20% and in order to compare the under achievement that occurred in the GP model, the results show that d−7 = 0.0000304 for a = 1, d−7 = 0.023 for a = 0.1, and d−7 = 0.034 for a = 0.01 which are rounded to zero.

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4.3

503

Media Campaign Strategy Using GP with Satisfaction Function

The model we are going to present is formulated as an extension of the model developed by Fernandez et al. [16]. Recently, Kaul et al. [21] present a WGP approach to multi-period media planning to determine optimal schedule of advertisements maximizing advertisement impressions and minimizing advertising expenditures. Aouni et al. [5] formulate a SGP model with satisfaction function for the optimal allocation of advertisements in different vehicles. The concept of “Media diet” was developed to assess each individual’s exposure to content in the media, based on the combination of media consumption and content. Media diet data usually consist of several questions like: – Consumption of the last issue, – Time since last consumption, – Number of saw/read/listened issues among the last five, for instance. The readership matrix’ task is to give an overview of what the entire readership population looks like and to ensure that each panel represents the readership on an average day. In order to create the matrix, the most commonly used variables are gender, age and reader frequency. For a representative individual, we suppose there exists an index I(j) of how much a person sees/listens/reads a particular vehicle j and we supposed to have n vehicles. For each fixed vehicle, the corresponding element in the matrix is a synthetic number in [0, 1] which represents a proxy of the above readership matrix and it can be interpreted as the averaged probability that an individual will be exposed to an advertisement placed in vehicle j. If xj advertising insertions are purchased in vehicle j, the average expected number of exposures per individual is I(j)xj. In this model the DM has two objectives: 1. The efficacy of a schedule x ¼ ðx1 ; x2 ; . . .; xn Þ is the number of exposures per n P I ð jÞxj . individual in the population, and this is obtained by 2. The total cost of schedule x ¼ ðx1 ; x2 ; . . .; xn Þ is

n P j¼1

The multi-criteria problem can be formulated as

j¼1

cðjÞxj .

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Table 7 Media consumption in Italy

Vehicle

Media consumptions

TV Radio Newspaper

94.79 67.28 52.95

Max Min

n X j¼1 n X

I ð jÞxj cðjÞxj

j¼1

Subject to x2D where the set D is general enough for covering a large set of restrictions found in problems of media campaign. Given two goals g1 (2,456) and g2 (904,000), we propose the following GP model with satisfaction function: Max Z ¼

2 X

 wiþ Fðdiþ Þ þ w i Fðdi Þ

i¼1

Subject to n X

þ I ð jÞxj þ d 1  d1 ¼ g1

j¼1 n X

þ cð jÞxj þ d 2  d2 ¼ g 2

j¼1

x2D 0  diþ  aivþ 0  diþ  aivþ

ði ¼ 1; 2Þ ði ¼ 1; 2Þ

Let us consider the following illustrative example based on real data from the Italian media market. Table 7 shows the real media diet in Italy according to Census data. Table 8 Prices’ list for an advertising slot

Vehicle

Prices’ list for an advertising slot

TV Radio Newspaper

73,600 11,200 54,801

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Table 8 shows average of official prices’ list for an advertising slot and provides the realizations of the random variable for different vehicles (TV, radio, newspaper). As in the previous example, let us assume the satisfaction function F to take the expression Fðx; aÞ ¼

1 : 1 þ a2 x 2

LINGO [27] provides the following optimal solution (x1, x2, x3) = (1, 35, 8) that is interpreted as maximize the advertising opportunities through radio, followed by newspaper and TV media.

5 Conclusions In the real world, strategic management decisions often imply to harmonize different needs and interests or to balance conflicting criteria. An important way to model such problems is through the use of a goal programming approach, which can combine the optimization with the DM desire to satisfy several goals simultaneously. The learning offered by GP models helps generating scenarios where the DM can interact and make changes to the model parameters to enhance the decision-making process. Indeed, the GP approach allows for a better modelling of real managerial situations: it is rare that criteria are to be minimized, rather the DM need to achieve certain objectives to satisfy all stakeholders’ perspectives. For instance in green supply chain management the DM aims at keeping the level of pollution below a certain sustainable threshold rather than purely minimizing. Is it realistic to reach a society with a zero level pollution? Similarly it is not possible to reset the production costs to zero aiming at delivering a differentiate product. Again in supply chain management, the simultaneous profit maximization and the minimization of inventory costs, or rejected rate or environmental impacts requires a multi-objective approach. This chapter drives the reader through the main features of the GP approach, and presents some relevant applications in strategic decision making.

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Modeling Highly Random Dynamical Infectious Systems Divine Wanduku

Abstract Random dynamical processes are ubiquitous in all areas of life:- in the arts, in the sciences, in the social sciences and engineering systems etc. The rates of various types of processes in life are subject to random fluctuations leading to variability in the systems. The variabilities give rise to white noise which lead to unpredictability about the processes in the systems. This chapter exhibits compartmental random dynamical models involving stochastic systems of differential equations, Markov processes, and random walk processes etc. to investigate random dynamical processes of infectious systems such as infectious diseases of humans or animals, the spread of rumours in social networks, and the spread of malicious signals on wireless sensory networks etc. A step-to-step approach to identify, and represent the various constituents of random dynamic processes in infectious systems is presented. In particular, a method to derive two independent environmental white noise processes, general nonlinear incidence rates, and multiple random delays in infectious systems is presented. A unique aspect of this chapter is that the ideas, mathematical modeling techniques and analysis, and the examples are delivered through original research on the modeling of vectorborne diseases of human beings or other species. A unique method to investigate the impacts of the strengths of the noises on the overall outcome of the infectious system is presented. Numerical simulation results are presented to validate the results of the chapter. Keywords Infection-free steady state · Stochastic stability · Basic reproduction number · Lyapunov functional technique · White noise intensity

D. Wanduku (B) Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3042, Statesboro, GA 30460, USA e-mail: [email protected]; [email protected] Tel:+14073009605 This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_17

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1 Introduction Compartmental dynamic models play increasing vital roles in investigating infectious disease systems such as human infectious disease systems, the infectious systems of the fauna and flora in an ecological setting, the infectious systems of wireless sensor network (WSN) involving the transmission of malicious signals (or worms) between the motes (or nodes) of the WSN, and the infectious social network systems of human beings involving the transmission of negative ideas and rumors etc. See for example [1–4]. These compartmental models represent dynamic relationships between various states of the population which generally form the different compartments. For instance, an SEIR compartmental dynamic model partitions the general population into four distinct states namely:- susceptible (S), exposed (E), infectious (I) and removal (R), where the susceptible state does not have the disease, the exposed state is infected but not infectious, the infectious class transmits the disease, and the removal class may represent recovered with temporary or permanent immunity, or simply removal by natural death or removal by some other characteristic that excludes the individuals from the disease transmission process etc. There are several other possible compartmentalisation of epidemic models such as SIS, SIR, SIRS, SEIRS, and SEIR models etc. [5–12] depending on the states or disease classes directly involved in the disease dynamics. Compartmentalizing the population involved in the disease dynamics allows for (1) easy mathematical representation of the features of the disease, and also (2) insight about the properties of the disease in each population state involved in the disease dynamics. For example, several studies that devote interest to SEIRS and SEIR models [5, 7, 10, 13, 14] which allow the inclusion of the compartment of individuals who are exposed to the disease, E, that is, infected but noninfectious individuals also allow for more insights about the disease dynamics during the incubation phase of the disease. For instance, in [7, 10] the existence of periodic solutions is investigated in the SEIRS epidemic models, in [15] the effects of seasonal changes on the disease dynamics are investigated in an SEIRS epidemic model. There are several different ways compartmental infectious dynamic models are improved to capture reality in a better way. Some models include time delays that occur in the disease dynamics such as the latency of the disease which is often taken to be the infected but noninfectious period of disease incubation, or the period of infectiousness. Some other studies include the time delay period of effective naturally acquired immunity conferred by the disease after recovery. See for example [9, 11, 16, 17]. Some other authors have studied one or more forms of these different types of delays jointly, but represented as two separate delay times [2, 4, 7, 18–20]. The occurrence of time delays in infectious dynamical systems may influence the dynamics of the disease in various important ways. For instance, in [7], the presence of delays in the compartmental infectious dynamic system creates periodic solutions for the infectious system. The occurrence of time delays can also destabilize infectionfree steady state populations leading to the existence of stable endemic steady states, thereby persisting the infection [21, 22].

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In more realistic infectious dynamic systems, the existence of noise and variability in the system is inevitable. This is because of the constant bombardment of the constituents of the infectious system by random environmental impacts which may influence the state of the system directly, or influence the driving parameters of the disease dynamics such as the disease transmission, natural death and recovery rates etc. For instance, [9, 11] used the gaussian white noise process to represent the variability in the natural death and disease transmission rates in the population, respectively. In [23], the presence of noise impacts the state of the system directly and it is represented by a white process that is proportional to the deviation of the state of the system from the endemic equilibrium state of the system. The occurrence of noise in the infectious dynamic system may give the dynamic system a more complex behavior beyond the average behavior determined by the deterministic version of the system. For example, an infection-free steady state which is found to be stable in the deterministic dynamic system, may no longer exist or no longer found stable in the stochastic version of the system. This is because the occurrence of noise in the system leads to fluctuations of the state of the system over time, and these fluctuations may destabilize the infection-free steady state population leading to the existence of a stable endemic steady state population, and consequently persist the disease in the population. The presence of noise in the system can also impose setbacks against disease eradication conditions for the system, such as when the threshold values for disease eradication are inflated by the intensities of the noises in the system. See for example, [9, 11, 12, 17, 24, 25]. Infectious disease transmission for human beings, most animals and plants, occur in two major ways namely:- (1) via direct contact with other humans such as through touch, airborne etc. for example, measles, and influenza etc., and (2) via a vector, for instance, malaria, and dengue fever etc. In WSN systems, disease transmission is considered the release and reception of malicious signals across different motes of the MSN, while for the infectious social network systems, infection transmission can be thought of as human contacts that involve the spread of negative ideas between two or more people. These modes of infection transmission for WSN and social network systems are analogous to the direct infectious contact modes of disease transmission for human beings, and therefore these infectious systems can be modeled in the same manner as infectious dynamic systems for human beings. Among the class of infectious diseases of human beings, vector-borne diseases exhibit several unique biological characteristics. For instance, the incubation of the disease requires two hosts - the vector and human hosts, which may be either directly involved in a single life cycle of the infectious agent consisting of two separate and independent segments of sub-life cycles1 that are completed separately in the two hosts, or directly involved in two separate and independent half-life cycles2 of the infectious agent in the hosts. Therefore, there exists a total latent time lapse of 1A

sub-life cycle may refer to a phase where the organism exhibits different behavioral characteristics. 2 A half-life is a series of intermediary developmental stages of the organism between the egg age and adult or sexual age of the organism.

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disease incubation which extends over the two segments of delay incubation times namely:- (1) the incubation period of the infectious agent ( or the half-life cycle) inside the vector, and (2) the incubation period of the infectious agent (or the other halflife cycle) inside the human being. For example, the dengue fever virus transmitted primarily by the Aedes aegypti and Aedes albopictus mosquito undergoes two delay incubation cycles:- (1) about 8–12 days incubation period inside the female mosquito vector, which starts immediately after biting an infectious human being and ingesting the dengue fever virus infected blood meal, and (2) another delay incubation period of about 2–7 days inside the human being when the hosting female infectious vector bites to acquire another blood meal from a susceptible human being, wherein the virus is successfully transferred from the infectious mosquito to the susceptible person. Indeed, for dengue fever transmission, a susceptible vector acquires infected blood meal from a dengue fever infectious person via a mosquito bite. The virus incubates in the mosquito for about 8–12 days, and at the end of the first incubation period the exposed mosquito becomes infectious. The virus is transferred to a susceptible human being after another successful mosquito bite, and it subsequently undergoes a second incubation phase in the exposed human being. The second incubation phase is mainly a viremia phase that involves the complete circulation of the virus in the human blood stream, and at the end of the phase the exposed person develops full blown fever. While the infectious dengue fever vector is known to stay infectious for the rest of the life span, it is important to note in modeling dengue fever dynamics that no relationship between vector survival and viral invasion or viral infection of the mosquito has been determined [26, 27]. For different type of vector-borne disease such as malaria which is transmitted by a parasite called plasmodium, the parasite undergoes a first half-life cycle called the sporogonic cycle inside the exposed female Anopheles mosquito lasting approximately 10–18 days after the first successful mosquito bite from a malaria infectious person. The parasite further completes the remaining half-life cycle called the exoerythrocytic cycle lasting about 7–30 days inside an exposed human being [26, 27], after the parasite is transmitted to a susceptible person following another successful infectious mosquito bite by the infectious female mosquito carrying the parasite. Several vector-borne diseases induce or confer natural immunity against the disease after infection and recovery. The effectiveness and duration of the natural protective immunity varies depending on the type of disease and also on other biological factors. For example, the exposure and successful recovery from one dengue fever viral strain confers lifelong immunity against the particular viral serotype [26]. The exposure and successful recovery from a malaria parasite, for example, falciparum vivae induces natural immunity against the disease which can protect against subsequent severe outbreaks of the disease. Moreover, the effectiveness and duration of the acquired immunity conferred in the case of malaria is determined by several different factors such as the species and the frequency of exposure to the parasites. On the spectrum of effectiveness of the naturally acquired immunity, the people living in malaria endemic zones offer the highest level of the acquired immunity, while pregnant women, children and people from areas of the world with low malaria transmission rates show the least level of immunity against the disease. Furthermore,

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it has also been determined that genetic inherited factors such sickle cell anemia, and duffy negative blood types produce natural resistance against certain species of the malaria parasite. See sources such as [27–29]. The interaction between susceptible, S, and infectious individuals, I , during the disease transmission process in some cases produces incidence rates of the disease which are qualitatively different in character from the simple most frequently used bilinear incidence rate given by the expression β S(t)I (t − T ) for vector-borne diseases, or the expression β S(t)I (t) for infectious diseases that involve direct humanto-human disease transmission, where the response rate of disease transmission with respect to the number of infectious individuals in the population over time is linear and unbounded. In these two expressions, β refers to the effective contact rate of the disease, and T is the incubation period for the vector-borne disease. There are other more realistic response behaviors for the disease transmission rate such as psychological, competition or overcrowding effects that occur upon the population in response to the increase in the number of infectious individuals, which produce behavioral changes among susceptible individuals that reduce or prevent disease transmission. These behavioral changes may include taking preventive measures such as the use of bed-nets to protect against vector-borne diseases like malaria and dengue fever, and the use of air pollution masks against airborne diseases like influenza, and also all other forms of preventive behaviors that limit contact between susceptible and infectious individuals. These behavioral response changes tend to produce incidence rates which are qualitatively nonlinear in character in response to the number of infectious individuals in the population. For instance, in [30–33] the authors consider a Holling S(t)I (t) that Type II functional response rate given by the expression β S(t)G(I (t)) = β1+αI (t) saturates for large values of I . In [32, 34, 35], a bounded Holling Type II response rate S(t)I p (t) with the expression given by β S(t)G(I (t)) = β1+αI p (t) , p ≥ 0 is used to represent the force of infection of the disease. In [36, 37], the nonlinear behavior of the incidence rate is represented by the general expression β S(t)G(I (t)) = β S p (t)I q (t), p, q ≥ 0. In [9, 30, 31, 33–35] the authors studied vector-borne diseases with several different nonlinear incidence rates for the disease. Other examples can be found in [9, 16, 30–38]. This chapter focuses on bringing understanding about modelling random compartmental infectious dynamic systems through original research that specializes on vector-borne diseases as an example of the different types of infectious diseases of human beings, most animals and plants. The techniques used in this study are applied in a general way to incorporate (1) the delays in a dynamic process, (2) the noise that originates from random environmental perturbations in the system. Furthermore, the ideas and techniques used in the study are applicable to all sorts of random dynamical problems in biology, engineering and the social sciences etc. and also suitable to all experts and audiences applying mathematical techniques to model dynamic processes in these different fields. Cooke [21] presented a general deterministic epidemic dynamic model for vector-borne diseases, wherein the bilinear incidence rate defined by the expression β S(t)I (t − T ) represents the number of new infections occurring per unit time during the disease transmission process. The derivation of this expression assumes

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homogenous mixing and the law of mass action, where the number of infectious vectors at time t that effectively transmit the infection to the susceptible class S, after the rate of β number of effective contacts per unit time per infective is proportional to the infectious human population, I, at the earlier time t − T. Various authors such as [20, 39] have also presented other types of dynamic models for vector-borne diseases containing delays. This study employs similar reasoning in [21], to derive the incidence rate of a general class of stochastic compartmental infectious dynamic models containing three random delays for vector-borne diseases. The compartmentalization of the infectious system is SEIRS. The three delays are classified under the two general delay-types namely:- the latency and the acquired immunity delay periods of the disease. Two of the delays represent the incubation period of the infectious agent inside the vector and the human host, and the third delay represents the period of effective naturally acquired immunity against the vector-borne disease. In addition, the general compartmental infectious dynamic system is driven by independent white noises which represent the variability in the natural death and disease transmission rates in the population. The class of disease dynamic models is expressed as a system of Ito-Doob stochastic differential equations. This work is presented as follows:- in Sect. 2, the epidemic dynamic model is derived. In Sect. 3, the model validation results are presented. In Sect. 4, the existence and asymptotic stochastic stability of the disease free equilibrium population is investigated. In Sect. 5, the asymptotic behaviors of the solutions of the stochastic system are characterized under the influence of different levels of the intensities of the white noises in the system. In Sect. 6, numerical simulation results are given.

2 Derivation of Stochastic and Deterministic Models In this section, a brief intuitive definition of a stochastic differential equation is presented, and the rest of the section will show how to derive a random compartmental dynamic model for an infectious disease using systems of stochastic differential equations that are derived as extensions to their corresponding deterministic systems of differential equations. Consider a deterministic dynamical system of differential equations which satisfies d x(t) = f (x(t), t)dt, x(t0 ) = x0 , x(t) ∈ Rd × [t0 , ∞),

(1)

where f is a continuous functions of x ≡ x(t) and t. The deterministic system can be interpreted as describing the rate of change with time of the state x of a system at any given time t ∈ [t0 , ∞), where the rate function f (x, t) describes how the state x is changing with time t. In this representation, the state of the system x(t) is a deterministic smooth function of time, and as a result the rate function f is also a smooth deterministic function.

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Since the presence of environmental fluctuations in almost all real life dynamical processes is inevitable as explained in the introduction, the system (1) can be viewed as the first approximation to reality, or in statistical language as “the mean of reality”. Therefore, it is correct to rewrite the system (1) as follows d x(t) = f (x(t), t)dt + [noise term], x(t0 ) = x0 , x(t) ∈ Rd × [t0 , ∞),

(2)

where the “[noise term]” should represent the random effects in the dynamical system. As it has been shown below, it turns out that a reasonable representation of the “[noise term]” is a Brownian motion w(t), where if we assume that there exists a continuous function g(x, t) that describes exactly how the random effects that are of Brownian motion type in the system affect the state x of the system over a small interval [t, t + dt] of length dt, then the new system obtained from (2) becomes d x(t) = f (x(t), t)dt + g(x(t), t)dw(t), x(t0 ) = x0 , x(t) ∈ Rd × [t0 , ∞), (3) where d x(t) and dw(t) are now interpreted as the following changes over the small interval [t, t + dt] of length dt: d x = x(t + dt) − x(t) and dw(t) = w(t + dt) − w(t), respectively. In the new system (3), the state x is no longer a deterministic function of time, but random since it depends on the Brownian motion w(t). Moreover, the rate function f is now called the drift, and the random effects g is called the diffusion of the system (3). The drift f (x, t) can be interpreted in one way as the velocity of motion of a particle along a Brownian motion path over a small interval [t, t + dt], where d x = x(t + dt) − x(t) is the displacement of the particle, and the diffusion g(x, t) is the intensity of the random fluctuations in the system in relation to the state x(t) at time t. The rest of this section will explain from the fundamental constituents of a random dynamical system, how to develop a system of stochastic differential equations to represent a random dynamical process in a system. The modeling technique presented below derives the stochastic model in two stages which are presented in two different interconnected subsections, first, in the absence of the noise in the system given in Sect. 2.1, and second, in the presence of noise in the system given in Sect. 2.2.

2.1 Deterministic Model A generalized class of stochastic SEIRS delayed epidemic dynamic models for vector-borne diseases is presented. The delays represent the incubation period of the infectious agents in the vector T1 , and in the human host T2 . The third delay represents the naturally acquired immunity period of the disease T3 , where the delays are random variables with density functions f T1 , t0 ≤ T1 ≤ h 1 , h 1 > 0, and f T2 , t0 ≤ T2 ≤ h 2 , h 2 > 0 and f T3 , t0 ≤ T3 < ∞. Furthermore, the joint density of T1 and T2 is given by f T1 ,T2 , t0 ≤ T1 ≤ h 1 , t0 ≤ T2 ≤ h 2 . Moreover, it is assumed that the random variables T1 and T2 are independent (i.e. f T1 ,T2 = f T1 . f T2 , t0 ≤ T1 ≤ h 1 , t0 ≤ T2 ≤ h 2 ).

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Indeed, the independence between T1 and T2 is justified from the understanding that the duration of incubation of the infectious agent for the vector-borne disease depends only on the suitable biological environmental requirements for incubation inside the vector and the human body which are unrelated. Furthermore, the independence between T1 and T3 follows from the lack of any real biological evidence to justify the interconnection between the incubation of the infectious agent inside the vector and the acquired natural immunity conferred to the human being. But T2 and T3 may be dependent as biological evidence suggests that the naturally acquired immunity is induced by exposure to the infectious agent. By employing similar reasoning in [9, 21, 31, 34], the expected incidence rate of the disease or force of infection of the disease at time t due to the disease transmission process between the infectious vectors and susceptible humans, S(t), is h given by the expression β t0 1 f T1 (s)e−μs S(t)G(I (t − s))ds, where μ is the natural death rate of individuals in the population, and it is assumed for simplicity that the natural death rate for the vectors and human beings are the same. Assuming exponential lifetime for the random incubation period T1 , the probability rate, 0 < e−μs ≤ 1, s ∈ [t0 , h 1 ], h 1 > 0, represents the survival probability rate of exposed vectors over the incubation period, T1 , of the infectious agent inside the vectors with the length of the period given as T1 = s, ∀s ∈ [t0 , h 1 ], where the vectors acquired infection at the earlier time t − s from an infectious human via for instance, biting and collecting an infected blood meal, and become infectious at time t. Furthermore, it is assumed that the survival of the vectors over the incubation period of length s ∈ [t0 , h 1 ] is independent of the age of the vectors. In addition, I (t − s), is the infectious human population at earlier time t − s, G is a nonlinear incidence function for the disease dynamics, and β is the average number of effective contacts per infectious individual per unit time. Indeed, the force of infection, h β t0 1 f T1 (s)e−μs S(t)G(I (t − s))ds signifies the expected rate of new infections at time t between the infectious vectors and the susceptible human population S(t) at time t, where the infectious agent is transmitted per infectious vector per unit time at the rate β. Furthermore, it is assumed that the number of infectious vectors at time t is proportional to the infectious human population at earlier time t − s. Moreover, it is further assumed that the interaction between the infectious vectors and susceptible humans exhibits nonlinear behavior, for instance, psychological and overcrowding effects, which is characterized by the nonlinear incidence function G. Therefore, the force of infection given by 

h1

β

f T1 (s)e−μs S(t)G(I (t − s))ds,

(4)

t0

represents the expected rate at which infected individuals leave the susceptible state and become exposed at time t. The susceptible individuals who have acquired infection from infectious vectors but are non infectious form the exposed class E. The population of exposed individuals at time t is denoted E(t). After the incubation period, T2 = u ∈ [t0 , h 2 ], of the infectious agent in the exposed human host, the individual becomes infectious, I (t),

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at time t. Applying similar reasoning in [18], the exposed population, E(t), at time t can be written as follows E(t) = E(t0 )e−μ(t−t0 ) p1 (t − t0 ) +

 t t0

β S(ξ)e−μT1 G(I (ξ − T1 ))e−μ(t−ξ) p1 (t − ξ)dξ,

(5) 

where p1 (t) =

0, t ≥ T2 , 1, t < T2

(6)

represents the probability that an individual remains exposed over the time interval [0, t]. It is easy to see from (5) that under the assumption that the disease has been in the population for at least a time t > maxt0 ≤T1 ≤h 1 ,t0 ≤T2 ≤h 2 (T1 + T2 ), in fact, t > h 1 + h 2 , so that all initial perturbations have died out, the expected number of exposed individuals at time t is given by  E(t) = t0

h2

 f T2 (u)

t t−u



h1

β

f T1 (s)e−μs S(v)G(I (v − s))e−μ(t−u) dsdvdu. (7)

t0

Similarly, for the removal population, R(t), at time t, individuals recover from the infectious state I (t) at the per capita rate α and acquire natural immunity. The natural immunity wanes after the varying immunity period T3 = r ∈ [t0 , ∞], and removed individuals become susceptible again to the disease. Therefore, at time t, individuals leave the infectious state at the rate αI (t) and become part of the removal population R(t). Thus, at time t the removed population is given by the following equation R(t) = R(t0 )e−μ(t−t0 ) p2 (t − t0 ) +



t

αI (ξ)e−μ(t−ξ) p2 (t − ξ)dξ,

(8)

t0



where p2 (t) =

0, t ≥ T3 , 1, t < T3

(9)

represents the probability that an individual remains naturally immune to the disease over the time interval [0, t]. But it follows from (8) that under the assumption that the disease has been in the population for at least a time t > maxt0 ≤T1 ≤h 1 ,t0 ≤T2 ≤h 2 ,T3 ≥t0 (T1 + T2 , T3 ) ≥ maxT3 ≥t0 (T3 ), in fact, the disease has been in the population for sufficiently large amount of time so that all initial perturbations have died out, then the expected number of removal individuals at time t can be written as  t  ∞ f T3 (r ) αI (v)e−μ(t−v) dvdr. (10) R(t) = t0

t−r

There is also constant birth rate B of susceptible individuals in the population. Furthermore, individuals die additionally due to disease related causes at the rate

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D. Wanduku

d

R

I

E

S

Finite incubation period inside the exposed human body =T2 Short or sufficiently long natural immunity period =T3

B

Fig. 1 The compartmental framework illustrates the transition rates between the states S, E, I, R of the system. It also shows the incubation delay T2 and the naturally acquired immunity T3 periods

d. A compartmental framework illustrating the transition rates between the different states in the system and also showing the delays in the disease dynamics is given in Fig. 1. It follows from (4), (7), (10) and the transition rates illustrated in the compartmental framework in Fig. 1 above, and also utilizing the following algorithm d X (t) = [all conversion rates into class X] − [all conversion rates from class X], dt (11) where X (t) ∈ {S(t), E(t), I (t), R(t)}, the family of SEIRS epidemic dynamic models for a vector-borne diseases in the absence of any random environmental fluctuations can be written as follows:   h1 f T1 (s)e−μs G(I (t − s))ds − μS(t) dS(t) = B − β S(t) t0   ∞ f T3 (r )I (t − r )e−μr dr dt, +α t0

(12)

  dE(t) = β S(t)

h1

f T1 (s)e−μs G(I (t − s))ds − μE(t)

t0



h2

−β t0



h1

f T2 (u)S(t − u)

f T1 (s)e

−μs−μu

 G(I (t − s − u))dsdu dt,

t0

(13)

Modeling Highly Random Dynamical Infectious Systems

  dI (t) = β

h2



h1

f T2 (u)S(t − u)

t0

519

f T1 (s)e−μs−μu G(I (t − s − u))dsdu

t0

−(μ + d + α)I (t)] dt,    ∞ −μs dR(t) = αI (t) − μR(t) − α f T3 (r )I (t − r )e dr dt,

(14) (15)

t0

where the initial conditions are given in the following: (S(t), E(t), I (t), R(t)) = (ϕ1 (t), ϕ2 (t), ϕ3 (t), ϕ4 (t)) , t ∈ (−∞, t0 ], ϕk ∈ C((−∞, t0 ], R+ ), ∀k = 1, 2, 3, 4, ϕk (t0 ) > 0, ∀k = 1, 2, 3, 4,

(16)

and C((−∞, t0 ], R+ ) is the space of continuous functions with the supremum norm ||ϕ||∞ = sup |ϕ(t)|.

(17)

t≤t0

2.2 Stochastic Model As earlier mentioned in the introduction, there are several different techniques to add gaussian white noise process into a dynamic system. One such techniques involves adding noise into the system as direct influence to the state of the system, where the impacts of the fluctuations in the system are (1) proportional to the state of the system, or (2) proportional to the deviation of the state of the system from a nonzero steady state etc. This type of noise in biological systems is commonly called demographic white noise, and the approach generally applied is called Langevian approach. See for example [40, 41] A second approach to adding noise into a dynamic system involves considering the influence of the random fluctuations on the driving parameters of the system such as the birth, death, recovery and disease transmission rates of an infectious system. This type of noise in biological systems is commonly called environmental white noise. Several investigators such as [6, 8, 11, 12, 17, 41] have used this approach to investigate biological dynamics. In this study, the second approach is utilized to model the effects of random environmental fluctuations that lead to variability in the disease transmission and natural death rates of the vector-borne disease. This approach entails the construction of a random walk process and application of the central limit theorem. This method is summarized below, and the reader is referred to [41] for a comprehensive development of the technique. For t ≥ t0 , let (, F, P) be a complete probability space, and Ft be a filtration (that is, sub σ- algebra Ft that satisfies the following: given t1 ≤ t2 ⇒ Ft1 ⊂ Ft2 ; E ∈ Ft and P(E) = 0 ⇒ E ∈ F0 ). The variability in the disease transmission and natural

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D. Wanduku

death rates are represented by independent white noise processes, and the rates are expressed as follows: μ → μ + σi ξi (t), ξi (t)dt = dwi (t), i = S, E, I, R, β → β + σβ ξβ (t) ξβ (t)dt = dwβ (t),

(18)

where ξi (t) and wi (t) represent the standard white noise and normalized wiener processes for the ith state at time t, with the following properties: w(0) = 0, E(w(t)) = 0, var (w(t)) = t. Furthermore, σi , i = S, E, I, R, represents the intensity value of the white noise process due to the natural death rate in the ith state, and σβ is the intensity value of the white noise process due to the disease transmission rate. The ideas behind the formulation of the expressions in (18) are given in the following. The constant parameters μ and β represent the natural death and disease transmission rates per unit time, respectively. In reality, random environmental fluc˜ Thus, the nattuations impact these rates turning them into random variables μ˜ and β. ural death and disease transmission rates over an infinitesimally small interval of time ˜ = βdt, ˜ [t, t + dt] with length dt is given by the expressions μ(t) ˜ = μdt ˜ and β(t) respectively. It is assumed that there are independent and identical random impacts acting upon these rates at times t j+1 over n subintervals [t j , t j+1 ] of length t = dtn , where t j = t0 + jt, j = 0, 1, . . . , n, and t0 = t. Furthermore, it is assumed that ˜ 0 ) = β(t) ˜ = βdt is also a μ(t ˜ 0 ) = μ(t) ˜ = μdt is constant or deterministic, and β(t constant. It follows that by letting the independent identically distributed random variables Z i , i = 1, . . . , n represent the random effects acting on the natural death rate, then it follows further that the rate at time tn = t + dt, that is, μ(t ˜ + dt) = μ(t) ˜ +

n 

Z j,

(19)

j=1

˜ + dt) can where E(Z j ) = 0, and V ar (Z j ) = σi2 t, i ∈ {S, E, I, R}. Note that β(t similarly be expressed as (19). And for sufficient large value of n, the summation in (19) converges in distribution by the central limit theorem to a random variable which is identically distributed as the wiener process σi (wi (t + dt) − wi (t)) = σi dwi (t), with mean 0 and variance σi2 dt, i ∈ {S, E, I, R}. It follows easily from (19) that μdt ˜ = μdt + σi dwi (t), i ∈ {S, E, I, R}.

(20)

Similarly, it can be easily seen that ˜ = βdt + σβ dwβ (t). βdt

(21)

Note that the intensities σi2 , i = S, E, I, R, β of the independent white noise ˜ = βdt + σβ ξβ (t) that processes in the expressions μ(t) ˜ = μdt + σi ξi (t) and β(t) ˜ represent the natural death rate, μ(t), ˜ and disease transmission rate, β(t), at time t, ˜ measure the average deviation of the random variable disease transmission rate, β,

Modeling Highly Random Dynamical Infectious Systems

521

and natural death rate, μ, ˜ about their constant mean values β and μ, respectively, over the infinitesimally small time interval [t, t + dt]. These measures reflect the force of the random fluctuations that occur during the disease outbreak at anytime, and which lead to oscillations in the natural death and disease transmission rates overtime, and consequently lead to oscillations of the susceptible, exposed, infectious and removal states of the system over time during the disease outbreak. Thus, in this study the words “strength” and “intensity” of the white noise are used synonymously. Also, the constructions “strong noise” and “weak noise” are used to refer to white noise with high and low intensities, respectively. Under the assumptions in the formulation of the natural death rate per unit time μ˜ as a Brownian motion process above, it can also be seen easily that under further assumption that the number of natural deaths N (t) over an interval [t0 , t0 + t] of ˜ = length t follows a poisson process {N (t), t ≥ t0 } with intensity of the process E(μ) μ, and mean E(N (t)) = E(μt) ˜ = μt, then the lifetime is exponentially distributed with mean μ1 and survival function S(t) = e−μt , t > 0.

(22)

It must be noted that despite the standard use of the formulation above for the sub-models (20)–(21) to represent environmental variability in biological systems, some authors such as [42] have raised issues about the biological justification for these models. Allen [42] has proposed an alternative extension of the sub-models in (20)–(21) obtained via a mean-reverting approach. Also, see [43]. Thus, substituting (18)–(22) into the deterministic system (12)–(15) leads to the following generalized system of Ito-Doob stochastic differential equations describing the dynamics of vector-borne diseases in the human population.   h1 f T1 (s)e−μs G(I (t − s))ds − μS(t) d S(t) = B − β S(t) t0   ∞ f T3 (r )I (t − r )e−μr dr dt +α t0



h1

−σ S S(t)dw S (t) − σβ S(t) 



h1

d E(t) = β S(t) 

h2



 f T1 (s)e−μs−μu G(I (t − s − u))dsdu dt

h1

f T2 (u)S(t − u)

t0

t0



−σ E E(t)dw E (t) + σβ S(t) 

h2

−σβ t0

(23)

f T1 (s)e−μs G(I (t − s))ds − μE(t)

t0

−β

f T1 (s)e−μs G(I (t − s))dsdwβ (t)

t0

h1

f T1 (s)e−μs G(I (t − s))dsdwβ (t)

t0



h1

f T2 (u)S(t − u)

f T1 (s)e−μs−μu G(I (t − s − u))dsdudwβ (t)

t0

(24)

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D. Wanduku

  d I (t) = β

h2



h1

f T2 (u)S(t − u)

t0

f T1 (s)e−μs−μu G(I (t − s − u))dsdu

t0

  −(μ + d + α)I (t) dt − σ I I (t)dw I (t) + σβ

h2

f T2 (u)S(t − u)

t0

 h1 × f T1 (s)e−μs−μu G(I (t − s − u))dsdudwβ (t) (25) t0    ∞ −μs f T3 (r )I (t − r )e dr dt − σ R R(t)dw R (t), d R(t) = αI (t) − μR(t) − α t0

(26) where the initial conditions are given in the following:- where ever necessary, we let h = h 1 + h 2 and define (S(t), E(t), I (t), R(t)) = (ϕ1 (t), ϕ2 (t), ϕ3 (t), ϕ4 (t)) , t ∈ (−∞, t0 ], ϕk ∈ C((−∞, t0 ], R+ ), ∀k = 1, 2, 3, 4, ϕk (t0 ) > 0, ∀k = 1, 2, 3, 4,

(27)

where C((−∞, t0 ], R+ ) is the space of continuous functions with the supremum norm (28) ||ϕ||∞ = sup |ϕ(t)|. t≤t0

Furthermore, the random continuous functions ϕk , k = 1, 2, 3, 4 are F0 − measurable, or independent of w(t) for all t ≥ t0 . Several epidemiological studies [7, 9, 10, 16, 44] have been conducted involving families of SIR, SEIRS, SIS etc. epidemic dynamic models, where the family type is determined by a set of general assumptions which characterize the nonlinear behavior of the incidence function G(I ) of the disease. Some general properties of the incidence function G assumed in this study include the following: Assumption 1 A1 G(0) = 0. A2 G(I ) is strictly monotonic on [0, ∞). A3 G (I ) < 0 ⇔ G(I ) is differentiable concave on [0, ∞). A4 lim I → G(I ) = C, 0 ≤ C < ∞ ⇔ G(I ) has a horizontal asymptote 0 ≤ C < ∞. A5 G(I ) ≤ I, ∀I > 0 ⇔ G(I ) is at most as large as the identity function f : I → I over the positive all I ∈ (0, ∞). An incidence function G that satisfies Assumption 1 A1–A5 can be used to describe the disease transmission process of a vector-borne disease scenario, where the disease dynamics is represented by the system (23)–(26), and the disease transmission rate between the vectors and the human beings initially increases or decreases for relatively small values of the infectious population size, and is bounded or steady for sufficiently large size of the infectious individuals in the population. It is noted that Assumption 1 is a generalization of some subcases of the

Modeling Highly Random Dynamical Infectious Systems

523

assumptions A1–A5 investigated in [7, 9, 16, 44]. Some examples of frequently used incidence functions in the literature that satisfy Assumption 1A1–A5 include: I (t) I (t) p , α > 0, G(I (t)) = 1+αI G(I (t)) = 1+αI 2 (t) , α > 0, G(I (t)) = I (t), 0 < p < 1 (t) and G(I ) = 1 − e−a I , a > 0. It can be observed that (24) and (26) decouple from the other equations for S and I in the system (23)–(26). It is customary to show the results for this study using the simplified system containing only the non-decoupled system equations for S and I , and then infer the results to the states E and R, since these states depend exclusively on S and I . Nevertheless, for convenience, the existence and stability results of the system (23)–(26) will be shown for the vector X (t) = (S(t), E(t), I (t)). The following notations will be used throughout this study: ⎧ ⎨ Y (t) = (S(t), E(t), I (t), R(t)) X (t) = (S(t), E(t), I (t)) ⎩ N (t) = S(t) + E(t) + I (t) + R(t).

(29)

3 Existence and Uniqueness of Solution It is important that the system of stochastic differential equations describing the disease dynamics in a human population has a unique positive solution in order to validate the model. The standard conditions for the existence of unique solution for the deterministic system (1) such as the requirements that the rate function f (x, t) be Lipschitz continuous with respect to x, uniformly continuous in t, and piecewise continuous in t are extended over to the stochastic system (2) as just one of the requirements for the drift f (x, t) and the diffusion g(x, t). A second requirement for stochastic systems is the linear growth condition. These conditions are presented in the following theorem. Theorem 1 Assume there exist two positive constants K 1 and K 2 such that (i) (Lipschitz condition) for all x, y ∈ Rd , and t ∈ [t0 , T ] | f (x, t) − f (y, t)|2



|g(x, t) − g(y, t)|2 ≤ K 1 |x − y|2 ,

(30)

(ii) (Linear growth condition) for all x ∈ Rd , and t ∈ [t0 , T ] | f (x, t)|2



|g(x, t)|2 ≤ K 2 (1 + |x|2 ).

(31)

Then there exists a unique second order solution process x(t) to the system (2). In most cases, the Lipschitz and linear growth conditions only justify the local existence of a unique solution process for the stochastic system. That is, for T ∞ in Theorem 1. Therefore, other functional analytic techniques involving stopping times and Lyapunov energy functions are used to extend the local solution globally, that

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is, for all t ∈ [t0 , ∞). In this section, it will be shown explicitly how to use the Lipschitz and linear growth conditions, and other functional analytic techniques to prove the existence and uniqueness of a solution to the stochastic system (23)–(26). The method used below is similar to my earlier studies [11, 17, 45]. It should be noted that the existence and qualitative behavior of the positive solutions of the system (23)–(26) depend on the sources (natural death or disease transmission rates) of variability in the system. As it is shown below, certain sources of variability lead to very complex uncontrolled behavior of the solutions of the system. The following Lemma describes the behavior of the positive local solutions for the system (23)–(26). This result will be useful in establishing the existence and uniqueness results for the global solutions of the stochastic system (23)–(26). Lemma 1 Suppose for some τe > t0 ≥ 0 the system (23)–(26) with initial condition in (27) has a unique positive solution Y (t) ∈ R4+ , for all t ∈ (−∞, τe ], then if N (t0 ) ≤ B , it follows that μ (a) if the intensities of the independent white noise processes in the system satisfy σi = 0, i ∈ {S, E, I } and σβ ≥ 0, then N (t) ≤ μB , and in addition, the set denoted by  B D(τe ) = Y (t) ∈ R4+ : N (t) = S(t) + E(t) + I (t) + R(t) ≤ , ∀t ∈ (−∞, τe ] μ

 B (−∞,τe ] 0, , (32) = B¯ 4 R+ , μ

is locally with respect to the system (23)–(26), where  self-invariant  (−∞,τe ] B ¯ BR4 , 0, μ is the closed ball in R4+ centered at the origin with radius μB + containing the local positive solutions defined over (−∞, τe ]. (b) If the intensities of the independent white noise processes in the system satisfy σi > 0, i ∈ {S, E, I } and σβ ≥ 0, then N (t) ≥ 0, for all t ∈ (−∞, τe ]. Proof It follows directly from (23)–(26) that when σi = 0, i ∈ {S, E, I } and σβ ≥ 0, then d N (t) = [B − μN (t) − d I (t)]dt (33) The result in (a) follows easily by observing that for Y (t) ∈ R4+ , the Eq. (33) leads to N (t) ≤ μB − μB e−μ(t−t0 ) + N (t0 )e−μ(t−t0 ) . And under the assumption that N (t0 ) ≤ μB , the result follows immediately. The result in (b) follows directly from Theorem 2. The following theorem presents the existence and uniqueness results for the global solutions of the stochastic system (23)–(26). Theorem 2 Given the initial conditions (27) and (28), there exists a unique solution process X (t, w) = (S(t, w), E(t, w), I (t, w))T satisfying (23)–(26), for all t ≥ t0 . Moreover, (a) the solution process is positive for all t ≥ t0 a.s. and lies in D(∞), whenever the intensities of the independent white noise processes in the system satisfy

Modeling Highly Random Dynamical Infectious Systems

525

σi = 0, i ∈ {S, E, I } and σβ ≥ 0. That is, S(t, w)  > 0, E(t, w) > 0, I (t, w) > (−∞,∞) ¯ 0, ∀t ≥ t0 a.s. and X (t, w) ∈ D(∞) = B 4 0, B , where D(∞) is defined R+ ,

μ

in Lemma 1, (32). (b) Also, the solution process is positive for all t ≥ t0 a.s. and lies in R4+ , whenever the intensities of the independent white noise processes in the system satisfy σi > 0, i ∈ {S, E, I } and σβ ≥ 0. That is, S(t, w) > 0, E(t, w) > 0, I (t, w) > 0, ∀t ≥ t0 a.s. and X (t, w) ∈ R4+ . Proof It is easy to see that the coefficients of (23)–(26) satisfy the local Lipschitz condition for the given initial data (27). Therefore there exist a unique maximal local solution X (t, w) = (S(t, w), E(t, w), I (t, w)) on t ∈ (−∞, τe (w)], where τe (w) is the first hitting time or the explosion time [46]. The following shows that X (t, w) ∈ D(τe ) almost surely, whenever σi = 0, i ∈ {S, E, I } and σβ ≥ 0, where D(τe (w)) is defined in Lemma 1 (32), and also that X (t, w) ∈ R4+ , whenever σi > 0, i ∈ {S, E, I } and σβ ≥ 0. Define the following stopping time 

τ+ = sup{t ∈ (t0 , τe (w)) : S|[t0 ,t] > 0, τ+ (t) = min(t, τ+ ), f or t ≥ t0 .

E|[t0 ,t] > 0, and I |[t0 ,t] > 0},

(34) and lets show that τ+ (t) = τe (w) a.s. Suppose on the contrary that P(τ+ (t) < τe (w)) > 0. Let w ∈ {τ+ (t) < τe (w)}, and t ∈ [t0 , τ+ (t)). Define 

V (X (t)) = V1 (X (t)) + V2 (X (t)) + V3 (X (t)), V1 (X (t)) = ln(S(t)), V2 (X (t)) = ln(E(t)), V3 (X (t)) = ln(I (t)), ∀t ≤ τ+ (t).

(35) It follows from (35) that d V (X (t)) = d V1 (X (t)) + d V2 (X (t)) + d V3 (X (t)),

(36)

where 1 1 1 d S(t) − (d S(t))2 S(t) 2 S 2 (t)   h1 B −β = f T1 (s)e−μs G(I (t − s))ds − μ S(t) t  ∞ 0 α f T3 (r )I (t − r )e−μr dr + S(t) t0

 h 1 2  1 2 1 2 −μs − σ S − σβ f T1 (s)e G(I (t − s))ds dt 2 2 t0  h1 f T1 (s)e−μs G(I (t − s))dsdwβ (t), (37) −σ S dw S (t) − σβ

d V1 (X (t)) =

t0

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D. Wanduku

1 1 1 d E(t) − (d E(t))2 E(t) 2 E 2 (t)   S(t) h 1 = β f T1 (s)e−μs G(I (t − s))ds − μ E(t) t0  h2 1 f T2 (u)S(t − u) −β E(t) t0  h1 f T1 (s)e−μs−μu G(I (t − s − u))dsdu ×

d V2 (X (t)) =

t0

 h 1 2 1 2 S 2 (t) 1 2 −μs f T1 (s)e G(I (t − s))ds − σ E − σβ 2 2 2 E (t) t0

 h 2 1 1 − σβ2 2 f T2 (u)S(t − u) 2 E (t) t0 2   h1 dt f T1 (s)e−μs−μu G(I (t − s − u))dsdu × t0

−σ E dw E (t) + σβ

S(t) E(t)



h1

f T1 (s)e−μs G(I (t − s))dsdwβ (t)

t0

 h2 1 f T2 (u)S(t − u) −σβ E(t) t0  h1 f T1 (s)e−μs−μu G(I (t − s − u))dsdudwβ (t), ×

(38)

t0

and 1 1 1 d I (t) − (d I (t))2 I (t) 2 I 2 (t)   h2 1 = β f T2 (u)S(t − u) I (t) t0  h1 f T1 (s)e−μs−μu G(I (t − s − u))dsdu − (μ + d + α) ×

d V3 (X (t)) =

t0

 h 2 1 1 f T2 (u)S(t − u) − σ 2I − σβ2 2 2 t0 2   h1 −μs−μu f T1 (s)e G(I (t − s − u))dsdu dt t0

−σ I dw I (t) + σβ 

h1

× t0

1 I (t)



h2

f T2 (u)S(t − u)

t0

f T1 (s)e−μs−μu G(I (t − s − u))dsdudwβ (t)

(39)

Modeling Highly Random Dynamical Infectious Systems

527

It follows from (36)–(39) that for t < τ+ (t), V (X (t)) − V (X (t0 )) ≥

 t

 −β

h1 t0

t0

1 2 σ 2 S 2 

f T1 (s)e−μs G(I (ξ − s))ds −

 h 1 1 dξ − σβ2 f T1 (s)e−μs G(I (ξ − s))ds 2 t0  t0   h2 1 −β + f T2 (u)S(ξ − u) E(ξ) t0 t  h1 × f T1 (s)e−μs−μu G(I (ξ − s − u))dsdu t0

2

 h 1 1 1 S 2 (ξ) − σ 2E − σβ2 2 f T1 (s)e−μs G(I (ξ − s))ds 2 2 E (ξ) t0

 h 2 1 1 − σβ2 2 f T2 (u)S(ξ − u) 2 E (ξ) t0 2   h1 −μs−μu dξ f T1 (s)e G(I (ξ − s − u))dsdu × t0



 h 2 1 1 −(3μ + d + α) − σ 2I − σβ2 f T2 (u)S(ξ − u) 2 2 t t0 2   h1 −μs−μu × f T1 (s)e G(I (ξ − s − u))dsdu dξ 

t0

+

t0



t0

+

 − σ S dw S (ξ) − σβ

t



h1



t0 t0

× +

f T1 (s)e−μs G(I (ξ − s))dsdwβ (ξ)  − σ E dw E (ξ) + σβ

t



h1



t0 t0

× −

t



f T1 (s)e−μs G(I (ξ − s))dsdwβ (ξ)

t0

+

 t t0

 ×

h1

t0



  h2 1 f T2 (u)S(ξ − u) σβ E(ξ) t0

h1

×

S(ξ) E(ξ)



f T1 (s)e−μs−μu G(I (ξ − s − u))dsdudwβ (ξ) −σ I dw I (ξ) + σβ

1 I (ξ)



h2 t0



f T2 (u)S(ξ − u)

 f T1 (s)e−μs−μu G(I (ξ − s − u))dsdudwβ (ξ) .

(40)

Taking the limit on (40) as t → τ+ (t), it follows from (34)–(35) that the left-hand side V (X (t)) − V (X (t0 )) ≤ −∞. This contradicts the finiteness of the right-handside of the inequality (40). Hence τ+ (t) = τe (w) a.s., that is, X (t, w) ∈ D(τe ), whenever

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σi = 0, i ∈ {S, E, I } and σβ ≥ 0, and X (t, w) ∈ R4+ , whenever σi > 0, i ∈ {S, E, I } and σβ ≥ 0. The following shows that τe (w) = ∞. Let k > 0 be a positive integer such that ||ϕ||  1 ≤ k, where ϕ = (ϕ1 (t), ϕ2 (t), ϕ3 (t)) , t ∈ (−∞, t0 ] defined in (27), and ||.||1 is the p − sum norm defined on R3 , when p = 1. Define the stopping time 

τk = sup{t ∈ [t0 , τe ) : ||X (s)||1 = S(s) + E(s) + I (s) ≤ k, s ∈ [t0 , t]} τk (t) = min(t, τk ).

(41)

It is easy to see that as k → ∞, τk increases. Set limk→∞ τk (t) = τ∞ . Then it follows that τ∞ ≤ τe a.s. We show in the following that: (1) τe = τ∞ a.s. ⇔ P(τe = τ∞ ) = 0, (2) τ∞ = ∞ a.s. ⇔ P(τ∞ = ∞) = 1. Suppose on the contrary that P(τ∞ < τe ) > 0. Let w ∈ {τ∞ < τe } and t ≤ τ∞ . Define  Vˆ1 (X (t)) = eμt (S(t) + E(t) + I (t)), (42) ∀t ≤ τk (t). The Ito-Doob differential d Vˆ1 of (42) with respect to the system (23)–(26) is given as follows: d Vˆ1 = μeμt (S(t) + E(t) + I (t))dt + eμt (d S(t) + d E(t) + d I (t))    ∞ = eμt B + α f T3 (r )I (t − r )e−μr dr − (α + d)I (t) dt

(43)

t0 μt

−σ S e S(t)dw S (t) − σ E eμt E(t)dw E (t) − σ I eμt I (t)dw I (t)

(44)

Integrating (44) over the interval [t0 , τ ], and applying some algebraic manipulations and simplifications it follows that V1 (X (τ )) = V1 (X (t0 )) + + +

 ∞ t0  τ t0

 B  μτ e − eμt0 μ 

f T3 (r )

t0

t0 −r

αeμξ I (ξ)dξ −

 τ τ −r

 αeμξ I (ξ)dξ dr −

 τ t0

d I (ξ)dξ

 −σ S eμξ S(ξ)dw S (ξ) − σ E eμξ E(ξ)dw E (ξ) − σ I eμξ I (ξ)dw I (ξ)

(45)

Removing negative terms from (45), it implies from (27) that B μτ e μ    ∞  t 0 + f T3 (r ) αeμξ ϕ3 (ξ)dξ dr

V1 (X (τ )) ≤ V1 (X (t0 )) +

t0

t0 −r

 τ  −σ S eμξ S(ξ)dw S (ξ) − σ E eμξ E(ξ)dw E (ξ) − σ I eμξ I (ξ)dw I (ξ) + t0

(46)

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But from (42) it is easy to see that for ∀t ≤ τk (t), ||X (t)||1 = S(t) + E(t) + I (t) ≤ V (X (t)).

(47)

Thus setting τ = τk (t), then it follows from (41), (46) and (47) that k = ||X (τk (t))||1 ≤ V1 (X (τk (t)))

(48)

Taking the limit on (48) as k → ∞ leads to a contradiction because the left-hand-side of the inequality (48) is infinite, but following the right-hand-side from (46) leads to a finite value. Hence τe = τ∞ a.s. The following shows that τe = τ∞ = ∞ a.s. Let w ∈ {τe < ∞}. It follows from (45)–(46) that B I{τe 0 that depends on S0∗ (in fact, kˆ1 = 6S0∗ and Kˆ 1 = 4 + S0∗ ), and under the assumptions that R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1, the drift part L V of the differential operator d V applied to V with respect to the stochastic dynamic

Modeling Highly Random Dynamical Infectious Systems

541

system (23)–(26) satisfies the following inequality:   L V (x, t) ≤ − φU 2 (t) + ψV 2 (t) + ϕW 2 (t) .

(83)

Proof The drift part L V of the differential operator d V applied to the Lyapunov functional defined in (78), (65) and (79) with respect to system (23)–(26) leads to the following: L V (x, t) = L V1 (x, t)  ∞ +2α f T3 (r )e−2μr W 2 (t)dr t0 +[2β S0∗  h1

(1 + c) + σβ2 (S0∗ )2 (4c + 2(1 − c)2 )]

f T1 (s)e−2μs G 2 (W (t))ds

×

t0

  + β S0∗ (4 + c) + β(S0∗ )2 (2 + c) + σβ2 (S0∗ )2 (4c + 10) ×  h2  h1 × f T2 (u) f T1 (s)e−2μ(s+u) G 2 (W (t))dsdu t0 t0  ∞ −2α f T3 (r )e−2μr W 2 (t − r )dr t0 −[2β S0∗  h1

(1 + c) + σβ2 (S0∗ )2 (4c + 2(1 − c)2 )]

f T1 (s)e−2μs G 2 (W (t − s))ds

×

t0

  − β S0∗ (4 + c) + β(S0∗ )2 (2 + c) + σβ2 (S0∗ )2 (4c + 10) ×  h2  h1 × f T2 (u) f T1 (s)e−2μ(s+u) G 2 (W (t − s − u))dsdu. (84) t0

t0

Under the assumptions for σi , i = S, E, I, β in Theorem 3(2), and for some suitable choice of the positive constant c, it follows from (68), (84), the statements of Assumption 1, A5 (i.e. G 2 (x) ≤ x 2 , x ≥ 0) and some further algebraic manipulations and simplifications that   L V (x, t) ≤ − φU 2 (t) + ψV 2 (t) + ϕW 2 (t) ,

(85)

where, φ = 2μ(1 − U0 ), ψ = 2μ(1 − V0 ) − 2μc



β(3S0∗ + 1) + σ 2E 1− 2μ

(86)

 ,

(87)

ϕ = 2(μ + d + α)(1 − R1 ) − c(3β S0∗ + β(S0∗ )2 + 4σβ2 (S0∗ )2 ) − 2c2 σβ2 (S0∗ )2 , (88)

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and R1 , V0 and U0 are defined in (80)–(82). In addition, under the assumptions of R1 , U0 , and V0 in the hypothesis of the theorem, and for a suitable choice of the positive constant c it follows that φ, ψ, and ϕ are positive constants and (83) follows immediately. The Lemma 4 asserts that there exists a positive-definite decrescent radially unbounded functional V defined in (78) which satisfy the conditions of the characterization in Theorem 5. This result implies that the zero steady state of the transformed system (61)–(63), and consequently the nonzero infection-free steady state E 0 of the original stochastic system (23)–(26) is stochastically asymptotically stable in the large. The following theorem formally states the asymptotic stability result for the disease free equilibrium E 0 , whenever it exists. Theorem 6 Suppose Theorem 3(2) and the hypotheses of Lemmas 2 and 4 are satisfied, then the disease-free equilibrium E 0 of the stochastic dynamic system (23)–(26) is stochastically asymptotically stable in the large in the set D(∞). Moreover, the steady state solution E 0 is exponentially mean square stable. Proof The result follows directly from Theorem 5. Moreover, the disease free equilibrium state is exponentially mean square stable from (Corollary 3.4, or Theorem 4.4 of [47]). The results in Lemma 4 and Theorem 6 hold the key to understand the underlying factors of the infectious dynamic system (23)–(26) that are controlling the eradication of disease from the system. Since the primary goal of this chapter is to provide mathematical techniques to model and to interpret random dynamical systems, the different observations will be titled below and elaborated on in the following:

4.2.1

The Disease Eradication Conditions

Theorem 6 signifies that in the absence of the noise in the system from the random fluctuations in the natural death rate of the susceptible population which is reflected by the condition that σ S = 0, then regardless of whether there is strong or weak noise in the system from the random fluctuations in (1) the natural death rates of the other disease related subclasses namely- the exposed, the infectious, and the removal populations, which is also reflected by the values of the intensities σi ≥ 0, i = E, I, R, or from random fluctuation in (2) the disease transmission rate of the vector-borne disease which is again reflected by the value of the intensity of the noise given by σβ2 ≥ 0, there is always an infection-free steady state for the population exhibiting the disease dynamics given by (23)–(26), where the infection-free steady state is given by E 0 . Moreover, the infection-free steady state is stochastically asymptotically stable in the large, whenever the following threshold conditions are satisfied, that is, R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1, where the threshold values R1 , U0 , and V0 are defined in (80)–(82). In other words, if the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 hold, then every sample path of the system (23)–(26) that starts in the neighborhood of the disease-free steady state E 0 has a high chance to stay in the neighborhood of E 0 ,

Modeling Highly Random Dynamical Infectious Systems

543

and ultimately becomes E 0 . That is, the sample paths of the disease related states E, I and R ultimately become zero, whenever the threshold conditions hold, and as a consequence, the disease is eliminated from the population. It should be noted that the threshold values R1 , U0 , and V0 defined in (80)–(82) are explicit in terms of the parameters of the system (23)–(26), and also computationally attractive, whenever the specific values for the parameters of the system are given. This observation suggests in theory that in a physical disease outbreak, if the physical characteristics of the disease scenario can be mathematically expressed in terms of the parameters of the system (23)–(26), wherein the threshold values, R1 , U0 , and V0 can be computed, then the disease eradication conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 can be easily checked. Moreover, the disease will be eradicated whenever the strength of the random fluctuations in the natural death rate of the susceptible individuals is very weak or unnoticeable, and this is also true regardless of whether the strengths of the random fluctuations in the natural death rate of the other disease related classes, or the strength of the fluctuation in the disease transmission rate are strong or weak. The threshold value R1 is called the basic reproduction number (also called the noise-modified basic reproduction number) for the disease dynamics described by the stochastic system (23)–(26), and it exists only under the condition that the intensities of the independent white noise processes in the system satisfy the assumptions that σi ≥ 0, i = E, I, R, β, and σ S = 0. This important threshold value is defined in theory as the expected number of secondary infectious cases that result from one infectious individual in a completely susceptible population. Moreover, this parameter signals the stability of the infection-free steady state, and consequently signal the eradication of disease from the system, whenever R1 ≤ 1. It also signals the existence of an endemic equilibrium, and consequently signals that the disease persists in the population, since an infectious steady state for the population exists, whenever the condition R1 > 1 holds. This parameter can also be modified by the presence of noise in the stochastic system, and as a result certain scenarios wherein the disease would be eradicated in the absence of noise in the system, would tend to favor the persistence of the disease, that is, the existence of a stable infectious state, whenever noise is introduced in the system. This is the case in this study as it is shown below. The noise modified basic reproduction number given by R1 =

kˆ1 σβ2 + 21 σ 2I α β S0∗ Kˆ 1 + + , (μ + d + α) (μ + d + α) (μ + d + α)

(89)

can be interpreted as follows:- firstly, since β is the average number of effective infections contacts that every infectious individual can make in the population, whenever β S ∗ Kˆ

0 1 homogeneous mixing is assumed, therefore the term (μ+d+α) represents a constant ˆ multiple K 1 of the disease transmission rate (also defined as the average number of new infections per unit time) given by the term β S0∗ in the infection-free steady state population N ∗ = S0∗ defined in (29) that contains just one infectious individual over 1 the average life-span of an infectious individual in the population given by (μ+d+α) .

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Note that in a population where people can either die naturally or die from disease related causes, or recover from the disease, the average life-spans is affected by these 1 . sources of mortality, and is given by (μ+d+α) α Furthermore, the term (μ+d+α) in (89) is the recovery rate in the infection-free steady state population N ∗ = S0∗ containing just one infectious individual over the average life-span of an infectious individual in the population. Let us note that the recovery rate α is a probabilistic rate defined as the probability of recovery from α represents the vector-borne disease per unit time per infective. The term (μ+d+α) influence of recovery on the basic reproduction number since the single infectious individual in the infection-free steady state population has a chance to survive from both natural and disease related deaths, and also fully recover from infection. kˆ1 σ 2 + 1 σ 2

β 2 I represents the influence of the noise in the system from the The last term (μ+d+α) disease transmission rate, and the natural death rate of the infectious population on the basic reproduction number. As this last term may no longer exist in the absence of noise in the system, so the name “modified-basic reproduction number” is used to describe the basic reproduction number for a stochastic dynamic system. It is also easy to see that the basic reproduction number in (89) is inflated by the noise term in the system, whenever the intensities σβ > 0 and σ I > 0. This observation is demonstrated in Fig. 2 of Sect. 6, and elaborated upon further in the next section. This observation also suggests that stronger noise in the system from the natural death rate of infectious individuals, or from the disease transmission rate which may lead to higher values of σβ > 0 and σ I > 0, may also inflate the basic reproduction number beyond the threshold bound R1 ≤ 1, thereby causing the disease to establish a stable endemic steady state in the population. The next sections considers different growth orders for the intensities of the white noise processes in the system, and examines how they affect the stochastic system.

4.2.2

The Effects of the Source and Intensity of White Noise

In addition to the existence results of Theorem 3, the results of Lemma 4 and Theorem 6 also suggest that the sources of the noises in the system, that is, from the disease transmission or natural death rates, and also the intensities of the white noises in the stochastic system (23)–(26), which are a result of the random fluctuations the disease dynamics exhibit direct consequences on the qualitative outcome of the vector-borne disease dynamics in the system with respect to the factors that determine disease eradication. To explain further, while Lemma 4 and Theorem 6 assert that the stochastic system (23)–(26) has a stochastically stable disease-free equilibrium, the result in Theorem 3(3) suggests the contrary. That is, Theorem 3(3) asserts that introducing the additional source of the random fluctuations in the system from the natural death rate of susceptible individuals in the population, then the stochastic system (23)–(26) no longer has a disease-free steady state wherein the disease can be eradicated, whenever the intensity of the white noise from the natural death rate of the

Modeling Highly Random Dynamical Infectious Systems

545

susceptible population σ S is significant, that is, whenever σ S > 0. Consequently, the disease can no longer be eradicated by applying the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1. The physical interpretation of this observation, in theory, is that stronger noise in the population from the natural deathrate of susceptible people in the population does no good to ease the disease control process in the population. This is justification for the obvious fact that when more people without the disease die from other causes apart from the disease, then only the disease related classes of people are left to continue to infect the remaining susceptible population. Furthermore, just like the basic reproduction number R1 , the other threshold values U0 and V0 defined in (80)–(82) also depend on the intensities of the white noise processes σi ≥ 0, i = E, I, R, β in the stochastic dynamic system (23)–(26). It can also be observed from (80)–(82) that high values of the intensities σi ≥ 0, i = E, I, R, β are associated with high values for the threshold values R1 , U0 , and V0 , and vice versa. This fact is much more visible in Fig. 2 of Sect. 6. Therefore, the intensities of the white noise processes in the system tend to inflate all the threshold values for disease eradication R1 , U0 and V0 of the stochastic dynamic system (23)–(26), and consequently exert constraints on the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 for disease eradication. In other words, stronger noises in the population from the natural deathrates of the disease related classes - exposed, infectious and removal populations, and also from the disease transmission rate tend to inflate the threshold values for disease eradication U0 , and V0 beyond the maximum threshold bounds R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1, and as a result, destabilize the infection-free steady state E 0 of the population, and lead to a stable endemic state for the disease in the population. Since it is obvious from the above discussion that the qualitative outcome of the disease dynamics represented in the stochastic dynamic model (23)–(26) such as the existence and stochastic stability of the disease-free equilibrium E 0 , and consequently disease eradication from the population depend on the intensities of the white noise processes in the system, it follows in the next section that the growth orders of the intensities of the white noise processes in the system are classified, and their impacts on the stability of the disease-free equilibrium, and also on disease eradication are examined. Moreover, numerical evidence for the effects of intensities of the white noise processes on the stochastic system (23)–(26) are presented in Sect. 6. As earlier mentioned in the introduction of this section, oftentimes the influence of the noise in a random dynamical system is more apparent in a comparative analysis to the stability results of the corresponding deterministic system. The following result characterizes the stability of the disease-free equilibrium of the stochastic system (23)–(26), whenever the intensities of the white noise processes in the system are so small that their existence can be ignored. It should be noted that the stability results that would be obtained under the assumption of negligible random fluctuations in the system are equivalent to the stability results obtained for the deterministic system (12)–(15) using the characterization in Theorem 4.

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Theorem 7 Let the hypotheses of Theorem 2, Theorem 3(1) and Lemma 3 be satisfied. There exists a Lyapunov functional V = V1 + V2 ,

(90)

where V1 ∈ C 2,1 (R3 × R+ , R+ ) is defined by (65) and V2 is defined as follows: 

∞

V2 (x, t) = 2α

f T3 (r )e

−2μr

t0



t

I 2 (v)dvdr

t−r

 h1  t f T1 (s)e−2μs G 2 (I (v))dvds +[2β S0∗ (1 + c) + σβ2 (S0∗ )2 (4c + 2(1 − c)2 )] t0 t−s   + β S0∗ (4 + c) + β(S0∗ )2 (2 + c) + σβ2 (S0∗ )2 (4c + 10) ×  h 2  h 1  t −2μ(s+u) × f T2 (u) f T1 (s)e G 2 (I (v − s))dvdsdu  +

t0 h2 t0



t0 h1

f T2 (u) f T1 (s)e

−2μ(s+u)



t−u t



G (I (v))dvdsdu . 2

(91)

t−s

t0

Furthermore, define R0 , U0 and V0 , as follows: R0 =

α β S0∗ Kˆ 0 + , (μ + d + α) (μ + d + α)

Uˆ 0 = and

μ 2β S0∗ + β + α + 2 K˜ (μ) 2

2μ

,

(2μ K˜ (μ)2 + α + β(2S0∗ + 1)) , Vˆ0 = 2μ

(92)

(93)

(94)

where, Kˆ 0 > 0 is a constant that depends only on S0∗ (in fact, Kˆ 0 = 4 + S0∗ ). Assume that R0 ≤ 1, Uˆ 0 ≤ 1, and Vˆ0 ≤ 1, then there exist positive constants φ1 , ψ1 , and ϕ1 , such that the differential operator V˙ applied to V with respect to the stochastic system (23)–(26) satisfies the following inequality:   V˙ (x, t) ≤ − φ1 U 2 (t) + ψ1 V 2 (t) + ϕ1 W 2 (t) .

(95)

Moreover, under the assumptions in the hypothesis of Theorem 3(1), the disease free equilibrium E 0 of the resulting system (23)–(26) is uniformly globally asymptotically stable in the set D(∞).

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Proof The result follows directly from the Proofs of Lemmas 3 and 4 by applying the conditions that σi = 0, i = S, E, I, β, and also applying the comparison stability results in Theorem 4, where from (86)–(88), φ1 = φ, ψ1 = ψ, and ϕ1 = ϕ. The discussion in Sect. 4.2.2 is carried on in the following subsection that expands on the results of the comparative analysis between the stability of the infection-free steady state of the system (23)–(26) in the presence, and also in absence of noise in the system. That is, the results given in Theorems 6, 7 are compared.

4.2.3

Stability in the Absence of Noise

Theorem 7 asserts that when the noise in the system from all sources, natural death or disease transmission rates, is very weak that can be ignored, that is, when σi → 0, i = S, E, I, R, β, then the behavior of the stochastic system (23)–(26) is equivalent to the behavior of the deterministic system (12)–(15). That is, there is an infectionfree steady state, E 0 , for the population which similarly to Theorem 6 is globally asymptotically stable, whenever the threshold conditions: R0 ≤ 1, Uˆ 0 ≤ 1 and Vˆ0 ≤ 1 hold, where the threshold values R0 , Uˆ 0 and Vˆ0 are defined in (92)–(94). In other words, the trajectories of the systems (23)–(26) and (12)–(15) in this case are the same, and continuously smooth all over time as there is less deflection by the noise in the system. Furthermore, all the trajectories that start in the neighborhood of the infection-free steady state E 0 certainly remain in the neighborhood of E 0 in a deterministic manner, and ultimately become the disease-disease free steady state. That is, the disease is eradicated from the system, whenever the threshold conditions R0 ≤ 1, Uˆ 0 ≤ 1 and Vˆ0 ≤ 1 hold. Furthermore, it should be noted similarly to Theorem 6 that the threshold values R0 , Uˆ 0 and Vˆ0 are explicit in terms of the parameters of the system (12)–(15), or equivalently in terms of the parameters of the system (23)–(26), whenever σi → 0, i = S, E, I, R, β, and they are also computationally attractive, whenever the specific values for the parameters of the system are given. This observation, in theory, implies that the conditions for disease eradication in a disease scenario which follows the disease dynamics (23)–(26), whenever σi → 0, i = S, E, I, R, β, are determined by verifying the threshold conditions R0 ≤ 1, Uˆ 0 ≤ 1 and Vˆ0 ≤ 1 for the specific values of the parameters of the system that correspond to the disease scenario. It is also easy to see from (89) and (92) that the two threshold values R0 and R1 satisfy R0 = R1 , whenever the conditions of Theorem 7 are satisfied. Therefore, these two threshold parameters R0 and R1 are the basic reproduction numbers for the system (23)–(26) without and with noise in the system, respectively. Moreover, R1 is a modification of R0 by adding the noise terms σβ and σ I . Thus, R0 is called the ordinary basic reproduction number for the system, while R1 is the noise-modified basic reproduction number for the disease dynamics. Also, comparing the threshold values for Theorems 7 and 6 defined in (92)–(94) and (80)–(82) respectively, it is easy to see that the basic reproduction numbers satisfy R0 ≤ R1 , and the other threshold values also satisfy U0 = Uˆ 0 and Vˆ0 ≤ V0 ,

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whenever the intensities of the white noise processes in the system (23)–(26) satisfy σi > 0, i = E, I, β. It is also easy to see that the intensities σi > 0, i = E, I, R, β of the corresponding white noise processes from the disease transmission rate, and the natural deathrates of the exposed, infections and removal classes inflate the threshold values R1 , U0 , and V0 defined in (80)–(82), but have no influence on the threshold values R0 , Uˆ 0 and Vˆ0 defined in (92)–(94). Moreover, one can see that when the intensities satisfy σi > 0, i = E, I, R, β, the threshold values - R0 , Uˆ 0 and Vˆ0 from (92)–(94) are smaller in magnitude than the corresponding threshold values R1 , U0 , and V0 . This implies that the threshold conditions R0 ≤ 1, Uˆ 0 ≤ 1 and Vˆ0 ≤ 1 are much more easily satisfied, when compared to the other threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 for the set of threshold values R1 , U0 , and V0 defined in (80)–(82). This observation implies that, the disease is more easily eradicated when there almost negligible noise in the system, than when the strength of the noise in the system is strong. Furthermore, this observation also suggests that the intensities of the random fluctuations in the disease dynamics (23)–(26) expressed as σi , i = E, I, R, β reflect the weights of the counter barrier effects exerted against the disease eradication process. In addition, comparing the results of Theorems 6 and 7, one can guess that the noise from the natural deathrates of the disease related classes-exposed, infectious and removal populations, and also the noise from the disease transmission rates do not influence a change on the existence and stability of the disease-free steady state E 0 , since the existence and stability of E 0 does not change under the conditions of both theorems, regardless of whether the intensities satisfy σi > 0, i = E, I, R, β, or the intensities satisfy σi → 0, i = E, I, R, β, provided that the threshold values satisfy R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1. This implies that the disease will be continuously eradicated from the system, regardless of the strengths of the noises from the natural deathrates of the disease related classes-exposed, infectious and removal populations, and also regardless of the strength of the noise from the disease transmission rate, provided that the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 are satisfied. The result in Theorem 7 also confirms the earlier observation from Theorem 6 that the source of the environmental white noise processes in the stochastic system owing to (1) the disease transmission rate, or owing to (2) the natural death rates of the different disease classes-S, E, I, R exhibit direct impacts on the disease eradication process from the population. This is because comparing the results of Theorem 7 and Theorem 3(3), it is easy to see that if the stochastic system is influenced by significant random fluctuations, where the source is from the natural death rate of susceptible individuals in the population, that is, σ S > 0, then by Theorem 3(3), there is no disease-free steady state for the population, and the disease cannot be eradicated by applying neither the threshold conditions (R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1) from Theorem 6, nor the threshold conditions (R0 ≤ 1, Uˆ 0 ≤ 1, and Vˆ0 ) from Theorem 7. More detailed discussions about the disease eradication process under the influence of various intensity levels of the white noise processes in the system are given in Sect. 5. Moreover, numerical evidence for the effects of the intensities of the white noise processes in the system on the disease eradication process are presented in Sect. 6.

Modeling Highly Random Dynamical Infectious Systems

4.2.4

549

Asymptotic Behavior When There is no Infection-Free Steady State

As earlier remarked in Remark 1, it is expected from the existence of solution results in Theorem 2(b), that the absence of a positively-self invariant space for the sample paths of the stochastic system (23)–(26), whenever the intensities satisfy σi > 0, ∀i ∈ {S, E, I, R} can lead to complex uncontrolled situations for the disease dynamics in the system. This fact is true as it will be shown later in this section. We recall that Theorem 3(3) asserts that the stochastic system (23)–(26) has no disease-free steady state for the population, whenever the intensity of the white noise process from the natural death rate of the susceptible population is significant, that is, σ S > 0. This study will be incomplete unless we know the fate of the disease dynamics in the human population, whenever there is no disease-free steady state wherein the disease can be eradicated. The general approach to investigate stochastic dynamic systems (55), whenever the infection-free steady state exists no where, involves estimating the expected distance between the trajectories of the stochastic system, and a potential disease-free steady state for the system. The estimate obtained generally informs us about the extend to which the noise from the natural deathrate of the susceptible class in the system (23)–(26) deviates the trajectories of the stochastic system from the potential disease-free steady state. Since the stochastic system (23)–(26) has the disease-free steady state E 0 = (S0∗ , 0, 0), S0∗ = μB from Theorem 3(1, 2), whenever σ S = 0, and loses the steady state, whenever σ S > 0, therefore E 0 is always a potential infection-free steady state for the system (23)–(26). The following result describes the oscillatory behavior of the trajectories of the stochastic system (23)–(26) in the neighborhood of the potential disease-free equilibrium E 0 obtained in Theorem 3[1.- 2.], whenever Theorem 3(3) is satisfied, that is, whenever the stochastic system (23)–(26) does not have a disease-free equilibrium. This result characterizes the expected average relative distance between the sample paths of the stochastic system (23)–(26) and the potential disease-free steady state E 0 . Moreover, this result builds the backbone to again more insights about the asymptotic oscillatory behavior of the stochastic system (23)–(26), whenever the system is subjected under the influence of various intensity levels of the white noise processes in the system which is discussed further in Sect. 5. Theorem 8 Let the hypothesis of Theorem 3(3) be satisfied. And define the following threshold values: R˜ 1 =

kˆ1 σβ2 + 21 σ 2I α β S0∗ Kˆ 1 + + , (μ + d + α) (μ + d + α) (μ + d + α) U˜ 0 =

μ 2β S0∗ + β + α + 2 K˜ (μ) 2

2μ

+

σ 2S , 2μ

(96)

(97)

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and

(2μ K˜ (μ)2 + α + β(2S0∗ + 1)) σ 2E + , V˜0 = 2μ 2μ

(98)

with some constants Kˆ 1 > 0, kˆ1 > 0 that depends on S0∗ (in fact, kˆ1 = 6S0∗ and Kˆ 1 = 4 + S0∗ ), Let X (t) = (S(t), E(t), I (t)) be a solution of the decoupled system from (23)–(26) with initial conditions (27). Assume that, R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1, then there exists a positive constant m > 0, such that the following inequality holds 1 lim sup E t→∞ t

 0

t

  3σ 2S (S0∗ )2 . (S(v) − S0∗ )2 + E 2 (v) + I 2 (v) dv ≤ m

(99)

Proof Let Theorem 3(3) be satisfied. Applying the differential operator d V to V defined in (78), and utilizing (67) and (68), it is easy to see that d V = L V dt − 2σ S (U (t) + V (t))(S0∗ + U (t))dw S (t) −2σ E (U (t)V (t) + (c + 1)V 2 (t))dw E (t) − 2σ I W 2 (t))dw I (t)  h1 ∗ f T1 (s)e−μs G(W (t − s))dsdwβ −2cσβ (S0 + U (t))V (t) t0

−2σ E [U (t) + (c + 1)V (t) + W (t)] ×  h2  h1 f T2 (u) f T1 (s)e−μ(s+u) (S0∗ + U (t − u))G(W (t − s − u))dsdudwβ (t) × t0

t0

(100) where for some positive constant valued function K˜ (μ), the drift part of (100), L V , satisfies the inequality   ˜ 2 (t) + ϕW ˜ 2 (t) + ψV ˜ 2 (t) , L V (x, t) ≤ − φU

(101)

where φ˜ = 2μ(1 − U˜ 0 )



ψ˜ = 2μ(1 − V˜0 ) − (2μ + σ 2E )c 1 −

β(3S0∗ + 1)



(2μ + σ 2E )

ϕ˜ = 2(μ + d + α)(1 − R˜ 1 ) − c(3β S0∗ + β(S0∗ )2 + 4σβ2 (S0∗ )2 ) − 2c2 σβ2 (S0∗ )2 . ∗

ˆ

(102) (103) (104)

β S K 1 +α+ 2 σ I Moreover, R˜ 1 = 0(μ+d+α) , where Kˆ 1 = 4 + S0∗ + 6 β1 σβ2 . Under the assumptions of R˜ 1 , U˜ 0 , and V˜0 in the hypothesis and for suitable choice of the positive constant ˜ ψ, ˜ and ϕ˜ are positive constants. Therefore, by integrating (100) c it follows that φ, from 0 to t on both sides and taking the expectation, it follows from (100)–(104) that 1

2

Modeling Highly Random Dynamical Infectious Systems E(V (t) − V (0)) ≤ −mE

 t  (S(v) − S0∗ )2 + E 2 (v) + I 2 (v) dv + 3σ 2S (S0∗ )2 t,

551

(105)

0

where V (0) is constant and

˜ ψ, ˜ ϕ). m = min(φ, ˜

(106)

Hence, diving both sides of (105) by t and m, and taking the limit supremum as t → ∞, then (99) follows immediately. The results of Theorem 8 are interpreted in the following. Theorem 8 signifies that under conditions that warrant the nonexistence of a disease free steady state for the stochastic system (23)–(26), the asymptotic expected average relative distance between the white noise influenced trajectories of the stochastic system and the potential disease-free steady state, E 0 obtained in Theorem 3(1, 2), does not exceed a constant multiple of the intensity, σ S , of the white noise process from the natural deathrate of the susceptible population, whenever the following threshold conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 are satisfied. That is, asymptotically, when the physical characteristics in a disease scenario allow significant random fluctuations in the natural deathrate of susceptible individuals which lead to a white noise process with intensity value σ S > 0, and consequently lead to the nonexistence or no where existence of an infection-free population steady state, and also allow physical conditions which mathematically can be represented by the parameters of the stochastic system (23)–(26), wherein the threshold values R˜ 1 , U˜ 0 , and V˜0 in (96)–(98) can be computed, it follows that if the following threshold conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 are satisfied, then the noise influenced population which is affected by the disease outbreak is expected to oscillate over time near the potential disease-free steady state population E 0 obtained in Theorem 3(1, 2). Furthermore, the size or amplitude of the oscillations is determined primarily by the size of the intensity, σ 2S , of the white noise process from the natural death rate of the susceptible individuals in the population. It is easily observed that for infinitesimally small values for the intensity of the white noise from the natural deathrate of the susceptible class, that is, for σ 2S → 0, it follows from (99) that all trajectories of the stochastic system (23)–(26) converge on average (in expectation) to the disease-free steady state E 0 asymptotically. In addition, for continuous increase in the values σ 2S , that is, for σ 2S → ∞, it is also easy to see from (99) that the average distance of the trajectories of the stochastic system from the potential infection-free steady state E 0 gets wider apart. This observation signifies that, the stronger the noise in the system from the natural deathrate of the susceptible population gets, the further and further away the system gets from the infection-free steady state wherein the disease can be eradicated. Also, the dependence of the size of the random oscillations of the state of the system near the infection-free steady state E 0 , and also the dependence of the magnitude of the threshold values R˜ 1 , U˜ 0 , and V˜0 on the intensities of the white noise processes in the system, σi , i = S, E, I, R, β, suggests as similarly remarked in Subsections 4.2.2 and 4.2.3, that the source (disease transmission or natural death

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rates) and intensity levels3 of the white noise processes in the system exert influence on the asymptotic oscillatory behavior of the system near the potential disease-free population steady state, E 0 , obtained in Theorem 3(1, 2). This is obvious as it is already remarked above that as σ 2S → ∞, the sample paths of the stochastic system get further and further away from E 0 . The examination of the influence of the source and intensity levels of the white noise processes on the oscillatory behavior of the system near E 0 is elaborated in Sect. 5. Moreover, numerical evidence for the effects of the intensity levels of the white noise processes in the system (23)–(26) on the oscillatory behavior of the system, such as population extinction etc., over time are presented in Sect. 6. Furthermore, as similarly remarked in Remark 4.2.3(2), comparing the threshold values from Theorems 7, 6, and 8, that is, R1 , U0 , V0 in (80)–(82), R0 , Uˆ 0 , Vˆ0 in (92)–(94), and R˜ 1 , U˜ 0 , V˜0 in (96)–(98), it is easy to see that R0 ≤ R1 = R˜ 1 , U0 = Uˆ 0 ≤ U˜ 0 and V0 ≤ Vˆ0 = V˜0 , whenever the intensity values of the white noise processes in the system (23)–(26) satisfy the condition σi > 0, i = S, E, I, R, β. It is also easy to see that the threshold value U˜ 0 in (97) has been further constrained by the assumption that σ S > 0, from the corresponding threshold value U0 = Uˆ 0 in (81) and (93). Meanwhile it was remarked in Remark 4.2.3(2) that the threshold values R1 , U0 and V0 from Theorem 6 would satisfy the threshold conditions R1 ≤ 1, U0 ≤ 1 and V0 ≤ 1 less easily compared to the set of threshold values R0 , Uˆ 0 and Vˆ0 from Theorem 7, it is easy to see that the threshold values R˜ 1 , U˜ 0 , and V˜0 from Theorem 8 would satisfy the threshold conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 also less easily when compared to he threshold values R1 , U0 and V0 from Theorem 6, and much less easily compared to the threshold values R0 , Uˆ 0 and Vˆ0 from Theorem 7. This observation suggests that the sources (natural death or disease transmission rates) and intensity levels of random fluctuations in the disease dynamics exhibit bearings on (1) the existence of a disease-free population steady state for the system (23)–(26), and also on (2) the disease eradication conditions for the system. For instance, adding the new source of random fluctuations due to the natural death rate of the susceptible population which leads to the white noise process with intensity σ S , then for infinitesimally small values for σ S , there exists a disease free population state, and the disease can be eradicated, whenever the conditions in Theorems 7 and 6 are satisfied. But for significant in magnitude values of the intensity σ S > 0, the additional source of the white noise from the random fluctuations in the natural deathrate of the susceptible individuals leads to a loss or nonexistence of the diseasefree population steady state. Moreover, in such events where the disease-free steady state ceases to exist, the solutions of the system (23)–(26) can oscillate closely to the potential disease-free population steady state E 0 obtained in Theorem 3(1, 2), provided that the conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 are satisfied, and the value of the intensity σ S is also relatively small, that is, σ S → 0. 3 The

intensity levels of the white noise processes in the system are described as infinitesimally small, significant in magnitude and small but not infinitesimally small, big in size and finite, and sufficiently large.

Modeling Highly Random Dynamical Infectious Systems

553

Furthermore, as similarly remarked in Remark 4.2.3(2), the results of Theorem 8 suggest that in a disease outbreak scenario that exhibits random fluctuations with high intensity values for the noise from the natural deathrate of susceptible population, that is, σ S > 0, and as a result does not allow the existence of an infection-free steady state population, but exhibit physical characteristics which can be represented mathematically by the parameters of the system (23)–(26), wherein the threshold values R˜ 1 , U˜ 0 , and V˜0 in (96)–(98), R1 , U0 , V0 in (80)–(82) and R0 , Uˆ 0 , Vˆ0 in (92)–(94) can all be computed, and satisfy the following relationship between the threshold values and also the threshold conditions given by R0 ≤ R1 = R˜ 1 ≤ 1, U0 = Uˆ 0 ≤ U˜ 0 ≤ 1 and V0 ≤ Vˆ0 = V˜0 ≤ 1, then the occurrence of the white noise processes from the random environmental fluctuations in all other sources namely:from (1) the natural deathrates of the exposed, infectious and removal populations, and also from (2) the disease transmission rate, exerts additional counter-positive constraints against the disease eradication process as determined by the relationships between the threshold values, and the threshold conditions R0 ≤ R1 = R˜ 1 ≤ 1, U0 = Uˆ 0 ≤ U˜ 0 ≤ 1 and V0 ≤ Vˆ0 = V˜0 ≤ 1. That is, whereas the disease can be eradicated much less rapidly when the disease scenario represented by (23)–(26) is controlled by the threshold values R1 , U0 , V0 in (80)–(82) than when it is controlled by the threshold values R0 , Uˆ 0 , Vˆ0 in (92)–(94), it follows that when the disease scenario is controlled by the threshold values R˜ 1 , U˜ 0 , and V˜0 in (96)–(98), the disease cannot be eradicated. Nevertheless, the disease population can be maintained close to a potential disease-free population steady state E 0 obtained in Theorem 3(1, 2), whenever the intensity σs is small, and the threshold conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 are satisfied. The oscillatory behavior of the system (23)–(26) relative to E 0 obtained in Theorem 3(1, 2) under the influence of various intensity levels of the white noise processes in the system is discussed further in Sect. 5.

5 Asymptotic Behavior of the System Subjected Under Various Orders for the Intensities of Noise This section exhibits the asymptotic properties of the stochastic system (23)–(26), whenever it is subjected under the direct influence of various growth orders of the intensities of the white noise processes in the system. In Sect. 4, several observations were made about the bearings of the source (natural death or disease transmission rates) and intensity levels of the white noise processes in the system on (1) the existence and stability of the disease free steady state population E 0 obtained in Theorem 3(1, 2), and consequently on (2) the disease eradication and also on (3) the expected distance between the solutions of the system (23)–(26) and the potential disease free equilibrium E 0 . In this section, several special disease scenarios are characterized to give more insight about the properties - (1), (2) and (3), with respect to the stochastic system (23)–(26).

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The special disease scenarios are determined by the qualitative behaviors of the intensities, σi2 , i = S, E, I, R, β, of the white noise processes originating from the natural death and disease transmission rates in the system. Furthermore, the qualitative character of the intensities of the white noise processes are classified in Hypothesis 1. The following definitions are helpful to understand the assumptions made about the intensity levels of the white noise processes described in Hypothesis 1: Definition 1 Given two real valued functions f and g, (1) if ∃k > 0, and n 0 , such that ∀n > n 0 , | f (n)| ≤ k|g(n)|, we say that f is big-o of g, and is denoted by f (n) = 0(g(n)) or f = 0(g). If f (n) → 0, as n → ∞, that is, f turns in the limit to a zero function for sufficiently large n, we write f = 0( ) or f (n) = 0( n1 ), for > 0. Also, if f (n) is a constant function as n → ∞, we write f (n) = 0(1). (2) if ∃k1 , k2 > 0, and n 0 , such that ∀n > n 0 , k1 |g(n)| ≤ | f (n)| ≤ k2 |g(n)|, we say that f is big-theta of g, and is denoted by f (n) = θ(g(n)). If f (n) → ∞ as n → ∞, we write f (n) = θ(n) or f = θ( 1 ), for > 0. The hypothesis that follows compares the intensity levels of the white noise processes from the following (1) the disease transmission rate, (2) the natural death rate of the susceptible population, and (3) the natural death rates of the three other subcategories - exposed, infectious and removal populations. Hypothesis 1 Using Definition 1, we assume that H1 : σβ → 0, σ S → 0, σi → 0, ∀i = E, I, R, ⇐⇒ σβ = 0( ), σ S = 0( ), σi = 0( ), ∀i = E, I, R; H2 : σβ < ∞, σ S → 0, σi → 0, ∀i = E, I, R, ⇐⇒ σβ = 0(1), σ S = 0( ), σi = 0( ), ∀i = E, I, R; H3 : σβ < ∞, σ S → 0, σi < ∞, ∀i = E, I, R, ⇐⇒ σβ = 0(1), σ S = 0( ), σi = 0(1), ∀i = E, I, R; H4 : (σβ → 0, or σβ < ∞) and σ S < ∞, and (σi → 0 or σi < ∞, ∀i = E, I, R), ⇐⇒ (σβ = 0( ), or σβ = 0(1)), and σ S = 0(1), and (σi = 0( ) or σi = 0(1), ∀i = E, I, R; H5 : σβ → ∞, σ S → 0, σi → 0, ∀i = E, I, R, ⇐⇒ σβ = θ( 1 ), σ S = 0( ), σi = 0( ), ∀i = E, I, R; H6 : σβ → ∞, σ S → 0, σi → ∞, ∀i = E, I, R, ⇐⇒ σβ = θ( 1 ), σ S = 0( ), σi = θ( 1 ), ∀i = E, I, R; H7 : (σβ → ∞, or σβ → 0, or σβ < ∞), and (σ S → ∞), and (σi → ∞, or σi < ∞, or σi → 0, ∀i = E, I, R), ⇐⇒ (σβ = θ( 1 ), or σβ = 0( ), or σβ = 0(1)), and (σ S = θ( 1 )), and (σi = θ( 1 ), or σi = 0(1), or σi = 0( ), ∀i = E, I, R). Hypothesis 1(H1 ) asserts that all the intensities, σi , i = S, E, I, R, β, of the white noise processes in the system continuously decrease in size to infinitesimally small values.

Modeling Highly Random Dynamical Infectious Systems

555

Hypothesis 1(H2 ) assumes that the intensity σβ of the white noise process from the disease transmission rate is finite in size, but the other white noise intensities σi , i = S, E, I, R from the natural deathrates of the susceptible, exposed and infectious populations continuously decrease in size to infinitesimally small values. Hypothesis 1(H3 ) assumes that the intensities σβ , σi , i = E, I, R of the white noise processes from the disease transmission rate, and the natural deathrates of the exposed, infectious and removal populations are finite in size, meanwhile the intensity, σ S , of the white noise process from the natural deathrate of the susceptible population continuously decreases in size to infinitesimally small values. Hypothesis 1(H4 ) asserts that the intensity, σβ , of the white noise process from the disease transmission rate either continuously decreases in size to infinitesimally small values or it is significant and finite in size. At the same time, the intensities, σi , i = E, I, R, of the white noise processes from the natural death rates in the exposed, infectious and removal populations are also either significant and finite in size, or they all continuously decrease in size to infinitesimally small values, meanwhile the intensity, σ S , of the white noise process from the natural deathrate of the susceptible population is significant and finite in size. Hypothesis 1(H5 ) states that the intensity, σβ , of the white noise process from the disease transmission rate continuously increases in size to sufficiently large values, whereas the intensities, σi , i = S, E, I, R, of the white noise processes from the natural deathrates of the susceptible, exposed, infectious and removal populations continuously decrease in size to infinitesimally small values. Hypothesis 1(H6 ) assumes that the intensities, σi , i = E, I, R, β, of the white noise processes from the disease transmission rate and the natural deathrates of the exposed, infectious and removal populations continuously increase in size to sufficiently large values, but the intensity, σ S , of the white noise process from the natural deathrate of the susceptible population continuously decreases in size to infinitesimally small values. Hypothesis 1(H7 ) states that the intensities, σi , i = E, I, R, β, of the white noise processes from the disease transmission rate and natural deathrates of the exposed, infectious and removal populations either continuously increase in size to sufficiently large values, or they are significant and finite in size, or they continuously decrease in size to infinitesimally small values, while the intensity, σ S , of the white noise process from the natural deathrate of the susceptible population continuously increases in size to sufficiently large values. The following results characterize the qualitative behavior of the solutions of the stochastic system (23)–(26) under the assumptions of Hypothesis 1. Theorem 9 If Hypothesis 1(H1 ) holds, then there exists a disease free equilibrium population E 0 = (S0∗ , 0, 0), S0∗ = μB for the stochastic system (23)–(26). Furthermore, there exists threshold values R1 , U0 and V0 define as follows: R1 =

α β S0∗ Kˆ 1 + α + , (μ + d + α) (μ + d + α)

(107)

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D. Wanduku

U0 = and V0 =

μ 2β S0∗ + β + α + 2 K˜ (μ) 2

2μ

,

(2μ K˜ (μ)2 + α + β(2S0∗ + 1)) , 2μ

(108)

(109)

with some constants Kˆ 1 > 0, kˆ1 > 0 that depends on S0∗ (in fact, kˆ1 = 6S0∗ and Kˆ 1 = 4 + S0∗ ), such that under the assumptions that R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1, the disease free equilibrium state is stochastically asymptotically stable in the large in D(∞). Moreover, it is mean square stable. Proof Let σβ = 0( ), σ S = 0( ), σi = 0( ), ∀i = E, I, R. It follows from the Theorem 3 that (23)–(26) has a disease free steady state given by E 0 = (S0∗ , 0, 0), S0∗ = μB . Furthermore, the rest of the result follows immediately from Lemma 4 and Theorem 6. Remark 3 Theorem 9 signifies that when the stochastic system (23)–(26) is continuously influenced by white noise processes from the disease transmission and natural death rates that have intensity values that continuously decrease in size to infinitesimally small values, the system has a disease-free steady state population E 0 , and the steady state is stochastically asymptotically stable in the large, whenever the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 given in (107)–(109) are satisfied. This result suggests that in a disease scenario where the disease outbreak results in random fluctuations in the disease transmission rate and in the natural death rates of all the subclasses-susceptible, exposed, infectious and removal populations, there exists a disease-free population steady state for the population given by E 0 = (S0∗ , 0, 0), S0∗ = μB , whenever the random environmental fluctuations in the disease transmission rate and also in the natural death processes have infinitesimally small intensity values. Furthermore, when the physical characteristics of the disease scenario can be mathematically represented by the parameters of the system (23)–(26), wherein the threshold values R1 , U0 and V1 from (107)–(109) can be computed, then the disease can be eradicated from the population, whenever the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 are satisfied. Theorem 10 If Hypothesis 1(H2 ) holds, then there exists a disease free equilibrium population E 0 = (S0∗ , 0, 0), S0∗ = μB for the stochastic system (23)–(26). Furthermore, there exists threshold values R1 , U0 and V0 define as follows: R1 =

kˆ1 σβ2 α β S0∗ Kˆ 1 + + , (μ + d + α) (μ + d + α) (μ + d + α) U0 =

μ 2β S0∗ + β + α + 2 K˜ (μ) 2

2μ

,

(110)

(111)

Modeling Highly Random Dynamical Infectious Systems

and V0 =

557

(2μ K˜ (μ)2 + α + β(2S0∗ + 1)) , 2μ

(112)

with some constant Kˆ 1 > 0 that depends on S0∗ (in fact, Kˆ 1 = 4 + S0∗ + β1 6σβ2 ) such that, under the assumptions that R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1, the disease free equilibrium state is stochastically asymptotically stable in the large in D(∞). Moreover, it is mean square stable. Proof Let σβ = 0(1), σ S = 0( ), σi = 0( ), ∀i = E, I, R. It follows from the Theorem 3 that (23)–(26) has a disease free steady state given by E 0 = (S0∗ , 0, 0), S0∗ = μB . Furthermore, the rest of the result follows immediately from Lemma 4 and Theorem 6. Theorem 11 If Hypothesis 1(H3 ) holds, then there exists a disease free equilibrium population E 0 = (S0∗ , 0, 0), S0∗ = μB for the stochastic system (23)–(26). Furthermore, there exists threshold values R1 , U0 and V0 define as follows: R1 =

kˆ1 σβ2 + 21 σ 2I α β S0∗ Kˆ 1 + + , (μ + d + α) (μ + d + α) (μ + d + α) U0 =

and V0 =

μ 2β S0∗ + β + α + 2 K˜ (μ) 2

2μ

,

(2μ K˜ (μ)2 + α + β(2S0∗ + 1)) σ 2E + , 2μ 2μ

(113)

(114)

(115)

with some constants Kˆ 1 > 0, kˆ1 > 0 that depend on S0∗ (in fact, kˆ1 = 6S0∗ and Kˆ 1 = 4 + S0∗ ), such that, under the assumptions that R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1, the disease free equilibrium state is stochastically asymptotically stable in the large in D(∞). Moreover, it is mean square stable. Proof Let σβ = 0(1), σ S = 0( ), σi = 0(1), ∀i = E, I, R. It follows from the Theorem 3 that (23)–(26) has a disease-free steady state given by E 0 = (S0∗ , 0, 0), S0∗ = B . Furthermore, the rest of the result follows immediately from Lemma 4 and μ Theorem 6. Remark 4 Theorems 10 and 11 signify that when the stochastic system (23)–(26) is continuously influenced by white noise processes from the disease transmission rate and natural deathrates where the intensity σβ of the white noise from the disease transmission rate is significant and finite in size, and the intensity σ S of the white noise from the natural deathrate of the susceptible population continuously decreases in size to infinitesimally small values, whereas the intensities, σi , i = E, I, R, of the white noise processes from the natural deathrates of the exposed, infectious and removal populations are either significant and finite in size, or they continuously decrease

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in sizes to infinitesimally small values, it follows that the system has a diseasefree steady state population E 0 , and the steady state is stochastically asymptotically stable in the large, whenever the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 in (110)–(112) and (113)–(115) are satisfied. This result suggests that in a disease scenario where the disease outbreak results in random fluctuations in the disease transmission and natural death rates, there exists a disease-free population steady state for the population given by E 0 = (S0∗ , 0, 0), S0∗ = μB , whenever the random environmental fluctuations in the disease transmission rate have significant and finite intensity values, and the environmental fluctuations in the natural deathrate of the susceptible population has infinitesimally small intensity values, regardless of whether there exists significant and finite in sizes, or there exists infinitesimally small intensity values for the random fluctuations from the natural deathrates of the exposed, infectious and removal populations. Furthermore, when the physical characteristics of the disease dynamics can be represented by the parameters of the system (23)–(26), wherein the threshold values R1 , U0 and V1 from (110)–(112) and (113)–(115) can be computed, then the disease can be eradicated from the population, whenever the threshold conditions R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 are satisfied. The following result characterizes the behavior of the solutions of the stochastic system (23)–(26) whenever the assumption of Hypothesis 1(H4 ) is satisfied. For simplicity, the results are presented only for the subcase when σβ = 0( ), σ S = 0(1), and σi = 0( ), ∀i = E, I, R. The other subcases can similarly be derived. Theorem 12 If Hypothesis 1(H4 ) holds, then there is no disease free equilibrium population state for the stochastic system (23)–(26). But, when σβ = 0( ), σ S = 0(1), and σi = 0( ), ∀i = E, I, R, there exists threshold values: R˜ 1 =

μ 2β S0∗ + β + α + 2 K˜ (μ) 2

(116)

σ 2S , 2μ

(117)

(2μ K˜ (μ)2 + α + β(2S0∗ + 1)) , V˜0 = 2μ

(118)

U˜ 0 = and

α β S0∗ Kˆ 1 + , (μ + d + α) (μ + d + α)

2μ

+

with some constant Kˆ 1 > 0 that depends on S0∗ (in fact, Kˆ 1 = 4 + S0∗ ), such that letting X (t) = (S(t), E(t), I (t)) be a solution of the decoupled system from (23)– (26) with initial conditions (27) then there exists a positive constant m > 0, such that the following inequality holds  t   3σ 2S (S0∗ )2 1 (S(v) − S0∗ )2 + E 2 (v) + I 2 (v) dv ≤ , (119) lim sup E m t→∞ t 0 whenever R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1.

Modeling Highly Random Dynamical Infectious Systems

559

Proof Let (σβ = 0( ), or σβ = 0(1)) and σ S = 0(1), and (σi = 0( ) or σi = 0(1), ∀i = E, I, R). It follows from the Theorem 3(3) that (23)–(26) does not have a disease free steady state. Furthermore, when σβ = 0( ), σ S = 0(1), and σi = 0( ), ∀i = E, I, R, the rest of the result follows immediately from Theorem 8. Remark 5 Theorem 12 signifies that when the stochastic system (23)–(26) is continuously influenced by the white noise processes from the disease transmission and natural death rates, where the intensity value σβ of the white noise process from the disease transmission process is significant and finite in size, or it continuously decreases in size to an infinitesimally small value, and the intensity value σ S of the white noise process from the natural deathrate in the susceptible population is significant and also finite in size, meanwhile the intensities, σi , i = E, I, R, of the white noise processes from the natural deathrates of the exposed, infectious and removal populations are either significant and finite in size, or they continuously decrease in size to infinitesimally small values, it follows that the system (23)–(26) does not have any disease-free steady state for the population. Nevertheless, the solutions of the system (23)–(26) continue to oscillate near the potential disease-free steady state E 0 = (S0∗ , 0, 0), S0∗ = μB obtained from Theorem 3(1, 2), whenever the threshold conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 in (116)–(118) are satisfied. Moreover, the size or amplitude of the oscillations of the solutions of (23)–(26) relative to the disease free steady state E 0 = (S0∗ , 0, 0), S0∗ = μB , is proportional to the size of the intensity σ S . This implies that small values of σ S result in small asymptotic expected average distance between the solutions of (23)–(26) and the potential disease free steady state E 0 = (S0∗ , 0, 0), S0∗ = μB and vice versa. This result suggests that in a disease scenario where the disease outbreak results in random fluctuations in the disease transmission and natural death rates, when the random environmental fluctuations in the natural deathrate of the susceptible population has a significant and finite intensity value σ S , the disease-free steady state population exists no where. Nevertheless, the states of the population ( which includes all subclasses- S, E, I, R) will oscillate over time near the potential disease-free steady state population E 0 = (S0∗ , 0, 0), S0∗ = μB , whenever the threshold conditions R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 in (116)–(118) are satisfied. Moreover, for small values of the intensity σ S , the size of the oscillations of the population relative to E 0 is small proportionately to the size of σ S . This implies that in this disease scenario, whereas the disease can not be eradicated by applying the threshold conditions in Theorem 6, it follows that when the intensity value, σ S , of the random fluctuations in the natural deathrate of the susceptible population is small in size, the population states affected by the disease outbreak can be contained closely to the potential disease-free steady state E 0 = (S0∗ , 0, 0), S0∗ = B , whenever the threshold values R˜ 1 , U˜ 0 , and V˜0 satisfy the threshold conditions μ R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1. In addition, the states of the population will continue to oscillate over time near the potential disease-free steady state E 0 , regardless of the size of the intensities σi , i = E, I, R, β of the random fluctuations in the disease transmission rate, or in the natural

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death rates of the exposed, infectious and removal individuals in the population, provided that the intensity values σi , i = E, I, R, β are small in magnitude. Theorem 13 If Hypothesis 1(H5 ) holds, then there exists a disease free equilibrium population E 0 = (S0∗ , 0, 0), S0∗ = μB for the stochastic system (23)–(26). But, the disease free equilibrium state is stochastically unstable in D(∞). Proof Let σβ = θ( 1 ), σ S = 0( ), σi = 0( ), ∀i = E, I, R. It follows from the Theorem 3(1, 2) that (23)–(26) has a disease free steady state given by E 0 = (S0∗ , 0, 0), S0∗ = μB . Furthermore, for any Lyapunov functional V , the differential operator L V associated with the Ito-Doob type system (23)–(26) has the following structure: 1 (120) L V (x) = Vt (x) + Vx (x)f(x) + g T (x)Vx x (x)g(x), 2 where f(x) and g (x) are vectors representing the drift and diffusion parts of the system (23)–(26). It is easy to see that under the assumption of σβ = θ( 1 ), the diffusion part g (x) = θ( 1 ). Consequently, it follows from (120) that L V (x) = θ( 1 ), ∀V . It follows further from Lyapunov stability comparative results that the steady state E 0 is stochastically unstable. Theorem 14 If Hypothesis 1(H6 ) holds, then there exists a disease-free equilibrium population E 0 = (S0∗ , 0, 0), S0∗ = μB for the stochastic system (23)–(26). But, the disease free equilibrium state is stochastically unstable in D(∞). Proof Let σβ = θ( 1 ), σ S = 0( ), σi = θ( 1 ), ∀i = E, I, R. It follows from the Theorem 3(1, 2) that (23)–(26) has a disease free steady state given by E 0 = (S0∗ , 0, 0), S0∗ = μB . Furthermore, for any Lyapunov functional V , the differential operator L V associated with the Ito-Doob type system (23)–(26) has the following structure: 1 (121) L V (x) = Vt (x) + Vx (x)f(x) + g T (x)Vx x (x)g(x), 2 where f(x) and g (x) are vectors representing the drift and diffusion parts of the system (23)–(26). It is easy to see that under the assumption of σβ = θ( 1 ), the diffusion part g (x) = θ( 1 ). Consequently, it follows from (121) that L V (x) = θ( 1 ), ∀V . Therefore, from Lyapunov stability comparative results, the steady state E 0 is stochastically unstable. Remark 6 Theorems 13 and 14 signify that when the stochastic system (23)–(26) is continuously influenced by white noise processes from the disease transmission and natural death rates, where the intensity value σ S of the white noise process from the natural deathrate of the susceptible population continuously decreases in size to infinitesimally small values, then the system has a disease free steady state population given by E 0 = (S0∗ , 0, 0), S0∗ = μB , regardless of the sizes of the intensities σi , i = E, I, R, β of the white noise processes from the disease transmission rate and natural deathrates of the exposed, infectious and removal populations. But the disease free steady state population E 0 that exists is clearly stochastically unstable,

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561

whenever the intensities σi , i = E, I, R, β of the white noise processes from the disease transmission rate or the natural deathrates of the exposed, infectious and removal populations continuously increase in size to sufficiently large values. This result suggests that in a disease scenario where the disease outbreak results in random fluctuations in the disease transmission rate and natural deathrates, there exists a disease free steady state for the population given by E 0 = (S0∗ , 0, 0), S0∗ = B , whenever the random environmental fluctuations in the natural deathrate of the μ susceptible population have infinitesimally small intensity values, regardless of the sizes of the intensity values of the random environmental fluctuations in the disease transmission rate and the natural deathrates of the exposed, infectious and removal individuals in the population. However, the sufficiently large intensity values for the random fluctuations in the disease transmission rate or the natural deathrates of the exposed, infectious and removal populations quickly “drive” all sample paths of the different population states away from the disease free population steady state E 0 , and consequently adversely favoring conditions that allow the disease to establish an endemic stable steady state or a stable significant number of the disease related classes-infectious, exposed and removal individuals in the population. This result further suggests that significantly high intensity values for random fluctuations in the disease transmission rate and the natural deathrates of the exposed, infectious and removal populations exert strong negative conditions against the disease eradication process. Numerical simulation results in Sect. 6 show that the high intensity values for the random fluctuations in the disease transmission rate or the natural deathrates of the exposed, infectious and removal populations lead to a general decrease in the average total population size over time, which in some cases may lead to the population extinction for sufficiently large intensity values. The following result describes the behavior of the solutions of the system (23)–(26) under the assumptions of Hypothesis 1(H7 ). For simplicity only the special case of σβ = 0(1), and σ S = θ( 1 ), and σi = θ( 1 ), ∀i = E, I, R. The results for the other cases can be similarly derived. Theorem 15 If Hypothesis 1(H7 ) holds, then there is no disease free equilibrium population for the stochastic system (23)–(26). Furthermore, when σβ = 0(1), σ S = θ( 1 ), and σi = θ( 1 ), ∀i = E, I, R, the system does not oscillate in the neighborhood of the potential disease free equilibrium E 0 = (S0∗ , 0, 0), S0∗ = μB obtained from Theorem 3(1, 2). Proof Let σβ = 0(1), and σ S = θ( 1 ), and σi = θ( 1 ), ∀i = E, I, R. It follows from the Theorem 3(3) that (23)–(26) does not have a disease free steady state. Furthermore, the rest of the result follows immediately from the Proof of Theorem 8. Remark 7 Theorem 15 signifies that when the stochastic system (23)–(26) is continuously influenced by the white noise processes from the disease transmission rate and natural deathrates, where the intensity value σ S of the white noise process from the natural deathrate of the susceptible population continuously increases in size to sufficiently large values, then it follows from Theorem 3(3) that the system (23)–(26) does not have a disease free steady state population. Moreover, the significantly large intensity values σi , i = E, I, R of the white noise processes from the

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natural deathrates of the exposed, infectious or removal populations lead to significantly large threshold values R˜ 1 , U˜ 0 , and V˜0 in (96)–(98), that violate the threshold conditions, R˜ 1 ≤ 1, U˜ 0 ≤ 1, and V˜0 ≤ 1 in Theorem 8. This implies that the solutions of the system (23)–(26) do no longer oscillate near the disease free steady state E 0 obtained from Theorem 3(1, 2). Furthermore, for σ S = θ( 1 ), it follows from (99) that the asymptotic expected average distance between the solutions of (23)–(26) and potential disease free equilibrium state E 0 obtained from Theorem 3(1, 2) is of the order θ( 1 ). This also implies that the sufficiently large intensity values of σ S lead to oscillations in the disease dynamics with very large oscillation sizes for the sample paths of the different states of the population over time. In addition, the sample paths of the different states of the population oscillate over time at significantly large distances away from the potential disease-free equilibrium state E 0 = (S0∗ , 0, 0), S0∗ = μB obtained from Theorem 3(1, 2). Thus, the disease can never be eradicated under the conditions of Theorem 15. Moreover, the numerical simulation results in Sect. 6 show that the high intensity values, σi , i = E, I, R, β, for the random fluctuations in the disease transmission rate or the natural deathrates of the susceptible, exposed, infectious and removal populations lead to a general decrease in the average total population size over time, which in some cases may lead to the population extinction for sufficiently large intensity values. This result suggests that in a disease scenario where the disease outbreak results in random fluctuations in the disease transmission rate and natural deathrates, it follows that when the random environmental fluctuations in the natural deathrate of the susceptible population exhibit sufficiently large intensity values, then the population does not exhibit a disease-free population steady state. Furthermore, when the intensity value of the random fluctuations in the natural death rates of the exposed, or infectious or removal populations is also sufficiently large in size, then the different states of the population oscillates over time at a farther distance away from the potential disease free population steady state E 0 = (S0∗ , 0, 0), S0∗ = μB wherein the disease would be eradicated, regardless of the size of the intensity value, σβ , of the random fluctuations in the disease transmission rate. This implies that in this disease scenario, the population experiencing the disease outbreak cannot contain the disease.

6 Example 6.1 Example 1: The Effect of the Intensity of the White Noise Process on Disease Eradication This example illustrates the results in Theorem 6 and Lemma 4, and also provides numerical evidence in support of the results in Sect. 5 that characterize the effects of the intensity of the white noise processes in the system originating from the

Modeling Highly Random Dynamical Infectious Systems Table 1 A list of specific values chosen for the system parameters for Example 6.1

563

Disease transmission rate

β

6.277E − 66

Constant birth rate

B

22.39 1000

Recovery rate Disease death rate Natural death rate

α d μ

5.5067E − 07 0.11838 0.6

random fluctuations in the disease dynamics on the stochastic asymptotic stability of the system in relation to the disease free equilibrium E 0 = (S0∗ , 0, 0), S0∗ = μB . Recall, Theorem 6 and Lemma 4 provide conditions for the threshold values R1 , U0 , and V0 defined in (80)–(82) which are sufficient for the stochastic stability of E 0 and consequently for disease eradication. For simplicity in this example, the following assumptions are considered:- (a1 ) there are no random fluctuations in the disease dynamics due to the natural death of susceptible individuals, that is, the intensity of the white noise due to the random fluctuations in the natural death of susceptible individuals σ S = 0. Indeed, from Theorem 6 and Lemma 4, there exists a stable disease free equilibrium E 0 , whenever σ S = 0 and the threshold values satisfy R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1. (a2 ) It is also assumed that the intensities of the white noise processes in the system due to the random fluctuations in the natural death and disease transmission processes for the other disease classes-exposed, infectious and removal are equal, that is, σ E = σ I = σ R = σβ = σ. The list of system parameter values in Table 1 are used to generate different values for R1 , U0 , and V0 under continuous changes in the values of σ = σ E = σ I = σ R = σβ . The Fig. 2 depicts the results for R1 and V0 . For U0 , it follows from Table 1 and (81) that U0 ≈ 1, where K˜ (μ) = 0.999991.

6.2 Example 2: Effect of the Intensity of White Noise on the Trajectories of the System The list of convenient choice of parameter values in Table 2 are used to generate the trajectories of the stochastic system (23)–(26) in order to (1) illustrate the impact of the source of the white noise process in the system (owing to the random fluctuations in the natural death or disease transmission processes) and also to (2) illustrate the effect of the intensity of the white noise process in the system on the trajectories of the different disease classes in the system, in order to uncover the overall behavior of the system over time.

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1.5 1.0 0.5 0.0

Basic reproduction ratio (R1)

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Intensity (σI = σβ = σ)

1.000005 0.999985

Threshold value (V0)

(b)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

Intensity (σE)

Fig. 2 a and b Show the values of the noise modified basic reproduction number, R1 , (defined in (80)) and the threshold parameter V0 (defined in (82)) over continuous changes in the values of the intensities of white noise processes due to random fluctuations in natural death and disease transmission processes of exposed, infectious and removal individuals, that is, σ = σ E = σβ = σ I = σ R . The curves in a and b show the values of R1 and V0 respectively. In addition, the broken horizontal lines depict the threshold mark, 1, for the threshold values R1 and V0 , where for the values of R1 and V0 below the threshold mark 1, the disease free equilibrium E 0 is stochastically asymptotically stable, and the disease can consequently be eradicated. It is easy to see that low values of σ ∈ [0, 0.7661] lead to R1 ≤ 1, and R1 > 1 other wise. For V0 , the low values of σ ∈ [0, 0.0045] lead to V0 ≤ 1, and V0 > 1 other wise. Therefore, values for R1 , U0 , and V0 that satisfy R1 ≤ 1, U0 ≤ 1, and V0 ≤ 1 are achieved for very low values of σ. This observation signifies that for a disease scenario where the physical processes lead to the specific parameter values defined in Table 1, the disease can only be eradicated when the random fluctuations in the disease dynamics exhibit very low intensity values of σ ∈ [0, 0.0045]. For any intensity values higher than 0.0045, the disease-free equilibrium E 0 is unstable, and this signifies that the disease outbreak becomes naturally uncontrollable and establishes a stable endemic population. Note that the observations of this example are consistent with the results in Theorems 9–11, and 13, 14

The Euer–Maruyama stochastic approximation scheme4 is used to generate trajectories for the different states S(t), E(t), I (t), R(t) over the time interval 4 A seed is set on the random number generator to reproduce the same sequence of random numbers

for the Brownian motion in order to generate reliable graphs for the trajectories of the system under different intensity values for the white noise processes, so that comparison can be made to identify differences that reflect the effect of intensity values.

Modeling Highly Random Dynamical Infectious Systems Table 2 A list of specific values chosen for the system parameters for Example 6.2

565

Disease transmission rate

β

0.6277

Constant birth rate

B

22.39 1000

Recovery rate Disease death rate Natural death rate Incubation delay time in vector Incubation delay time in host Immunity delay time

α d μ T1 T2 T3

0.05067 0.01838 0.002433696 2 units 1 unit 4 units

Table 3 Shows the intensity values of the white noise processes in the system and the corresponding sample means for the trajectories of the S, E, I, R states generated over time t ∈ [0, 4] in Example ¯ E, ¯ I¯, R¯ respectively 2. The sample means for S, E, I, R are denoted S, S¯ E¯ I¯ R¯ σi , i = S, E, I, R, β Figure # σi = O( ), i = S, E, I, R, β σi = O( ), i = S, E, I, R, and σβ = 0.5 σi = O( ), i = S, E, I, R, and σβ = 9 σi = 0.5, i = E, I, R, and σ S = σβ = O( ) σi = 0.5, i = S, E, I, R, and σβ = O( ) σi = 9, i = S, E, I, R, and σβ = O( ) σi = 0.5, i = S, E, I, R, and σβ = 0.5 σi = 9, i = S, E, I, R, and σβ = 9

Figure 3 Figure 4

10.06048 10.04129

4.979256 4.978257

5.704827 5.687113

1.975407 1.973783

Figure 5

9.681482

4.906452

5.385973

1.94617

Figure 6

10.06048

4.715779

5.42661

1.845652

Figure 7

9.553725

4.692877

5.42661

1.845652

Figure 8

1.980488

0.8066963

1.200498

0.240599

Figure 9

9.529665

4.687529

5.406493

1.843888

Figure 10

1.88787

0.4633994

0.8659143

0.2315721

[0, T ], where T = max(T1 + T2 , T3 ) = 4. The special nonlinear incidence functions aI , a = 0.05 in [44] is utilized to generate the numeric results. FurtherG(I ) = 1+I more, the following initial conditions are used ⎧ S(t) = 10, ⎪ ⎪ ⎨ E(t) = 5, ∀t ∈ [−T, 0], T = max(T1 + T2 , T3 ) = 4. I (t) = 6, ⎪ ⎪ ⎩ R(t) = 2,

(122)

The sample means for the sample paths of the S, E, I, R states generated over time t ∈ [0, T ] are summarized in Table 3, and will be used to compare the effect of the intensity values of the white noise processes in the system on the trajectories of the system over time.

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(c) Sample path for S(t) 5.00 4.99 4.97 4.96

10.00

1

2

3

4

0

1

2

3

time(t)

time(t)

(e) Sample path for I(t)

(f) Sample path for R(t)

4

1.97 1.93

1.95

R(t)

1.99

5.5 5.6 5.7 5.8 5.9 6.0

0

I(t)

4.98

E(t)

10.08 10.04

S(t)

10.12

(d) Sample path for E(t)

0

1

2

3

4

0

time(t)

1

2

3

4

time(t)

Fig. 3 c, d, e and f show the trajectories of the disease classes (S, E, I, R) respectively, when there are only infinitesimally small random fluctuations in the disease dynamics, that is, when the intensities of the white noise processes in the system due to random fluctuations in the natural death and disease transmission processes in all the classes (S, E, I, R) are described as follows: σ S = σ E = σβ = σ I = σ R = 0( )

The following observations can be made from Table 3: Remark 8 (1) When σi = O( ), i = S, E, I, R, there is moderate decrease in the average val¯ E, ¯ I¯, R¯ of S, E, I, R from the trajectories in Figs. 3, 4 and 5 as σβ increases ues S, from σβ = O( ) to σβ = 9. ¯ E, ¯ I¯, R¯ (2) When σβ = O( ), there is a sharp decrease in the average values S, of S, E, I, R from the trajectories in Fig. 3, and Figs. 7, 8 as σi , i = S, E, I, R increases from σi = O( ), i = S, E, I, R to σi = 9, i = S, E, I, R. (3) When all the σi ’s, that is, σi , i = S, E, I, R, β equally increase together from σi = O( ), i = S, E, I, R, β to σi = 9, i = S, E, I, R, β, there is a sharper

Modeling Highly Random Dynamical Infectious Systems

567

(g) Sample path for S(t) 5.00 4.99 4.97 4.96

10.00

1

2

3

4

0

1

2

3

time(t)

time(t)

(i) Sample path for I(t)

(j) Sample path for R(t)

4

1.97 1.93

1.95

R(t)

1.99

5.5 5.6 5.7 5.8 5.9 6.0

0

I(t)

4.98

E(t)

10.08 10.04

S(t)

10.12

(h) Sample path for E(t)

0

1

2

time(t)

3

4

0

1

2

3

4

time(t)

Fig. 4 g, h, i and j show the trajectories of the disease classes (S, E, I, R) respectively, when there are only infinitesimally small random fluctuations in the disease dynamics from the natural death of the classes (S, E, I, R), that is, when σ S = σ E = σ I = σ R = 0( ), but there are random fluctuations in the disease transmission process with low intensity value of σβ = 0.5

¯ E, ¯ I¯, R¯ of S, E, I, R from the trajectories decrease in the average values S, in Figs. 3, and 9, 10. (4) When σ S = σβ = O( ), there is no change in the average value S¯ of S and there is ¯ I¯, R¯ of E, I, R from the trajectories moderate decrease in the average values E, in Figs. 3 and 6 as σi , i = E, I, R increases from σi = O( ), i = E, I, R to σi = 0.5, i = E, I, R. The Figs. 3, 4 and 5 can be used to examine the effect of increasing the intensity value of the white noise process, σβ , originating from the random fluctuations in the disease transmission process on the trajectories for (S, E, I, R) in the absence of any significant random fluctuations in the disease dynamics due to the natural death process for all the disease classes (S, E, I, R), that is, σi = ( ), i = S, E, I, R. It can be

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(l) Sample path for E(t)

4.85

4.90

E(t)

4.95

9.8 10.0 9.6

4.80

9.2

9.4

1

2

3

4

0

1

2

3

time(t)

time(t)

(m) Sample path for I(t)

(n) Sample path for R(t)

4

4.8

1.88

5.2

I(t)

R(t)

5.6

1.96

6.0

2.00

0

1.92

S(t)

5.00

(k) Sample path for S(t)

0

1

2

3

4

0

1

2

3

4

time(t)

time(t)

Fig. 5 k, l, m and n show the trajectories of the disease classes (S, E, I, R) respectively, when there are only infinitesimally small random fluctuations in the disease dynamics from the natural death of the classes (S, E, I, R), that is, when σ S = σ E = σ I = σ R = 0( ), but there are random fluctuations in the disease transmission process with high intensity value of σβ = 9. In addition, the broken line on the sample path for S(t) in k depicts the S-coordinate S0∗ = μB = 9.199999 for the disease free steady state E 0 = (S0∗ , 0, 0, 0), S0∗ =

B μ,

E 0∗ = 0, I0∗ = 0, R0∗ = 0

observed from Fig. 3 that when the intensity value σβ is infinitesimally small, that is, σβ = 0( ), no significant oscillations occur over time on the trajectories for S, E, I, R in (a), (b), (c) and (d) respectively. Furthermore, for significant but low intensity values5 for σβ , that is, σβ = 0.5, Fig. 4 shows that some significant oscillations occur on the trajectories for the susceptible (g) and infectious (i) populations. Moreover, the size of the oscillations observed on the trajectories for the susceptible (g) and infectious population (i) seem to be small in value over time compared to Fig. 5. In addition, no significant oscillations are observed on the trajectories for the exposed 5 That

is, σβ = O(1).

Modeling Highly Random Dynamical Infectious Systems

569

(h) and removal (j) populations. In Fig. 5, with an increase in the intensity value for σβ to σβ = 9, more disease classes exhibit significant oscillations on their trajectories, for instance, more significant sized oscillations are observed on the trajectory of one additional class- exposed population (l) than is observed in Fig. 4h. Moreover, it appears that the high intensity value σβ = 9 has increased the size of the oscillations in the susceptible (k) and infectious (m) populations, and further deviating the trajectories of the system away from the noise-free state in Fig. 3. In addition, the trajectories for the states- (S, E, I ) in Fig. 5k, l, m respectively, oscillate near the disease free state E 0 = (S0∗ , 0, 0, 0), where S0∗ = μB = 9.199999, E 0∗ = 0, I0∗ = 0, R0∗ = 0. One can also observe from Table 3 and Remark 8 that for σi = O( ), i = S, E, I, R, the average values of S, E, I, R over time on the trajectories in Figs. 3, 4 and 5 decrease continuously with increase in the intensity value of σβ from σβ = O( ) to σβ = 9. These observations related to the oscillatory behavior of the system, for example, comparing the trajectory of S in Figs. 3c, 4g and 5k, and also comparing the trajectory for I in Figs. 3e, 4i and 5m suggest that continuously increasing the intensity value for σβ tends to increase the oscillatory behavior of the trajectories of the system that results in an average decrease in the size of the susceptible, exposed, infectious and removal populations over time. Furthermore, the size of the oscillations in the system is proportional to the size of the intensity values of the white noise process as remarked for Theorem 8. Figures 3, 6, and 7 can be used as an example to examine the effect of the intensity of the white noise process, σi , i = S, E, I, R, originating from the random fluctuations in the natural death process of each class-S, E, I, R, on the trajectories of the system, in the absence of any significant fluctuation in the disease dynamics owing to the disease transmission process, that is, σβ = O( ). For example, to examine the effect of σβ for the susceptible class, S, on the trajectories of the stochastic stochastic system, observe that in Fig. 6, when σ S = σβ = O( ) and σi = 0.5, i = E, I, R, no significant oscillations occur on the trajectories of S in Fig. 6o and also on Fig. 3c. Furthermore, when σ S is increased to σ S = 0.5, Fig. 7s depicts significant sized oscillations on the trajectory of S. Moreover, the trajectory for S oscillates near the disease free steady state S0∗ = 9.199999. It can be further observed using Table 3 and Remark 8 that no major differences have occurred on the trajectories of the other states E, I, R in both Figs. 6p, q, r and 7t, u, v respectively. In addition, it can be seen from Table 3 and Remark 8 that when σβ = O( ), the increase in the intensity value of σ S from σ S = O( ) to σ S = 0.5 on average leads to a decrease in the susceptible population size over time in Fig. 7s than it is observed in Fig. 6o and 3c. These observations suggest that in the absence of random fluctuations in the disease dynamics from the disease transmission process, that is, σβ = O( ), the intensity of the white noise process, σ S , owing to the natural death of the susceptible class S, (1) exhibits a significant effect primarily on its trajectory, and (2) the effect of increasing the intensity value6 of σ S leads to an oscillatory behavior on the trajectory of S that decreases the susceptible population averagely over time. Note that a similar

6 That

is σ S = θ( 1 ).

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(o) Sample path for S(t) 5.0

E(t)

4.8

10.08 10.00

4.4

4.6

10.04

1

2

3

4

0

1

2

3

time(t)

(q) Sample path for I(t)

(r) Sample path for R(t)

4

2.00

time(t)

1.70

1.80

R(t)

5.0 5.2 5.4 5.6 5.8 6.0

I(t)

0

1.90

S(t)

10.12

(p) Sample path for E(t)

0

1

2

time(t)

3

4

0

1

2

3

4

time(t)

Fig. 6 o, p, q and r show the trajectories of the disease classes (S, E, I, R) respectively, when there are significant small random fluctuations in the disease dynamics from the natural death process of exposed, infectious and removal classes, with intensity value σ E = σ I = σ R = 0.5, but there are only infinitesimally small fluctuations in the disease dynamics due to the disease transmission and natural death processes of susceptible individuals, that is, σ S = σβ = 0( )

numerical and graphical diagnostic approach can be used to examine the effects of the other classes E, I, R, whenever σβ = O( ). Figures 3, 7 and 8 can be used to examine the effect of increasing the intensity value of the white noise process originating from the natural death, σi , i = S, E, I, R, in the absence of any significant random fluctuations in the disease dynamics from the disease transmission process, that is, when σβ = O( ). Figure 7s, t, u, v show that the trajectories for S, E, I, R respectively, oscillate near the disease free equilibrium E 0 = (9.199999, 0, 0, 0) over time when the intensity value is increased from σi = O( ), i = S, E, I, R to σi = 0.5, i = S, E, I, R than is observed in the Fig. 3c, d, e, f. Furthermore, the oscillations on the trajectories seem to be small in size over time. When the intensity value, σi , i = S, E, I, R, is further increased to

Modeling Highly Random Dynamical Infectious Systems

571

(t) Sample path for E(t)

4.8 4.4 1

2

3

4

0

1

2

3

time(t)

time(t)

(u) Sample path for I(t)

(v) Sample path for R(t)

4

1.90 1.70

1.80

R(t)

5.0 5.2 5.4 5.6 5.8 6.0

2.00

0

I(t)

4.6

E(t)

9.4 9.0

S(t)

9.8

5.0

10.2

(s) Sample path for S(t)

0

1

2

3

4

0

1

2

3

4

time(t)

time(t)

Fig. 7 s, t, u and v show the trajectories of the disease classes (S, E, I, R) respectively, when there are significant but small random fluctuations in the disease dynamics from the natural death process in all the disease classes- susceptible, exposed, infectious and removal classes with low intensity value of σ S = σ E = σ I = σ R = 0.5, but there are infinitesimally small fluctuations in the disease dynamics from the disease transmission process, that is, σβ = 0( ). In addition, the broken line on the sample path for S(t) in s depicts the S-coordinate S0∗ = μB = 9.199999 for the disease free steady state E 0 = (S0∗ , 0, 0, 0), S0∗ =

B μ,

E 0∗ = 0, I0∗ = 0, R0∗ = 0

σi = 9, i = S, E, I, R, the oscillations on the trajectories in Fig. 8 w, x, y, z, appear to have increased in size. Furthermore, Table 3 and Remark 8 show that the oscillations lead to a decrease in the average values of S, E, I, R over time, and further away from the disease free state of S0∗ = 9.199999. Moreover, the population rapidly becomes extinct over time. These observations suggest that the increase7 in the intensity value of the white noise due to natural death in all classes, σi , i = S, E, I, R, in the population (1) leads to an increase in the oscillatory behavior of the system 7 That

is, σi = θ( 1 ), i = S, E, I, R.

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Fig. 8 w, x, y and z show the trajectories of the disease classes (S, E, I, R) respectively, when there are significant and large random fluctuations in the disease dynamics from the natural death process in all the disease classes- susceptible, exposed, infectious and removal classes with sufficiently high intensity value of σ S = σ E = σ I = σ R = 9, but there are infinitesimally small fluctuations in the disease dynamics from the disease transmission process, that is, σβ = 0( ). In addition, the broken line on the sample paths for S(t), E(t), I (t) and R(t) depict the S, E, I, R-coordinates S0∗ = B ∗ ∗ ∗ ∗ ∗ μ = 9.199999, E 0 = 0, I0 = 0, R0 = 0 for the disease free steady state E 0 = (S0 , 0, 0, 0), S0 =

B ∗ ∗ ∗ μ , E 0 = 0, I0 = 0, R0 = 0. w, x, y and z also show that the population goes extinct over time due to the high intensity of the white noise

which decreases the population size averagely over time and also (2) leads to population extinction over time. Note that this observation is consistent with the results of Theorem 15. Figures 3, 9 and 10 can be used to examine the effect of increasing the intensity values, σi , i = S, E, I, R, β, of all the white noise processes in the system on the trajectories of the system. Figure 9a1, b1, c1, d1 show that the trajectories for S, E, I, R respectively oscillate near the disease free steady state E 0 = (9.199999, 0, 0, 0) over

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(b1) Sample path for E(t)

4.8 4.6

E(t)

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Fig. 9 a1, b1, c1 and d1 show the trajectories of the disease classes (S, E, I, R) respectively, when there are significant but small random fluctuations in the disease dynamics from the natural death process in all the disease classes- susceptible, exposed, infectious and removal classes with low intensity value of σ S = σ E = σ I = σ R = 0.5, and there are also significant fluctuations in the disease dynamics from the disease transmission with a low intensity value of σβ = 0.5. In addition, the broken line on the sample path for S(t) in a1 depicts the S-coordinate S0∗ = μB = 9.199999 for the disease free steady state E 0 = (S0∗ , 0, 0, 0), S0∗ =

B μ,

E 0∗ = 0, I0∗ = 0, R0∗ = 0

time when the intensity value is increased from σi = O( ), i = S, E, I, R, β to σi = 0.5, i = S, E, I, R, β than it is observed in Fig. 3c, d, e, f. Furthermore, the oscillations of the trajectories seem to be small in size compared to the corresponding trajectories in Fig. 10. When the intensity values of σi , i = S, E, I, R, β are further increased to σi = 9, i = S, E, I, R, it can be seen from Fig. 10e1, f1, g1, h1, Table 3 and Remark 8 that the oscillations increase in size and lead to a sharp decrease in the average values of S, E, I, R on their trajectories over time, and also further deviating the average susceptible population size away from the disease free

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Fig. 10 e1, f1, g1 and h1 show the trajectories of the disease classes (S, E, I, R) respectively, when there are significant and large random fluctuations in the disease dynamics from the natural death process in all the disease classes- susceptible, exposed, infectious and removal classes with a sufficiently high intensity value of σ S = σ E = σ I = σ R = 9, and there are also significant fluctuations in the disease dynamics from the disease transmission process with a sufficiently high intensity value of σβ = 9. In addition, the broken line on the sample paths for S(t), E(t), I (t) and R(t) depict the S, E, I, R-coordinates S0∗ = μB = 9.199999, E 0∗ = 0, I0∗ = 0, R0∗ = 0 for the dis-

ease free steady state E 0 = (S0∗ , 0, 0, 0), S0∗ = μB , E 0∗ = 0, I0∗ = 0, R0∗ = 0. Furthermore, e1, f1, g1 and h1 show that the population goes extinct over time due to the high intensity of the white noise

state of S0∗ = 9.199999. Moreover, the population rapidly becomes extinct over time. These observations suggests that the increase in the intensity value of the white noise processes in the system due to the random fluctuations in the disease dynamics originating from the disease transmission and natural death processes for all disease classes in the population leads to (1) an increase in the oscillatory behavior of the system which decreases the average total population size over time, and also leads to (2) the rapid extinction of the population over time.

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It can also be observed by comparing Figs. 8w, x, y, z, and 10e1, f1, g1, h1, that for a fixed value of σi = 9, i = S, E, I, R, if σβ increases from σβ = O( ) in Fig. 8w, x, y, z to σβ = 9 in Fig. 10e1, f1, g1, h1, then the population more rapidly becomes extinct than it is observed in Fig. 8w, x, y, z. Indeed, in Fig. 8w, x, y, z, the trajectories for the susceptible S, exposed E, infectious I and Removal R states go extinct at approximately the following times t = 2, t = 1.8, t = 2 and t = 1.8 respectively. Meanwhile, in Fig. 10e1, f1, g1, h1, the trajectories for susceptible S, exposed E, infectious I and Removal R go extinct earlier at the approximate times t = 1.5, t = 1, t = 1 and t = 1.4 respectively. Note that these observations are consistent with the results of Theorem 15. The following pairs of figures:- (Figs. 4 g, h, i, j and 5k, l, m, n) and (Figs. 7s, t, u, v and 8w, x, y, z), can be used with reference to Fig. 3, to examine and compare the two major sources of random fluctuations in the disease dynamics namely-natural death and disease transmission processes, in order to determine the source which has stronger effect on the trajectories of the system, whenever the intensity values of the white noise processes increase in value. In the absence of random fluctuations in the natural death process, that is, σi = O( ), i = S, E, I, R, as the intensity value of σβ is increased from σβ = 0.5 to σβ = 9, the pair of figures (Figs. 4g, h, i, j and 5k, l, m, n) show an increase in the oscillatory behavior on the trajectories of the system which is more significant in size for the S and I classes over time. Furthermore, the oscillatory behavior leads to a decrease in the average susceptible and infectious populations over time than it is observed in Fig. 3c and e respectively, as shown in Table 3 and Remark 8. Moreover, the general disease population does not go extinct over time. Meanwhile, in the absence of random fluctuations in the disease transmission process, that is, σβ = O( ), the increase in the intensity value of σi , i = S, E, I, R from σi = 0.5, i = S, E, I, R to σi = 9, i = S, E, I, R, the pair of figures (Figs. 7s, t, u, v and 8w, x, y, z) show very strong increase in the oscillatory behavior on the trajectories of the system which is significant in all the states- S, E, I and R. Furthermore, from Table 3 and Remark 8, it can be seen that the oscillatory behavior of the system leads to a rapid decrease in the average values of all the states-S, E, I and R over time, with the mean susceptible population size deviating much further away from the disease free steady state S0∗ = 9.199999, than it is observed in Fig. 3. Moreover, the disease population goes extinct over time with the increase in the intensity value of σi , i = S, E, I, R.

7 Conclusion This chapter elaborates on a step-to-step approach to identify, and mathematically represent the various constituents of random dynamic processes in infectious dynamical systems such as systems wherein infectious diseases are transmitted between humans, animals and plants, or systems wherein bad rumors and negative ideas can

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spread socially between human beings, or wireless sensory network systems wherein malicious signals can be transmitted between the motes of the network. The mathematical modeling technique and method of analyses are delivered through a step-to-step modeling and analysis of a generalized class of stochastic SEIRS epidemic dynamic models with nonlinear incidence rates, three distributed delays and random perturbations for vector-borne diseases such as malaria and dengue fever. Moreover, the models in the class are perturbed by random environmental fluctuations from (1) the disease transmission rate between susceptible and infectious individuals, and also from (2) the natural deathrates of the sub-categories - susceptible, exposed, infectious and removal individuals of the population. In addition, the random fluctuations in the disease dynamics are incorporated into the epidemic dynamic models via independent white noise or Wiener processes. Also, the three delays are random variables. Whereas, two of the delays represent the incubation periods of the infectious agent in the vector and human host, the third delay represents the period of effective naturally acquired immunity against the disease which is conferred to individuals after recovery from infection. The class of epidemic dynamic models is represented as a system of Ito-Doob type stochastic differential equations with a general nonlinear incidence function G. The nonlinear incidence function G can be used to characterize the disease transmission rates for disease scenarios that exhibit a striking initial increase or decrease in disease transmission rates that becomes constant or bounded, whenever the infectious population size is large. The existence of a unique global positive solution is exhibited for the stochastic dynamic system by applying the standard Lipschitz and linear growth conditions locally, and then extending the solutions globally using stopping times and energy functions. Moreover, a positive self-invariant set for the stochastic dynamic system is presented. Detailed results for the asymptotic behavior of the stochastic dynamic system are presented namely:- (1) the existence and asymptotic stochastic stability of a feasible disease-free equilibrium of the stochastic system, and (2) the asymptotic oscillatory character of the solutions of the stochastic system near a potential diseasefree equilibrium. The threshold values for the stochastic stability of the disease free steady state, and for disease eradication, such as the basic reproduction number for the disease dynamics are computed. The analytic asymptotic results for the epidemic dynamic system suggest that the sources (disease transmission or natural death rates) and size of the intensity values of the white noise processes in the system exhibit direct consequences on the overall asymptotic behavior of the epidemic dynamic model with respect to the feasible or potential disease free population steady state for the epidemic dynamic model, and consequently on disease eradication. This observation leads to further thorough examination of the asymptotic properties of the stochastic epidemic dynamic system under various intensity levels of the white noise processes in the system. In addition, two numerical simulation examples are presented to justify the results from the analyses.

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References 1. Kawachi, K.: Deterministic models for rumor transmission. Nonlinear Anal.: R. Word Appl. 9, 1989–2028 (2008) 2. Keshri, N., Mishra, B.: Two time-delay dynamic model on the transmission of malicoius signals in wireless sensor network. Chaos, Soliton Fractals 68, 151–158 (2014) 3. Leclerc, M., Dore, T., Gilligan, C.A., Lucas, P., Filipe, J.A.N.: Estimating the delay between host infection and disease (incubation period) and assessing its significance to the epidemiology of plant diseases. PLoS ONE 9(1) (2014) 4. Zhang, Z., Yang, H.: Stability and Hopf bifurcation in a delayed SEIRS worm model in computer network. Math. Probl. Eng. 2013, 9 (2013) 5. De la Sena, M., Alonso-Quesadaa, S., Ibeasb, A.: On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules. Appl. Math. Comput. 270, 953–976 (2015) 6. Du, N.H., Nhu, N.N.: Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises. Appl. Math. Lett. 64, 223–230 (2017) 7. Jianga, Z., Mab, W., Wei, J.: Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model. Math. Comput. Simul. 122, 35–54 (2016) 8. Liu, Q., Chen, Q.: Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Physica A 428, 140–153 (2015) 9. Liu, Qun, Jiang, Daqing, Shi, Ningzhong, Hayat, Tasawar, Alsaedi, Ahmed: Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence. Commun. Nonlinear Sci. Numer. Simul. 40, 89–99 (2016). November 10. Mateusa, J.P., Silvab, C.M.: Existence of periodic solutions of a periodic SEIRS model with general incidence. Nonlinear Anal.: R. World Appl. 34, 379–402 (2017) 11. Wanduku, D.: Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbation. Appl. Math. Comput. 294, 49–76 (2017) 12. Wanduku, D., Ladde, G.S.: Fundamental properties of a two-scale network stochastic human epidemic dynamic model. Neural, Parallel, Sci. Comput. 19, 229–270 (2011) 13. De la Sen, M., Alonso-Quesada, S., Ibeas, A.: On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules. Appl. Math. Comput. 270, 953–976 (2015) 14. Mateus, J.P., Silva, C.M.: A non-autonomous SEIRS model with general incidence rate. Appl. Math. Comput. 247, 169–189 (2014) 15. Bai, Z., Zhou, Y.: Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate. Nonlinear Anal.: R. World Appl. 13(3), 1060–1068 (2012) 16. Kyrychko, Y.N., Blyussb, K.B.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate. Nonlinear Anal.: R. World Appl. 6(3), 495–507 (2005) 17. Wanduku, D., Ladde, G.S.: Global properties of a two-scale network stochastic delayed human epidemic dynamic model. Nonlinear Anal.: R. World Appl. 13, 794–816 (2012) 18. Cooke, K.L., van den Driessche, P.: Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 35(2), 240–260 (1996). Dec 19. Gao, S., Teng, Z., Xie, D.: The effects of pulse vaccination on SEIR model with two time delays. Appl. Math. Comput. 201(12), 282–292 (2008) 20. Sampath Aruna Pradeep, B.G., Ma, W.: Global stability analysis for vector transmission disease dynamic model with non-linear incidence and two time delays. J. Interdiscip. Math. 18(4) (2015) 21. Cooke, K.L.: Stability analysis for a vector disease model. Rocky Mt. J. Math. 9(1) 31–42 (1979) 22. Takeuchi, Y., Ma, W., Beretta, E.: Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal. 42, 931–947 (2000) 23. Beretta, E., Kolmanovskii, V., Shaikhet, L.: Stability of epidemic model with time delay influenced by stochastic perturbations. Math. Comput. Simul. 45, 269–277 (1998)

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24. Zhou, Y., Zhang, W., Yuan, S., Hu, H.: Persistence and extinction in stochastic sirs models with general nonlinear incidence rate. Electron. J. Differ. Equ. 2014(42), 1–17 (2014) 25. Zhu, L., Hu, H.: A stochastic SIR epidemic model with density dependent birth rate. Adv. Differ. Equ. 2015, 330 (2015) 26. http://www.who.int/denguecontrol/human/en/ 27. https://www.cdc.gov/malaria/about/disease.html 28. Doolan, D.L., Dobano, C., Baird, J.K.: Acquired immunity to malaria. Clin. Microbiol. Rev. 22(1), 13–36 (2009) 29. Hviid, L.: Naturally acquired immunity to Plasmodium falciparum malaria. Acta Trop. 95(3), 270–275 (2005). October 30. Capasso V, Serio G.A.: A generalization of the Kermack-Mckendrick deterministic epidemic model. Math. Biosci. 42, 43 (1978) 31. Huo, H.-F., Ma, Z.-P.: Dynamics of a delayed epidemic model with non-monotonic incidence rate. Commun. Nonlinear Sci. Numer. Simul. 15(2), 459–468 (2010) 32. Xiao, D., Ruan, S.: Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 208(2), 419–429 (2007). Aug 33. Xue, Y., Duan, X.: Dynamic analysis of an sir epidemic model with nonlinear incidence rate and double delays. Int. J. Inf. Syst. Sci. 7(1), 92–102 (2011) 34. Capasso, V.: Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, vol. 97 (1993) 35. Muroya, Y., Enatsu, Y., Nakata, Y.: Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate. J. Math. Anal. Appl. 377(1), 1–14 (2011) 36. Korobeinikov, A., Maini, P.K.: A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. Eng. 1(1), 57–60 (2004) 37. Liu, W.M., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359–380 (1987) 38. Liu, W.M., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359 (1987) 39. Chiyaka, C. et al.: transmission model of endemic human malaria in a partially immune population. Math. Comput. Model. 46, 806–822 (2007) 40. Allen, E.J., Allen, L.J.S., Arciniega, A., Greenwood, P.: Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl. 26, 274–297 (2008) 41. Ladde, A.G., Ladde, G.S.: An Introduction to Differential Equations: Stochastic Modelling, Methods and Analysis, vol. 2. World Scientific Publishing, Singapore (2013) 42. Allen, E.J.: Environmental variability and mean-reverting processes. Discret. Contin. Dyn. Syst. 21, 2073–2089 (2016) 43. Cai, Y., jiao, J., Gui, Z., liu, Y. et al.: Environmental variability in a stochastic epidemic model. Appl. Math. Compuat. 329, 210–226 (2018) 44. Moghadas, S.M., Gumel, A.B.: Global Statbility of a two-stage epidemic model with generalized nonlinear incidence. Math. Comput. Simul. 60, 107–118 (2002) 45. Wanduku, D., Ladde, G.S.: Global analysis of a stochastic two-scale network human epidemic dynamic model with varying immunity period. (Accepted (2013) and to appear in J. Appl. Math. Phys.) 46. Xuerong, M.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing Ltd., Sawston (2008) 47. Mao, X.: Stochastic Differential Equations and Application, 2nd edn. Woodhead Publishing, Sawston (2007) 48. Murray, M., li, Z., Sastry, S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, LLC, Boca Raton (1994) 49. Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Berlin (2013) 50. Wanduku, D., Ladde, G.S.: Global stability of two-scale network human epidemic dynamic model. Neural, Parallel, Sci. Comput. 19, 65–90 (2011)

On Weighted Convergence of Double Singular Integral Operators Involving Summation Gümrah Uysal and Hemen Dutta

Abstract This chapter consists of five sections. First section is devoted to introduction part in which the description of the problem is presented and theoretical background is given. In the second section, the preliminary concepts which are utilized in the sequel are introduced. Then, the conditions under which double singular integral operators involving summation are well-defined in the space of Lebesgue measurable functions defined on different sets are presented. In the third section, Fatou type convergences of handled operators are discussed. In the fourth section, the rate of convergences with respect to obtained approximations in the preceding section are established. In the last section, we present some concluding remarks. Keywords Pointwise convergence · Fatou type convergence · Integral operators involving summation · Generalization of Lebesgue point 2010 Mathematics Subject Classification 41A35 · 41A25 · 47G10

1 Introduction Analyzing the attitudes of integral type operators in the limit position is quite appropriate technique while working on approximation of functions having relatively worst properties, such as non-differentiability and non-integrability. In this technique, the operator which is used as an approximation tool has a kernel function and usually this function is constructed as an approximate identity. Approximate identities have very specific and sophisticated properties similar to Dirac’s δ−function (see [13]). It is G. Uysal (B) Division of Technology of Information Security, Department of Computer Technologies, Karabuk University, 78050 Karabuk, Turkey e-mail: [email protected] H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, India e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_18

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well known that approximate identity concept is originated from Dirac’s δ−function. For some detailed information, we refer the reader to the monograph by Butzer and Nessel [11]. The usage of Gauss–Weierstrass and Poisson kernels, which are the good examples for approximate identity concept, is an indisputable truth from a literature standpoint. We refer the readers who are interested in learning the approximation properties of integral operators with the indicated kernels to [38]. The singularity of a kernel function, which sometimes refers to approximate identity type kernels too, directly makes the integral operator singular. Therefore, the integral operator is called singular integral operator. Using singular integrals in approximation theory came out from the construction of Fourier series of the functions. The works by Fatou [14] and Lebesgue [22] should be mentioned as pilot works among the oldest studies around this idea. Fourier series are highly popular tools of many branches of science, such as physics, mathematics and engineering. Emphasizing the role of Fourier series and indirectly, singular integral operators in science, these operators have a large scope of applications including magnetic resonance imaging and image processing. Some applications can be found in [10, 12]. Miscellaneous type linear integral operators’ convergences have been studied in various directions for different purposes throughout the years: almost everywhere convergence of singular integral operators [11, 31], Fatou type convergence of integral operators with approximate identity type kernels [35], a family of two-parameter singular integral operators [16, 29, 40], a family of m−singular integral operators constructed by using the mth finite differences [24], singular integral operators with convolution type kernels in different settings [7, 23, 37], singular integral operators with certain non-convolution type kernels [8] and singular integral operators with non-isotropic kernels [2]. Singular integral operators are also used for weighted convergence analysis in many function spaces, such as weighted Lebesgue spaces. In the spaces of this kind, one needs a weight function, which is usually denoted by w, ϕ or ρ up to desire, in order to make the space elements be integrable. For instance, one may consider a pointwise approximation problem established in the weighted Lebesgue     ϕ space of measurable functions, L 1 (R) , consisting of f : R → R for which  ϕf  is integrable on R in the sense of Lebesgue. Here, ϕ : R → R\{0} is a measurable weight function equipped with some extra properties up to quality of the problem. It should be noted that if f is naturally integrable, then ϕ can be easily taken as 1 to simplify the solution. In this case the role of original kernel function does not change. On the other hand, if f is not integrable, then the multiplication of kernel function and weight function may serve us as a new kernel function throughout the solution of the problem. For some advanced approximation results in the framework of this idea, we refer the reader to [1, 3, 24, 42]. Natanson’s famous lemma, which was included in the monograph by Natanson [28], was generalized in some studies, such as [16, 40, 41] and became a useful tool to prove the convergence theorems where it was applicable. This type lemmas are used to show boundedness of the integral satisfying some certain conditions and give an upper bound to it. In the mentioned

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works above, the authors first proved Natanson-type lemmas then used them at some stage of the proof of the convergence theorem as in [28]. Now, let us consider the following setting of nonlinear integral operators which was first considered by Musielak [25]:  Tη ( f ; y) =

Kη (t − y, f (t))dt, y ∈ G, η ∈ ,

(1)

G

where G is a locally compact Abelian group on which Haar measure is defined and  is a non-empty set of indices with a topology. The operators of type (1) and more advanced type operators were widely examined in the monograph by Bardaro et al. [9] using the method that originated from [25]. This method is, roughly speaking, described as setting the Lipschitz condition on the kernel function Kη with respect to second variable. If the variable y is not fixed within (1), then two-parameter nonlinear singular integral operators are obtained. This case was studied by Swiderski and Wachnicki [39]. It is well known that the special case of (1) is obtained by taking Kη (., f (.)) = Lη (.) f (.) , where Lη was a linear kernel which simultaneously made the operator be linear. If this linear case is ignored, then the nonlinear singular integral operators turn into certain type nonlinear singular integral operators. These operators are used in many areas, such as Fourier analysis and sampling theory. For some studies concerning various type nonlinear singular integral operators, we refer the reader to [5, 15, 26]. In [41], Taberski generalized his previous works by giving some further results on the pointwise approximation of functions f ∈ L 1 (R) using the bivariate integral operators given in the following setting:  f (u, v)Kλ (u − x, v − y) dvdu, (x, y) ∈ R,

Lλ ( f ; x, y) =

(2)

R

where R = −π, π × −π, π and Kλ (u, v) stands for a kernel function enriched with some properties for any fixed λ ∈ , where  is a given non-empty set of indices with accumulation point λ0 . The papers [33, 34] and [30] based on [41], are devoted to the study of pointwise convergence of the operators of type (2) on three dimensional sets which also consist characteristic points (x0 , y0 ) of different types. Then, Musielak [27] investigated the convergence conditions of the bivariate counterparts of the operators of type (1) in some modular function spaces. As concerns the study of double integral operators, we refer the reader to [20, 21, 42–44]. In the current manuscript, we generalize some of the results in [43]. Incorporating the terminology designated in [41] with the summation included integral operators studied in, such as [4, 6], the following singular integral operators are obtained: Tη ( f ; x, y) =

n   m=1 D

f m (t, s)Kη,m (t − x, s − y)dsdt, (x, y) ∈ D,

(3)

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where D = (a, b) × (c, d), n ≥ 1 is a finite natural number and η ∈ ,  is a nonempty index set consisting of the non-negative real numbers η on which the notion η0 denotes either its accumulation point or ∞. Here, f : D ⊆ R2 → R is a planemeasurable function in the sense of Lebesgue on D, f m denotes mth power of the function f , and Kη,m : R2 → R+ 0 for each fixed m = 1, 2, . . . , n and for each fixed η ∈ , stands for a kernel function satisfying some properties for any fixed η ∈  and ϕ ϕ for any fixed m = 1, 2, . . . , n. Further, let f ∈ L 1 (D)  , where L 1 (D) is the space   of all measurable functions f : D → R for which  ϕf  are integrable in the sense of Lebesgue in D. Here, ϕ : R2 → R+ is a Lebesgue measurable weight function on ϕ R2 satisfying additional properties. In other words, if f ∈ L 1 (D) , then the norm of that function (see, e.g., [24, 42]) satisfies f L ϕ1 (D) =

    f (t, s)     ϕ (t, s)  dsdt < ∞. D

For n = 1, the operators of type (3) are in linear form. On the other hand, if we take kernel functions equal to each other, that is, Kη = Kη,m for all m = 1, 2, . . . , n and f is integrable in the sense of Lebesgue, then the operators of type (3) reduces to the general setting of the operators of type (2) which was previously considered by many authors (see, for example, [30, 33, 41, 42]). The operators of type (3) may also be seen as a special case of nonlinear integral operators. In this chapter, Fatou type weighted pointwise convergences are obtained for two cases: D = (a, b) × (c, d) stands for any bounded rectangular region in R2 and D = R2 . Also, the order of convergences are computed with respect to these results.

2 Preliminaries Now, we start by giving the following characterization of μ−generalized Lebesgue point of the function g ∈ L 1 (D) in view of the μ−generalized Lebesgue point definition given in [30]. An equivalent version of this definition can be found in [43]. Definition 1 Let D = (a, b) × (c, d) be any bounded rectangular region in R2 . A μ−generalized Lebesgue point of function g ∈ L 1 (D) is a point (x0 , y0 ) ∈ D at which the relation 1 lim (h,k)→(0,0) μ1 (h)μ2 (k)

±h  ±k 

|g (t + x0 , s + y0 ) − g (x0 , y0 )| dsdt = 0, 0 0

holds, that is, relation (4) consists of the following four relations:

(4)

On Weighted Convergence of Double Singular Integral Operators …

1 lim (h,k)→(0,0) μ1 (h)μ2 (k)

1 lim (h,k)→(0,0) μ1 (h)μ2 (k)

1 (h,k)→(0,0) μ1 (h)μ2 (k)

h k |g (t + x0 , s + y0 ) − g (x0 , y0 )| dsdt = 0,

(4a)

|g (t + x0 , s + y0 ) − g (x0 , y0 )| dsdt = 0,

(4b)

|g (t + x0 , s + y0 ) − g (x0 , y0 )| dsdt = 0,

(4c)

|g (t + x0 , s + y0 ) − g (x0 , y0 )| dsdt = 0,

(4d)

0 0

0 k −h 0

h 0

lim

1 lim (h,k)→(0,0) μ1 (h)μ2 (k)

583

0 −k

0 0 −h −k

h hold, where μ1 (h) = ρ1 (u)du > 0, 0 < h ≤ b − a and ρ1 (u) is Lebesgue inte0

k grable and non-negative function on [0, b − a] , and μ2 (k) = ρ2 (v)dv > 0, 0 < 0

k ≤ d − c and ρ2 (v) is Lebesgue integrable and non-negative function on [0, d − c] . For the case D = R2 , it is enough to suppose that 0 < h, k ≤ δ ∗ , where δ ∗ is a certain positive real number. Definition 2 (see [17, 41]) If the function g : R2 → R is bimonotonically increasing on [α1 , α2 ] × [β1 , β2 ] ⊂ R2 , then the following equality β2 α2   g(t, s) var (g; [α1 , α2 ; β1 , β2 ]) = α1 β1

= g(α1 , β1 ) − g(α1 , β2 ) − g(α2 , β1 ) + g(α2 , β2 ) holds. Similarly, if the function g : R2 → R is bimonotonically decreasing on [α1 , α2 ] × [β1 , β2 ] ⊂ R2 , then the following equality var (g; [α1 , α2 ] × [β1 , β2 ]) =

β2 α2  

g(t, s)

α1 β1

= g(α1 , β2 ) − g(α1 , β1 ) − g(α2 , β2 ) + g(α2 , β1 ) holds. Definition 3 (Class Aϕ ) Let  be a non-empty index set consisting of the nonnegative real numbers η on which the notion η0 denotes either its accumulation

584

G. Uysal and H. Dutta

point or ∞. Further, Kη,m : R2 → R+ 0 is a Lebesgue integrable kernel function on its domain for each fixed η ∈  and for each fixed m = 1, 2, . . . , n, and Lebesgue measurable weight function ϕ : R2 → R+ satisfies the following submultiplication property for each fixed m = 1, 2, . . . , n: ϕm (u + t, v + s) ≤ ϕm (u, v)ϕm (t, s), ∀(u, v) ∈ R2 and ∀(t, s) ∈ R2 .

(5)

Suppose that μ−generalized Lebesgue point set of ϕ is non-empty. If Kη,m satisfies the following properties, then it belongs to Class Aϕ . a: At each μ−generalized Lebesgue point (x0 , y0 ) ∈ D of ϕ,        m m  lim ϕ (t, s)Kη,m (t − x, s − y)dsdt − Cm ϕ (x0 , y0 ) = 0,  (x,y,η)→(x0 ,y0 ,η0 )   D

where Cm , m = 1, 2, . . . , n, are certain positive constants, holds. b: At each μ−generalized Lebesgue point (x0 , y0 ) ∈ D of ϕ and for every ξ > 0, the following holds:  lim

(x,y,η)→(x0 ,y0 ,η0 )

 sup

√ ξ≤ t 2 +s 2

n 

 ϕ (x + t, y + s)Kη,m (t, s) m

= 0.

m=1

c: At each μ−generalized Lebesgue point (x0 , y0 ) ∈ D of ϕ and for every ξ > 0, the following holds: ⎡ lim

(x,y,η)→(x0 ,y0 ,η0 )

⎢ ⎣

n  m=1

 √ ξ≤ t 2 +s 2

⎤ ⎥ ϕm (x + t, y + s)Kη,m (t, s)dsdt ⎦ = 0.

d: ϕm (t, s)Kη,m (t, s) is monotonically increasing on (−∞, 0] and monotonically decreasing on [0, ∞) with respect to t for all values of s and ϕm (t, s)Kη,m (t, s) is monotonically increasing on (−∞, 0] and monotonically decreasing on [0, ∞) with respect to s for all values of t for each fixed η ∈  and for each fixed m = 1, 2, . . . , n. Similarly, ϕm (t, s)Kη,m (t, s) is bimonotonically increasing with respect to (t, s) on [0, ∞) × [0, ∞) and (−∞, 0] × (−∞, 0] and bimonotonically decreasing with respect to (t, s) on (−∞, 0] × [0, ∞) and [0,  (−∞, 0] for each fixed η ∈  and for each fixed m = 1, 2, . . . , n.  ∞) × e: ϕKη,m  L 1 (R2 ) is finite for all η ∈  and for each fixed m = 1, 2, . . . , n. Throughout this chapter Kη,m (t, s) belongs to Class Aϕ . Remark 1 It is well-known that some of the frequently used weight functions, such as exponential function, satisfies inequality (5). Therefore, it would be better to generalize a characterization of a weight function analogues to that of done for

On Weighted Convergence of Double Singular Integral Operators …

585

kernel functions. For the ideas incorporated to obtain the conditions in Definition 3, we refer the reader to [1, 4, 24, 42]. Especially, we refer the reader to see p. 299 in [24] for a very sophisticated charaterization of a weight function. On the other hand, for some functions satisfying (5), we refer the reader to [18]. Now, we give a lemma concerning well-definiteness of the operators of type (3). ϕ

ϕ

Lemma 1 If f ∈ L 1 (D), then the operators Tη ( f ; x, y) ∈ L 1 (D) and the inequality   Tη ( f ; x, y)

ϕ L 1 (D)

≤

n 

  Bm ϕKη,m  L 1 (R2 ) f L ϕ1 (D)

m=1

(for D = (a, b) × (c, d) , which denotes any bounded rectangular region in R2 , the numbers Bm denote Vm and for D = R2 , the numbers Bm denote Ym in the above expression) holds provided f is bounded on D. Proof Let D = (a, b) × (c, d) be any bounded rectangular region in R2 . Further, we define the functions g m : R2 → R, m = 1, 2, . . . , n by  g (u, v) = m

f m (u, v), (u, v) ∈ D, 0, other wise.

We may easily get the following inequality:   Tη ( f ; x, y)

 ϕ L 1 (D ) =

D

 = D

 ≤ D

    n     1   m f (t, s)Kη,m (t − x, s − y)dsdt  d yd x   ϕ(x, y)   m=1 D     n    1   m (t, s)K g (t − x, s − y)dsdt  d yd x  η,m  ϕ(x, y)   m=1 R2 ⎛ ⎞ n     1 ⎜ g m (t, s) Kη,m (t − x, s − y)dsdt ⎟ ⎝ ⎠ d yd x. ϕ(x, y) m=1

R2

  We may denote the supremum values of  f m−1  on D as Vm for each m. Therefore, using Fubini’s Theorem (see [32], p. 77) and inequality (5), we proceed as follows:   Tη ( f ; x, y)

ϕ L 1 (D ) ≤

=

=

n 

 Vm

m=1

R2

n 

 Vm

m=1

R2

n 



m=1

Vm R2

⎞ ⎛     f (t + x, s + y)   d yd x ⎠ dsdt  Kη,m (t, s) ⎝   ϕ(x, y) D

⎞ ⎛     f (t + x, s + y)  ϕ(t + x, s + y)   ⎠ Kη,m (t, s) ⎝  ϕ(t + x, s + y) d yd x dsdt  ϕ(x, y) D

⎞ ⎛     f (t + x, s + y)  ϕ(t + x, s + y)   d yd x ⎠ dsdt Kη,m (t, s) ⎝  ϕ(t + x, s + y)  ϕ(x, y) D

586

G. Uysal and H. Dutta ≤

n 



m=1

=

n  m=1

ϕ(t, s)Kη,m (t, s)dsdt

Vm

    f (u, v)     ϕ(u, v)  dvdu D

R2

  Vm ϕKη,m  L (R2 ) f L ϕ (D) . 1 1

Thus, the proof is completed for the case D = (a, b) × (c, d). Now, let D = R2 . Similar operations yield   Tη ( f ; x, y)

ϕ

L 1 (R2 )

≤

n 

 Ym

m=1

≤

n  m=1

=

n 

R2

 Ym R2

⎞     f (t + x, s + y)   d yd x ⎠ dsdt  Kη,m (t, s) ⎝   ϕ(x, y) ⎛

R2

    f (u, v)    ϕ(t, s)Kη,m (t, s)dsdt  ϕ(u, v)  dvdu R2

  Ym ϕKη,m  L 1 (R2 ) f L ϕ1 (R2 ) ,

m=1

  where the numbers Ym denote the supremum values of  f m−1  on R2 for each m. Hence, the proof is completed. 

3 Fatou Type Convergence The following theorem gives a Fatou type convergence of the integral operators of ϕ type (3) at μ-generalized Lebesgue point of f ∈ L 1 (D), where D = (a, b) × (c, d) , which denotes any bounded rectangular region in R2 . Using Fatou type convergence, the sensitivity of the pointwise convergence will be increased via taking limit as (x, y, η) tends to (x0 , y0 , η0 ) on the set denoted by Z , where Z can be seen as the comfort zone for the approximation. In other words, if the point (x0 , y0 ) is not fixed within the operators, such as of type (3), then the quality of the analysis increases (for further details, see also [14, 35, 41]). Theorem 1 Suppose that ϕf is bounded on D. If (x0 , y0 ) is a common μ−generalized ϕ Lebesgue point of f ∈ L 1 (D) and ϕ, then lim

Tη ( f ; x, y) =

(x,y,η)→(x0 ,y0 ,η0 )

on any set Z on which the function

n  m=1

Cm f m (x0 , y0 )

On Weighted Convergence of Double Singular Integral Operators … n 

ϕm (x, y)

m=1

⎧ 0 +δ 0 +δ y ⎪ ⎨x ⎪ ⎩

587

ϕm (t − x, s − y)Kη,m (t − x, s − y)ρ1 (|x0 − t|) ρ2 (|y0 − s|) dsdt

x0 −δ y0 −δ x 0 +δ

ϕm (t − x, 0)Kη,m (t − x, 0)ρ1 (|x0 − t|) dt

+ 2μ2 (|y0 − y|) x0 −δ y 0 +δ

ϕm (0, s − y)Kη,m (0, s − y)ρ2 (|y0 − s|) ds

+ 2μ1 (|x0 − x|) y0 −δ

 +4ϕ (0, 0)Kη,m (0, 0)μ1 (|x0 − x|) μ2 (|y0 − y|) , m

where 0 < δ
0, there exists δ > 0 such that for every h and for every k for which 0 < h, k ≤ δ < 21 min {b − a, d − c} holds, we have the following inequality by (4c): y0 x 0 +h 

x0 y0 −k

   f (t, s) f (x0 , y0 )    ϕ(t, s) − ϕ (x , y )  dsdt ≤ εμ1 (h)μ2 (k). 0 0

(6)

Now, we write Tη ( f ; x, y) −

n  m=1

Cm f m (x0 , y0 ) =

n  

f m (t, s)Kη,m (t − x, s − y)dsdt −

m=1 D

n 

Cm f m (x0 , y0 ) .

m=1

Addition and subtraction of the following expression  n  f m (x0 , y0 ) ϕm (t, s)Kη,m (t − x, s − y)dsdt m (x , y ) ϕ 0 0 m=1 D

to the right hand side of the above equality, we have the following inequality   n      Cm f m (x0 , y0 ) Tη ( f ; x, y) −   m=1

588

G. Uysal and H. Dutta

  n   ≤  m=1

    m m   m  f (t, s) f , y (x ) 0 0     ϕm (t, s) − ϕm (x , y )  ϕ (t, s)Kη,m (t − x, s − y)dsdt  0 0  D       n  m    f (x0 , y0 )   m m    + ϕ (t, s)Kη,m (t − x, s − y)dsdt − Cm ϕ (x0 , y0 )  ϕm (x , y )   0 0   m=1 D

= I1 + I2 . In view of condition (a) of Class Aϕ , I2 tends to zero as (x, y, η) tends to (x0 , y0 , η0 ). The integral I1 can be written in the form:

I1 =

⎧ n ⎪ ⎨   ⎪

m=1 ⎩D\B

δ

⎫   ⎪ ⎬  f m (t, s) f m (x0 , y0 )   − m +  m ⎪ ϕ (x0 , y0 )  ⎭ ϕ (t, s) Bδ

× ϕ (t, s)Kη,m (t − x, s − y)dsdt m

= I11 + I12 , where

  Bδ = (t, s) : (t − x0 )2 + (s − y0 )2 < δ 2 , (x0 , y0 ) ∈ D .

Now, using the assumptions given by 0 < |x0 − x| < can define the following set:

δ 2

and 0 < |y0 − y| < 2δ , we

 δ2 Nδ = (x, y) : (x − x0 )2 + (y − y0 )2 < , (x0 , y0 ) ∈ D . 2 Comparing the sets Bδ and Nδ gives the relation R2 \Bδ ⊆ R2 \Dδ , where  δ2 Dδ = (t, s) : (t − x)2 + (s − y)2 < , (x, y) ∈ Nδ . 2 Therefore, we have the following inequality  I11 ≤

sup √

√δ ≤ 2

u 2 +v 2

n  m=1

 !   fm m  ϕ (x + u, y + v)Kη,m (u, v)  m  ϕ 

  m  f (x0 , y0 )   (b − a)(d − c) . +  m ϕ (x , y )  0

L 1 (D)

0

Consequently, by condition (b) of class Aϕ , and I11 tends to zero as (x, y, η) tends to (x0 , y0 , η0 ).

On Weighted Convergence of Double Singular Integral Operators …

589

Now, we prove that I12 tends to zero as (x, y, η) tends to (x0 , y0 , η0 ). Using identity (v) in p. 13 in [36] and by boundedness of ϕf , there exists a number P, which denotes maximum of supremum values of

I12

fm , ϕm

m = 1, 2, . . . , n on D, such that

 n     f (t, s) f (x0 , y0 )  m  ≤P  ϕ (t, s) − ϕ (x , y )  ϕ (t, s)Kη,m (t − x, s − y)dsdt 0 0 m=1 Bδ

holds for I12 . Thus, by (5), we have I12 ≤ P

n 

ϕm (x, y)

m=1

⎧ ⎪ ⎨x0+δ y0 ⎪ ⎩

x0 +

x0 y0 −δ

⎫  ⎪ y0 ⎬  f (t, s) f (x0 , y0 )  m  ϕ (t − x, s − y) −  ⎪ ϕ (x0 , y0 )  ⎭ ϕ (t, s)

x0 −δ y0 −δ

× Kη,m (t − x, s − y)dsdt ⎧ ⎫  ⎪ n ⎨ x0 y0+δ x0+δ y0+δ ⎪ ⎬  f (t, s)  f (x0 , y0 )  m  ϕ (t − x, s − y) − +P ϕm (x, y) +  ⎪ ⎪ ϕ (x0 , y0 )  ⎩ ⎭ ϕ (t, s) m=1 x0 −δ y0

x0

y0

× Kη,m (t − x, s − y)dsdt =P

n 

ϕm (x, y)(I121 + I122 + I123 + I124 ).

m=1

Let us consider the integral I121 . Let us define the function F (t, s) by  t y0   f (u, v) f (x0 , y0 )   dvdu. − F (t, s) =  ϕ(u, v) ϕ (x0 , y0 )  x0 s

In view of inequality (6), the following expression |F (t, s)| ≤ εμ1 (t − x0 ) μ2 (y0 − s) ,

(7)

where 0 < t − x0 ≤ δ and 0 < y0 − s ≤ δ, holds. By Theorem 2.6 in [41], we can write y0 x 0 +δ 

I121 = (L) x0 y0 −δ

   f (t, s) f (x0 , y0 )  m   ϕ (t, s) − ϕ (x , y )  ϕ (t − x, s − y)Kη,m (t − x, s − y)dsdt 0 0

y0 x 0 +δ 

ϕm (t − x, s − y)Kη,m (t − x, s − y)d [−F (t, s)] ,

= (LS) x0 y0 −δ

590

G. Uysal and H. Dutta

where LS denotes Lebesgue–Stieltjes integral. Applying bivariate integration by parts (see Theorem 2.2, p. 100 in [41] or [19]) to the Lebesgue–Stieltjes integral, we have   x0 +δ y0      m |I121 | =  ϕ (t − x, s − y)Kη,m (t − x, s − y)d [−F (t, s)]    x0 y0 −δ  y0 x 0 +δ 

  |F (t, s)| dϕm (t − x, s − y)Kη,m (t − x, s − y)

≤ x 0 +δ

+

x0 y0 −δ

  |F (t, y0 − δ)| dϕm (t − x, y0 − δ − y)Kη,m (t − x, y0 − δ − y)

x0

y0 +

  |F (x0 + δ, s)| dϕm (x0 − x + δ, s − y)Kη,m (x0 − x + δ, s − y)

y0 −δ

+ |F (x0 + δ, y0 − δ)| ϕm (x0 + δ − x, y0 − δ − y) Kη,m (x0 + δ − x, y0 − δ − y) .

If we apply inequality (7) to the last inequality and change the variables, then we have y x0 +δ−x 0 −y 

  μ1 (t + x − x0 ) μ2 (y0 − s − y) dϕm (t, s) Kη,m (t, s)

|I121 | ≤ ε x0 −x y0 −δ−y x0 +δ−x 

+ εμ2 (δ)

  μ1 (t + x − x0 ) dϕm (t, y0 − δ − y) Kη,m (t, y0 − δ − y)

x0 −x y 0 −y

  μ2 (y0 − s − y) dϕm (x0 + δ − x, s) Kη,m (x0 + δ − x, s)

+ εμ1 (δ) y0 −δ−y

+ εμ1 (δ)μ2 (δ)ϕm (x0 + δ − x, s) Kη,m (x0 + δ − x, y0 − δ − y) . Since monotonicity properties of ϕm Kη,m is not obviously seen from the last inequality, in order to use condition (d) , we may enlarge the expression on the right-hand side by using monotonic and bimonotonic variation functions. After that, by the aid of the method of integration by parts, we get y x0 +δ−x 0 −y 

|I121 | ≤ −ε

⎛ ⎝

x0 −x y0 −δ−y

× {μ1 (t − x0 +

x)}t

x0 +δ−x

s 

t

y0 −δ−y

⎞ ϕm (u, v) Kη,m (u, v)⎠

{μ2 (y0 − s − y)}s dsdt

On Weighted Convergence of Double Singular Integral Operators …

"x

y x0 +δ−x 0 −y 

−ε

#

0 +δ−x



591

ϕ (u, y0 − δ − y) Kη,m (u, y0 − δ − y) m

t

x0 −x y0 −δ−y

× {μ1 (t − x0 + x)}t {μ2 (y0 − s − y)}s dsdt ⎞ ⎛ y x0 +δ−x 0 −y  s  ⎝ −ε ϕm (x0 − x + δ, v) Kη,m (x0 − x + δ, v)⎠ y0 −y−δ

x0 −x y0 −δ−y

× {μ1 (t − x0 + x)}t {μ2 (y0 − s − y)}s dsdt y x0 +δ−x 0 −y 

−ε

ϕm (x0 − x + δ, y0 − δ − y) Kη,m (x0 − x + δ, y0 − δ − y) x0 −x y0 −δ−y

× {μ1 (t − x0 + x)}t {μ2 (y0 − s − y)}s dsdt = ε ( j1 + j2 + j3 + j4 ) . Using condition (d) of Class Aϕ and by Definition 2, we get x0 +δ−x 

0

j1 + j2 + j3 + j4 = −

ϕm (t, s)Kη,m (t, s) x0 −x y0 −δ−y

× {μ1 (t + x − x0 )}t {μ2 (y0 − s − y)}s dsdt x0 +δ−x  y0 −y $

+ x0 −x

% ϕm (t, s)Kη,m (t, s) {μ1 (t + x − x0 )}t −2ϕm (t, 0)Kη,m (t, 0)

0

× {μ2 (y0 − s − y)}s dsdt. Hence, the following inequality holds for I121 (for the similar situation, see [30, 41]): y0 x 0 +δ 

|I121 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (t − x0 ) ρ2 (y0 − s) dsdt x0 y0 −δ x 0 +δ

+ 2εμ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0) ρ1 (t − x0 ) dt. x0

592

G. Uysal and H. Dutta

Similar computations for I122 , I123 and I124 yield: x0 y0 |I122 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (x0 − t) ρ2 (y0 − s) dsdt x0 −δ y0 −δ

x0 + 2εμ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0) ρ1 (x0 − t) dt x0 −δ y0

+ 2εμ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y) ρ2 (y0 − s) ds y0 −δ

+ 4εϕm (0, 0)Kη,m (0, 0) μ1 (|x0 − x|) μ2 (|y0 − y|) , x0 y0 +δ |I123 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (x0 − t) ρ2 (s − y0 ) dsdt x0 −δ y0 y 0 +δ

+ 2εμ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y) ρ2 (s − y0 ) ds, y0

x 0 +δ 0 +δ y

|I124 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (t − x0 ) ρ2 (s − y0 ) dsdt. x0

y0

Hence, the following inequality is obtained for I12 : n 

|I12 | ≤ εP

ϕm (x, y)

m=1

⎧ x +δ y +δ ⎪ ⎨ 0 0 ⎪ x0 −δ y0 −δ ⎩

ϕm (t − x, s − y)Kη,m (t − x, s − y)

×ρ1 (|x0 − t|) ρ2 (|y0 − s|) dsdt

x 0 +δ

+ 2μ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0)ρ1 (|x0 − t|) dt x0 −δ y 0 +δ

+ 2μ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y)ρ2 (|y0 − s|) ds y0 −δ

 +4ϕ (0, 0)Kη,m (0, 0)μ1 (|x0 − x|) μ2 (|y0 − y|) . m

On Weighted Convergence of Double Singular Integral Operators …

593

Also, the last inequality is obtained for other cases of the assumptions |x0 − x| < 2δ and |y0 − y| < 2δ . The remaining part of the proof is clear by the hypotheses. Thus, the proof is completed.  Now, we give a corollary similar to that of almost everywhere convergence theorems (see, for example, [11, 24, 29]) by fixing the point (x0 , y0 ) within the operators, such as of type (3). In the following result D = (a, b) × (c, d) denotes any bounded rectangular region in R2 . Corollary 1 Suppose that ϕf is bounded on D. If (x0 , y0 ) is a common μ−generalized ϕ Lebesgue point of f ∈ L 1 (D) and ϕ, then lim Tη ( f ; x0 , y0 ) =

η→η0

n 

Cm f m (x0 , y0 )

m=1

on any set Z on which the function n δ δ 

ϕm (t, s)Kη,m (t, s)ρ1 (|t|) ρ2 (|s|) dsdt,

m=1 −δ −δ

where 0 < δ
0, there exists δ > 0 such that for every h and for every k for which 0 < h, k ≤ δ < δ ∗ , holds, we have the following inequality by (4c): y0 x 0 +h 

x0 y0 −k

   f (t, s) f (x0 , y0 )    ϕ(t, s) − ϕ (x , y )  dsdt < εμ1 (h)μ2 (k). 0 0

(8)

Now, we write Tη ( f ; x, y) −

n 

Cm f m (x0 , y0 ) =

m=1

n   m=1

f m (t, s)Kη,m (t − x, s − y)dsdt −

n 

Cm f m (x0 , y0 ) .

m=1

R2

Addition and subtraction of the following expression  n  f m (x0 , y0 ) ϕm (t, s)Kη,m (t − x, s − y)dsdt m (x , y ) ϕ 0 0 m=1 R2

to the right hand side of the above equality, we have the following inequality:   n      Cm f m (x0 , y0 ) Tη ( f ; x, y) −   m=1

  n   ≤  m=1 2

    m m   m  f (t, s) f , y (x ) 0 0     ϕm (t, s) − ϕm (x , y )  ϕ (t, s)Kη,m (t − x, s − y)dsdt  0 0  R       n  m    f (x0 , y0 )   m m    + ϕ (t, s)Kη,m (t − x, s − y)dsdt − Cm ϕ (x0 , y0 )  ϕm (x , y )   0 0  2  m=1 R

= I1 + I2 . In view of condition (a) of Class Aϕ , I2 tends to zero as (x, y, η) tends to (x0 , y0 , η0 ).

On Weighted Convergence of Double Singular Integral Operators …

595

The integral I1 can be written in the form:

I1 =

⎧ n ⎪ ⎨  ⎪

m=1 ⎩ 2 R \Bδ

⎫   ⎪ ⎬  f m (t, s) f m (x0 , y0 )   − m +  m ⎪ ϕ (x0 , y0 )  ⎭ ϕ (t, s) Bδ

× ϕ (t, s)Kη,m (t − x, s − y)dsdt m

= I11 + I12 , where

  Bδ = (t, s) : (t − x0 )2 + (s − y0 )2 < δ 2 , (x0 , y0 ) ∈ R2 .

Now, using the assumptions given by 0 < |x0 − x| < can define the following set:

δ 2

and 0 < |y0 − y| < 2δ , we

 δ2 Nδ = (x, y) : (x − x0 )2 + (y − y0 )2 < , (x0 , y0 ) ∈ R2 . 2 Comparing the sets Bδ and Nδ gives the relation R2 \Bδ ⊆ R2 \Dδ , where  δ2 Dδ = (t, s) : (t − x)2 + (s − y)2 < , (x, y) ∈ Nδ . 2 Therefore, we have the following inequality  I11 ≤

sup √

√δ ≤ 2

u 2 +v 2



+ √

√δ ≤ 2

u 2 +v 2

n  m=1

   fm  ϕ (x + u, y + v)Kη,m (u, v)  m  ϕ  m

L 1 (R2 )

 n  m   f (x0 , y0 )  m    ϕm (x , y )  ϕ (x + u, y + v)Kη,m (u, v)dvdu.

m=1

0

0

Consequently, by conditions (b) and (c) of class Aϕ , and I11 tends to zero as (x, y, η) tends to (x0 , y0 , η0 ). Now, we prove that I12 tends to zero as (x, y, η) tends to (x0 , y0 , η0 ). Using identity (v) in p. 13 in [36] and by boundedness of ϕf , there exists a number P ∗ , which denotes maximum of supremum values of that I12

fm , ϕm

m = 1, 2, . . . , n on R2 , such

 n     f (t, s) f (x0 , y0 )  m  ≤P  ϕ (t, s) − ϕ (x , y )  ϕ (t, s)Kη,m (t − x, s − y)dsdt 0 0 m=1 ∗

Bδ

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holds for I12 . Thus, by (5), we have n 

I12 ≤ P ∗

ϕm (x, y)

m=1

⎧ ⎪ ⎨x0+δ y0 ⎪ ⎩

x0 +

x0 y0 −δ

⎫  ⎪ y0 ⎬  f (t, s) f (x0 , y0 )  m  ϕ (t − x, s − y) −  ⎪ ϕ (x0 , y0 )  ⎭ ϕ (t, s)

x0 −δ y0 −δ

× Kη,m (t − x, s − y)dsdt ⎧ ⎫  ⎪ n ⎨ x0 y0+δ x0+δ y0+δ ⎪ ⎬  f (t, s)  f (x0 , y0 )  m  ϕ (t − x, s − y) − ϕm (x, y) + + P∗  ⎪ ⎪ ϕ (x0 , y0 )  ⎩ ⎭ ϕ (t, s) m=1 x0 −δ y0

x0

y0

× Kη,m (t − x, s − y)dsdt = P∗

n 

ϕm (x, y)(I121 + I122 + I123 + I124 ).

m=1

Let us consider the integral I121 . Let us define the function G (t, s) by  t y0   f (u, v) f (x0 , y0 )   G (t, s) =  ϕ(u, v) − ϕ (x , y )  dvdu. 0 0 x0 s

In view of inequality (8), the following expression |G (t, s)| ≤ εμ1 (t − x0 ) μ2 (y0 − s) ,

(9)

where 0 < t − x0 ≤ δ and 0 < y0 − s ≤ δ, holds. By Theorem 2.6 in [41], we can write y0 x 0 +δ 

I121 = (L) x0 y0 −δ

   f (t, s) f (x0 , y0 )  m   ϕ (t, s) − ϕ (x , y )  ϕ (t − x, s − y)Kη,m (t − x, s − y)dsdt 0 0

y0 x 0 +δ 

ϕm (t − x, s − y)Kη,m (t − x, s − y)d [−G (t, s)] ,

= (LS) x0 y0 −δ

where LS denotes Lebesgue–Stieltjes integral. Applying bivariate integration by parts (see Theorem 2.2, p. 100 in [41]) to the Lebesgue–Stieltjes integral, we have   x0 +δ y0      |I121 | =  ϕm (t − x, s − y)Kη,m (t − x, s − y)d [−G (t, s)]    x0 y0 −δ  y0 x 0 +δ 

≤ x0 y0 −δ

  |G (t, s)| dϕm (t − x, s − y)Kη,m (t − x, s − y)

On Weighted Convergence of Double Singular Integral Operators … x 0 +δ

+

597

  |G (t, y0 − δ)| dϕm (t − x, y0 − δ − y)Kη,m (t − x, y0 − δ − y)

x0

y0 +

  |G (x0 + δ, s)| dϕm (x0 − x + δ, s − y)Kη,m (x0 − x + δ, s − y)

y0 −δ

+ |G (x0 + δ, y0 − δ)| ϕm (x0 + δ − x, y0 − δ − y) Kη,m (x0 + δ − x, y0 − δ − y) .

If we apply inequality (9) to the last inequality and change the variables, then we have y x0 +δ−x 0 −y 

  μ1 (t + x − x0 ) μ2 (y0 − s − y) dϕm (t, s) Kη,m (t, s)

|I121 | ≤ ε x0 −x y0 −δ−y x0 +δ−x 

+ εμ2 (δ)

  μ1 (t + x − x0 ) dϕm (t, y0 − δ − y) Kη,m (t, y0 − δ − y)

x0 −x y 0 −y

+ εμ1 (δ)

  μ2 (y0 − s − y) dϕm (x0 + δ − x, s) Kη,m (x0 + δ − x, s)

y0 −δ−y

+ εμ1 (δ)μ2 (δ)ϕm (x0 + δ − x, s) Kη,m (x0 + δ − x, y0 − δ − y) . Similar evaluations as in Theorem 1 yield the following inequality for I121 : y0 x 0 +δ 

|I121 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (t − x0 ) ρ2 (y0 − s) dsdt x0 y0 −δ x 0 +δ

+ 2εμ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0) ρ1 (t − x0 ) dt. x0

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Similar computations for I122 , I123 and I124 yield: x0 y0 |I122 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (x0 − t) ρ2 (y0 − s) dsdt x0 −δ y0 −δ

x0 + 2εμ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0) ρ1 (x0 − t) dt x0 −δ y0

+ 2εμ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y) ρ2 (y0 − s) ds y0 −δ

+ 4εϕm (0, 0)Kη,m (0, 0) μ1 (|x0 − x|) μ2 (|y0 − y|) , x0 y0 +δ |I123 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (x0 − t) ρ2 (s − y0 ) dsdt x0 −δ y0 y 0 +δ

+ 2εμ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y) ρ2 (s − y0 ) ds, y0

x 0 +δ 0 +δ y

|I124 | ≤ ε

ϕm (t − x, s − y)Kη,m (t − x, s − y) ρ1 (t − x0 ) ρ2 (s − y0 ) dsdt. x0

y0

Hence, the following inequality is obtained for I12 : |I12 | ≤ εP ∗

n 

ϕm (x, y)

m=1

⎧ x +δ y +δ ⎪ ⎨ 0 0 ⎪ x0 −δ y0 −δ ⎩

ϕm (t − x, s − y)Kη,m (t − x, s − y)

×ρ1 (|x0 − t|) ρ2 (|y0 − s|) dsdt

x 0 +δ

+ 2μ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0)ρ1 (|x0 − t|) dt x0 −δ y 0 +δ

+ 2μ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y)ρ2 (|y0 − s|) ds y0 −δ

 +4ϕ (0, 0)Kη,m (0, 0)μ1 (|x0 − x|) μ2 (|y0 − y|) . m

Also, the last inequality is obtained for other cases of the assumptions |x0 − x| < and |y0 − y| < 2δ .

δ 2

On Weighted Convergence of Double Singular Integral Operators …

Therefore, we get

599

  n      m Cm f (x0 , y0 ) Tη ( f ; x, y) −   m=1

      n  m    f (x0 , y0 )   m m    ≤ ϕ (t, s)K (t − x, s − y)dsdt − C ϕ , y (x ) η,m m 0 0   ϕm (x , y )   0 0  2  m=1 R   n   fm m   + sup ϕ (x + u, y + v)K (u, v) η,m  ϕm  √ √δ ≤ u 2 +v 2 m=1 L 1 (R2 ) 2   m   f (x0 , y0 )  n  +  m ϕm (x + u, y + v)Kη,m (u, v)dvdu ϕ (x0 , y0 )  m=1 √ √δ ≤ 2

+ εP ∗

n 

ϕm (x, y)

m=1

u 2 +v 2

⎧ x +δ y +δ ⎪ ⎨ 0 0 ⎪ x0 −δ y0 −δ ⎩

ϕm (t − x, s − y)Kη,m (t − x, s − y)

×ρ1 (|x0 − t|) ρ2 (|y0 − s|) dsdt

x 0 +δ

+ 2μ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0)ρ1 (|x0 − t|) dt x0 −δ y 0 +δ

+ 2μ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y)ρ2 (|y0 − s|) ds y0 −δ

 +4ϕ (0, 0)Kη,m (0, 0)μ1 (|x0 − x|) μ2 (|y0 − y|) , m

where P ∗ denotes maximum of supremum values of Thus, the claim easily follows.

fm , ϕm

m = 1, 2, . . . , n on R2 . 

Corollary 2 Suppose that ϕf is bounded on R2 . If (x0 , y0 ) is a common μ− ϕ generalized Lebesgue point of f ∈ L 1 (R2 ) and ϕ, then lim Tη ( f ; x0 , y0 ) =

η→η0

n 

Cm f m (x0 , y0 )

m=1

on any set Z on which the function n δ δ 

ϕm (t, s)Kη,m (t, s)ρ1 (|t|) ρ2 (|s|) dsdt,

m=1 −δ −δ

where for certain positive number δ ∗ , 0 < δ < δ ∗ , is bounded as η tends to η0 .

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4 Rate of Pointwise Convergence Theorem 3 Assume that the hypotheses of Theorem 1 (or Theorem 2) are satisfied. Let ⎧ x +δ y +δ ⎪ n ⎨ 0 0 m  ϕ (t − x, s − y)Kη,m (t − x, s − y) m (η, δ, x, y) = ϕ (x, y) x0 −δ y0 −δ ⎪ ⎩ m=1 ×ρ1 (|x0 − t|) ρ2 (|y0 − s|) dsdt x 0 +δ

+ 2μ2 (|y0 − y|)

ϕm (t − x, 0)Kη,m (t − x, 0)ρ1 (|x0 − t|) dt x0 −δ y 0 +δ

+ 2μ1 (|x0 − x|)

ϕm (0, s − y)Kη,m (0, s − y)ρ2 (|y0 − s|) ds y0 −δ

 +4ϕ (0, 0)Kη,m (0, 0)μ1 (|x0 − x|) μ2 (|y0 − y|) , m

for a positive number δ 0 satisfying 0 < δ < δ 0 , and the following conditions are satisfied for m = 1, 2, . . . , n: i: (η, δ, x, y) tends to zero as (x, y, η) tends to (x0 , y0 , η0 ) for some δ > 0. ii: As (x, y, η) tends to (x0 , y0 , η0 ), we have        m m  ϕ (t, s)Kη,m (t − x, s − y)dsdt − Cm ϕ (x0 , y0 ) = o((η, δ, x, y)).    D

iii: For every ξ > 0,  sup

√ ξ≤ t 2 +s 2

n 

 ϕ (x + t, y + s)Kη,m (t, s) = o((η, δ, x, y)) m

m=1

as (x, y, η) tends to (x0 , y0 , η0 ). iv: For every ξ > 0,  n  m=1

ϕm (x + t, y + s)Kη,m (t, s)dsdt = o((η, δ, x, y))

√ ξ≤ t 2 +s 2

as (x, y, η) tends to (x0 , y0 , η0 ). ϕ Then, at each μ-generalized Lebesgue point of f ∈ L 1 (D) and ϕ, we have

On Weighted Convergence of Double Singular Integral Operators …

601

  n      m Cm f (x0 , y0 ) = o((η, δ, x, y)). Tη ( f ; x, y) −   m=1

as (x, y, η) tends to (x0 , y0 , η0 ). Proof The result is obvious by the hypotheses of Theorem 1 (or Theorem 2).



Remark 2 Similar theorem concerning rate of convergence may be given by replacing little o notation “o(.)” by big O notation “O(.)”.

5 Concluding Remarks In this chapter, we proved some theorems on different types of pointwise convergence for the family of nonlinear bivariate singular integral operators of type (3). For this aim, we used bivariate analogues of some notions given in one variable case, such as monotonicity, method of integration by parts. By using these concepts a special class of kernel functions, called Class Aϕ , is defined. Therefore, main results are presented as Theorems 1 and 2. Some corollaries showing the differences between Fatou type convergence which is based on parameters and pointwise convergence which is associated with almost everywhere convergence are given. By using these theorems, we obtained the rate of pointwise convergence.

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Circular-Like and Circular Elements in Free Product Banach ∗-Algebras Induced by p-Adic Number Fields Q p Over Primes p Ilwoo Cho

Abstract In this paper, we study weighted-circular, and circular elements in a certain free product Banach ∗-probability space (LS, τ 0 ) induced by measurable functions on p-adic number fields Q p , for primes p. To do that, we first constructand-consider weighted-semicircular, and semicircular elements in (LS, τ 0 ). From our (weighted-)semicircular elements, we establish (weighted-)circular elements and study their free distributions by computing joint free moments of them and their adjoints. The circular law is re-characterized by joint free moments of our circular elements and their adjoints. More interestingly, our weighted-circularity dictated by p-adic analysis is fully characterized by weights of weighted-semicircular elements containing number-theoretic data obtained from fixed primes p. Keywords Free probability · p-Adic number fields · Banach ∗-probability spaces · Weighted-semicircular elements · Semicircular elements · Weighted-circular elements · Circular elements 1991 Mathematics Subject Classification 05E15 · 11G15 · 11R47 · 11R56 · 46L10 · 46L54 · 47L30 · 47L55

1 Introduction In [3, 9], we constructed-and-studied weighted-semicircular elements and semicircular elements induced by p-adic number fields Q p , for all p ∈ P, where P is the set of all primes in the set N of all natural numbers. The main purpose of this paper is to study circular elements, and circular-like elements, called weighted-circular elements, in a certain Banach ∗-probability space induced from p-adic analysis on {Q p } p∈P . The free distributions of such operators, characterized by their joint free I. Cho (B) Department of Mathematics and Statistics, Saint Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA 52803, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_19

605

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moments of them and their adjoints, are the main results of this paper. From these, one can not only re-characterize the well-known circular law, but also characterize the circular-like laws induced from p-adic analysis in terms of weights containing number-theoretic data in primes p. Recall and remark that the circular law induced from the semicircular law is well-known in free probability theory; in particular, the circularity is regarded as a special case of R -diagonality (e.g., see [1, 12–14]). So, here, we are more interested in “circular-like” laws induced from our weighted-circular elements generated by the weighted-semicircular elements of [3, 9]. Especially, our “weighting” is determined by number-theoretic information dictated by given primes. Thus, the main results illustrate close connections among number theory, operator theory, operator algebra theory, representation theory, via free probability theory. For the self-contained-ness of text, we briefly consider the main results of [3, 9] in the first half of the paper. And then we establish-and-study our circular elements, and weighted-circular elements.

1.1 Preview and Motivation We have considered how primes (or prime numbers) act on operator algebras. The relations between primes and operators have been studied in various different approaches. For instance, we studied how primes act on certain von Neumann algebras generated by p-adic and Adelic measure spaces (e.g., [4, 6]). Meanwhile, in [5], primes are regarded as linear functionals acting on arithmetic functions. In such a case, one can understand arithmetic functions as Krein-space operators under certain representations (e.g., see [8]).  Also, in [2, 7], we considered free-probabilistic structures on Hecke algebras H G L 2 (Q p ) , for primes p. Such series of research are motivated by well-known number-theoretic results (e.g., [10, 11, 17]). In [9], the author and Jorgensen constructed weighted-semicircular elements, and corresponding semicircular elements in a certain Banach ∗-algebra LS p induced from the ∗-algebra M p consisting of measurable functions on a p-adic number fields Q p , for a fixed prime p ∈ P. The construction of weighted-semicircular elements, itself, was one of the main results of [9]. Based on the construction of [9], for any fixed prime p, one can obtain weighted-semicircular elements Q p, j in Banach ∗-probability spaces LS p ( j) (on the Banach ∗-algebra LS p ), for all j ∈ Z. In [3], the author constructed the free product Banach ∗-probability space LS  of the system LS p ( j) p∈P, j∈Z of [9], over both primes and integers, and studied weighted-semicircular elements Q p, j ’s, and the corresponding semicircular elements  p, j ’s in LS, as free generators of LS. The free distributions of free reduced words in Q p, j ’s and those in  p, j ’s are considered there. Here, we will use the same frameworks of [3] to study circular-like, and circular elements induced from p-adic analysis.

Circular-Like and Circular Elements in Free Product Banach …

607

Recall that if s1 and s2 are semicircular elements, and if they are free from each other, then a circular element c is defined to be an operator, √ 1 c = √ (s1 + is2 ) , with i = −1 in C. 2 So, it is natural to construct our circular elements from our free semicircular family { p, j ∈ LS p ( j)} p∈P, j∈Z in LS. Motivated by circularity, to generate our weighted-circular elements, we use the free weighted-semicircular family, {Q p, j ∈ LS p ( j)} p∈P, j∈Z in LS (See below).

1.2 Overview In Sect. 2, we briefly introduce backgrounds of our works. In Sect. 3 through 7, we construct our Banach ∗-probability space (LS, τ 0 ), and weighted-semicircular elements induced from p-adic analysis on Q p , for primes p. In Sect. 8, we determine the maximal free weighted-semicircular family Q, and the maximal free semicircular family  in the Banach ∗-algebra LS. In Sect. 9, circular elements in LS generated by the free semicircular family  of Sect. 8 are studied. In particular, we concentrate on studying free distributions of our circular elements C by computing joint free moments of {C, C ∗ }. The free distributions of such circular elements C are fully characterized by the free moments of C, those of C ∗ , and non-vanishing mixed free moments of {C, C ∗ }. Under the identically-free-distributedness, one can re-characterize the circular law in terms of our joint free moments. However, by the universality of the circular law, the freedistributional data of (any) circular elements are already known (in terms of joint free cumulants) under R-diagonality. So, more interestingly, we study the weightedcircularity dictated by p-adic analysis. In Sect. 10, we consider p-adic-analytic, free-probabilistic information of certain circular-like elements generated by our weighted-semicircular elements. From our free weighted-semicircular family Q (in the sense of Sect. 8) of LS, we construct so-called the Adelic weighted-circular elements, and study free distributions of such elements in terms of the joint free moments of them and their adjoints in LS. In Sect. 11, we study all our processes and main results in Sects. 8, 9 and 10 conversely. It is shown that whenever two primes and two integers are given with an additional condition, one can have the corresponding weighted-circular law. As a special case, we show that, whenever two distinct primes are given, they induce the circular law free-probabilistically.

608

I. Cho

2 Preliminaries In this section, we briefly introduce backgrounds of our proceeding works.

2.1 Free Probability Readers can study free probability theory from [16, 18] (and the cited papers therein). Free probability is understood as the noncommutative operator-algebraic version of classical measure theory and statistics. The classical independence is replaced by the freeness, by replacing measures on sets to linear functionals on algebras. It has various applications not only in pure mathematics (e.g., [15]), but also in related scientific topics (e.g., [4–6, 8, 9]). Here, we will use combinatorial free probability theory of Speicher (e.g., [16]). In particular, in the text, without introducing detailed definitions and combinatorial backgrounds, free moments and free cumulants of operators will be computed; and the free product of ∗-probability spaces in the sense of [16, 18] is considered. For more about circularity and R-diagonality, see [1, 12–14].

2.2

p-Adic Analysis on Q p

In this section, we provide a main motivation of our free-probabilistic models on the ∗-algebra M p of measurable functions on Q p . For more about p-adic, and Adelic analysis, see [17]. Let p ∈ P, and let Q p be the p-adic number field. Under the p-adic addition, and the p-adic multiplication, the set Q p forms a ring algebraically (e.g., [17]). It is equipped with the non-Archimedean norm |.| p , which is the p-norm on the set Q of all rational numbers defined by  a 1   |x| p =  p k  = k , b p p whenever x = p k ab in Q, where k, a ∈ Z, and b ∈ Z \ {0}. For instance,    8    = 2 3 · 3  2 and

 1  1 1 = 3 = , 3 2 2 8

  8     = 3−1 · 8 = 1 = 3, 3 3 3−1 3

Circular-Like and Circular Elements in Free Product Banach …

and

609

  8   = 1, whenever q ∈ P \ {2, 3}. 3 q

Topologically, the p-adic number field Q p is the p-norm closure of Q. So, under topology, it forms a Banach space (e.g., [17]). Let’s understand the Banach ring Q p as a measure space,   Q p = Q p , σ (Q p ), μ p , where σ (Q p ) is the σ -algebra of Q p consisting of all μ p -measurable subsets, where μ p is the left-and-right additive invariant Haar measure on Q p satisfying μ p (Z p ) = 1, where Z p is the unit disk of Q p , consisting of all p-adic integers. Moreover, if we define (1) Uk = p k Z p = { p k x ∈ Q p : x ∈ Z p }, for all k ∈ Z (with U0 = Z p ), then these μ p -measurable subsets Uk ’s of (1) satisfy Q p = ∪ Uk , k∈Z

and μ p (Uk ) =

1 = μ p (x + Uk ) , for all k ∈ Z, pk

(2)

and · · · ⊂ U2 ⊂ U1 ⊂ U0 = Z p ⊂ U−1 ⊂ U−2 ⊂ · · ·. In fact, the family {Uk }k∈Z forms a basis of the Banach topology for Q p (e.g., [17]). Define now subsets ∂k of Q p by ∂k = Uk \ Uk+1 , for all k ∈ Z.

(3)

We call such μ p -measurable subsets ∂k , the kth boundaries of Uk in Q p , for all k ∈ Z. By (2) and (3), one obtains that Q p =  ∂k ,

(4)

k∈Z

and μ p (∂k ) = μ p (Uk ) − μ p (Uk+1 ) = for all k ∈ Z, where  means the disjoint union.

1 1 − k+1 , pk p

610

I. Cho

Now, let M p be the algebra,   M p = C {χ S : S ∈ σ (Q p )} ,

(5)

  where χ S are the usual characteristic functions of S ∈ σ Q p . Then the set M p of (5) forms a well-defined ∗-algebra over C with its adjoint, ⎛

⎞∗



⎝



tS χS ⎠ =

de f

S∈σ (G p )

tS χS ,

S∈σ (G p )

where t S ∈ C, having their conjugates t S in C. Let f ∈ M p . Then one can define the p-adic integral by ⎛





⎝

Qp

⎞



t S χ S ⎠ dμ p =

S∈σ (Q p )

t S μ p (S).

(6)

S∈σ (Q p )

Note that, by (4), if S ∈ σ (Q p ), then there exists a subset  S of Z, such that  S = { j ∈ Z : S ∩ ∂ j = ∅},

(7)

satisfying 

 Qp

χ S dμ p = =



Q p j∈ S



χ S∩∂ j dμ p

  μp S ∩ ∂ j

j∈ S

by (6) ≤

 j∈ S

     1 1 μp ∂ j = − j+1 , pj p j∈

(8)

S

by (4) and (6), for all S ∈ σ (Q p ), where  S is in the sense of (7). More precisely, one can get the following proposition. Proposition 1 Let S ∈ σ (Q p ), and let χ S ∈ M p . Then there exist r j ∈ R, such that 0 ≤ r j ≤ 1 in R, for all j ∈  S , and

 Qp

χ S dμ p =

 j∈ S

 rj

1 1 − j+1 j p p

(9)

 .

Circular-Like and Circular Elements in Free Product Banach …

611

Proof The existence of r j ’s and the p-adic integral in (9) are obtained by (8). See [3, 9] for more details. 

3 Analysis on M p Note first that the ∗-algebra M p is commutative, and hence, (traditional, or usual “noncommutative”) free probability is meaningless (or, it is not needed for studying analysis) on M p , for p ∈ P, because the freeness on this commutative structure, M p , is trivial. However, we are not interested at this moment in the free-probabilistic structures of M p ; we are interested in certain analytic data will be used later to establish (weighted-)semicircular elements expressed as forms of free-probabilistic data. So, as in [9], we use “free-probabilistic models” on M p to construct-and-study our weighted-semicircularity and semicircularity by using concepts and tools from free probability theory under commutativity. So, without loss of generality, we use terminology and computation techniques from free probability theory “non-traditionally” here. Note also that, later, in Sect. 6, we construct “traditional” free-probabilistic structures (as in [3, 15, 18]) from our “non-traditional” free-probabilistic models of Sects. 3, 4 and 5. Throughout this section, fix a prime p ∈ P, and the corresponding p-adic number field Q p . Let M p be the ∗-algebra (5) consisting of μ p -measurable functions on Q p . Let Uk be the basis elements (1) of the topology for Q p with their boundaries ∂k of (3), i.e., (10) Uk = p k Z p , for all k ∈ Z, and ∂k = Uk \ Uk+1 , for all k ∈ Z. Define a linear functional ϕ p : M p → C by the p-adic integration (6),  ϕp ( f ) =

Qp

f dμ p , for all f ∈ M p .

(11)

Then, by (11), one obtains     1 1 1 ϕ p χU j = j , and ϕ p χ∂ j = j − j+1 , p p p for all j ∈ Z, by (2) and (4).   Definition 1 The pair M p , ϕ p is called the p-adic free probability space for p ∈ P, where ϕ p is the linear functional (11) on M p . Remark that it is not a “traditional” free probability space, but, for our purposes, we call it p-adic “free probability space,” “non-traditionally.”

612

I. Cho

Now, let ∂k be the kth boundary Uk \ Uk+1 of Uk in Q p , for all k ∈ Z. Then, for k1 , k2 ∈ Z, one obtains that χ∂k1 χ∂k2 = χ∂k1 ∩∂k2 = δk1 ,k2 χ∂k1 , and hence,

    ϕ p χ∂k1 χ∂k2 = δk1 ,k2 ϕp χ∂k1 = δk1 ,k2

−

1 p k1

(12)



1

pk1 +1

,

where δ is the Kronecker delta. Proposition 2 Let ( j1 , . . . , j N ) ∈ Z N , for N ∈ N. Then N

χ∂ jl = δ( j1 ,..., jN ) χ∂ j1 in M p ,

(13)

l=1

and hence,

 ϕp



N

 = δ( j1 ,..., jN )

χ∂ jl

l=1



where δ( j1 ,..., jN ) =

1 1 − j +1 j 1 p p1 

N −1

δ jl , jl+1

l=1

 ,

  δ jN , j1 . 

Proof The proof of (13) is done by induction on (12).   Thus, one can get that, for any S ∈ σ Q p , ϕ p (χ S ) =



 rj

j∈ S

1 1 − j+1 pj p

 ,

(14)

by (9) and (13), where 0 ≤ r j ≤ 1 are in the sense of (9) for all j ∈ Z, and  S is in the sense of (7).   Also, if S1 , S2 ∈ σ Q p , then ⎛ χ S1 χ S2 = ⎝



⎞⎛ χ S1 ∩∂k ⎠ ⎝

k∈ S1

=





⎞ χ S2 ∩∂ j ⎠

j∈ S2

  χ S1 ∩∂k χ S2 ∩∂ j

(k, j)∈ S1 × S2

=



δk, j χ(S1 ∩S2 )∩∂ j

(k, j)∈ S1 × S2

=



j∈ S1 ,S2

χ(S1 ∩S2 )∩∂ j ,

(15)

Circular-Like and Circular Elements in Free Product Banach …

613

where  S1 ,S2 =  S1 ∩  S2 , by (12) and (13). Thus, there exist w j ∈ R, such that 0 ≤ w j ≤ 1, for all j ∈  S1 ,S2 , and





ϕ p χ S1 χ S2 =





wj

j∈ S1 ,S2

1 1 − j+1 j p p

(16)  ,

  by (15), for all S1 , S2 ∈ σ Q p .   In (16), definitely, if  S1 ,S2 is empty, then ϕ p χ S1 χ S2 = 0. Inductively, we obtain the following generalized result under induction.   Proposition 3 Let Sl ∈ σ (Q p ), and let χ Sl ∈ M p , ϕ p , for l = 1, . . . , N , for N ∈ N. Let N

 S1 ,...,SN = ∩  Sl in Z, l=1

where  Sl are in the sense of (7), for l = 1, . . . , N . Then there exist r j ∈ R, such that (17) 0 ≤ r j ≤ 1 in R, for j ∈  S1 ,...,SN , 

and ϕp

N

χ Sl

l=1

 =

 j∈ S1 ,...,S N

 rj

1 1 − j+1 pj p

Proof The proof of (17) is done by induction on (16).

 . 

The above free-moment formula (17), provides a universal tool to compute the (non-traditional)  free-distributional data of elements in our p-adic free probability  space M p , ϕ p .

  4 Representations of M p , ϕ p   Fix a prime p in P, and let M p , ϕ p be the p-adic free probability space (nontraditionally). By understanding Q p as a measure space, construct the L 2 -space H p of Q p ,     de f H p = L 2 Q p , σ (Q p ), μ p = L 2 Q p , (18) over C. Then this L 2 -space H p of (18) is a well-defined Hilbert space equipped with its inner product 2 ,

614

I. Cho de f

h 1 , h 2 2 =

 Qp

h 1 h ∗2 dμ p ,

(19)

for all h 1 , h 2 ∈ H p . Definition 2 We call the Hilbert space H p of (18), the p-adic Hilbert space. By the very construction (18) of the p-adic Hilbert space H p , our ∗-algebra M p acts on H p , via an algebra-action α p , α p ( f ) (h) = f h, for all h ∈ H p ,

(20)

for all f ∈ M p . i.e., for any f ∈ M p , the image α p ( f ) is a multiplication operator on H p with its symbol f contained in the operator algebra B(H p ) of all bounded linear operators on H p . p

Notation Denote α p ( f ) of (20) by α f , for all f ∈ M p . Also, for convenience, denote   p p αχS simply by α S , for all S ∈ σ Q p . For instance, p

αUk = αχpU = α p (χUk ), k

and

  p α∂k = αχp∂ = α p χ∂k , k

for all k ∈ Z, where Uk and ∂k are in the sense of (10) in Q p .



By (20), the linear morphism α p is a well-determined ∗-algebra-action of M p p acting on H p because the images α f form well-determined ∗-homomorphism on H p , for all f ∈ M p . Indeed, p

α f1 f2 (h) = f 1 f 2 h = f1 ( f 2 h) p p p = f 1 α f2 (h) = α f1 α f2 (h), for all h ∈ H p , for all f 1 , f 2 ∈ M p ; and 

p

α f (h 1 ), h 2

 2

 = f h 1 , h 2 2 = Q p f h 1 h ∗2 dμ p   = Q p h 1 f h ∗2 dμ p = Q p h 1 (h 2 f ∗ )∗ dμ p    p = Q p h 1 ( f ∗ h 2 )∗ dμ p = h 1 , α f ∗ (h 2 ) , 2

for all h 1 , h 2 ∈ H p , implying that: 

p

αf

∗

= α f ∗ , for all f ∈ M p ,

where 2 is the inner product (19) on H p .

Circular-Like and Circular Elements in Free Product Banach …

615

Proposition 4 The linear morphism α p of (20) is a well-defined ∗-algebra action of M p acting on H p . Equivalently, the pair (H p , α p ) is a Hilbert-space representation of M p . Proof It is proven by the discussions in the very above paragraphs. See [3] for more details.    Definition 3 The Hilbert-space representation H p , α p is said to be the p-adic representation of M p . Depending on the p-adic representation (H p , α p ) of M p , one can construct the C -algebra M p in the operator algebra B(H p ). ∗

Definition 4 Let M p be the operator-norm closure of M p in the operator algebra B(H p ), i.e.,     de f p (21) Mp = α p Mp = C α f : f ∈ Mp , in B(H p ), where X mean the operator-norm closures of subsets X of B(H p ). ∗ Then this C ∗ -subalgebra M  p ) is called the p-adic C -algebra of the  p of B(H p-adic free probability space M p , ϕ p .

5 Analysis on M p   Throughout this section, let’s fix a prime p ∈ P, and let M p , ϕ p be the corre sponding p-adic free probability space. Let H p , α p be the p-adic representation  of M p , and let M p be the corresponding p-adic C ∗ -algebra (21) of M p , ϕ p . We here consider suitable (non-traditional) free-probabilistic models on M p (in p the similar sense of Sect. 4). In particular, we are interested in a system {ϕ j } j∈Z of linear functionals on M p , determined by the jth boundaries {∂ j } j∈Z of Q p . p Define a linear functional ϕ j : M p → C by a linear morphism,  de f  p ϕ j (a) = αap (χ∂ j ), χ∂ j 2 ,

(22)

for all a ∈ M p , for all j ∈ Z, where 2 is the inner product (19) on the p-adic Hilbert space H p of (18). Remark that if a ∈ M p , then a=



t S χ S in M p

S∈σ (Q p )

(with t S ∈ C), where for M p .



is finite or infinite (limit of finite) sum(s) under C ∗ -topology

616

I. Cho p

Definition 5 Let j ∈ Z, and let ϕ j be the linear functional (22) on the p-adic   p C ∗ -algebra M p . Then the pair M p , ϕ j is said to be the jth p-adic C ∗ -probability space (non-traditionally). So, one can get the system p

{(M p , ϕ j ) : j ∈ Z} of jth p-adic C ∗ -probability spaces. take the corresponding jth p-adic C ∗ -probability space  j ∈ Z, and  Now, fix   p p M p , ϕ j . For S ∈ σ Q p , and a generating operator α S ∈ M p , one has that    p  p p ϕ j (α S ) = α S (χ∂ j ), χ∂ j 2 = χ S∩∂ j , χ∂ j 2   = χ S∩∂ j χ∂∗j dμ p = χ S∩∂ j χ∂ j dμ p Qp

Qp

by (19)  =

Qp

  χ S∩∂ j dμ p = μ p S ∩ ∂ j



= rS

1 1 − j+1 pj p

 ,

(23)

  for some 0 ≤ r S ≤ 1 in R, for S ∈ σ Q p .     p p p Proposition 5 Let S ∈ σ Q p , and α S = αχS ∈ M p , ϕ j , for a fixed j ∈ Z. Then there exists r S ∈ R, such that (24) 0 ≤ r S ≤ 1 in R, and p ϕj

 p n  αS = rS



1 1 − j+1 pj p

 , f orall n ∈ N.

p

Proof Remark that the generating operator α S is a projection in M p , in the sense that:  p 2  p ∗ p α S = α S = α S , in M p , since

and

∗  p ∗  p p α S = α (χ S ) = α p (χ S∗ ) = α p (χ S ) = α S ,  p 2 p α S = α p (χ S2 ) = α p (χ S ) = α S .

Circular-Like and Circular Elements in Free Product Banach …

So,

617

 p n p α S = α S , for all n ∈ N.

Thus, for any n ∈ N, we have p ϕj

 p n  p p αS = ϕ j (α S ) = r S



1 1 − j+1 pj p

 ,

for some 0 ≤ r S ≤ 1 in R, by (23).



As corollaries of (24), we obtain the following results. Corollary 1 Let Uk and ∂k are in the sense of (10), for all k ∈ Z. Then p ϕj

 p n  αUk =

and p

ϕj



1 pj

−

0

 p n  α∂k = δ j,k



1 p j+1

if k ≤ j otherwise,

1 1 − j+1 pj p

(25)



for all n ∈ N, for k ∈ Z.



6 Semigroup C ∗ -Subalgebras S p of M p Let M p be the p-adic C ∗ -algebra for an arbitrarily fixed p ∈ P, as in Sects. 4 and 5. Take operators p (26) Pp, j = α∂ p ∈ M p , j

p

where ∂ j are the jth boundaries (10) of Q p , for all j ∈ Z. Then these operators Pp, j of (26) are projections on the p-adic Hilbert space H p in M p , i.e., ∗ 2 Pp, j = Pp, j = Pp, j in M p , for all j ∈ Z. Indeed,    ∗ p ∗ Pp, = α p χ∂∗p = α p (χ∂ jp ) = Pp, j , j = α∂ p j

j

and

     2 p 2 = α p χ∂2p = α p χ∂ jp = Pp, j , Pp, j = α∂ p j

for all j ∈ Z.

j

618

I. Cho

We now restrict our interests to these projections Pp, j of (26). Definition 6 Fix p ∈ P. Define S p by the C ∗ -subalgebra     S p = C ∗ {Pp, j } j∈Z = C {Pp, j } j∈Z of M p ,

(27)

where Pp, j are projections (26), for all j ∈ Z. We call this C ∗ -subalgebra S p , the p-adic boundary (C ∗ -)subalgebra of M p . The p-boundary subalgebra S p of the p-adic C ∗ -algebra M p satisfies the following structure theorem. Proposition 6 Let S p be the p-adic boundary subalgebra (27) of the p-adic C ∗ -algebra M p . Then   ∗-iso ∗-iso S p = ⊕ C · Pp, j = C⊕|Z| ,

(28)

j∈Z

in M p . Proof The proof of (28) is done by the mutually-orthogonality of the projections {Pp, j } j∈Z of (26) in M p . Indeed, they satisfy   Pp, j1 Pp, j2 = α p χ∂ jp ∩∂ jp 1

2

  = α p χδ j1 , j2 ∂ jp = δ j1 , j1 Pp, j1 , 1

in M p . See [9] for details.

 p

Define now linear functionals ϕ j on S p by p

p

ϕ j = ϕ j |S p on S p ,

(29)

p

where ϕ j in the right-hand side of (29) are the linear functionals (22) on the p-adic C ∗ -algebra M p , for all j ∈ Z. So, one can get the C ∗ -probabilistic models,   p S p , ϕ j , for all j ∈ Z.

(30)

NotationFor convenience, we denote the (non-traditional) C ∗ -probability spaces  p  S p , ϕ j of (30) simply by S p, j , for all j ∈ Z (for all p ∈ P).

Circular-Like and Circular Elements in Free Product Banach …

619

7 Weighted-Semicircular Elements Let M p be the p-adic C ∗ -algebra, and let S p be the boundary subalgebra (27) of M p , satisfying the structure theorem (28), i.e.,   ∗-iso S p = C ∗ {Pp, j } j∈Z = C⊕|Z| , p

where Pp, j are projections α∂ p of (61) on H p , for all p ∈ P, and j ∈ Z. j

Fix p ∈ P. Recall that the generating projections Pp, j of S p satisfy  1 1 p ϕ j Pp, j = j − j+1 , ∀ j ∈ Z, p p

(31)

by (24) and (25). Now, let φ be the Euler totient function, defined to be an arithmetic function φ : N → C,

(32)

satisfying φ(n) = |{k ∈ N : k ≤ n, gcd(n, k) = 1}| , for all n ∈ N, where gcd means the greatest common divisor. In other words, the arithmetic function φ of (32) counts how many co-primes of a natural number n has, for all n ∈ N. It is well-known that    1 1− , φ(n) = n

q∈P, q|n q for all n ∈ N, where “q | n” means “q divides n,” or “q is a divisor of n.” Thus, one has   1 , ∀ p ∈ P, φ( p k ) = p k − p k−1 = p k 1 − p for all k ∈ N, by (32). So, we have that   p ϕ j Pp, j = p1j − =

p p j+1



1 p j+1

1−



1 p

=



=

1 pj

  1 − 1p

φ( p) , p j+1

by (31) and (33), for all Pp, j ∈ S p . More generally,

(33)

620

I. Cho

 p ϕ j Pp,k = δ j,k



 φ( p) , ∀ p ∈ P, k ∈ Z. p j+1

(34) p

Now, for a fixed prime p, define new linear functionals τ j on S p , by linear morphisms satisfying that 1 p p ϕ , on S p , (35) τj = φ( p) j p

for all j ∈ Z, where ϕ j are in the sense of (29). Then one obtains new (non-traditional) C ∗ -probability spaces, p

{S p ( j) = (S p , τ j ) : p ∈ P, j ∈ Z},

(36)

p

where τ j are in the sense of (35). Proposition 7 Let S p ( j) = (S p , τ j ) be a C ∗ -probability space (36), and let Pp,k be generating operators of S p ( j), for p ∈ P, j ∈ Z. Then p

δ j,k p n  τ j Pp,k = j+1 , for all n ∈ N. p

(37)

Proof The free-moment formula (37) is proven by (34) and (35). Indeed, since Pp,k are projections in S p ( j),  p n  p = τ j Pp,k = δ j,k τ j Pp,k



1 p j+1

 ,

by (25), for all n ∈ N, for all p ∈ P, j ∈ Z.



7.1 Semicircular and Weighted-Semicircular Elements Let (A, ϕ) be an arbitrary topological ∗-probability space (C ∗ -probability space, or W ∗ -probability space, or Banach ∗-probability space, etc.), equipped with a topological ∗-algebra A (C ∗ -algebra, resp., W ∗ -algebra, resp., Banach ∗-algebra, etc.), and a bounded linear functional ϕ on A. As usual, if an operator a ∈ A is regarded as an element of (A, ϕ), we call a, a free random variable of (A, ϕ). Definition 7 Let a be a self-adjoint free random variable in (A, ϕ). It is said to be even in (A, ϕ), if all odd free moments of a vanish, i.e.,   ϕ a 2n−1 = 0, for all n ∈ N.

(38)

Let a be a “self-adjoint,” and “even” free random variable of (A, ϕ) satisfying (38). Then it is said to be semicircular in (A, ϕ), if

Circular-Like and Circular Elements in Free Product Banach …

ϕ(a 2n ) = cn , for all n ∈ N,

621

(39)

where cn are the nth Catalan numbers, cn =

1 n+1



2n n

 =

1 (2n)! (2n)! , = n + 1 (n!)2 n!(n + 1)!

for all n ∈ N. It is well-known that, if kn (...) is the free cumulant on A in terms of a linear functional ϕ (in the sense of [16]), then a self-adjoint free random variable a is semicircular in (A, ϕ), if and only if ⎛

⎞

kn ⎝a, a, ......, a ⎠ =   



1 if n = 2 0 otherwise,

(40)

n-times

for all n ∈ N (e.g., see [16]). The above equivalent free-distributional data (40) of the semicircularity is obtained by the Möbius inversion of [16]. Thus, the semicircular free random variables a of (A, ϕ) can be re-defined by the self-adjoint free random variables a satisfying the free-cumulant characterization (40). Motivated by (40), one can define the weighted-semicircularity. Definition 8 Let a ∈ (A, ϕ) be a self-adjoint free random variable. It is said to be weighted-semicircular in (A, ϕ) with its weight t0 (in short, t0 -semicircular), if there exists t0 ∈ C× = C \ {0}, such that ⎛

⎞

kn ⎝a, a, ...., a ⎠ =   



t0 if n = 2 0 otherwise,

(41)

n-times

for all n ∈ N, where kn (...) is the free cumulant on A in terms of ϕ. By the definition (41), and by the Möbius inversion of [16], we obtained the following free-moment characterization (42) of the weighted-semicircularity (41): A self-adjoint free random variable a is t0 -semicircular in (A, ϕ), if and only if there exists t0 ∈ C× , such that n ϕ(a n ) = ωn t02 c n2 , (42) 

with ωn =

1 if n is even 0 if n is odd,

for all n ∈ N, where cm are the mth Catalan numbers for all m ∈ N.

622

I. Cho

Indeed, if a self-adjoint element a satisfies the free-moment formula (42), then  

kn (a, . . . , a) =

  |V | 

ϕ a μ(π, 1n )

V ∈π

π∈N C(n)

by the Möbius inversion of [16], where N C(n) is the lattice of all noncrossing partitions over {1, . . . , n}, and μ is the Möbius functional in the incidence algebra acting on {N C(n)}∞ n=1 , in the sense of [16] 



=

  |V | 

ϕ a μ(π, 1n )

V ∈π

π∈N Ce (n)

where N Ce (n) is the subset of the lattice N C(n), consisting of all noncrossing partitions whose blocks have even number of entries 



= by (42) =

n 2

t0

π∈N Ce (n)

⎛

n 2

= t0 ⎝

V ∈π



⎪ ⎩

c |V |

μ(π, 1n )





V ∈π



⎧ 2 ⎪ ⎨ t02 = t0

2



π∈N Ce (n)

=

μ(π, 1n )

t0 c |V |

π∈N Ce (n)





|V | 2

2

c |V |

V ∈π

2

⎞ μ(π, 1n )⎠

if n = 2

n

t02 (0) = 0 otherwise,

for all n ∈ N, by (39) and (40). Therefore, the free-moment formula (42) holds, if and only if the free-cumulant formula (41) holds. Therefore, we use the t0 -semicircularity definition (41) and the characterization (42) alternatively.

7.2 Tensor Product Banach ∗-Algebra LS p  p Let S p (k) = S p , τk be a kth p-adic boundary probability space, for p ∈ P, k ∈ Z. Define now a bounded linear transformations c p and a p “acting on the p-adic boundary subalgebra S p of M p ,” by linear morphisms satisfying,

Circular-Like and Circular Elements in Free Product Banach …

  c p Pp, j = Pp, j+1 , and

623

(43)

  a p Pp, j = Pp, j−1 ,

on S p , for all j ∈ Z. By the definition (43), the linear transformations c p and a p are bounded under the operator-norm induced by the C ∗ -norm on S p . So, the linear transformations c p and a p are regarded as Banach-space operators on S p , by regarding   S p as a Banach space. i.e., c p and a p are elements of the operator space B S p consisting of all bounded operators on S p . Definition 9 The Banach-space operators c p and a p of (43) are called the p-creation, respectively, the p-annihilation on S p , for p ∈ P. Define a new Banach-space  operator l p ∈ B S p , by (44) l p = c p + a p on S p . We call it the p-radial operator on S p . Let l p be the p-radial operator c p + a p of (44) on S p . Construct a Banach algebra L p by L p = C[l p ] in B(S p ), (45) where Y mean the operator-norm-topology closures of all subsets Y of B(S p ). By the definition (45), L p is a well-defined Banach algebra as a closed subspace of the operator space B(S p ). On this Banach algebra L p , define the adjoint (∗) by ∞  k=0

sk l kp ∈ L p −→

∞ 

sk l kp ∈ L p ,

(46)

k=0

where sk ∈ C with their conjugates sk ∈ C. Then, equipped with the adjoint (46), this Banach algebra L p of (45) forms a Banach ∗-algebra. Definition 10 Let L p be a Banach ∗-algebra (45) for p ∈ P. We call it the p-radial (Banach-∗-)algebra on S p . Let L p be the p-radial algebra (45) on S p . Construct now the tensor product Banach ∗-algebra LS p by (47) LS p = L p ⊗C S p , where ⊗C means the tensor product of Banach ∗-algebras. Take now a generating element l kp ⊗ Pp, j , for some k ∈ N0 = N ∪ {0}, and j ∈ Z, where Pp, j are in the sense of (26) in S p , with axiomatization: l 0p = 1S p , the identity operator on S p ,

624

I. Cho

in B(S p ), satisfying   1S p Pp, j = Pp, j , for all Pp, j ∈ S p , for all j ∈ Z. Note that, by (47) and (28), the elements l kp ⊗ Pp, j generate LS p , under linearity, because k  l p ⊗ Pp, j = l kp ⊗ Pp, j , for all k ∈ N0 , and j ∈ Z, for p ∈ P. We concentrate on such generating operators of LS p . Define a linear morphism E p : LS p → S p by a linear transformation satisfying that:  j+1 k+1 k  p E p l p ⊗ Pp, j = k l kp (Pp, j ), [2] + 1 for all k ∈ N0 , j ∈ Z, where for all k ∈ N0 ; for example,

k  2

(48)

is the minimal integer greater than or equal to k2 ,

$ % $ % 4 3 =2= . 2 2

By the cyclicity (45) of the tensor factor L p of LS p , and by the structure theorem (28) of another tensor factor S p , the above morphism E p of (48) is a well-defined bounded surjective linear transformation. Now, consider how our p-radial operator l p = c p + a p of (44) works on S p . Observe first that: if c p and a p are the p-creation, respectively, the p-annihilation on S p , then     c p a p Pp, j = Pp, j = a p c p Pp, j , for all j ∈ Z, p ∈ P, and hence c p a p = 1S p = a p c p on S p .

(49)

Lemma 1 Let c p , a p be the p-creation, respectively, the p-annihilation on S p . Then n n   cnp a np = c p a p = 1S p = a p c p = a p c p , and cnp1 a np2 = a np2 cnp1 on S p , for all n, n 1 , n 2 ∈ N0 .

(50)

Circular-Like and Circular Elements in Free Product Banach …

625



Proof The formulas in (50) hold by (49). By (50), one can get that n n    k n−k  n cpap , l np = c p + a p =

(51)

k=0

with c0p = 1S p = a 0p , for all n ∈ N, where

  n! n , ∀k ≤ n ∈ N0 . = k k!(n − k)!

Thus, one obtains the following proposition. Proposition 8 Let l p ∈ L p be the p-radial operator on S p . Then does not contain 1S p -term, and l 2m−1 p 

l 2m p

contains its 1S p

2m -term, m

(52)

 · 1S p , for all m ∈ N.

(53)

Proof The proofs of (52) and (53) are done by the straightforward computations under (50) and (51). See [3, 9] for more details. 

7.3 Weighted-Semicircular Elements Q p, j in LS p Fix p ∈ P, and let LS p = L p ⊗C S p be the tensor product Banach ∗-algebra (47), and let E p be the linear transformation (48) from LS p onto S p . Throughout this section, let (54) Q p, j = l p ⊗ Pp, j ∈ LS p , for j ∈ Z, where Pp, j are the projections (26) generating S p . Observe that n  Q np, j = l p ⊗ Pp, j = l np ⊗ Pp, j ,

(55)

for all n ∈ N, for all j ∈ Z. If Q p, j ∈ LS p is in the sense of (54) for j ∈ Z, then Ep



Q np, j



 j+1 n+1 n    p = E p l p ⊗ Pp, j =  n  l np Pp, j , +1 2

by (48) and (55), for all n ∈ N.

(56)

626

I. Cho

0 Now, for a fixed j ∈ Z, define a linear functional τ p, j on LS p by p

0 τ p, j = τ j ◦ E p on LS p , p

(57)

p

where τ j = φ(1p) ϕ j is in the sense of (35). p 0 By the bounded-linearity of both τ j and E p , the morphism τ p, j of (57) is a bounded   0 linear functional on LS p . So, the pair LS p , τ p, j forms a (non-traditional) Banach ∗-probability space. By (56) and (57), if Q p, j is in the sense of (54), then    ( p j+1 )n+1 p  n n 0 τ p, j Q p, j = [ n ]+1 τ j l p (Pp, j ) , 2

(58)

for all n ∈ N.

  0 Theorem 1 Let Q p, j = l p ⊗ Pp, j ∈ LS p , τ p, j , for a fixed j ∈ Z. Then Q p, j is   0 p 2( j+1) -semicircular in LS p , τ p, j . More precisely, one obtains that  n   2( j+1)  n2 0 n , τ p, j Q p, j = ωn c 2 p

(59) 0, p, j

for all n ∈ N, where ωn are in the sense of (42). Equivalently, if kn (...) is the free 0 cumulant on LS p in terms of the linear functional τ p, j of (58) on LS p , then ⎛ ⎞  2( j+1) if n = 2 p ⎟ 0, p, j ⎜ kn (60) ⎝ Q p, j , Q p, j , . . . , Q p, j ⎠ = 0 otherwise,    n-times

for all n ∈ N. Proof The free-moment formula (59) is obtained by (52), (53) and (58). And the free-cumulant formula (60) is obtained by (59) under the Möbius inversion of [16]. See [3, 9] for details.  The above theorem shows that the jth generating operator Q p, j in the p-adic radial projection algebra LS p is p 2( j+1) -semicircular in the Banach ∗-probability space,   0 LS p ( j) = LS p , τ p, j , for j ∈ Z. Corollary 2 Let Q p, j = l p ⊗ Pp, j ∈ LS p , for p ∈ P. then it is p 2( j+1) -semicircular in LS p ( j), for all j ∈ Z. Proof The proof is done by (41), (42), (59) and (60).



Circular-Like and Circular Elements in Free Product Banach …

627

8 Semicircularity on LS For all p ∈ P, j ∈ Z, let

  0 LS p ( j) = LS p , τ p, j

(61)

be a (non-traditional) Banach ∗-probability space of the tensor product Banach 0 ∗-algebra LS p of (47), and the linear functional τ p, j of (57). Definition 11 We call such Banach ∗-probability spaces LS p ( j) of (61), the jth p-(adic-)filtered probability (Banach-∗-)space, for all p ∈ P, j ∈ Z. Let Q p,k = l p ⊗ Pp,k be the kth generating elements of the jth p-filtered probability space LS p ( j) of (61), for all k ∈ Z, for fixed p ∈ P, j ∈ Z. Then the “ jth” generating element Q p, j of LS p ( j) is p 2( j+1) -semicircular in LS p ( j) by (59) and (60). More precisely, if {Q p,k }k∈Z are generating elements of the jth p-filtered probability space LS p ( j), then   kn0, p, j Q p,k , . . . , Q p,k =



δ j,k p 2( j+1) if n = 2 0 otherwise,

(62)

   n  n  0 2( j+1) 2 n c2 , τ p, j Q p,k = δ j,k ωn p

and

for all p ∈ P, j ∈ Z, for all n ∈ N, where  ωn =

1 if n is even 0 if n is odd,

for all n ∈ N. More precisely, the jth generating element Q p, j is p 2( j+1) -semicircular in LS p ( j), while the kth generating elements Q p,k have the zero-distribution in LS p ( j), whenever k = j in Z. For the family 

   0 LS p ( j) = LS p , τ p, j : p ∈ P, j ∈ Z

of jth p-filtered probability spaces of (61), one can define the free product Banach ∗-probability space, denote

LS =

  de f LS, τ 0 =



p∈P, j∈Z

  0 LS p , τ p, j .

as in [16, 18], with LS =



p∈P, j∈Z

LS p , and τ 0 =



p∈P, j∈Z

0 τ p, j.

(63)

628

I. Cho

Note that the Banach ∗-probability space LS of (63) is a well-determined “traditional” free probability space, as a noncommutative topological ∗-probability space (e.g., see [3]). For more about (free-probabilistic) free product of free probability spaces, see [16, 18].   Definition 12 The Banach ∗-probability space LS = LS, τ 0 of (63) is called the free Adelic filterization. Let LS be the free Adelic filterization (63). Then, by (62), we obtain a subset   Q = Q p, j = l p ⊗ Pp, j ∈ LS p ( j) p∈P, j∈Z of LS. Since each Q p, j is taken from the free blocks LS p ( j) of LS, they are p 2( j+1) semicircular elements in LS, for all p ∈ P, j ∈ Z, because    n  0 n( j+1) n c2 , τ 0 Q np, j = τ p, j Q p, j = ωn p for all n ∈ N, by (63). Furthermore, since all entries Q p, j of the family Q are taken from the mutuallydistinct free blocks {LS p ( j)} p∈P, j∈Z of LS, these entries Q p, j are mutually free from each other in the free Adelic filterization LS. Recall that a subset S = {at }t∈ of an arbitrary (topological or pure-algebraic) ∗-probability space (A, ϕ) is said to be a free family, if, for any pair (t1 , t2 ) ∈ 2 of “distinct” elements t1 and t2 of a countable index set , the corresponding free random variables at1 and at2 are free in (A, ϕ) (e.g., [16, 17]). Definition 13 Let S = {at }t∈ be a free family in an arbitrary topological ∗-probability space (A, ϕ). This family S is said to be a free semicircular family, if it is a free family, and each element at of S is semicircular, for all t ∈ . Similarly, the family S is called a free weighted-semicircular family, if it is a free family, and each element at of S is rt -semicircular for some rt ∈ C× , for t ∈ . So, by the construction (63) of the free Adelic filterization LS, we obtain the following result. Theorem 2 Let LS be the free Adelic filterization (63), and let Q = {Q p, j ∈ LS p ( j)} p∈P, j∈Z ⊂ LS,

(64)

where LS p ( j) are the jth p-filtered probability spaces, the free blocks, of LS. Then this family Q of (64) is a free weighted-semicircular family in LS. Proof Let Q be a subset (64) in LS. Then as we discussed above, all elements Q p, j of Q are p 2( j+1) -semicircular in LS, for all p ∈ P, j ∈ Z. Also, they are mutually free from each other in LS, because all entries Q p, j are contained in the mutually distinct free blocks LS p ( j) of LS, for all p ∈ P, j ∈ Z. Therefore, the family Q of (64) is a free weighted-semicircular family in LS. See [3] for more details. 

Circular-Like and Circular Elements in Free Product Banach …

629

Now, take elements 1

de f

 p, j =

p j+1

Q p, j , ∀ p ∈ P, j ∈ Z,

(65)

in LS, where Q p, j ∈ Q, where Q is the free weighted-semicircular family (64) in the free Adelic filterization LS. Then, by the self-adjointness of Q p, j , these operators  p, j of (65) are self-adjoint in LS, because p j+1 ∈ Q ⊂ R× in C× , satisfying p j+1 = p j+1 , for all p ∈ P, j ∈ Z. Also, one obtains the following free-cumulant computation; if kn0 (...) is the free cumulant on LS with respect to the linear functional τ 0 on LS, then kn0







 p, j , . . . ,  p, j =

 =

p

1

1



Q p, j , . . . , j+1 Q p, j p j+1 p n   1 kn0, p, j Q p, j , . . . , Q p, j , j+1

kn0, p, j

(66)

by the bimodule-map property of free cumulant (e.g., [16]), for all n ∈ N, where 0, p, j kn (...) mean the free cumulants (60) on the free blocks LS p ( j) with respect to 0 the linear functionals τ p, j on LS p . By the free-distributional data (66), we obtain the following result. Theorem 3 Let  p, j = p 1j+1 Q p, j ∈ LS p ( j) be free random variables (65) of the free Adelic filterization LS, for all p ∈ P, j ∈ Z. Then  p, j are semicircular in LS, and the family    =  p, j ∈ LS p ( j) : p ∈ P, j ∈ Z forms a free semicircular family in LS. Proof Consider that kn0







 p, j , . . . ,  p, j =

1 p j+1

n

  kn0, p, j Q p, j , . . . , Q p, j

by (66)

=

⎧ 2  0, p, j  1 ⎪ ⎪ Q p, j , Q p, j ⎨ p j+1 k2  ⎪ ⎪ ⎩

1 p j+1

n

0, p, j

kn



 Q p, j , . . . , Q p, j = 0

by the p 2( j+1) -semicircularity of Q p, j ∈ Q in LS.

if n = 2 otherwise,

(67)

630

I. Cho

( =

1 p j+1

0

2 

p j+1

2

=1

if n = 2 otherwise,

(68)

for all n ∈ N. By the free-cumulant computation (68), the self-adjoint free random variables  p, j ∈ LS p ( j) are semicircular in LS, for all p ∈ P, j ∈ Z. Thus, the family  of (67) forms a free family in LS, because all elements  p, j are the scalar-multiples of Q p, j ∈ Q, contained in mutually-distinct free blocks LS p ( j) of LS, for all j ∈ Z, p ∈ P. Therefore, this family  is a free semicircular family in LS. 

9 Circular Elements of LS Induced by  Let LS = (LS, τ 0 ) be our free Adelic filterization (63), and let  = { p, j ∈ LS p ( j) : p ∈ P, j ∈ Z}

(69)

be the free semicircular family (67) in LS. From the family  of (69), we construct corresponding circular elements in LS, and study their operator-theoretic properties. As we discussed in Sect. 1, the circular law, the free distributions of circular elements, is already well-studied. However, here, we compute the joint free moments of our circular elements to re-characterize the circular law from p-adic analysis over p ∈ P.

9.1 Circular Elements Let (A, ϕ) be a topological ∗-probability space, and let s1 and s2 be semicircular elements in (A, ϕ). Definition 14 Let s1 and s2 be semicircular elements in (A, ϕ), and assume that s1 and s2 are free in (A, ϕ). Then the circular element c generated by s1 and s2 is defined to be a free random variable, √ 1 c = √ (s1 + is2 ) , with i = −1 in C, 2

(70)

in (A, ϕ). Such a circular element c of (70) is R-diagonal in the sense of [1, 12–14]. So, the joint free cumulants

Circular-Like and Circular Elements in Free Product Banach …

631

 )     (r , . . . , rn ) ∈ {1, ∗}n , kn cr1 , . . . , crn  1 for all n ∈ N of c are well-known as its free-distributional data under R-diagonality, where kn (...) is the free cumulant on A in terms of the linear functional ϕ on A. Proposition 9 (See [1, 12–14]) Let c be a circular element (70) in (A, ϕ). Then the only non-vanishing “mixed” free cumulants of {c, c∗ } are k2 (c, c∗ ) = 1 = k2 (c∗ , c), where kn (...) is the free cumulant on A in terms of ϕ.

(71) 

In the above proposition, the “mixed” free cumulants mean the joint free cumulants of {c, c∗ }, always having c and c∗ together in kn (cr1 , . . . , crn ), for all mixed (r1 , . . . , rn ) in {1, ∗}, for all n ∈ N \ {1}. Recall that joint free cumulants covers not only mixed cases, but also free cumulants of c, and those of c∗ , too. For instance, one can have the following non-mixed case; k2 (c, c) = ϕ(c2 ) − ϕ(c)2  1 1  = ϕ s12 + is1 s2 + is2 s1 + s22 − (ϕ(s1 + is2 ))2 2 2  1 2 1 2 2 ϕ(s1 ) + ϕ(s2 ) − (0) = 2 2 1 = (c1 + c1 ) = 1. 2 So, in Sect. 9, we are interested in the joint free moments of such a circular element c obtained from our p-adic settings under identically-free-distributedness (See Sect. 9.3 below).

9.2 Circular Elements in LS Let LS = (LS, τ 0 ) be the free Adelic filterization, and let  be the free semicircular family (69) of LS. Recall that all entries  p, j of  are induced from the p 2( j+1) semicircular elements Q p, j in the free weighted-semicircular family Q of LS, i.e.,  p, j =

1 Q p, j , for all p ∈ P, j ∈ Z. p j+1

Take arbitrary pairs W = ( p1 , p2 ) ∈ P 2 , and J = ( j1 , j2 ) ∈ Z2 .

(72)

632

I. Cho

If either W is alternating in P, or J is alternating in Z, then two elements S1 =  p1 , j2 , S2 =  p2 , j2 ∈ 

(73)

are free in LS, by (67). Proposition 10 Let S1 and S2 be in the sense of (73) in LS, and assume that either W or J is alternating in P, respectively, in Z, where W and J are in the sense of (72). Then the element √ 1 C = √ (S1 + i S2 ) , with i = −1 in C 2

(74)

is a circular element in LS. Proof The circularity of the operator C of (74) is clear by the very definition (70) of circular elements, because the freeness of two semicircular elements S1 and S2 of (73) is guaranteed in the free Adelic filterization LS.  The above proposition shows that from our free semicircular family  of (69) one can construct infinitely many corresponding circular elements in LS. Definition 15 Let C be circular elements (74) in LS generated by the free semicircular family  of (69). Then we call such operators C, the Adelic circular elements (in LS) to emphasize how it is constructed. In this section, we concentrate on computing free moments of the Adelic circular elements C of (74). First, consider the following: Let N1

N2

k=1

l=1

T1 = Q npkk , jk and T2 = Q qml l,il

(75)

are free reduced words of the radial-Adelic probability space LS, for p1 , . . . , p N1 , q1 , . . . , q N2 ∈ P, and j1 , . . . , j N1 , i 1 , . . . , i N2 ∈ Z, and n 1 , . . . , n N1 , m 1 , . . . , m N2 ∈ N, for N1 , N2 ∈ N, where Q pk , jk , Q ql ,il ∈ Q are our weighted-semicircular elements generating LS, for all k = 1, . . . , N1 , and l = 1, . . . , N2 , where Q is our free weighted-semicircular family (64) in LS.

Circular-Like and Circular Elements in Free Product Banach …

633

Assume that either 

   p1 , . . . , p N1 , or j1 , . . . , j N1

is alternating, and either     q1 , . . . , q N2 , or i 1 , . . . , i N2 is alternating, and hence, the operators T1 and T2 of (75) form free reduced word in LS. Assume further that either p N1 = q1 in P, or j N1 = i 1 in Z. Then, we have either  or

 p1 , . . . , p N 1 , q 1 , . . . , q N 2 , 

j1 , . . . , j N1 , i 1 , . . . , i N2



is alternating in P, respectively, in Z. It shows that a free (non-reduced, or reduced) word T1 T2 becomes again a free “reduced” word in LS. (Remark that, we cannot guarantee the free word T2 T1 forms a free reduced word in LS. But, under above assumptions, T1 T2 is a free reduced word in LS.) Then one has that   nN mN τ (T1 T2 ) = τ Q np11 , j1 · · · Q p N1 , jN Q qm11,i1 · · · Q q N 2i N 1 1 2 2     n  N1 n1 0 0 = τ p1 , j1 Q p1 , j1 · · · τ p N , jN Q p N , jN 1 1 1 1     m  N2 m1 0 0 · τq1 ,i1 Q q1 ,i1 · · · τq N ,i N Q q N , jN 2

2

= τ (T1 )τ (T2 ) = τ (T2 )τ (T1 ).

2

2

(76)

Recall-and-remark-again that even though T1 T2 forms a free reduced word in LS, it is possible that T2 T1 is not a free reduced word (i.e., it can be just a free non-reduced word) in LS. In such a case, τ (T2 T1 ) = τ (T1 T2 ), which guarantees that τ (T2 T1 ) = τ (T2 )τ (T1 ), in general,

634

I. Cho

in C. So, without free-reduced-word-ness of T1 T2 , we cannot guarantee the equalities in (76), in general. Proposition 11 Let T1 and T2 be in the sense of (75) in the free Adelic filterization LS. If T1 and T2 are free reduced words, and if T1 T2 forms again a free reduced word in LS as above, then τ (T1 )τ (T2 ) = τ (T1 T2 ) = τ (T2 )τ (T1 ).

(77)

If T1 T2 is not a free reduced word in LS, then the above formula (77) does not hold in general. Proof Suppose (i) T1 and T2 are free reduced words in the sense of (75), and (ii) T1 T2 forms a new free reduced word in LS. Then, by (76), the formula (77) holds. Assume now that the condition (ii) does not hold. For instance, let T1 = Q 2p1 , j1 Q 2p2 , j2 and T2 = Q 2p2 , j2 Q 2p3 , j3 be free reduced words in LS. It is easy to check that not only T1 T2 is not a free reduced word in LS, but also, T1 T2 = Q 2p1 , j1 Q 4p2 , j2 Q 2p3 , j3 in LS, as a free reduced word. In this case, one has    2( j +1) 2( j +1) 2( j +1) 2( j +1) p2 2 p3 3 τ (T1 ) τ (T2 ) = p1 1 p2 2 2( j1 +1) 4( j2 +1) 2( j3 +1) p2 p3 ,

= p1 while

2( j1 +1)

τ (T1 T2 ) = p1



4( j2 +1)

p2

 2( j ) c2 p3 2

2( j1 +1) 4( j2 +1) 2( j3 +1) p2 p3 ,

= 2 p1 and hence,

τ (T1 T2 ) = τ (T1 )τ (T2 ). Thus, if T1 T2 is not a free reduced word in LS, then the formula (77) does not hold in general.  As a corollary of the above proposition, one obtains the following result. N1

N2

k=1

l=1

Corollary 3 Let W1 = npkk , jk and W2 = qml l,il be free random variables in LS. Assume that both W1 and W2 be free reduced words in LS, moreover, suppose a free word W1 W2 forms again a free reduced word in LS. Then τ (W1 W2 ) = τ (W1 )τ (W2 ) = τ (W2 )τ (W1 ).

(78)

Circular-Like and Circular Elements in Free Product Banach …

635

If W1 W2 is not a free reduced word in LS, then the formula (78) does not hold in general. Proof Suppose W1 and W2 are free reduced words, and W1 W2 is again a free reduced word in LS. By the very construction of our semicircular elements;  p, j =

1 Q p, j , for all p ∈ P, j ∈ Z, p j+1

where Q p, j ∈ Q are the p 2( j+1) -semicircular elements of LS; there exist r1 , r2 ∈ Q, such that τ (W1 W2 ) = r1r2 τ (T1 T2 ) = (r1 τ (T1 )) (r2 τ (T2 )) = τ (W1 ) τ (W2 ) = τ (W2 )τ (W1 ), where T1 and T2 are in the sense of (75), satisfying (77). Therefore, the formula (78) holds, whenever W1 W2 is again a free reduced word in LS. Similarly, one can verify that if W1 W2 is a free non-reduced word in LS, then the formula (78) does not hold in general, with the same arguments with the above proposition.  Now, let

 1  C = √  p1 , j1 + i p2 , j2 ∈ LS 2

be an Adelic circular element (74), where either ( p1 , p2 ) or ( j1 , j2 ) is alternating in P, respectively, in Z. Here, the condition; either ( p1 , p2 ) is alternating, or ( j1 , j2 ) is alternating (because C is circular); guarantees that p2 = p1 in P, respectively, j2 = j1 in Z. Therefore, both  p1 , j1  p2 , j2 and  p2 , j2  p1 , j1 are free reduced words with their length-2 of LS. More precisely, np11 , j1 np22 , j2 and np22 , j2 np11 , j1 are free reduced words with their lengths-2 in LS, for all n 1 , n 2 ∈ N.

(79)

636

I. Cho

Corollary 4 Let  p1 , j1 ,  p2 , j2 ∈  be our semicircular elements of LS, where either ( p1 , p2 ), or ( j1 , j2 ) is an alternating pair in P, respectively, in Z. Then       τ 0 np11 , j1 np22 , j2 = τ 0 np11 , j1 τ 0 np22 , j2 = ωn 1 ωn 2 c n21 c n22     = τ 0 np22 , j2 τ 0 np11 , j1   = τ 0 np22 , j2 np11 , j1 ,

(80)

for all n 1 , n 1 ∈ N. Proof The proof of (80) is done by (78) and (79). In particular, the second equality  of (80) holds by the semicircularity of  pl , jl ∈ , for all l = 1, 2. Let C be an Adelic circular element (74) in LS. Motivated by both (78) and (80), one can get that  n  1 n 0  τ  p1 , j1 + i p2 , j2 √ 2 ⎛ ⎞ n     n 1 = √ τ0 ⎝

Uil ⎠ l=1 2 n (i 1 ,...,i n )∈{1,2}

  τ 0 Cn =



where Uil is either  p1 , j1 , or i p2 , j2  =

1 √ 2

+ n * n     n 0 k n−k n−k τ  p1 , j1 i  p2 , j2 k k=0

by (71), (78) and (80)  =

1 √ 2

+ n * n   n  n−k  0  k  0  n−k  τ p1 , j1  p1 , j1 τ p2 , j2  p2 , j2 i k k=0

with identities:

  τ p0l , jl 0pl , jl = 1, for all l = 1, 2,

and axiomatization: ω0 = 1 then it goes to

 =

1 √ 2

+ n * n   n n−k (i )ωk ωn−k c k2 c n−k , 2 k k=0

by the semicircularity of  p1 , j1 and  p2 , j2 , for all n ∈ N.

(81)

Circular-Like and Circular Elements in Free Product Banach …

637

  Theorem 4 Let C = √12  p1 , j1 + i p2 , j2 be an Adelic circular element of the free Adelic filterization LS generated by our free semicircular family  of (69). Then +  * n     n 1 n  n n−k τ C = √ , (82) (i )ωk ωn−k c k2 c n−k 2 k 2 k=0

for all n ∈ N, with axiomatization: ω0 = 1. Proof The free-moment formula (82) is proven by (81) with help of (71), (78) and (80).  Observe now that if C is an Adelic circular element (74) in LS, then the adjoint C ∗ is  1  C ∗ = √  p1 , j1 − i p2 , j2 2 in LS, by the self-adjointness of our semicircular elements  pl , jl ∈ , for all l = 1, 2. Similar to (82), one obtains the following free-distributional data of C ∗ . Theorem 5 Let C ∗ be the adjoint of an Adelic circular element C of (74) in LS. Then +  * n     ∗ n 1 n  n  3(n−k)  ωk ωn−k c k2 c n−k τ (C ) = √ , (83) i 2 k 2 k=0

for all n ∈ N, with axiomatization: ω0 = 1. Proof The proof of (83) is similar to that of (82). Indeed, with same argument, one can have that +  * n     ∗ n  1 n  n  τ (C ) = √ , (84) (−i)n−k ωk ωn−k c k2 c n−k 2 k 2 k=0

for all n ∈ N, with axiomatization: ω0 = 1. Note that (−i)n = (−1)n i n = (i 2 )n i n = i 3n , for all n ∈ N. Therefore, in the formula (84), (−i)n−k = i 3(n−k) , for all k ≤ n ∈ N ∪ {0}. Therefore, the formula (84) is identical to (83).



By (82) and (83), we obtain the free-distributional data of C and those of C ∗ . Then how about the mixed free moments of C and C ∗ ?

638

I. Cho

By the circularity (70), if C is an Adelic circular element (74) in LS, and if kn0 (...) is the free cumulant on LS in terms of the linear functional τ 0 on LS, then the only non-vanishing mixed free cumulants are     k20 C ∗ , C = 1 = k20 C, C ∗ ,

(85)

by (71). Therefore, one can get that, for any (t1 , . . . , tn ) ∈ Nn , and (r1 , . . . , rn ) ∈ {1, ∗}n , for n ∈ N,

 

  τ 0 C r1 t1 C r2 t2 . . . C rn tn =

π∈N C(n)



kπ,|V |

V ∈π

by the Möbius inversion of [16], where   ri t 0 0 ri1 ti1 , . . . , C |V | i|V | kπ,|V | = k|V | C is a block-depending free cumulant of V in a noncrossing partition π (e.g., [16]), whenever   V = i 1 , . . . , i |V | in π, and hence, it goes to =

 π∈N C2 (n)





kπ,|V |

V ∈π

by (85), because of the circularity of C, where ⎫  ⎧  for any blocks V of π, ⎪ ⎪ ⎪  ⎪ ⎬ ⎨  |V | = 2, where |V | , N C2 (n) = π ∈ N C(n)  ⎪ ⎪  means the cardinality of V, ⎪ ⎪ ⎭ ⎩  as sets and hence =

 θ∈N C2alt (n)





kθ,|B| ,

B∈θ

by (85), where kθ,|B| is in the above sense, and

(86)

(87)

Circular-Like and Circular Elements in Free Product Banach …

⎫  ⎧  for any block B of θ, ⎪ ⎪  ⎪ ⎪ ⎬ ⎨  k = k2τ (C ∗ , C), , N C2alt (n) = θ ∈ N C2 (n)  θ,|B| or ⎪ ⎪  ⎪ ⎪ ⎭ ⎩  kθ,|B| = k τ (C, C ∗ ) 2

639

(88)

where N C2 (n) is in the sense of (86). Lemma 2 Let C be an Adelic circular element (74) in LS. Then     τ C r1 t1 C r2 t2 . . . C rn tn =  N C2alt (n) ,

(89)

for (r1 , . . . , rn ) ∈ {1, ∗}n , and (t1 , . . . , tn ) ∈ Nn , where N C2alt (n) is in the sense of (88), for all n ∈ N, where |Y | mean the cardinalities of sets Y. Proof The proof of the free-distributional data (89) of C and C ∗ is done by (87) and (88). Indeed, one obtains that   τ C r1 t1 . . . C rn tn =

 θ∈N C2alt (n)





kθ,|B| ,

B∈θ

by (87), where N C2alt (n) is a subset (88) of N C2 (n) of (86). By (85), the only non-vanishing mixed free cumulants of C and C ∗ are   k20 C ∗ , C = 1 = k20 (C, C ∗ ). Thus kθ,|B| = 1, for all B ∈ θ, for all θ ∈ N C2alt (n). Therefore, the formula (89) is equivalent to (87).



Remark that the above result (89) implies of course that if N C2alt (n) is empty, then the corresponding mixed free moment is vanishing. Therefore, we obtain the following result. Proposition 12 Let C be an Adelic circular element in LS. Then the only nonvanishing mixed free moments of C and C ∗ are either     τ (C ∗ C)n , or τ (CC ∗ )n ,

(90)

for all n ∈ N. Proof It is not difficult to verify that the only nonvanishing mixed free moments of our circular element C and its adjoint C ∗ would be   τ (C ∗ C)(C ∗ C) . . . (C ∗ C) ,

640

I. Cho

or

  τ (CC ∗ )(CC ∗ ) . . . (CC ∗ ) ,

by (89). Indeed, if not, the subset N C2alt (n) of (88) is empty in the lattice N C(n) of all noncrossing partitions over {1, . . . , n}, for n ∈ N. Clearly, if n is odd, then N C2alt (n) is empty, and even though n is even, N C2alt (n) is empty in N C2 (n), whenever there exists at least one ti0 , such that ti0 = 1, in N, or there exists at least one pair (ri0 ,  ri0 +1 ), such that ri0 = ri0 +1 in {1, ∗}. By (90), we focus on computing the non-vanishing mixed free moments     τ (C ∗ C)n , and τ (CC ∗ )n , for all n ∈ N, to determine the non-vanishing mixed free-distributional data of C and C ∗ .   Theorem 6 Let C = √12  p1 , j1 + i p2 , j2 be an Adelic circular element (74) of the free Adelic filterization LS. Then n    ∗ n    n τ (C C) = ck cn−k = τ 0 (CC ∗ )n , k 0

k=0

for all n ∈ N. Proof Let C be given as above, and let Sl =  pl , jl , for all l = 1, 2. Then

C ∗C =

1 2

=

1 2

(S1 − i S2 ) (S1 + i S2 ) 

 S12 + i S1 S2 − i S2 S1 + S22 ,

and thus,    1  τ 0 (C ∗ C)n = n τ 0 (S12 + i S1 S2 − i S2 S1 + S22 )n 2  1  = n τ 0 (S12 + S22 )n + [Rest Terms] 2 n  1  = n τ 0 S12 + S22 2 because of (80), and the evenness of semicircular elements S1 and S2 =

n       1  n τ 0 S12k τ 0 S22(n−k) n 2 k=0 k

(91)

Circular-Like and Circular Elements in Free Product Banach …

641

by (78) and (80) =

n   1  n ck cn−k 2n k=0 k

by the semicircularity of S1 and S2 , for all n ∈ N. Similarly, one can get that   n    1  = τ 0 (C ∗ C)n , τ 0 (CC ∗ )n = n τ 0 S12 + S22 2 for all n ∈ N. Therefore, the free-distributional data (91) holds true.



So, from the circularity (74) of our Adelic circular element C, we obtain the full free-distributional information (82), (83) and (91) of C by computing joint-freemoments of {C, C ∗ }. Let’s summarize the full characterization of the free-momental free distributions of Adelic circular elements.   Corollary 5 Let C = √12  p1 , j1 + i p2 , j2 be an Adelic circular element (74) of the free Adelic filterization LS generated by the free semicircular family  of (69). Then the free distribution of C is characterized by the free-moment formulas as follows; +   * n    n 1 n  n τ C = √ , ωk ωn−k (i n−k )c k2 c n−k 2 k 2 k=0 and

and

  τ (C ∗ )n =



1 √ 2

+ n * n    3(n−k)  n ωk ωn−k i c k2 c n−k , 2 k

(92)

k=0

n       1  n τ (C ∗ C)n = n ck cn−k = τ (CC ∗ )n , 2 k=0 k

for all n ∈ N, and all other mixed free moments of C and C ∗ vanish. Proof The free-distributional data (92) of {C, C ∗ } in LS are obtained by (82), (83) and (91). 

642

I. Cho

9.3 Free Distributions of Circular Elements: The Circular Law Let (Al , ϕl ) be topological ∗-probability spaces, for l = 1, 2, and let al ∈ (Al , ϕl ), for l = 1, 2. We say these two free random variables a1 and a2 are identically free-distributed, if     ϕ1 a1r1 t1 a1r2 t2 . . . a1rn tn = ϕ2 a2r1 t1 a2r2 t2 . . . a2rn tn , equivalently,

(93)

    kn1 a1r1 t1 , . . . , a1rn tn = kn2 a2r1 t1 , . . . , a2rn tn ,

for all (r1 , . . . , rn ) ∈ {1, ∗}n , and (t1 , . . . , tn ) ∈ Nn , for all n ∈ N, where knl (...) are the free cumulants on Al in terms of ϕl , for l = 1, 2. By the definition (70) of circularity, and by the universality of the semicircular law, all circular elements are identically free-distributed from each other in the sense of (93). Therefore, by our main result (92) of Sect. 9.2, we obtain the following free-momental free-distributional characterization of circular elements universally. Theorem 7 Let c = √12 (s1 + is2 ) be a circular element in an arbitrary topological ∗-probability space (A, ϕ), where s1 and s2 are two free, semicircular elements in (A, ϕ). Then the free distribution of c is characterized by +  * n     n 1 n  n ϕ c = √ , ωk ωn−k (i n−k )c k2 c n−k 2 k 2 k=0 and

and

  ϕ (c∗ )n =



1 √ 2

+ n * n   n 3(n−k) )c k2 c n−k , ωk ωn−k (i 2 k

(94)

k=0

n       1  n ck cn−k = ϕ (cc∗ )n , ϕ (c∗ c)n = n 2 k=0 k

for all n ∈ N. And all other mixed free moments of {c, c∗ } vanish. Proof The proof of the free-moment formulas in (94) is done by the identically freedistributedness of circular elements, and our Adelic circular element C of (74) in the free Adelic filterization LS. See (92). 

Circular-Like and Circular Elements in Free Product Banach …

643

10 Weighted-Circular Elements of LS Induced by Q The free distributions of Adelic circular elements C of (74) is completely characterized by (92) in the free Adelic filterization LS. And the free distributions of the Adelic circular elements re-characterize the circular law by (94) under identicallyfree-distributedness (93). So, under universality, the free-distributional data (92) and (94) may not be so interesting after all, because the circular law is already well-known. (However, the interesting point here is how we re-characterize the circular law from “Adelic circular” elements.) Thus, in the rest of this paper, we study circular-like elements of LS, instead of circular elements. In particular, we are interested in certain operators generated by the free weightedsemicircular family Q of (64), Q = {Q p, j ∈ LS p ( j) : p ∈ P, j ∈ Z}

(95)

in LS. Recall that every element Q p, j in the family Q of (95) is a p 2( j+1) -semicircular element in LS, satisfying   n  τ Q np, j = ωn p 2( j+1) 2 c n2 , equivalently, knτ







Q p, j , . . . , Q p, j =

(96)

p 2( j+1) if n = 2 0 otherwise,

for all n ∈ N, for all p ∈ P, j ∈ Z. Definition 16 Let Q be the free weighted-semicircular family (95) of LS, and let Q pl , jl ∈ Q, for l = 1, 2. Define operators X by X=/



1 2( j1 +1)

p1

2( j2 +1)

+ p2

 Q p1 , j1 + i Q p2 , j2 ,

(97)

where p1 , p2 ∈ P, and j1 , j2 ∈ Z, satisfying either ( p1 , p2 ) or ( j1 , j2 ) is alternating in P, respectively, in Z. We call this operator X of (97), the Adelic weighted-circular (in short, Adelic w-circular) element of LS generated by Q p1 , j1 , Q p2 , j2 ∈ Q. By definition, one can understand our p 2( j+1) -semicircular elements Q p, j ∈ Q as follows: (98) Q p, j = p j+1  p, j , ∀ p ∈ P, j ∈ Z, in LS, where  p, j are semicircular elements in the free semicircular family  of (69). So, by (97) and (98), one has that an Adelic w-circular element X is identified with

644

I. Cho

X =

/



1 2( j1 +1)

2( j2 +1)

p1

+ p2

/

p11

* =

j +1

2( j1 +1)

p1

2( j2 +1)

+ p2

   j +1 j +1 p11  p1 , j1 + i p22  p2 , j2

 p1 , j1 +

/

+

j +1

p22 2( j1 +1)

p1

2( j2 +1)

+ p2

(99)

 p2 , j2 ,

in LS. Notation For convenience, j +1

/

pl l 2( j +1) p1 1

+

denote

= βl , for all l = 1, 2.

2( j +1) p2 2

and

1

/

2( j1 +1)

p1

(100)

denote 2( j2 +1)

+ p2

= β0 .

Remark that these quantities β0 , β1 and β2 are contained in R+ , the subset of R consisting of all positive real numbers.  So, by using the notation (100), the equality (99) can be re-written by   X = β0 Q p1 , j1 + i Q p2 , j2 = β1  p1 , j1 + iβ2  p2 , j2 .

(101)

By (101), we also regard our Adelic w-circular elements (97) as operators generated by the free semicircular family  of (69) in LS. It means that one can use same techniques and tools used in Sect. 9 to compute free-distributional data of our Adelic w-circular elements.   Theorem 8 Let X = β0 Q p1 , j1 + i Q p2 , j2 be an Adelic w-circular element (97) in the free Adelic filterization LS = (LS, τ 0 ), induced by Q p1 , j1 = Q p2 , j2 ∈ Q, where β0 is in the sense of (100). Then τ (X ) = 0

n

  n β1k β2n−k (i n−k )ωk ωn−k c k2 c n−k k=0 2 k

n

 n n

= β0

   n n−k (i )wk,1 wk,2 , k=0 k

where

k( j1 +1)

wk,1 = ωk p1 and

  c k2 = τ 0 Q kp1 , j1 ,

(n−k)( j2 +1)

wk,2 = ωn−k p2

  n−k 0 Q c n−k = τ p2 , j2 , 2

(102)

Circular-Like and Circular Elements in Free Product Banach …

645

for k = 0, 1, . . . , n, for all n ∈ N, where βl are in the sense of (100), for all l = 0, 1, 2. Furthermore, the formula (102) can be re-stated by 

τ0 X

n



=

β0n

* n    n k=0

k

ωk ωn−k



k( j +1) (n−k)( j2 +1) p1 1 p2

+

 c k2 c n−k 2

,

(102)

for all n ∈ N. Proof Let X be an Adelic w-circular element (97) in LS. Then one can regard X as X = β1  p1 , j1 + iβ2  p2 , j2 , by (101), where βl are in the sense of (100), for all l = 1, 2. So,  n  τ 0 (X n ) = τ 0 β1  p1 , j1 + iβ2  p2 , j2 n     k  0  n−k  n τ p2 , j2 iβ2  p2 , j2 = τ p01 , j1 β1  p1 , j1 k k=0

under the same argument of the proof of (82) n    n

=

k=0

k

β1k β2n−k (i n−k )ωk ωn−k c k2 c n−k 2

by the semicircularity of  pl , jl ∈ , for l = 1, 2 =

n    n k=0

=

n    n k=0

* =

k

β0n

k

  k( j +1) β0k β0n−k (i n−k ) ωk p1 1 c k2   (n−k)( j2 +1) · ωn−k p2 c n−k 2   k( j +1) β0n (i n−k ) ωk p1 1 c k2

n    n k=0

k

  (n−k)( j2 +1) · ωn−k p2 c n−k 2   k( j +1) (i n−k ) ωk p1 1 c k2   (n−k)( j2 +1) ωn−k p2 , c n−k 2

for all n ∈ N. Therefore, the formula (102) holds.

646

I. Cho

Moreover, by (100), we have ⎛ β0n

⎞n

= ⎝/

1 2( j1 +1)

p1 and



2( j1 +1)

p1

k( j1 +1)



⎠ =

p1

2( j2 +2)

2( j1 +1)

p1

(n−k)( j2 +1)

p2

2( j +1) 2( j +1) p1 1 + p2 2

n

2

2( j2 +1)

+ p2

2( j2 +1)

p2

2( j1 +1)

+ p2

+ n2

1 p1

 k2 

2( j1 +1)

p1

β1k β2n−k = =

2( j2 +1)

+ p2

*

,

 n−k 2

2( j2 +1)

+ p2

k( j1 +1) (n−k)( j2 +1) p2 ,

= β0n p1

for all k, n ∈ N0 . Therefore, the refined formula (102) holds too, by (102).



Note that the proofs of (102) and (102) is done as above by (101). In other words, since one can regard our Adelic w-circular elements X of (97) (generated by Q) as an operator β1  p1 , j1 + β2  p2 , j2 (generated by ) in LS, it is okay to use-and-apply computing techniques and tools of Sect. 9 for the proof of (102). This processes illustrate again the connection between our weighted-semicircular elements in Q, and the corresponding semicircular elements in . Similar to (102) and (102) , we obtain the following result.   Theorem 9 Let X = β0 Q p1 , j1 + i Q p2 , j2 be an Adelic w-circular element (97) in LS, where β0 is in the sense of (100). Then τ 0 ((X ∗ )n ) = =

  n β1k β2n−k (i 3(n−k) )c k2 c n−k k=0 2 k

n

β0n

 n

   n 3(n−k) )w w (i k,1 k,2 , k=0 k

where

k( j1 +1)

wk,1 = ωk p1 and

  c k2 = τ 0 Q kp1 , j1 ,

(n−k)( j2 +1)

wk,2 = ωn−k p2

(103)

  n−k 0 Q , c n−k = τ p , j 2 2 2

for k = 0, 1, . . . , n, for all n ∈ N, where βl are in the sense of (100), for l = 0, 1, 2. Furthermore, the formula (103) can be re-stated as follows; ⎛ ⎞ n       ∗ n n k( j +1) (n−k)( j +1) n 1 2 ωk ωn−k p1 c k c n−k ⎠ , τ (X ) = β0 ⎝ p2 k 2 2 k=0

for all n ∈ N.

(103)

Circular-Like and Circular Elements in Free Product Banach …

647

Proof Let X be an Adelic w-circular element (97) in the free Adelic filterization LS. Then, by (101), X = β1  p1 , j1 + iβ2  p2 , j2 , and hence,

X ∗ = β1  p1 , j1 − iβ2  p2 , j2 ,

because βl ∈ R× , satisfying βl = βl in C, for all l = 1, 2, where βl are in the sense of (100), and  pl , jl ∈ , for l = 1, 2. So, one can get that     τ 0 (X ∗ )n = τ 0 (β1  p1 , j1 − iβ2  p2 , j2 )n n        n (−i)n−k τ p01 , j1 β1k kp1 , j1 τ p02 , j2 β2n−k n−k = p2 , j2 k k=0 n    n  3(n−k)  k n−k i β1 β2 ωk ωn−k c k2 c n−k = 2 k k=0 n      3(n−k)  k  n k( j +1) β0 ωk p1 1 c k2 = i k k=0   (n−k)( j2 +1) · ωn−k p2 , c n−k 2 for all n ∈ N. So, the free-distributional data (103) holds. By (103), the refined formula (103) is obtained by the similar proof of (102) .  Also, one can get the following joint free-momental information of our Adelic w-circular elements X of (97). Again by (101), to compute the joint free moments of X and X ∗ , one can use the same techniques to prove (91): With the similar arguments from the proofs of (85) and (86), one can verify that the only non-vanishing mixed free cumulants of X and X ∗ are   k20 X ∗ , X , and k20 (X, X ∗ ), by (101). Lemma 3 Let X be an Adelic w-circular element (97) in the free Adelic filterization LS. Then the only non-vanishing mixed free cumulants of {X, X ∗ } are   k20 X ∗ , X = 1 = k20 (X, X ∗ ), where kn0 (...) is the free cumulant on LS in terms of τ 0 . Proof Let X be in the sense of (97) in LS, i.e., X = β1  p1 , j1 + iβ2  p2 , j2 ,

(104)

648

I. Cho

where βl are in the sense of (100), for l = 1, 2. For convenience, let’s denote  pl , jl by Sl , for all l = 1, 2. Then     k20 X ∗ , X = τ 0 X ∗ X − τ 0 (X ∗ )τ 0 (X ) by the Möbius inversion of [16]

= τ 0 (X ∗ X )

by the semicircularity (or the even-ness) of the summands S1 and S2 of X   = τ 0 β12 S12 + iβ1 β2 S1 S2 − iβ2 β1 S2 S1 + β22 S22 by (102) and (103)

    = β12 τ 0 S12 + β22 τ 0 S22

by the freeness on S1 and S2 , and by the semicircularity of them = β12 c1 + β22 c1 = β12 + β22 . Remark that, by (100), one has 2( j1 +1)

β12 + β22 =

p1

2( j1 +1)

p1

2( j2 +1)

2( j2 +1)

+ p2

+

p2

2( j1 +1)

p1

2( j2 +1)

+ p2

= 1.

Similarly, one can get that     k20 X, X ∗ = τ 0 X X ∗ = β22 + β12 = 1. Therefore, the free-probabilistic information (104) holds. Meanwhile, by (71), (97) and (101), all other mixed free cumulants of {X, X ∗ } vanish.  By (104), one can use similar arguments and techniques used in Sect. 9 to compute mixed free moments of our Adelic w-circular elements and their adjoints.   Theorem 10 Let X = β0 Q p1 , j1 + i Q p2 , j2 be an Adelic w-circular element (97) of LS, where β0 is in the sense of (100). Then ∗

τ ((X X ) ) = 0

n

  n β12k β22(n−k) ck cn−k k=0 k

n

 n 2n

= β0

   n w w k,1 k,2 , k=0 k

(105)

Circular-Like and Circular Elements in Free Product Banach …

and

649

    τ 0 (X X ∗ )n = τ 0 (X ∗ X )n ,

where

2k( j1 +1)

wk,1 = p1 and

2(n−k)( j2 +1)

wk,2 = p2

  ck = τ Q 2k p1 , j1

  , cn−k = τ Q 2(n−k) p2 , j2

for all k = 0, 1, . . . , n, for all n ∈ N, where βl are in the sense of, for all l = 0, 1, 2. Meanwhile, all other mixed free moments of X and X ∗ vanish. Furthermore, the formula (105) is refined by * n   +  n  ∗ n 2k( j1 +1) 2(n−k)( j2 +1) 2n p2 ck cn−k , τ (X X ) = β0 p1 k

(105)

k=0

for all n ∈ N. Proof Let X be an Adelic w-circular element (97) of LS, satisfying that X = β1  p1 , j1 + β2  p2 , j2 , by (101), where βl are in the sense of (100), for l = 1, 2. As before, we denote  pl , jl by Sl , for all l = 1, 2. Then    n  τ 0 (X ∗ X )n = τ 0 β12 S12 + β22 S22 by (104), as in the proof of (91) =

n    n

k

k=0

τ0

 k  0  2 2 n−k  β12 S12 τ β2 S2

by (78), (80) and (101) =

n    n k=0

by the semicircularity of S1 and S2

k

β12k β22(n−k) ck cn−k

650

I. Cho

=

n    n k=0

k *

= β02n

   2k( j +1) 2(n−k)( j2 +1) cn−k β02k β02(n−k) p1 1 ck p2

n    n

k

k=0

2k( j +1) p1 1 ck



2(n−k)( j2 +1) p2 cn−k

+ 

,

for all n ∈ N. Similarly, we obtain that τ 0 ((X X ∗ )n ) =

  n β12k β22(n−k) ck cn−k k=0 k

n

 = β02n

    n 2k( j1 +1) 2(n−k)( j2 +1) p p c c , k n−k 1 2 k=0 k

n

for all n ∈ N. Therefore, we obtain the formula (105). And, by (105), the formula (105) , too. From the above three theorems, we obtain a full free-momental characterization of free distributions of our Adelic w-circular elements by (102), (103) and (105). We summarize these characterizations in the following corollary.   Corollary 6 Let X = β0 Q p1 , j1 + i Q p2 , j2 be the Adelic w-circular element induced 2( j +1) by pl l -semicircular elements Q pl , jl ∈ Q, for l = 1, 2, in the free Adelic filterization LS, where β0 is in the sense of (100). Then the free distribution of X is determined by * n   +  n  n k( j1 +1) (n−k)( j2 +1) 0 n n−k p2 c k2 c n−k τ X = β0 , (i )ωk ωn−k p1 2 k k=0

and * n   +  n  ∗ n k( j1 +1) (n−k)( j2 +1) n 3(n−k) τ (X ) = β0 )ωk ωn−k p1 p2 c k2 c n−k , (i 2 k 0

k=0

for all k = 0, 1, . . . , n, for all n ∈ N, and the only non-vanishing mixed free moments of X and X ∗ are     n n 2k( j1 +1) 2(n−k)( j2 +1) 2n 0 ∗ n p2 ck cn−k τ ((X X ) ) = β0 p1 k=0 k = τ 0 ((X X ∗ )n ) , for all k = 0, 1, . . . , n, where

Circular-Like and Circular Elements in Free Product Banach …

β0 = /

1 2( j +1) p1 1

+

2( j +1) p2 2

651

in R× ,

for all n ∈ N. Proof The free distribution of X represented by the above three formulas is obtained  by (102) , (103) and (105) .

11 From Numbers to the Circular Law In this section, we will use the same notations and terminology introduced in Sect. 10. As we have seen in Sects. 9 and 10, if X is an Adelic w-circular elements in the free Adelic filterization LS, then the free distribution of X is characterized by the weights of weighted-semicircular elements generating X. In other words, our weighted-circular laws are characterized by given primes and integers from the circular law. Now, let (106) p1 , p2 ∈ P, and j1 , j2 ∈ Z, satisfying either p1 = p2 in P, or j1 = j2 in Z. For all k ≤ n in N0 , define positive quantities ( p , p2 : j1 , j2 )

denote

qk≤n = qk≤n1 by

∈ R+ ,

(107)

k( j1 +1) (n−k)( j1 +1) p2 .

qk≤n = p1

Also, our main results (102) , (103) and (105) show the existence of certain Adelic w-circular elements from the integers (106) whose free-distributional data are determined by the quantities (107) under certain rules. Theorem 11 Let p1 , p2 ∈ P and j1 , j2 ∈ Z satisfy (106). Then there exists a free j ,j random variable X p11 , p22 (in a topological ∗-probability space), such that the free j1 , j2 j ,j distribution of X p1 , p2 is determined by the nth free moments of X p11 , p22 , * n   +  n n n−k β0 ; ωk ωn−k (i )qk≤n c k2 c n−k 2 k k=0

∗  j ,j and the nth free moments of the adjoint X p11 , p22 ,

652

I. Cho

β0n

* n    n k=0

k

+ ωk ωn−k (i

3(n−k)

)qk≤n c k2 c n−k 2

;

∗  j ,j j ,j and the only non-vanishing mixed free moments of X p11 , p12 and X p11 , p22 , * n   +  n   2n 2 qk≤n ck cn−k , β0 k k=0

where β0 = /

1 2( j +1) p1 1

+

2( j +1) p2 2

in R× ,

and qk≤n are in the sense of (107) for all k ≤ n ∈ N0 . Proof Now, let p1 , p2 ∈ P, and j1 , j2 ∈ Z, satisfying the condition (106), and let (p ,p : j , j ) qk≤n = qk≤n1 2 1 2 be in the sense of (107), for all k ≤ n ∈ N0 . For the given pl and jl , one can construct the semicircular elements Sl =  pl , jl ∈  in the free Adelic filterization LS. From these, we define the corresponding free random variable X in LS by   j +1 j +1 (108) X = β0 p11 S1 + i p22 S2 ∈ LS. Then, from the very construction, this free random variable X is our Adelic wcircular element in LS, whose free distribution is fully determined by (102) , (103) , and (105) . j ,j Therefore, if we take X p11 , p22 as the Adelic w-circular element X of (108), then this is the very free random variable induced from such p1 , p2 , j1 and j2 satisfying the above free-distributional data.  The above theorem shows that, whenever two primes and two integers, satisfying the condition (106), are taken, one can have the corresponding weighted-circular laws. The following corollary is a direct consequences of the above theorem. Corollary 7 Let p1 = p2 ∈ P. Then there exists a free random variable X p1 , p2 (in a topological ∗-probability space) such that the free distribution of X p1 , p2 is the circular law. Proof Suppose p1 and p2 are two distinct primes in P. Then one can obtain the weighted-semicircular elements Sl = Q pl ,−1 ∈ Q, for all l = 1, 2, in the free Adelic filterization LS. Since p1 = p2 in P, the operators S1 and S2 are free from each other. Thus, one can have the corresponding Adelic w-circular element,

Circular-Like and Circular Elements in Free Product Banach …

653

 1  X = β0 (S1 + i S2 ) = √  p1 ,−1 + i p2 ,−1 ∈ LS, 2   because Q pl ,−1 are pl2(−1+1) = 1 -semicircular in LS, and hence, it is semicircular in LS, satisfying Q pl ,−1 =  pl ,−1 in LS, for all l = 1, 2. Equivalently, this operator X becomes an Adelic circular element in LS, whose free distribution is nothing but the very circular law.  The above corollary shows that, whenever two distinct primes are given, one can establish the corresponding circular law.

References 1. Boedihardjo, M., Dykemma, K.: On Algebra-Valued R-Diagonal Elements (2016). Preprint arXiv:1512.06321v2 2. Cho, I.: Representations and corresponding operators induced by Hecke algebras. Complex Anal. Oper. Theo. 10(3), 437–477 (2016) 3. Cho, I.: Free semicircular families in free product Banach ∗ -algebras induced by p-adic number fields over primes p. Complex Anal. Oper. Theory 507–565 (2017) 4. Cho, I.: On dynamical systems induced by p -adic number fields. Opusc. Math. 35(4), 445–484 (2015) 5. Cho, I.: Free distributional data of arithmetic functions and corresponding generating functions. Complex Anal. Oper. Theory 8(2), 537–570 (2014) 6. Cho, I.: Dynamical systems on arithmetic functions determined by prims. Banach J. Math. Anal. 9(1), 173–215 (2015) 7. Cho, I., Gillespie, T.: Free probability on the Hecke algebra. Complex Anal. Oper. Theory 9(7), 1491–1531 (2015) 8. Cho, I., Jorgensen, P.E.T.: Krein-space operators induced by Dirichlet characters. Special issues: Contemporary Mathematics: Commutative and Noncommutative Harmonic Analysis and Applications, pp. 3–33. American Mathematical Society (2014) 9. Cho, I., Jorgensen, P.E.T.: Semicircular elements induced by p -adic number fields. Opuscu. Math. (2017) (to appear) 10. Gillespie, T.: Superposition of Zeroes of automorphic l -functions and functoriality. Ph.D. thesis, University of Iowa (2010) 11. Gillespie, T.: Prime number theorems for Rankin–Selberg L functions over number fields. Sci. China Math. 54(1), 35–46 (2011) 12. Kemp, T., Speicher, R.: Strong Haagerup inequalities for free R-diagonal elements. J. Funct. Anal. 251(1), 141–173 (2007) 13. Larsen, F.: Powers of R-diagonal elements. J. Oper. Theory 47(1), 197–212 (2002) 14. Nica, A., Shlyakhtenko, D., Speicher, R.: R-diagonal elements and freeness with amalgamation. Canad. J. Math. 53, 355–381 (2001) 15. Radulescu, F.: Random matrices, amalgamated free products and subfactors of the C ∗ -algebra of a free group of nonsingular index. Invent. Math. 115, 347–389 (1994) 16. Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Am. Math. Soc. Mem. 132(627) (1998)

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17. Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-Adic Analysis and Mathematical Physics. Series on Soviet & East European Mathematics, vol. 1. World Scientific, Singapore (1994). ISBN: 978-981-02-0880-6 18. Voiculescu, D., Dykemma, K., Nica, A.: Free Random Variables. CRM Monograph Series, vol. 1 (1992)

On Statistical Deferred Weighted B-Convergence S. K. Paikray and Hemen Dutta

Abstract This chapter consists of five sections. The first section is introductory in which a brief description of the development of the topic has been presented. In the second section some basic definitions related to deferred weighted B-mean method together with the regularity condition has been discussed. Moreover, based on the regular methods a theorem has been proved showing the relation between convergence and summability via our proposed summability mean. In the third section, based on our proposed method, we have established a Korovkin-type approximation theorem for the functions of two variables defined on a Banach space and presented an illustrative example to show that our method is a non-trivial extension of some traditional and statistical versions of certain approximation theorems which were demonstrated in earlier works. In section four, we have established another result for the rate of the deferred weighted B-statistical convergence for the same set of functions via the modulus of continuity. Finally, in the concluding section, we have considered a number of fascinating special cases (remarks) and observations in support of our definitions and of the outcomes presented in this chapter. Keywords Statistical convergence · Statistical deferred weighted B-summability · Deferred weighted B-statistical convergence · Rate of convergence · Korovkin-type approximation theorems · Positive linear operators and Banach space 2010 Mathematics Subject Classication Primary: 40A05 · 41A36 · Secondary: 40G15

S. K. Paikray (B) Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India e-mail: [email protected] H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_20

655

656

S. K. Paikray and H. Dutta

1 Introduction The family of convergent infinite series is a very small part of the set of all infinite series. With the rapid growth of mathematical analysis, it was observed that many series diverges in the Cauchy’s sense behaved regularly with regard to algebraic operations very much in the same way as the convergent ones. At this stage it was thought that, however obvious and natural the Cauchy’s definition of assigning a sum to an infinite series may be, a definition of this kind in all circumstances must be considered as an arbitrary one, and it might be replaced by quite different definitions. During the nineteenth century, many mathematicians like Abel, Borel, Cesàro, Euler, Haudörff, Hölder, Riesz, Nörlund etc. developed new methods to associate sum in a reasonable way to divergent series of slowly oscillatory type. In 1931, Agnew [1] introduced the preliminary idea of deferred Cesàro summability mean and later, he also proposed the deferred Nörlund mean. The idea of statistical convergence was introduced and studied by Fast [7] and Steinhaus [27]. Recently, statistical convergence has been an energetic research area due essentially to the fact that it is more general than classical convergence and such theory is investigated in the study in the areas of (for instance) Fourier Analysis, Number Theory, Functional Analysis and Approximation Theory. For more details, see the recent works [2, 5, 6, 9, 17, 21, 23–25]. Let K ⊆ N and also let K(n) = {k : k  n

k ∈ K}

and

and suppose that |K(n)| be the cardinality of K(n). Then the natural density of K is defined by 1 |{k : k  n n→∞ n

d(K) = lim

and

k ∈ K}|,

provided that the limit exists. A given sequence (xn ) is statistically convergent (or stat-convergent) to  if, for each  > 0, K = {k : k ∈ N

and

|xk − |  }

has zero natural density (see [7, 27]). That is, for each  > 0, we have d(K ) = lim

n→∞

1 |{k : k  n n

and

Here, we write stat lim xn = . n→∞

|xk − |  }| = 0.

On Statistical Deferred Weighted B-Convergence

657

We present below an example to illustrate that every convergent sequence is statistically convergent but the converse is not true. Example 1 Let x = (xn ) be a sequence defined by xn =

⎧ ⎨

(n = m 2 ; m ∈ N)

1 2

⎩ n 3 −1

(otherwise).

n 3 +1

Here, the sequence (xn ) is statistically convergent to 1 even if it is not classically convergent. In 2009, Karakaya and Chishti [11], introduced the fundamental concept of weighted statistical convergence and later the definition was modified by Mursaleen et al. (see [18]). Suppose that ( pk ) be a sequence of nonnegative numbers such that Pn =

n 

pk ( p0 > 0; n ∈ N).

k=0

Then, upon setting tn =

n 1  pk x k Pn k=0

(n ∈ N0 := N ∪ {0}),

the given sequence (xn ) is weighted statistically convergent (or stat N¯ -convergent) to a number  if, for each  > 0, the following set: {k : k  Pn

pk |xk − |  }

and

has zero weighted density (see [18]). That is, for each  > 0, lim

n→∞

1 |{k : k  Pn Pn

and

pk |xk − |  }| = 0.

Here, we write stat N¯ lim xn = . In the similar lines, replacing tn by tn with tn =

n 1  pn−k xk Pn k=0

(n ∈ N0 := N ∪ {0}),

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S. K. Paikray and H. Dutta

it becomes statistical Nörlund mean or weighted statistical mean and accordingly the above definition reduces to statistically Nörlund convergent or simply weighted statistically convergent. In the year 2013, Belen and Mohiuddine [2] established a new technique for weighted statistical convergence in terms of the de la Vallée Poussin mean and it was subsequently investigated further by Braha et al. [6] as the n -weighted statistical convergence. Very recently, a certain class of Nörlund type equi-statistical convergence and associated Korovkin-type approximation theorems involving algebraic functions have been introduced by Srivastava et al. (for details, see [23]). Suppose X and Y are two sequence spaces and let A = (an,k ) be a non-negative regular matrix. If for every xk ∈ X the series, An x =

∞ 

an,k xk ,

k=1

converges for all n ∈ N and the sequence (An x) ∈ Y , then we say that the matrix A maps X into Y . By the symbol (X, Y ), we denote the set of all matrices which map X into Y . Next, as regards to regularity condition, a matrix A is said to be regular, if lim An x = 

n→∞

whenever

lim xk = .

k→∞

We recall here, the well-known Silverman–Toeplitz theorem (see details [4]), which asserts that A = (an,k ) is regular if and only if the following conditions hold true: ∞  |an,k | < ∞; (i) sup n→∞

k=1

(ii) lim an,k = 0 for each k; n→∞

(iii) lim

n→∞

∞ 

an,k = 1.

k=1

Freedman and Sember [8] extended the definition of statistical convergence by considering the non-negative regular matrix A = (an,k ), which he called it A-statistical convergence. Let for any non-negative regular matrix A, we say that a sequence (xn ) is said to be A-statistically convergent (or statA -convergent) to a number  if, for each  > 0, dA (K ) = 0. That is, for each  > 0, we have lim

n→∞

 k:|xk −|

an,k = 0.

On Statistical Deferred Weighted B-Convergence

659

Here, we write statA lim xn = . Subsequently, the concept of A-statistical convergence was extended by Kolk [12] to B-statistical convergence with respect to FB -convergence (or B-summable) due to Steiglitz (see [26]). Suppose that B = (Bi ) be a sequence of infinite matrices with Bi = (bn,k (i)). Then a sequence (xn ) is said to be B-summable to the value B − lim(xn ), if lim (Bi x)n = lim

n→∞

n→∞

∞ 

bn,k (i)(x)k = B lim (xn ) uniformly for i n→∞

k=0

(n, i ∈ N0 := N ∪ {0}).

The method (Bi ) is regular if and only if the following conditions hold true (see, for details, [3, 26]): (i) B = sup

∞ 

|bn,k (i)| < ∞;

n,i→∞ k=0

(ii) lim bn,k (i) = 0 uniformly in i, n→∞

(iii) lim

n→∞

∞ 

for each k ∈ N;

bn,k (i) = 1 uniformly in i.

k=0

Let K = {ki } ⊂ N (ki < ki+1 ) for all i. The B-density of K is defined by dB (K) = lim

n→∞

∞ 

bn,k (i) = b uniformly in i,

k=0

provided the limit exists. Let R+ be the set of all regular methods B with bn,k (i)  0 (∀ n, k, i). Also, let B ∈ R+ , then we say that a sequence (xn ) is B-statistically convergent (or statB convergent) to a number  if, for every  > 0, we have dB (K ) = 0. That is, for each  > 0, we have,  lim n→∞

bn,k (i) = 0 uniformly in i.

k:|xk −|

Here, we write statB lim xn = .

660

S. K. Paikray and H. Dutta

Quite recently, Mohiuddine [16] introduced the notion of weighted A-summability by using a weighted regular summability matrix. He also provided the definitions of statistically weighted A-summability and weighted A-statistical convergence. In particular, he proved a Korovkin type approximation theorem under the consideration of statistically weighted A-summable sequences of real or complex numbers. Subsequently, Kadak et al. [10] has investigated the statistical weighted B-summability by using a weighted regular matrix to establish some approximation theorems. Here, we introduce the notion of (presumably new) deferred weighted B-statistical convergence to establish certain new approximation results.

2 Definitions and Regular Methods In this section, we introduce some definitions related to our proposed study. Also, we present here certain inclusion relations with regard to regular methods. Let (an ) and (bn ) be sequences of non-negative integers satisfying the following conditions: (a) an < bn (n ∈ N) and (b) lim bn = ∞. n→∞

The above conditions (a) and (b) are known as the regularity conditions for the deferred Nörlund mean (see Agnew [1]). We next suppose that ( pn ) and (qn ) are the sequences of non-negative real numbers such that bn 

Pn =

pm .

m=an +1

Before presenting our definitions, we recall the deferred Nörlund mean (σn ) σn =

bn 1  pb −m xm . Pn m=a +1 n n

A given sequence (xn ) is said to be deferred Nörlund summable (or c D(N , p) summable) to  if, lim σn = .

n→∞

Here, we write

c D(N , p) lim xn = . n→∞

On Statistical Deferred Weighted B-Convergence

661

We denote the set of all sequences that are deferred Nörlund summable by c D(N , p) . Definition 1 A sequence (xn ) is deferred Nörlund B-summable (or deferred weighted B-summable) to , if the B-transform of (xn ) is deferred weighted summable to the same number , that is, lim Bn(an ,bn ) (x) =

n→∞

bn ∞  1  pb −m bm,k (i)xk =  uniformly in i. Pn m=a +1 k=1 n

(1)

n

Here, we write [D(N )B ; p] lim xn = . n→∞

We denote the set of all sequences that are deferred Nörlund B-summable by [D(N )B ; p]. Remark 1 If, pbn −m = 1, and B = I, then the Bn(an ,bn ) (x) mean reduces to the deferred Cesàro mean (see [9]). Definition 2 Let B = (bn,k (i)) and let (an ) and (bn ) be sequences of non-negative integers. The matrix B = (Bi ) is said to be a deferred Nörlund regular matrix (or deferred weighted regular method) if, Bx ∈ c D(N , p) (∀ xn ∈ c) with

c D(N , p) lim Bi xn = B lim(xn )

and we denote it by B ∈ (c : c D(N , p) ). This means that Bn(an ,bn ) (x) exists for each n ∈ N, xn ∈ c and Bn(an ,bn ) (x) →  whenever xn → . We denote R+D(w) as the set of all deferred Nörlund regular matrices (methods). As a characterization of the deferred Nörlund regular methods, we present the following theorem. Theorem 1 Let B = (bn,k (i)) be a sequence of infinite  matrices, and let (an ) and (bn ) be sequences of non-negative integers. Then B ∈ c : c D(N , p) if and only if ∞  1 sup n,i k=1 Pn

lim

n→∞

 b  n      pbn −m bm,k (i) < ∞;   

(2)

m=an +1

bn 1  pb −m bm,k (i) = 0 uniformly in i (for each k, i ∈ N) Pn m=a +1 n n

(3)

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S. K. Paikray and H. Dutta

and lim

n→∞

bn ∞  1  pb −m bm,k (i) = 1 uniformly in i. Pn m=a +1 k=1 n

(4)

n

Proof Assume that (2)–(4) hold true and that xn →  (n → ∞). Then for each  > 0, there exists m 0 ∈ N such that |xn − |   (n > m 0 ). Thus, we have     bn ∞     1 |Bn(an ,bn ) (x) − | =  pbn −m bm,k (i)xk −  P  n m=an +1 k=1   ⎛ ⎞   bn bn ∞ ∞      1  1 pbn −m bm,k (i)(xk − ) +  ⎝ pbn −m bm,k (i) − 1⎠ =  Pn   Pn m=an +1 k=1 m=an +1 k=1         bn bn ∞ ∞      1   1     pbn −m bm,k (i)(xk − ) + ||  pmbn −m bm,k (i) − 1  P P    n m=an +1 k=1  n m=an +1 k=1         bn−2 bn bn ∞     1    1  pbn −m bm,k (i)(xk − ) +  pbn −m bm,k (i)(xk − 1)   P P n n     m=an +1 k=1 m=an +1 k=bn−1     b ∞ n    1  pbn −m bm,k (i) − 1 +||    Pn m=an +1 k=1

 1 Pn

bn 

bn−2

 sup |xk − | k

   1 +||   Pn

k=1

pbn −m bm,k (i) + 

m=an +1

bn ∞   m=an +1 k=1

   pbn −m bm,k (i) − 1 . 

1 Pn

bn ∞  

pbn −m bm,k (i)

m=an +1 k=1

Taking n → ∞ and using (3) and (4), we get   bn ∞  1      pbn −m bm,k xk −   ,   Pn  m=an +1 k=1

which implies that lim

n→∞

bn ∞  1  pb −m bm,k (i)xk =  = lim (xn ) uniformly in i (i  0), n→∞ Pn m=a +1 k=1 n n

since  > 0 is arbitrary. Conversely, let B ∈ (c : c D(N , p) ) and xn ∈ c. Then, since Bx exists, we have the inclusion:

On Statistical Deferred Weighted B-Convergence

663

(c : c D(N , p) ) ⊂ (c : ∞ ). Clearly, there exists a constant M such that   bn ∞  1      pbn −m bm,k (i)  M (∀ m, n, i)   Pn  m=an +1 k=1

and the corresponding series,   bn ∞  1      pbn −m bm,k (i)   Pn  m=an +1 k=1

converges uniformly in i for each n. Therefore, (2) is valid. We now consider the sequence x (n) = (xk(n) ) ∈ c0 defined by xk(n) =

⎧ ⎨1 ⎩

0

(n = k) (n = k).

for all n ∈ N and y = (yn ) = (1, 1, 1, ...) ∈ c. Then, since Bx (n) and By are belong  to c D(N , p) , thus (3) and (4) are fairly obvious. Next for statistical version, we present below the following definitions. Definition 3 Let B ∈ R+D(w) , and let (an ) and (bn ) be sequences of non-negative integers and also let K = (ki ) ⊂ N (ki ≤ ki+1 ) for all i. Then the deferred Nörlund B-density (or deferred weighted B-density) of K is defined by

B d D(N ) (K) = lim

n→∞

bn  1  pb −m bm,k (i) uniformly in i, Pn m=a +1 k∈K n n

provided that the limit exists. A sequence (xn ) is said to be deferred Nörlund B-statistical convergent (or deferred weighted B-statistical convergent) to a number  if, for each  > 0, we have B d D(N ) (K ) = 0 uniformly in i,

where K = {k : k ∈ N and |xk − |  }. Here, we write

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S. K. Paikray and H. Dutta

statBD(N ) lim (xn ) = . n→∞

Definition 4 Let B ∈ R+D(w) , and let (an ) and (bn ) be sequences of non-negative integers. A sequence (xn ) is said to be statistically deferred Nörlund B-summable (or statistically deferred weighted B-summable) to a number  if, for each  > 0, we have d(E  ) = 0 uniformly in i. Here, we write stat D(N ) lim (xn ) =  or stat lim Bn(an ,bn ) x = . n→∞

n→∞

We now prove the following theorem which determines a inclusion relation between the deferred Nörlund B-statistically convergence and the statistically deferred Nörlund B-summability. Theorem 2 Let a sequence (xn ) is deferred Nörlund B-statistically convergent to a number , then it is statistically deferred Nörlund B-summable to the same number , but the converse is not true. Proof Let (xn ) be deferred Nörlund B-statistical convergent to , we have B d D(N ) (K ) = 0 uniformly in i,

where K = {k : k ∈ N and |xk − |  }. Thus, we have     bn ∞     1     (an ,bn )  (xn ) −  =  pbn −m bm,k (i)(xk − ) Bn P  n m=an +1 k=1          b bn ∞ ∞ n   1    1        pbn −m bm,k (i) (xk − ) + ||  pbn −m bm,k (i) − 1  Pn m=an +1 k=1  Pn m=an +1 k=1       bn   1     pm bm,k (i) (xk − )  Pn m=an +1 k∈K          bn bn ∞ ∞    1    1   +  pm bm,k (i) (xk − ) +  pbn −m bm,k (i) − 1 P P  n m=an +1 k ∈/ K   n m=an +1 k=1   sup |xk − | k

1  Pn

bn 

k∈K m=an +1

pbn −m bm,k (i) + 

1 Pn

bn 



m=an +1 k ∈ / K

pbn −m bm,k (i)

On Statistical Deferred Weighted B-Convergence    1 +||   Pn

665

   pbn −m bm,k (i) − 1 → 0 (n → ∞),  m=an +1 k∈K bn 



which implies that Bn(an ,bn ) (xn ) →  (n → ∞). Thus, the sequence (xn ) is deferred Nörlund B-summable to the number  and hence the sequence (xn ) is statistically deferred Nörlund B-summable to the same number .  In order to prove that the converse is not true, we consider an Example (below). Example 2 Let us consider the sequence of infinite matrices B = (Bi ) = (bn,k (i)) given by (see [10])

1 (i  k  i + n) n+1 Bi = 0 (otherwise). We suppose also that an = 2n − 1, bn = 4n − 1 and pn = 1. It can be easily seen that B ∈ R+D(w) . We also consider the sequence (xn ) by

xn =

0 1

(n is even) (n is odd).

(5)

Since Pn = 2n, we get bn 4n−1 i+n 4n−1 ∞  1  1  1  1  1 → 0. pbn −m bm,k (i)xk = xk = Pn m=a +1 k=1 2n m=2n n + 1 k=i 2n m=2n 2 n

Clearly, the sequence (xn ) is neither convergent nor statistically convergent. However, the sequence (xn ) is deferred Nörlund B-summable to 0, so it is statistically deferred Nörlund B-summable to 0, but the sequence (xn ) is not deferred Nörlund B-statistically convergent.

3 A Korovkin-Type Theorem A few researchers worked so far toward extending or generalizing the Korovkin-type approximation theorems in many ways based on several alternative aspects, including (for example) function spaces, abstract Banach lattices, Banach algebras, and Banach spaces. This theory is significantly valuable in Real Analysis, Functional Analysis, Harmonic Analysis, Measure Theory, Probability Theory, Summability Theory, Partial Differential Equations, and so on. But the primary applications of

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the Korovkin-type approximation theorems are concerned with constructive approximation theory which uses such results as a significant tool. Indeed, even today, the improvement of Korovkin-type approximation theory is a long way from being completed satisfactorily. For further details and results related to the Korovkin-type approximation theorems and other related developments, we choose to refer the interested reader to the recent works [5, 6, 20, 22, 23]. The main objective of this section is to extend the notion of statistical convergence by the help of the deferred Nörlund regular technique and to show, how this technique leads to a number of results based upon approximation of functions of two variables over a Banach space C(D). Moreover, we establish some important approximation theorems related to the statistical deferred Nörlund B-summability and deferred Nörlund B-statistical convergence, which will effectively extend and improve most (if not all) of the existing results depending upon the choice of sequences of the deferred Nörlund B mean. Based upon the proposed methodology and techniques, we intend to estimate the rate of convergence and investigate the Korovkin-type approximation results. In fact, we extend here the result of Kadak et al. [10] by using the notion of the statistical deferred Nörlund B-summability and establish the following theorem. Let D be any compact subset of the real two-dimensional space. We denote by C(D), the space of all continuous real valued functions on D defined by   1 D = (x, y) ∈ R2 : x ∈ [0, A], y ∈ [0, A − x], 0 < A  3 and equipped with norm:  f C(D) = sup{| f (x, y)| : (x, y) ∈ D}, f ∈ C(D). Let T : C(D) → C(D) be a positive linear operator (that is, f  0 implies T ( f )  0). Also, we use the notation T ( f ; x, y) for the values of T ( f ) at the a point (x, y) ∈ D. Theorem 3 Let B ∈ R+D(w) , and let (an ) and (bn ) be a sequences of non-negative integers. Let Tn (n ∈ N) be a sequence of positive linear operators from C(D) into itself and let f ∈ C(D). Then stat D( N¯ ) lim Tn ( f (s, t); x, y) − f (x, y)C(D) = 0, n

f ∈ C(D)

(6)

if and only if stat D( N¯ ) lim Tn ( f i (s, t); x, y) − f (x, y)C(D) = 0, (i = 0, 1, 2, 3), n

where f 0 (s, t) = 1,

f 1 (s, t) =

s , 1−s−t

f 2 (s, t) =

t 1−s−t

(7)

On Statistical Deferred Weighted B-Convergence

 f 3 (s, t) =

and

667

2

s 1−s−t

 +

t 1−s−t

2 .

Proof Since each of the functions f i (s, t) ∈ C(D), (i = 0, 1, 2, 3) the following implication: (3.1) =⇒ (3.2) is fairly obvious. In order to complete the proof of the Theorem 3, we first assume that (7) hold true. Let f ∈ C(D), ∀(x, y) ∈ D. Since f (x, y) is bounded on D, then there exists a constant K > 0 such that | f (x, y)|  K

(∀ x, y ∈ D),

which implies that | f (s, t) − f (x, y)|  2K

(s, t, x, y ∈ D).

(8)

Clearly, f is a continuous function on D, for a given  > 0, there exists δ = δ() > 0 such that | f (s, t) − f (x, y)| <  whenever     s x  0, we choose  > 0, such that 0 <  < r . Then, upon setting Pn = |{n : n  N

and

|Ln ( f (s, t); x, y) − f (x, y)|  r }|

and   Pi,n =  n : n  N

and

 r −   (i = 0, 1, 2, 3). |Ln ( f i (s, t); x, y) − f i (x, y)|  4N 

On Statistical Deferred Weighted B-Convergence

669

We easily find from (12) that, Pn 

3 

Pi,n .

i=0

Thus, we have  Pi,n C(D) Pn C(D)  . n n i=0 3

(13)

Finally, by using the above assumption about the implication in (7) and Definition 4, the right-hand side of (13) is seen to tend to zero (n → ∞). Consequently, we get stat D(N ) lim Tn ( f (s, t); x, y) − f (x, y)C(D) = 0. n→∞

Hence, the implication (6) holds true. The proof of Theorem 3 is thus completed.  Remark 2 If we put, B = I and pbn −m = 1 (∀ n) in our Theorem 3, then we obtain statistical deferred Cesàro-summability version of Korovkin type approximation theorem (see [9]). Next, we recall the generating function of bivariate non-tensor type Meyer-König and Zeller operators of the Bernstein power series of two variables (see [14, 15, 19]). Let us consider the sequence of bivariate non-tensor generalized linear positive operators as follows: Ln ( f (s, t); x, y) =

∞ p n (s, t)x k y l ∞   k,l k=0 l=0

(s, t; x, y)

 f

ck,l,n ak,l,n , ak,l,n + ck,l,n + qn ak,l,n + ck,l,n + qn



(14) n (s, t)  0 ∀ (s, t) ∈ D, and f ∈ C(D). Based on the double indexed funcwhere pk,l n ), the generating function n (s, t; x, y) is given by tion sequence ( pk,l

n (s, t; x, y) =

∞  ∞ 

n pk,l (s, t)x k y l (∀ (s, t) ∈ D).

k=0 l=0

We also suppose that the following conditions hold true:

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S. K. Paikray and H. Dutta

(i) (ii) (iii) (iv)

(s, t; x, y) = (1 − x − y)n+1 (s, t; x, y); n+1 n+1 n n (s, t) = qn+1 pk,l (s, t) and ak,l+1,n pk,l+1 (s, t) = qn+1 pk,l (s, t); ak+1,l,n pk+1,l qn+1 qn → ∞, qn → 1 and qn = 0; ak+1,l,n − ak,l,n+1 = φn and ck,l+1,n − ck,l,n+1 = ψn ,

where φn  u  ∞, ψn  v  ∞ and a0,l,n = ck,0,n = 0. The power series that defines Ln given by ([10]) may not converges. So we will consider the subspace ξL = { f ∈ C(D) : ∀ (x, y) ∈ D, Ln ( f (s, t); x, y) < ∞} and we write Ln : ξL → C(D) as a positive linear operator on ξL . We also observed that Ln ( f 0 ; x, y) = 1, Ln ( f 1 ; x, y) =

x y qn+1 qn+1 , Ln ( f 2 ; x, y) = qn 1 − x − y qn 1 − x − y

and Ln ( f 3 ; x, y) =

qn+1 qn+2 qn2

qn+1 ψn x 2 + y2 x qn+1 φn y y + + + . 1 − x − y qn2 1−x −y (1 − x − y)2 qn2 (1 − x − y)

Moreover, for the effect of the test functions f i (i = 0, 1, 2, 3) and the motivations of using this Korovkin set, we have, for the nodes given by s=

ck,l,n ak,l,n and t = , ak,l,n + ck,l,n + qn ak,l,n + ck,l,n + qn

the denominators of f 1 (s, t) =

an,k,l ck,l,n and f 2 (s, t) = qn qn

are independent of k and l, respectively. Example 3 Let Tn : ξL → C(D) be a positive linear operators defined by Tn ( f ; x, y) = (1 + xn )Ln ( f ; x, y),

(15)

where (xn ) be a sequence defined as in Example 2. It is clear that the sequence (Tn ) satisfies the conditions (7) of our Theorem 3, thus we obtain:

On Statistical Deferred Weighted B-Convergence

671

stat D(N ) lim Tn (1; x, y) − 1C(D) = 0, n       s x   ; x, y − stat D(N ) lim Tn = 0, n 1−s−t 1 − x − y C(D)       t y   ; x, y − stat D(N ) lim Tn =0 n 1−s−t 1 − x − y C(D) and     2  2 2  2    s t x y   stat D(N ) lim Tn + ; x, y − +  n   1−s−t 1−s−t 1−x −y 1−x −y

= 0.

C(D )

Therefore, from Theorem 3, we have stat D(N ) lim Tn ( f (s, t); x, y) − f (x, y)C(D) = 0, n

f ∈ C(D).

However, since (xn ) is not statistical weighted B-summable, so the result of Kadak et al. ([10], p. 85 Theorem 3) does not hold true for our operators defined by (15). Moreover, since (xn ) is statistical deferred weighted B-summable, therefore, we conclude that our Theorem 3 works for the operators which we consider here.

4 Rate of the Deferred Weighted B-Statistical Convergence In this section, we compute the rate of the deferred weighted (Nörlund) B-statistical convergence of a sequence of positive linear operators of functions of two variables defined on C(D) into itself with the help of the modulus of continuity. We first present the following definition. Definition 5 Let B ∈ R+D(w) , (an ) and (bn ) be sequences of non-negative integers. Also let (u n ) be a positive non-decreasing sequence. We say that a sequence (xn ) is deferred Nörlund B-statistical convergent to a number  with the rate o(u n ) if, for each  > 0, we have bn   1 pbn −m bm,k (i) = 0 uniformly in i, n→∞ u n Pn m=a +1 k∈K

lim

n



where K = {k : k ∈ N and |xk − |  }. Here, we write xn −  = statBD(N ) − o(u n ).

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We now state and prove a Lemma as follows. Lemma 1 Let (u n ) and (vn ) be two positive non-decreasing sequences. Assume that B ∈ R+D(w) , (an ) and (bn ) be sequences of non-negative integers, and let x = (xn ) and y = (yn ) be two sequences such that xn − 1 = statBD(N ) − o(u n ) and yn − 2 = statBD(N ) − o(vn ). Then each of the following assertions hold true: (i) (xn − 1 ) ± (yn − 2 ) = statBD(N ) − o(wn ); (ii) (xn − 1 )(yn − 2 ) = statBD(N ) − o(u n vn ); (iii) γ(xn − 1 ) = statBD(N ) − o(u n ) (for any scalar γ); √ (iv) |xn − 1 | = statBD(N ) − o(u n ), where wn = max{u n , vn }. Proof In order to prove the assertion (i) of Lemma 1, we define the following sets for  > 0 and x ∈ D:   Nn =  k : k ∈ N and | (xk + yk ) − (1 + 2 )|    ,     N0;n =  k : k ∈ N and |xk − 1 |   2 and     N1,n =  k : k ∈ N and |yk − 2 |  . 2 Clearly, we have Nn ⊆ N0,n ∪ N1,n which implies, for n ∈ N, that lim

n→∞

1 Pn

bn 



m=an +1 k∈Nn

pbn −m bm,k (i)  lim

n→∞

1 Pn

bn 



pbn −m bm,k (i)

m=an +1 k∈N0,n

+ lim

n→∞

1 Pn

bn 



pbn −m bm,k (i).

m=an +1 k∈N1,n

(16)

On Statistical Deferred Weighted B-Convergence

673

Moreover, since wn = max{u n , vn },

(17)

by (16), we get lim

n→∞

bn 

1 wn Pn



pbn −m bm,k (i)  lim

n→∞

m=an +1 k∈Nn

1 u n Pn

+ lim

n→∞

bn 



pbn −m bm,k (i)

m=an +1 k∈N0,n

1 vn Pn

bn 



pbn −m bm,k (i).

m=an +1 k∈N1,n

(18) Also, by applying the Theorem 3, we obtain bn   1 pbn −m bm,k (i) = 0 uniformly in i, n→∞ wn Pn m=a +1 k∈N

lim

n

(19)

n

which proves the assertion (i) of Lemma 1. The other assertions (ii)–(iv) of Lemma 1 are similar to (i), so it is not difficult to prove these assertions along similar lines. This, evidently completes the proof Lemma 1.  Recalling that the modulus of continuity of a function of two variables f (x, y) ∈ C(D) is defined by  ω( f ; δ) =

sup

(s,t,x,y)∈D

  2 2 | f (s, t) − f (x, y)| : (s − x) + (t − y)  δ

(δ > 0),

(20) which implies ⎡ | f (s, t) − f (x, y)|  ω ⎣ f ;



s x − 1−s−t 1−x −y



2 +

t y − 1−s−t 1−x −y

2

⎤ ⎦.

(21) Now we present a theorem to get the rates of deferred Nörlund B-statistical convergence with the help of the modulus of continuity in (20). Theorem 4 Let B ∈ R+D(w) , and (an ) and (bn ) be sequences of non-negative integers. Let Tn : C(D) → C(D) be sequences of positive linear operators. Also, let (u n ) and (vn ) are the positive non-decreasing sequences. Suppose that the following conditions are satisfied:

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S. K. Paikray and H. Dutta

(a) Tn (1; x, y) − 1C(D) = statBD(N ) − o(u n ); (b) ω( f, λn ) = statBD(N ) − o(vn ) on D, where  λn = Tn (ϕ2 (s, t), x, y)C B (D)  with

ϕ(s, t) =

x s − 1−s−t 1−x −y

2

 +

y t − 1−t −s 1−x −y

2 .

Then, for all f ∈ C B (D), the following assertion holds true: Tn ( f (s, t); x, y) − f (x, y)C B (D) = statBD( N¯ ) − o(wn ),

(22)

where (wn ) is given by (17). Proof Let f ∈ C(D) and (x, y) ∈ D. Using (21), we have |Tn ( f ; x, y) − f (x, y)|  Tn (| f (s, t) − f (x, y)|; x, y) + | f (x, y)||Tn (1; x, y) − 1|, ⎞ ⎛ $% &2 % &2 y s x t + 1−s−t − 1−x−y ⎟ ⎜ 1−s−t − 1−x−y  Tn ⎜ + 1; x, y ⎟ ⎠ ω( f, δ) + H |Tn (1; x, y) − 1| ⎝ δ   1  Tn (1; x, y) + 2 Tn (ϕ(s, t); x, y) ω( f, δ) + H |Tn (1; x, y) − 1|, δ

where H =  f C(D) . Taking the supremum over (x, y) ∈ D on both sides, we have 

 1  T (ϕ(s, t); x, y) +  T (1; x, y) − 1 + 1 n n C( D ) C( D ) δ2 +H Tn (1; x, y) − 1C(D) .

Tn ( f ; x, y) − f (x, y)C(D)  ω( f, δ)

Now, putting δ = λn =

(

Tn (ϕ2 ; x, y), we get

  Tn ( f ; x, y) − f (x, y)C(D)  ω( f, λn ) Tn (1; x, y) − 1C(D) + 2 + H Tn (1; x, y) − 1C(D)  ω( f, λn )Tn (1; x, y) − 1C(D) + 2ω( f, λn ) + H Tn (1; x, y) − 1C(D) .

So, we have Tn ( f ; x, y) − f (x, y)C(D)    μ ω( f, λn )Tn (1; x, y) − 1C(D) + ω( f, λn ) + Tn (1; x, y) − 1C(D) ,

On Statistical Deferred Weighted B-Convergence

675

where μ = max{2, H }. For a given  > 0, we consider the following sets:   (23) Hn = n : n ∈ N and Tn ( f ; x, y) − f (x, y)C(D)   ;    ; (24) H0,n = n : n ∈ N and ω( f, λn )Tn ( f ; x, y) − f (x, y)C(D)  3μ    (25) H1,n = n : n ∈ N and ω( f, λn )  3μ and H2,n

   = n : n ∈ N and Tn (1; x, y) − 1C(D)  . 3μ

(26)

Finally, in view of the conditions (a) and (b) of Theorem 4 in conjunction with Lemma 1, the last inequalities (23)–(26) lead us to the assertion (22) of Theorem 4. This completes the proof of Theorem 4. 

5 Concluding Remarks In this concluding section of our investigation, we present several further remarks and observations concerning to the various results which we have proved here. Remark 3 Let (xn )n∈N be the sequence given in Example 2. Then, since stat D(N ) lim xn → 0 on n→∞

C(D),

we have stat D(N ) lim Tn ( f i ; x, y) − f i (x, y)C(D) = 0 n→∞

(i = 0, 1, 2, 3).

(27)

f ∈ C(D),

(28)

Therefore, by applying Theorem 3, we write stat D(N ) lim Tn ( f ; x, y) − f (x, y)C(D) = 0, n→∞

where s t f 2 (s, t) = 1−s−t 1−s−t 2  2  t s + . and f 3 (s, t) = 1−s−t 1−s−t f 0 (s, t) = 1, f 1 (s, t) =

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However, since (xn ) is not ordinarily convergent and so also it does not converge uniformly in the ordinary sense. Thus, the classical Korovkin theorem does not work here for the operators defined by (15). Hence, this application clearly indicates that our Theorem 3 is a non-trivial generalization of the classical Korovkin-type theorem (see [13]). Remark 4 Let (xn )n∈N be the sequence as given in Example 2. Then, since stat D(N ) lim xn → 0 on C(D), n→∞

so (27) holds. Now by applying (27) and our Theorem 3, condition (28) holds. However, since (xn ) does not statistical Nörlund B-summable, so we can say that the result of Kadak et al. ([10], p. 85, Theorem 3) does not hold true for our operator defined in (15). Thus, our Theorem 3 is also a non-trivial extension of Kadak et al. [[10], 20 p. 85, Theorem 3]. Based upon the above results, it is concluded here that our proposed method has successfully worked for the operators defined in (15) and therefore, it is stronger than the classical (ordinary) and statistical version of the Korovkin type approximation theorem (see [10, 13, 18]) established earlier. Remark 5 Let us suppose we replace the conditions (a) and (b) in our Theorem 4 by the following condition: |Tn ( f i ; x, y) − f i (x, y)|C(D) = statBD(N ) − o(u ni )

(i = 0, 1, 2, 3). (29)

Now, we can write Tn (ϕ2 ; x, y) = F

3 

Tn ( f i (s, t); x, y) − f i (x, y)C(D) ,

(30)

i=0

where   4M F =  + M + 2 , (i = 0, 1, 2, 3). δ It now follows from (29), (30) and Lemma 1 that ( λn = Tn (ϕ2 ) = statBD(N ) − o(dn ) on C(D), where o(dn ) = max{u n 0 , u n 1 , u n 2 , u n 3 }. Hence, clearly, we get ω( f, δ) = statBD(N ) − o(dn ) on C(D).

(31)

On Statistical Deferred Weighted B-Convergence

677

By using (31) in Theorem 4, we immediately see for all f ∈ C(D) that Tn ( f ; x, y) − f (x, y) = statBD(N ) − o(dn ) on C(D).

(32)

Therefore, if we use the condition (29) in Theorem 4 instead of conditions (a) and (b), then we obtain the rates of the statistical deferred Nörlund B-summability of the sequence (Tn ) of positive linear operators in Theorem 3.

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21. Srivastava, H.M., Et, M.: Lacunary statistical convergence and strongly lacunary summable functions of order α. Filomat 31, 1573–1582 (2017) 22. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: A certain class of weighted statistical convergence and associated Korovkin type approximation theorems for trigonometric functions. Math. Methods Appl. Sci. 41, 671–683 (2018) 23. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. (RACSAM). 112, 1487–1501 (2018) 24. Srivastava, H.M., Mursaleen, M., Alotaibi, A.M., Nasiruzzaman, Md, Al-Abied, A.A.H.: Some approximation results involving the q-Szász-Mirakjan-Kantorovich type operators via Dunkl’s generalization. Math. Methods Appl. Sci. 40, 5437–5452 (2017) 25. Srivastava, H.M., Mursaleen, M., Khan, A.: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model 55, 2040–2051 (2012) 26. Steiglitz, M.: Eine verallgemeinerung des begriffs der fastkonvergenz. Math. Jpn. 18, 53–70 (1973) 27. Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials Roberto B. Corcino

Abstract In this chapter, a certain variation of Bernoulli and Euler numbers and polynomials is introduced by means of polylogarithm, particularly, the polyBernoulli and Euler numbers and polynomials. Furthermore, a certain generalization of poly-Bernoulli and poly-Euler numbers and polynomials is defined by means of multiple polylogarithm. Common properties shared by the family of Bernoulli and Euler numbers and polynomials are discussed including recurrence relations, explicit formulas and several identities expressing these generalizations in terms of the other special numbers and functions (e.g. Stirling numbers and their generalizations). Keywords Bernoulli numbers · Euler numbers · Poly-Bernoulli numbers · Poly-Euler numbers · Polylogarithm · Multiple polylogarithm · Stirling numbers · r -Whitney numbers

1 Introduction In the early of sixteenth century, Pierre de Fermat (1601–1665) had tried to evaluate the sum s p (n) n−1  s p (n) = k p, (1) k=1

which is useful in computing the area under the curve f (x) = x p by its rectangular approximations. According to history, specifically, on the wonderful beginnings of integration [28], Johann Faulhaber (1580–1635) was the first to express s p (n) as polynomial in n but only for p = 0, 1, 2, . . . , 17. Clearly, it was already known to Jacob Bernoulli (1654–1705) some few expressions of s p (n) as polynomial in n. Based on these expressions, Bernoulli had discovered that R. B. Corcino (B) Department of Mathematics, Cebu Normal University, 6000 Cebu City, Philippines e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_21

679

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R. B. Corcino

s p (n) =

p  p! Bk n p+1−k , k! ( p + 1 − k)! k=0

(2)

where Bk are the well-known Bernoulli numbers. It had been observed that Bk = 0 when k = 2i + 1, for i = 1, 2, . . . The first few terms of Bernoulli numbers are given by B0 = 1, B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42. It can be observed that these numbers form a sequence of rational numbers. In the history of mathematics, Bernoulli numbers have possessed many interesting arithmetic properties. These numbers have appeared in several known and important mathematical expressions and identities. For instance, the formulas for finding the values of zeta functions, Euler–Maclaurin summation and L-functions contain the Bernoulli numbers. The following are some known properties of Bernoulli numbers: (i)

generating function ∞  n=0

(ii)

Bn

z zn = z , |z| < 2π; n! e −1

(3)

 n−1  1  n+1 Bk ; k n + 1 k=0

(4)

recurrence relation Bn = −

(iii)

explicit formula Bn =

(iv)

  m m n 1  j (−1) j j m + 1 m=0 j=0 n 

(5)

matrix representation 1 2  1  n 0 (−1)  Bn =  (n − 1)!  0  .. .  0

1 3

1 2 0 .. . 0

. . . n1 1 ... 1 − 1 33 . . . nn−1 . . . 2 2 .. . . .. . . .  n−1 0 . . . n−2 1 4

1  n+1 

    n . 2  ..  .  n 

1 n

(6)

n−2

More properties can be found in [1, 6, 9, 26, 37, 38, 44]. It is worth-mentioning that, based on the history of mathematics, the sequence of Bernoulli numbers was

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independently discovered by the Japanese mathematician, Seki Takakazu (1642– 1708), and the Indian mathematician, Srinivasa Ramanujan (1887–1920). By proper application of the generating function method, one can easily obtain different forms of formulas and relations for the numbers. It would then be convenient to introduce and define other variations and generalizations of Bernoulli numbers through generating functions analogous to that in (3). For instance, the Bernoulli polynomials can be defined by means of the following generating function, which is the product of the function in (3) and the function e zx : ∞ 

Bk (x)

k=0

ze zx zk = z , |z| < 2π, k! e −1

(7)

Obviously, B0 (x) = 1 and when x = 0, (7) reduces to (3). Some properties of Bernoulli polynomials can easily be obtained using the generating function method. For example, differentiating both sides of (7) with respect x yields ∞ ∞   z d zk z k+1 Bk (x) = z e zx z = Bk (x) dx k! e −1 k! k=0 k=0

=

∞ ∞   z k+1 zk = (k + 1)Bk (x) k Bk−1 (x) . (k + 1)! k! k=0 k=0

Comparing the coefficients of

zk k!

gives

d Bk (x) = k Bk−1 (x), dx

(8)

which is an interesting differential equation involving Bernoulli polynomials. When k = 1, d Bd1x(x) = B0 (x) = 1. Integrating both sides gives  B1 (x) =

d x = x + c.

When x = 0, c = B1 (0) = B1 = −1/2. Hence, 1 B1 (x) = x − . 2 Moreover, the following are some few values of Bernoulli polynomials B2 (x) = x 2 − x +

1 6

3 1 B3 (x) = x 3 − x 2 + x 2 2

(9) (10)

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1 30 5 4 5 3 1 5 B5 (x) = x − x + x − x 2 3 6

B4 (x) = x 4 − 3x 3 + x 2 −

(11) (12)

Bernoulli has observed that n(n − 1) 1 = (B2 (n) − B2 ) 2 2 1 n(n − 1)(2n − 1) = (B3 (n) − B3 ) s2 (n) = 6 3 s1 (n) =

and has made the following generalization s p (n) =

n−1 

kp =

k=0

 1  B p+1 (n) − B p+1 . p+1

Another direct consequence of (7) is to replace the variable x with 1 − x. This gives ∞  k=0

Bk (1 − x)

ze z(1−x) ze z e−zx zk = z = z k! e −1 e −1 ∞

 (−z)e(−z)x zk = = −z (−1)k Bk (x) . (e − 1) k! k=0 Again, comparing the coefficients of

zk k!

gives the following interesting relation

Bk (1 − x) = (−1)k Bk (x). The following are another forms of identities for Bernoulli polynomials which are immediate consequences of (7): (−1)k Bk (−x) = Bk (x) + kx k−1 |B2k (x)| < B2k , k = 1, 2, . . . , 0 < x < 1 Bk (1/2) = −(1 − 21−k )Bk k = 0, 1, . . .   j n  j 1  (x + k)n . (−1)k Bn (x) = j + 1 k k=0 j=0 The proofs are left as an exercise for the readers. One can find more properties from [1, 6, 9, 26, 37, 38, 44].

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We can derive another expression for Bernoulli polynomials in terms of Bernoulli numbers. This can be obtained using the method of generating function. That is, ∞ 

Bk (x)

k=0

Comparing the coefficients of

z zk = z e zx k! e −1 ∞

∞

 zk  (x z)k = Bk k! k! k=0 k=0

  ∞ k   zk k Bi x k−i . = i k! k=0 i=0 zk k!

gives

Bk (x) =

k    k i=0

i

Bi x n−i .

(13)

We observe that relation is an expression of Bk (x) as polynomial in x. The above differential equation also implies that the Bernoulli polynomials are classified as Appell polynomials. It is known that if pn (x) are Appell polynomials, then the following statements are equivalent: • For n = 1, 2, 3, . . . ,

d pn (x) = npn−1 (x) dx

and p0 (x) is a non-zero constant; • For some sequence {cn } of scalars with c0 = 0, n    n ck x n−k ; pn (x) = i k=0

(14)

• For the same sequence of scalars, pn (x) =

∞  ck k=0

k!

xn

(15)

pk (x)y n−k .

(16)

D

k

where D = ddx ; • For n = 0, 1, 2, . . . , pn (x + y) =

n    n k=0

k

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(See [17, 20, 33] for more discussions on Appell polynomials) Hence, Eqs. (16) and (8) imply the following addition formula Bk (x + y) =

n    n k=0

k

Bk (x)y n−k .

Moreover, Eq. (14) and the explicit formula in (13) imply that the desired scalar cn is just the Bernoulli numbers Bn . Thus, (15) gives the following relation

∞  Bk k D xn. Bn (x) = k! k=0 It is left to the readers to verify that this equation gives (9)–(12). Leonard Euler (1707–1783) had studied Bernoulli numbers for many years and produced numerous results on these numbers. This made him known as the “godfather” of Bernoulli numbers (see [6]). Parallel to the origin of the Bernoulli numbers, Euler had also the desire to evaluate the alternating sum A p (n) =

n 

(−1)n−i k p ,

(17)

k=1

by expressing it as polynomial in n. With this desire, Euler had introduced the numbers, which are now known as the Euler numbers. These numbers, denoted by E n , can be defined by means of the following generating function ∞ 

En

n=0

2 zn = z , |z| < π. n! e + e−z

(18)

Euler numbers E n are equal to zero when n is odd. The following are the first few values of Euler numbers with even indices: E 0 = 1, E 2 = −1, E 4 = 5, E 6 = −61, E 8 = 1385, E 10 = −50521. The Euler numbers have possessed several properties including the explicit formulas:  k (−1) j (k − 2 j)2n+1 , i 2 = −1 kikk 2 j k=1 j=0   k j n   K 1 δn, mkm − = (2n)! k1 , . . . , kn (2 j)! 0≤k ,...,k ≤n j=1

E 2n = i E 2n

2n+1 k  

1

n

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and the recurrence relation E 2n = −

n−1    2n k=0

2k

E 2k ,

(19)

(see [42, 44]). One can easily generate the first values of Euler numbers using (19). Moreover, the Euler numbers satisfy the following power series expansion ∞

 1 π 2n+1 E 2n = (−1)k 2n+2 2 (2n)! (2k + 1)2n+1 k=0 and matrix representation

E 2n

 1  2!  1  4!   . n = (−1) (2n)!  ..  1   (2n−2)!  1 (2n)!

1 1 2!

.. .

0 1 .. .

1 1 (2n−4)! (2n−6)! 1 1 (2n−2)! (2n−4)!

... ... .. . ... ...

0 0 .. .

 0  0  ..  .  1  1

(20)

1 2! 1 4! 4!

A straight forward generalization of Euler numbers, denoted by E n (x), is defined in polynomial form by means of the following generating function ∞ 

E k (x)

n=0

zk 2e x z = z . k! e +1

(21)

The polynomials E k (x) are called Euler polynomials. Clearly, E 0 (x) = 1 and E k = 2k E k (1/2). We can also derive some identities for Euler polynomials using the generating function method, which are parallel to those of Bernoulli polynomials. For example, differentiating both sides of (21) with respect x yields ∞ ∞   d zk z k+1 2 E k (x) E k (x) = z e zx z = dx k! e +1 k! k=0 k=0

=

∞ ∞   z k+1 zk = (k + 1)E k (x) k E k−1 (x) . (k + 1)! k! k=0 k=0

Comparing the coefficients of

zk k!

gives

d E k (x) = k E k−1 (x), dx

(22)

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R. B. Corcino

which is analogous to the above differential equation involving Bernoulli polynomials. When k = 1, d Ed1x(x) = E 0 (x) = 1. Integrating both sides gives  E 1 (x) =

d x = x + c.

When x = 1/2, c = E 1 (1/2) − 1/2 = E 1 /2 − 1/2 = −1/2. Hence, 1 E 1 (x) = x − . 2 The next Euler polynomials are given by E 2 (x) = x 2 − x 3 1 E 3 (x) = x 3 − x 2 + 2 4 E 4 (x) = x 4 − 2x 3 + x. Moreover, replacing the variable x with 1 − x yields ∞  k=0

E k (1 − x)

2e z(1−x) 2e z e−zx zk = z = z k! e +1 e +1 ∞

=

Again, comparing the coefficients of

 2e(−z)x zk = (−1)k E k (x) . −z e +1 k! k=0 zk k!

gives the following interesting relation

E k (1 − x) = (−1)k E k (x). Also, the Euler polynomials satisfy the following relations and explicit formula: E n (x + 1) + E n (x) = 2x n n    n E k (z)E n−k (w) = 2(1 − w − z)E n (z + w) + 2E n+1 (z + w) k k=0   j n  1  k j (x + k)n . (−1) E n (x) = j 2 k j=0 k=0 It is left to the readers to prove these identities.

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We can derive another expression for Euler polynomials in terms of Euler numbers. This can be obtained using the method of generating function. That is, ∞ 

2k E k (x)

k=0

2 e−z zk = 2z e2zx −z k! e +1 e 2 e(2x−1)z + e−z ∞

∞

 zk  ((2x − 1)z)k = Ek k! k! k=0 k=0

∞ k     k zk k−i (2x − 1) E i . = i k! k=0 i=0 =

Comparing the coefficients of

zk k!

ez

gives

k   1  k (2x − 1)k−i E i . E k (x) = k 2 i=0 i

We recall that the Genocci numbers G k are defined by ∞  k=0

Gk

2z zk = z . k! e +1

Hence, ∞  k=0

E k (x)

2z −1 zx zk = z z e k! e +1 ∞

2z  z k−1 e z + 1 k=0 k! ∞

∞

 (x z)k zk 1  = Gk z k=0 k! k! k=0

  ∞ k zk 1  k k−i . Gi x = z k=0 i=0 i k!

=

This implies that k  

∞   k zk z k+1 G i x k−i = . (k + 1)E k (x) i (k + 1)! k! k=0 k=0 i=0

∞ 

(23)

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Thus, we have k E k−1 (x) =

k    k i=0

i

G i x k−i .

(24)

Euler polynomials can also be expressed in terms of some combinatorial numbers and special functions. The following are few examples of these expressions:  n  n   −1 (k) j S(n, k)(x)k− j E n (x) = j j=0 k= j  x  2 Bn (x) − 2n Bn , E n−1 (x) = n 2  where S(n, k) are the Stirling numbers of the second kind, xk are the generalized binomial coefficients, (x)k is the falling factorial of x of degree k and Bn (x) are the above Bernoulli polynomials. The above differential equation involving Euler polynomials also implies that the Euler polynomials are classified as Appell polynomials. Hence, Eqs. (14), (16) and (22) imply the following formulas E n (x + y) =

n    n k=0

n    n ck x n−k . E n (x) = k k=0

k

E k (x)y n−k

The explicit formula in (23) and (24) would possibly suggest the value of the scalar cn , which can be used to compute E n (x) using the following formula E n (x) =

∞  ck k=0

k!

D

k

xn,

which is obtained from (15). It is left to the readers to identify the appropriate value of ck . In this chapter, a special type of variation and generalization of Bernoulli and Euler numbers and polynomials will be discussed. This type of variation and generalization is based on the recently introduced poly-Bernoulli and poly-Euler numbers and polynomials.

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2 Poly-Bernoulli Numbers and Polynomials with Parameters a, b, c The poly-Bernoulli numbers were first introduced by Kaneko [27] in 1997. These numbers were defined in terms of the following polylogarithm: Definition 1 For any integer k ∈ Z, the kth polylogarithm Lik (z) is defined by ∞  zn . nk n=0

Lik (z) =

(25)

When z = 1, the kth polylogarithm gives the Riemann zeta function. That is, Lik (1) = ζ(k) =

∞  1 . nk n=0

Also, when k = 1, the 1st polylogarithm yields the natural logarithmic function as follows: Li1 (z) = − log(1 − z). This special case of the polylogarithm motivates the construction of poly-Bernoulli numbers in the sense that Li1 (1 − e−x ) = x. Now, we are ready to define the poly-Bernoulli numbers of Kaneko [27]. Definition 2 ([27]) The poly-Bernoulli numbers, denoted by Bn(k) , are defined by ∞

Lik (1 − e−x )  (k) x n . = Bn 1 − e−x n! n=0

(26)

One can easily verify that Bn(1) = Bn (1). It is known that the polylogarithm can alternatively be defined as 

z

Lik+1 (z) = 0

Lik (t) dt. t

(27)

Hence, the left-hand side of (26) can be written in the form of iterated integrals as follows: ex

1 1 + ex

 0

x

1 1 + ex



x 0

...

1 1 + ex

 0

x

∞

n  1 (k) x . d x . . . d x = B n 1 + ex n! n=0

(28)

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More properties of poly-Bernoulli numbers are established in [8–10, 26, 27, 29–32, 40] including the following explicit formulas: Bn(k)



1 = n+1

Bn(−k) =

min(n,k) 

  n−1   n B (k) Bn(k−1) − m−1 m m=1

( j!)2 S(n + 1, j + 1)S(k + 1, j + 1), n, k ≥ 0

j=0

Bn(−k) = (−1)n

n 

(−1)m m!S(n, m)(m + 1)k .

m=0

Since the Stirling numbers of the second kind S(n, k) count the number of ways to partition an n-set into k nonempty subsets, the last two explicit formula can be used to interpret Bn(−k) in terms of set partition and permutation. The following are some known combinatorial interpretations of Bn(−k) : (i) the number of permutations of the set {0, 1, 2, . . . , n + k + 1} with the ascendingto-max property, i.e. if two elements with consecutive values are both in the first n + 1 positions (resp. in the last k + 1 positions) then they should follow each other directly or have consecutive positions in the permutations [7]; (ii) the number of permutations π ∈ Sn+k such that −n ≤ i − π(i) ≤ k [7]; (iii) the number of binary lonesum matrices of size n × k where the binary lonesum matrix is a binary matrix which can be uniquely reconstructed from its row and column sums [9]; (iv) the number of n × k 01 matrices such that none of the matrices 

   11 11 , 10 11

(29)

appear as submatrices [7]. Parallel to Bernoulli numbers, poly-Bernoulli numbers can also be extended to polynomial case, which we can call as poly-Bernoulli polynomials. Let us formally define these polynomials as follows. Definition 3 ([21]) The poly-Bernoulli polynomials, denoted by Bn(k) (x), are defined by ∞ Lik (1 − e−t ) xt  (k) t n (30) e = Bn (x) . 1 − e−t n! n=0 Clearly, Bn(k) (0) = Bn(k) and (−1)n Bn(1) (−x) = Bn (x). It is natural to establish the properties for Bn(k) (x) parallel to those of poly-Bernoulli numbers. It is left to the readers as an exercise.

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Another variation of generalizing Bernoulli numbers is by introducing additional parameters. Luo et al. [34] introduced such generalization for Bernoulli polynomials by adding two parameters a and b, which is given by   ∞  b  t t n  xt t ln < 2π, , e = B (x; a, b) n bt − a t n!  a n=0

(31)

such that when a = 1 and b = e, Bn (x; 1, e) = Bn (x). Parallel to this generalization, the poly-Bernoulli polynomials can also be extended by introducing three parameters a, b and c. Jolany et al. [21] defined such generalization for poly-Bernoulli numbers and polynomials, which is given in the following definition. Definition 4 ([21]) The generalized poly-Bernoulli polynomials with three parameters a, b, c, denoted by Bn(k) (x; a, b, c), are defined by ∞

Lik (1 − (ab)−t ) xt  (k) tn . c = B (x; a, b, c) n bt − a −t n! n=0

(32)

The following theorem contains an explicit formula for Bn(k) (x; a, b, c), which is established in [15]. Theorem 1 ([15]) For k ∈ Z, n ≥ 0, we have Bn(k) (x; a, b, c) =

  m  1 j m (x ln c − j ln a − ( j + 1) ln b)n . (−1) k j (m + 1) m=0 j=0 n 

(33) Proof Using the following expression ∞

∞

 (1 − (ab)−t )m−1  (1 − (ab)−t )m Lik (1 − (ab)−t ) −t −t =b =b bt − a −t mk (m + 1)k m=1 m=0   m ∞   1 −t j m e− jt ln(ab) =b (−1) k (m + 1) j m=0 j=0   m ∞   1 j m e−t ( j ln a+( j+1) ln b) , = (−1) k j (m + 1) m=0 j=0 the generating function in (32) yields ∞  n=0

Bn(k) (x; a, b, c)

tn Lik (1 − (ab)−t ) xt ln c Lik (1 − (ab)−t ) xt c = e = t −t n! b −a bt − a −t

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R. B. Corcino

  m  1 j m et (x− j ln a−( j+1) ln b) = (−1) k (m + 1) j m=0 j=0 ⎛ ⎞   ∞ ∞ m n    m 1 j n⎠ t ⎝ (x ln c − j ln a − ( j + 1) ln b) = (−1) j (m + 1)k j=0 n! n=0 m=0 ∞ 

By comparing the coefficients of

tn n!



on both sides, we obtain (33).

To establish possible combinatorial meanings for Bn(k) (x; a, b, c), it is necessary to express these polynomials in terms of some combinatorial numbers like binomial coefficients, factorials and Stirling numbers and other special functions and numbers. The following theorem contains such identities. Theorem 2 ([15]) For any positive numbers a, b, c and any real numbers x with k ∈ Z and n ≥ 0, the generalized poly-Bernoulli polynomials satisfy the following identities:   n (k) Bn−l (−m log c; a, b)x (m) (34) l m=0 l=m    ∞  n  n (k) (k) l l Bn−l (0; a, b)(x)m (log c) (35) Bn (x; a, b, c) = m l m=0 l=m n−m    ∞   n−m  l + s (k) n  l (k) Bn−m−l (0; a, b)Bm(s) (x log c) (36) Bn (x; a, b, c) = l+s  s m l m=0 l=0 n  n s     s (k) m (−λ)s− j Bn−m Bn(k) (x; a, b, c) = ( j; a, b)Hm(s) (x log c; λ) s j (1 − λ) m=0 j=0 Bn(k) (x; a, b, c) =

∞  n 

(log c)l S(l, m)

(37) where x (m) and (x)m are the rising and falling factorials of x of degree n, respectively, and 

t et − 1

s e xt =

∞ 

Bn(s) (x)

n=0

tn and n!



1−λ et − λ

s e xt =

∞ 

Hn(s) (x; λ)

n=0

Proof For relation (34), note that (32) can be written as ∞  n=0

Bn(k) (x; a, b, c)

Lik (1 − (ab)−t ) tn = (1 − (1 − e−t log c ))−x n! bt − a −t

tn . n!

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Using Newton’s Binomial Theorem and the exponential generating function for S(n, m) ∞  (et − 1)m tn , S(n, m) = n! m! n=0 we have ∞ 

 ∞  Lik (1 − (ab)−t )  x + m − 1 = (1 − e−t log c )m m n! bt − a −t m=0

t Bn(k) (x; a, b, c)

n=0 ∞ 

n

(et log c − 1)m Lik (1 − (ab)−t ) −mt log c e m! bt − a −t m=0 ∞

∞

∞    (t log c)n tn (m) (k) = x S(n, m) Bn (−m log c; a, b) n! n! m=0 n=0 n=0     ∞ ∞  n   tn n (k) (log c)l S(l, m) Bn−l (−m log c; a, b)x (m) = n! l n=0 m=0 l=m =

x (m)

Comparing coefficients completes the proof of (34). For relation (35), we can write (32) as ∞ 

Bn(k) (x; a, b, c)

n=0

Lik (1 − (ab)−t ) t log c tn = ((e − 1) + 1)x n! bt − a −t

Using Newton’s Binomial Theorem and the exponential generating function for S(n, m), we have ∞ 

Bn(k) (x; a, b, c)

n=0 ∞ 

∞   Lik (1 − (ab)−t )  x tn (et log c − 1)m = m n! bt − a −t m=0

(et log c − 1)m Lik (1 − (ab)−t ) m! bt − a −t m=0

∞

∞ ∞ n    (t log c)n t = (x)m S(n, m) Bn(k) (0; a, b) n! n! m=0 n=0 n=0     ∞ ∞  n   tn n (k) (log c)l S(l, m) Bn−l (0; a, b)(x)m = n! l n=0 m=0 l=m =

(x)m

Comparing coefficients completes the proof of (35).

694

R. B. Corcino

For relation (36), Eq. (32) can be written as ∞ 

Bn(k) (x; a, b, c)

n=0

=

∞  n=0

∞ 

tn = n!



(et − 1)s s!

∞ 

t n+s S(n + s, s) (n + s)!



t s e xt log c (et − 1)s

Lik (1 − (ab)−t ) bt − a −t



s! ts

∞

 (k) t n s! Bn (0; a, b) m! n! t s

t (s) Bm (x log c)

m=0



m

n=0

∞ ∞

  (s) s! t n+s t m (k) t n−m = S(n + s, s) Bm (x log c) Bn−m (0; a, b) (n + s)! m! (n − m)! t s n=0 m=0 n=m ⎧ ⎫ ∞ ⎨ ∞ n−m n−m−l m s! ⎬   t t l+s t (s) (k) Bm (x log c)Bn−m−l (0; a, b) = S(l + s, s) ⎩n=m (l + s)! (n − m − l)! m! t s ⎭ m=0 l=0 ⎫ ⎧ n−m  ∞ ⎨ n   n−m ⎬ tn  n  (k) (s) l = l+s  S(l + s, s)Bn−m−l (0; a, b)Bm (x log c) ⎭ n! ⎩ m n=0

m=0

l=0

s!

Comparing coefficients completes the proof of (36). For relation (37), note that (32) can be written as ∞ 

Bn(k) (x; a, b, c)

n=0

tn = n!



(1 − λ)s xt log c e (et − λ)s



(et − λ)s (1 − λ)s



Lik (1 − (ab)−t ) bt − a −t



⎞

⎛ s   n −t  s Li (1 − (ab) ) t k ⎝ = Hn(s) (x log c; λ) e jt ⎠ (−λ)s− j t − a −t j n! b n=0 j=0 ∞

∞

  s  s   1 tn tn s− j (s) (k) (−λ) = Hn (x log c; λ) Bn ( j; a, b) (1 − λ)s j=0 j n! n! n=0 n=0 s   n   ∞    s n 1 tn (k) s− j (−λ) Hm(s) (x log c; λ)Bn−m = ( j; a, b) s (1 − λ) j=0 j n! m n=0 m=0 ⎞ ⎛ n  ∞ n s      tn s m s− j (k) (s) ⎠ ⎝ (−λ) . = B ( j; a, b)H (x log c; λ) n−m m (1 − λ)s j=0 j n! n=0 m=0 ∞ 

Comparing coefficients completes the proof of (37).



In the next theorem, the polynomials Bn(k) (x; a, b, c) are expressed as polynomial in x. However, we need to introduce certain generalization of poly-Bernoulli numbers, denoted by Bi(k) (a, b), which is equivalent to Bi(k) (0; a, b, c). More precisely, the numbers Bi(k) (a, b) are defined as coefficients of the following generating function

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

695

∞

Li(k) (1 − (ab)−t )  (k) tn . = B (a, b) n bt − a −t n! n=0 Now, we are ready to introduce the following theorems that appeared in [15]. Theorem 3 ([15]) The generalized poly-Bernoulli polynomials satisfy the following relation n    n Bn(k) (x; a, b, c) = (ln c)n−i Bi(k) (a, b)x n−i . (38) i i=0 Proof Using (32), we have ∞  n=0

Bn(k) (x; a, b, c)

∞  Li(k) (1 − (ab)−t ) xt tn tn xt ln c (k) = c = e B (a, b) n n! bt − a −t n! n=0

= =

∞  n  (xt ln c)n−i

tn , n!

i=0

ti i!

(n − i)!

 n tn n−i (k) n−i (ln c) Bi (a, b)x . i n!

n=0 i=0 n  ∞   n=0

Comparing the coefficients of

Bi(k) (a, b)

we obtain the desired result.



Note that, when a = c = e and b = 1, Definition 3 reduces to ∞

Li(k) (1 − e−t ) xt  (k) tn . e = B (x; e, 1, e) n 1 − e−t n! n=0

(39)

where Bn(k) (x; e, 1, e) is equivalent to Bn(k) (x), the poly-Bernoulli polynomials. The following theorem gives a relation between Bn(k) (x; a, b, c) and Bn(k) (x). Theorem 4 ([15]) The generalized poly-Bernoulli polynomials satisfy the following relation   x ln c − ln b . (40) Bn(k) (x; a, b, c) = (ln a + ln b)n Bn(k) ln a + ln b Proof Using (32), we have ∞  Li(k) (1 − (ab)−t ) xt ln c tn e Bn(k) (x; a, b, c) = t n! b (1 − (ab)−t ) n=0 − e−t ln ab ) 1 + e−t ln ab  n  ∞  t n (k) x ln c − ln b = (ln a + ln b) Bn . ln a + ln b n! n=0 =e

Comparing the coefficients of

tn , n!

x ln c−ln b ln ab t

ln ab Li(k) (1

we obtain the desired result.



696

R. B. Corcino

The following theorem contains a derivative formula for generalized polyBernoulli polynomials. Theorem 5 ([15]) The generalized poly-Bernoulli polynomials satisfy the following relation d (k) (41) B (x; a, b, c) = (n + 1)(ln c)Bn(k) (x; a, b, c). d x n+1 Proof Differentiating both sides of (32) with respect to x yields ∞  t (ln c)Li(k) (1 − (ab)−t ) xt ln c d (k) tn Bn (x; a, b, c) = e dx n! bt − a −t n=0 ∞ ∞   d (k) t n−1 tn Bn (x; a, b, c) = (ln c)Bn(k) (x; a, b, c) . dx n! n! n=0 n=0

Hence, ∞  n=0

∞

 1 d (k) tn tn (ln c)Bn(k) (x; a, b, c) . Bn+1 (x; a, b, c) = n + 1 dx n! n! n=0

Comparing the coefficients of

tn , n!

we obtain the desired result.



The following corollary immediately follows from Theorem 5 by taking c = e. For brevity, let us denote Bn(k) (x; a, b, e) by Bn(k) (x; a, b). Corollary 6 ([15]) The generalized poly-Bernoulli polynomials are Appell polynomials in the sense that d (k) B (x; a, b) = (n + 1)Bn(k) (x; a, b). d x n+1

(42)

Consequently, using the characterization of Appell polynomials in (16) [33, 41, 43], the following addition formula can easily be obtained. Corollary 7 ([15]) The generalized poly-Bernoulli polynomials satisfy the following addition formula Bn(k) (x

+ y; a, b) =

n    n i=0

i

Bi(k) (x; a, b)y n−i .

However, we can derive the addition formula for Bn(k) (x; a, b, c) as follows

(43)

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials ∞ 

Bn(k) (x + y; a, b, c)

n=0

697

Li(k) (1 − (ab)−t ) (x+y)t tn = c n! bt − a −t Li(k) (1 − (ab)−t ) xt yt c c bt − a −t ∞

∞

n   tn (k) nt = Bn (x; a, b, c) (y ln c) n! n! n=0 n=0

  ∞ n   tn n n−i (k) (y ln c) Bi (x; a, b, c) . = i n! n=0 i=0 =

Comparing the coefficients of

tn n!

yields the following result.

Theorem 8 ([15]) The generalized poly-Bernoulli polynomials satisfy the following addition formula Bn(k) (x

n    n (ln c)n−i Bi(k) (x; a, b, c)y n−i . + y; a, b, c) = i i=0

Note that when x = 0, the above addition formula yields Bn(k) (y; a, b, c)

n    n (k) (ln c)i Bn−i = (0; a, b, c)y i i i=0

which is exactly the formula in Theorem 3 that expresses Bn(k) (y; a, b, c) as polyno (k) mial in y with the numbers ni (ln c)i Bn−i (0; a, b, c) as coefficients. Being classified as Appell polynomials, the generalized poly-Bernoulli polynomials Bn(k) (x; a, b) must possess the following properties n    n ci x n−i = i i=0 ∞

 ci Di x n . Bn(k) (x; a, b, c) = i! i=0

Bn(k) (x; a, b, c)

for some scalar ci = 0. It is then necessary to find the sequence {cn }. However, using Theorem 3, ci = Bi(k) (a, b), which implies the following theorem. Theorem 9 The generalized poly-Bernoulli polynomials satisfy the following formula ∞

 B (k) (a, b) i Bn(k) (x; a, b, c) = Di x n . i! i=0

698

R. B. Corcino

For example, when n = 3, we have B3(k) (x; a, b, c)

∞

 B (k) (a, b) i i D x3 = i! i=0 B0(k) (a, b) 3 B1(k) (a, b) 1 3 B2(k) (a, b) 2 3 x + D x + D x 0! 1! 2! B (k) (a, b) 3 3 D x + 3 3! = B0(k) (a, b)x 3 + 3B1(k) (a, b)x 2 + 3B2(k) (a, b)x + B3(k) (a, b). =

Sasaki [39] defined another version of generalized poly-Bernoulli numbers and polynomials using the concept of generalized Dirichlet L-function.

3 Poly-Euler Numbers and Polynomials with Parameters a, b, c Ohno and Sasaki [35] were the first to introduce the poly-Euler numbers. Analogous to the definition of poly-Bernoulli numbers, poly-Euler numbers are also defined in terms of polylogarithm. Definition 5 ([35]) The poly-Euler numbers, denoted by E n(k) , are defined by ∞

Lik (1 − e−4t )  (k) t n = . En 4t cosh t n! n=0

(44)

Parallel to Euler numbers, poly-Euler numbers can also be extended to polynomial case, which we can call as poly-Euler polynomials. Now, let us formally define these polynomials as follows. Definition 6 The poly-Euler polynomials, denoted by E n(k) (x), are defined by ∞

Lik (1 − e−4t ) xt  (k) t n e = E n (x) . 4t cosh t n! n=0

(45)

One can easily verify that E n(1) (x) = n E n−1 (x). Further generalization and other properties of poly-Euler numbers and polynomials including their relations with other special numbers and functions are found in [3–5, 8, 9, 11, 12, 18]. In 2012, Jolany et al. [24] introduced certain generalization of poly-Euler polynomials. It is defined by adding three more parameters a, b, c.

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

699

Definition 7 ([24]) The generalized poly-Euler polynomials E n(k) (x; a, b, c) with parameters a, b and c are defined by ∞

2Lik (1 − (ab)−t ) xt  (k) tn c = E n (x; a, b, c) . −t t a +b n! n=0

(46)

Note that the poly-Euler polynomials in [4, 36] can be deduced from (46) by replacing t with 4t and taking x = 1/2. Moreover, when x = 0, (46) gives E n(k) (0; a, b, c) = E n(k) (a, b) where

∞

2Lik (1 − (ab)−t )  (k) tn , = E (a, b) n a −t + bt n! n=0

and when a = 1 and b = c = e with E n(k) (x; 1, e, e) = E n(k) (x), we obtain Eq. (45). Theorem 10 ([15]) The generalized Poly-Euler polynomials with parameters a, b, c have the following explicit formula   j n  m   2(−1)m− j+i j × E n(k) (x; a, b, c) = jk i m=0 j=0 i=0 × (x ln c − (m − j + i) ln a − (m − j + i + 1) ln b)n . Proof ∞  n=0

E n(k)

2Lik (1 − (ab)−t ) xt tn = t c n! b ((ab)−t + 1)

 

  1 − (ab)−t m ! " −t m −t m = 2b (−1) (ab) c xt k m m≥0 m≥0 = b−t

  j m   2(−1)m− j+i j jk

m≥0 j=0 i=0

=

  j m   2(−1)m− j+i j m≥0 j=0 i=0

=

jk

i

  j m   2(−1)m− j+i j m≥0 j=0 i=0

=

i

jk

i

(ab)−t (m− j+i) c xt

e−t (m− j+i) ln ab e−t ln b e xt ln c

et[x ln c−(m− j+i) ln a−(m− j+i+1) ln b]

  j n  m   2(−1)m− j+i j × i jk n≥0 m≥0 j=0 i=0 × {t[x ln c − (m − j + i) ln a − (m − j + i + 1) ln b]}n

tn . n!

700

R. B. Corcino

By comparing the coefficient of t n /n!, we obtain the desired explicit formula.



The next theorem contains identities, which are parallel to those in Theorem 2. Theorem 11 ([15]) For any positive numbers a, b, c and any real numbers x with k ∈ Z and n ≥ 0, the generalized poly-Euler polynomials satisfy the following relation   ∞  n  n (k) (log c)l S(l, m) E n−l (−m log c; a, b)x (m) (47) l m=0 l=m   ∞  n  n (k) (k) l (log c) S(l, m) E n−l (0; a, b)(x)m (48) E n (x; a, b, c) = l m=0 l=m n−m    ∞   n−m  l + s (k) n  l (k) E n−m−l (0; a, b)Bm(s) (x log c) (49) E n (x; a, b, c) = l+s  s m l m=0 l=0 n  n s     s (k) m (−λ)s− j E n−m E n(k) (x; a, b, c) = ( j; a, b)Hm(s) (x log c; λ). s j (1 − λ) m=0 j=0

E n(k) (x; a, b, c) =

(50) Proof For relation (47), note that (46) can be written as ∞ 

E n(k) (x; a, b, c)

n=0

2Li(k) (1 − (ab)−t ) tn = (1 − (1 − e−t log c ))−x n! a −t + bt

Using Newton’s Binomial Theorem and the exponential generating function for S(n, m), we have ∞ 

 ∞  2Li(k) (1 − (ab)−t )  x + m − 1 = (1 − e−t log c )m n! a −t + bt m m=0

t E n(k) (x; a, b, c)

n=0 ∞ 

n

(et log c − 1)m 2Li(k) (1 − (ab)−t ) −mt log c e m! a −t + bt m=0 ∞

∞

∞    (t log c)n tn (m) (k) = x S(n, m) E n (−m log c; a, b) n! n! m=0 n=0 n=0     ∞ ∞  n   tn n (k) = (log c)l S(l, m) E n−l (−m log c; a, b)x (m) n! l n=0 m=0 l=m =

x (m)

Comparing coefficients completes the proof of (47).

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

701

For relation (48), we can write (46) as ∞ 

E n(k) (x; a, b, c)

n=0

2Li(k) (1 − (ab)−t ) t log c tn = ((e − 1) + 1)x n! a −t + bt

Using Newton’s Binomial Theorem and the exponential generating function for S(n, m), we have ∞ 

E n(k) (x; a, b, c)

n=0 ∞ 

∞   2Li(k) (1 − (ab)−t )  x tn (et log c − 1)m = m n! a −t + bt m=0

(et log c − 1)m 2Li(k) (1 − (ab)−t ) m! a −t + bt m=0 ∞

∞

∞    (t log c)n tn (k) = (x)m S(n, m) E n (0; a, b) n! n! m=0 n=0 n=0 ∞ n    ∞   tn n (k) (log c)l S(l, m) E n−l (0; a, b)(x)m = n! l n=0 m=0 l=m =

(x)m

Comparing coefficients completes the proof of (48). For relation (49), Eq. (46) can be written as ∞ 

t E n(k) (x; a, b, c)

n!

n=0

=

∞  n=0

=

n

∞ 

 =

t n+s S(n + s, s) (n + s)! S(n + s, s)

n=0

⎧

t n+s (n + s)!

(et − 1)s s!

∞ 



t s e xt log c (et − 1)s

t (s) Bm (x log c)

m=0 n=m

(s)

2Li(k) (1 − (ab)−t ) a −t + bt

∞ 

m!

m=0

∞ ∞  

m



Bm (x log c)

t (k) E n (0; a, b)

n=0

tm m!

(k)

E n−m (0; a, b)

n

n!





s! ts

s! ts

t n−m (n − m)!



s! ts

⎫ t l+s t n−m−l t m s! ⎬ (s) (k) Bm (x log c)E n−m−l (0; a, b) = S(l + s, s) ⎩ (l + s)! (n − m − l)! m! t s ⎭ m=0 n=m l=0 ⎫ ⎧ n−m  ∞ ⎨ n   n−m ⎬ tn  n  (k) (s) l = l+s  S(l + s, s)E n−m−l (0; a, b)Bm (x log c) ⎭ n! ⎩ m ∞ ⎨ ∞ n−m  

n=0

m=0

l=0

s!

Comparing coefficients completes the proof of (49).

702

R. B. Corcino

For relation (50), note that (46) can be written as ∞ 

E n(k) (x; a, b, c)

n=0

tn = n!



(1 − λ)s xt log c e (et − λ)s



(et − λ)s (1 − λ)s



2Li(k) (1 − (ab)−t ) a −t + bt



⎞

⎛ s   n −t  s 2Li (1 − (ab) ) t (k) ⎝ = (−λ)s− j Hn(s) (x log c; λ) e jt ⎠ −t + bt j n! a n=0 j=0 ∞

∞

  s  s   1 tn tn s− j (s) (k) (−λ) = Hn (x log c; λ) E n ( j; a, b) (1 − λ)s j=0 j n! n! n=0 n=0 s   ∞  n     s n 1 tn (k) s− j (s) (−λ) H = (x log c; λ)E ( j; a, b) n−m m m (1 − λ)s j=0 j n! n=0 m=0 ⎛ ⎞ n  s   ∞ n n    s m s− j (k) (s) ⎝ ⎠t . (−λ) = E ( j; a, b)H (x log c; λ) n−m m (1 − λ)s j=0 j n! n=0 m=0 ∞ 



Comparing coefficients completes the proof of (50).

In the following theorem, the polynomials E n(k) (x; a, b, c) are expressed in terms of certain generalization of poly-Euler numbers E i(k) (a, b), i = 0, 1, . . . , n. Theorem 12 The generalized poly Euler polynomials satisfy the following relation E n(k) (x; a, b, c) =

n    n (ln c)n−i E i(k) (a, b)x n−i i i=0

(51)

Proof Using (46), we have ∞  n=0

∞  2Lik (1 − (ab)−t ) xt tn xt ln c (k) = c = e E (a, b) n n! a −t + bt n! n=0

t E n(k) (x; a, b, c)

n

= =

∞  n  (xt ln c)n−i

Comparing the coefficients of

tn , n!

i=0

ti i!

(n − i)!

 tn n (k) . (ln c)n−i E i (a, b)x n−i n! i

n=0 i=0 n  ∞   n=0

E i(k) (a, b)

we obtain the desired result.



Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

703

The next theorem contains a relation between E n(k) (x; a, b, c) and E n(k) (x). Theorem 13 The generalized poly Euler polynomials satisfy the following relation E n(k) (x; a, b, c)

= (ln a + ln b)

n

E n(k)



x ln c + ln a ln a + ln b

 (52)

Proof Using (46), we have ∞ 

E n(k) (x; a, b, c)

n=0

2Lik (1 − (ab)−t ) xt ln c tn = −t e n! a (1 + (ab)t ) − e−t ln ab ) 1 + et ln ab   ∞  x ln c + ln a t n . = (ln a + ln b)n E n(k) ln a + ln b n! n=0 = 2e

Comparing the coefficients of

tn , n!

x ln c+ln a ln ab t

ln ab Lik (1

we obtain the desired result.



The next theorem contains a differential equation involving the generalized polyEuler polynomials. Theorem 14 The generalized poly-Euler polynomials satisfy the following relation d (k) E (x; a, b, c) = (n + 1)(ln c)E n(k) (x; a, b, c) d x n+1

(53)

Proof Differentiating both sides of (46) with respect to x yields ∞  2t (ln c)Lik (1 − (ab)−t ) xt ln c d (k) tn E n (x; a, b, c) = e dx n! (a −t + bt ) n=0 ∞ ∞   d (k) t n−1 tn E n (x; a, b, c) = (ln c)E n(k) (x; a, b, c) . dx n! n! n=0 n=0

Hence, ∞  n=0

∞

 1 d (k) tn tn E n+1 (x; a, b, c) = (ln c)E n(k) (x; a, b, c) . n + 1 dx n! n! n=0

Comparing the coefficients of

tn , n!

we obtain the desired result.



The following corollary contains a differential equation involving the polynomials E n(k) (x; a, b), which is an immediate consequence of Theorem 14 by taking c = e. The polynomials E n(k) (x; a, b) are equivalent to E n(k) (x; a, b, e).

704

R. B. Corcino

Corollary 15 The generalized poly-Euler polynomials are Appell polynomials in the sense that d (k) E (x; a, b) = (n + 1)E n(k) (x; a, b) (54) d x n+1 The next corollary contains an addition formula, which immediately follows from Corollary 6 using the characterization of Appell polynomials [33, 41, 43]. Corollary 16 The generalized poly-Euler polynomials with parameters a, b satisfy the following addition formula E n(k) (x

+ y; a, b) =

n    n i=0

i

E i(k) (x; a, b)y n−i

(55)

Taking x = 0 in formula (55) and using the fact that E n(k) (0; a, b) = E n(k) (a, b), Corollary 16 gives formula (51) in Theorem 12 with c = e. Theorem 17 The generalized poly-Euler polynomials with parameters a, b, c satisfy the following addition formula E n(k) (x + y; a, b, c) =

n    n (ln c)n−i E i(k) (x; a, b, c)y n−i . i i=0

(56)

Proof ∞  n=0

E n(k) (x + y; a, b, c)

2Li(k) (1 − (ab)−t ) (x+y)t tn = c n! a −t + bt 2Li(k) (1 − (ab)−t ) xt yt c c a −t + bt ∞

∞

n   tn (k) nt = E n (x; a, b, c) (y ln c) n! n! n=0 n=0

n   ∞   n tn (k) (y ln c)n−i E i (x; a, b, c) . = i n! n=0 i=0 =

Comparing the coefficients of

tn n!

yields the desired result.



Being classified as Appell polynomials, the generalized poly-Euler polynomials E n(k) (x; a, b) must possess the following properties

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

E n(k) (x; a, b)

=

n    n i=0

E n(k) (x; a, b) =

705

i

∞  ci i=0

i!

ci x n−i

Di

xn.

for some scalar ci = 0. It is then necessary to find the sequence {cn }. However, using Theorem 12, ci = E i(k) (a, b), which implies the following theorem. Theorem 18 The generalized poly-Euler polynomials satisfy the following formula E n(k) (x; a, b)

∞

 E (k) (a, b) i i D xn. = i! i=0

For example, when n = 3, we have E 3(k) (x; a, b, c)

∞

 E (k) (a, b) i i D x3 = i! i=0 E 0(k) (a, b) 3 E 1(k) (a, b) 1 3 E 2(k) (a, b) 2 3 x + D x + D x 0! 1! 2! E (k) (a, b) 3 3 + 3 D x 3! = E 0(k) (a, b)x 3 + 3E 1(k) (a, b)x 2 + 3E 2(k) (a, b)x + E 3(k) (a, b). =

4 Multi Poly-Bernoulli Numbers and Polynomials with Parameters a, b, c The multi poly-Bernoulli numbers was first introduced by Imatomi et al. [19] using the concept of multiple polylogarithm also known as multiple zeta values, which is given by  z mr Li(k1 ,k2 ,...,kr ) (z) = . (57) k1 k2 kr 0m r >0

m k11 m k12 . . . m rkr

.

(60)

Parallel to the definition of generalized poly-Bernoulli polynomials with parameters a, b, c in (4), we have the following generalization of multi poly-Bernoulli numbers in polynomial form. Definition 9 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b, c, denoted by Bn(k1 ,k2 ,...,kr ) (x; a, b, c), are defined by ∞

Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) r xt  (k1 ,k2 ,...,kr ) tn . c = B (x; a, b, c) n (bt − a −t )r n! n=0

(61)

When x = 0, the numbers Bn(k1 ,k2 ,...,kr ) (a, b) := Bn(k1 ,k2 ,...,kr ) (0; a, b, c) are called multi poly-Euler numbers with parameters a, b, which are given by ∞

Li(k1 ,k2 ,...,kr ) (1 − (ab)−t )  (k1 ,k2 ,...,kr ) tn . = B (a, b) n (bt − a −t )r n! n=0

(62)

One can easily prove the following theorem using the same argument in deriving the identities in Theorem 2. Theorem 21 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b, c satisfy the following identities.

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

707

  ∞  n  n l = (r log c) S(l, m) × l m=0 l=m

Bn(k1 ,k2 ,...,kr ) (x; a, b, c)

(k1 ,k2 ,...,kr ) × Bn−l (−m log c; a, b)(x)(m)   ∞  n  n (k1 ,k2 ,...,kr ) l Bn (x; a, b, c) = (r log c) S(l, m) × l m=0 l=m (k1 ,k2 ,...,kr ) (0; a, b)(x)(m) × Bn−l n−m  ∞   n−m  n  l (k1 ,k2 ,...,kr ) Bn (x; a, b, c) =   S(l + s, s)× m l=0 l+s l m=0 (k1 ,k2 ,...,kr ) × Bn−m−l (0; a, b)Bm(s) (xr log c) n  s   n   s m (k1 ,k2 ,...,kr ) (−λ)s− j × (x; a, b, c) = Bn s j (1 − λ) m=0 j=0 (k1 ,k2 ,...,kr ) × Bn−m ( j; a, b)Hm(s) (xr log c; λ).

The next theorem contains an explicit formula for Bn(k1 ,k2 ,...,kr ) (x; a, b, c). Theorem 22 ([13]) For k ∈ Z, n ≥ 0, we have Bn(k1 ,k2 ,...,kr ) (x; a, b, c) =



1

× . . . m rkr   m r −r  j mr − r × (r x − j ln a − ( j + 1) ln b)n . (−1) j j=0

k1 k2 m r >···>m 1 >0 m 1 m 2

Proof Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) = b−r t (bt − a −t )r

=b =

1

m r >···>m 1 >0

m k11 m k22 . . . m rkr



1

m r −r 

m r >···>m 1 >0

m k11 m k22 . . . m rkr

j=0

So, we get Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) xr t ln c e (bt − a −t )r



(1 − (ab)−t )m r −r

m r >···>m 1 >0

m k11 m k22 . . . m rkr

  m r − r − jt ln(ab) (−1) e j j=0

m r −r 



−r t



j



(−1)



j

 m r − r −t ( j ln a+( j+1) ln b) e . j

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R. B. Corcino



1

m r −r 

m r >···>m 1 >0

m k11 m k22 . . . m rkr

j=0

=

=

∞ 

⎛

.

1

m r >···>m 1 >0

m k11 m k22 . . . m rkr

  mr − r ×. (−1) j j=0

× (r x − j ln a − ( j + 1) ln b)n

By comparing the coefficients of

tn n!

  m r − r t (r x− j ln a−( j+1) ln b) e j

m r −r 



⎝

n=0

(−1) j

j

 tn n!

on both sides, the proof is completed.



The next theorem contains an expression of Bn(k1 ,k2 ,...,kr ) (x; a, b, c) as polynomial in x. Theorem 23 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b, c satisfy the following relation Bn(k1 ,k2 ,...,kr ) (x; a, b, c) =

n    n

i

i=0

(r ln c)n−i Bi(k1 ,k2 ,...,kr ) (a, b)x n−i

(63)

Proof Using (61), we have ∞ 

Bn(k1 ,k2 ,...,kr ) (x; a, b, c)

n=0

= e xr t ln c

∞ 

Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) r xt tn = c n! (bt − a −t )r

Bn(k1 ,k2 ,...,kr ) (a, b)

n=0

= =

∞  n  (xr t ln c)n−i

tn n!

Bi(k1 ,k2 ,...,kr ) (a, b)

(n − i)!

 tn n n−i (k1 ,k2 ,...,kr ) n−i (r ln c) Bi . (a, b)x i n!

n=0 i=0 n  ∞   n=0

i=0

ti i!

Comparing the coefficients of

tn , n!

we obtain the desired result.



Note that, when a = c = e and b = 1, Definition 9 reduces to ∞

Li(k1 ,k2 ,...,kr ) (1 − e−t ) r xt  (k1 ,k2 ,...,kr ) t n e = Bn (x) . (1 − e−t )r n! n=0

(64)

In the following theorem, the polynomials Bn(k1 ,k2 ,...,kr ) (x; a, b, c) are expressed in terms of Bn(k1 ,k2 ,...,kr ) (x).

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

709

Theorem 24 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b, c satisfy the following relation Bn(k1 ,k2 ,...,kr ) (x; a, b, c)

= (ln a + ln b)

n

Bn(k1 ,k2 ,...,kr )



xr ln c − r ln b ln a + ln b

 (65)

Proof Using (61), we have ∞ 

Bn(k1 ,k2 ,...,kr ) (x; a, b, c)

n=0

=e =

xr ln c−r ln b t ln ab

∞ 

Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) xr t ln c tn = e n! br t (1 − (ab)−t )r

−t ln ab ) ln ab Li(k1 ,k2 ,...,kr ) (1 − e −t ln ab 1+e

(ln a + ln b)

n

Bn(k1 ,k2 ,...,kr )



n=0

Comparing the coefficients of

tn , n!

xr ln c − r ln b ln a + ln b



tn . n! 

we obtain the desired result.

The next theorem contains a differential equation involving the generalized polyBernoulli polynomials with parameters a, b, c. Theorem 25 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b, c satisfy the following differential equation d (k1 ,k2 ,...,kr ) B (x; a, b, c) = (n + 1)(r ln c)Bn(k1 ,k2 ,...,kr ) (x; a, b, c) d x n+1

(66)

Proof Using (61), we have ∞  t (r ln c)Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) xr t ln c d (k1 ,k2 ,...,kr ) tn Bn (x; a, b, c) = e dx n! (bt − a −t )r n=0 ∞ ∞   d (k1 ,k2 ,...,kr ) t n−1 tn Bn = (x; a, b, c) (r ln c)Bn(k1 ,k2 ,...,kr ) (x; a, b, c) . dx n! n! n=0 n=0

Hence, ∞  n=0

∞

 1 d (k1 ,k2 ,...,kr ) tn tn Bn+1 (x; a, b, c) = (r ln c)Bn(k1 ,k2 ,...,kr ) (x; a, b, c) . n + 1 dx n! n! n=0

Comparing the coefficients of

tn , n!

we obtain the desired result.



The following corollary immediately follows from Theorem 25 by taking c = e1/r .

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R. B. Corcino

Corollary 26 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b are Appell polynomials in the sense that d (k1 ,k2 ,...,kr ) B (x; a, b, e1/r ) = (n + 1)Bn(k1 ,k2 ,...,kr ) (x; a, b, e1/r ) d x n+1

(67)

The following corollary contains an addition formula for the generalized multi poly-Bernoulli polynomials, which can be obtained using the characterization of Appell polynomials [33, 41, 43]. Corollary 27 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b satisfy the following addition formula Bn(k1 ,k2 ,...,kr ) (x + y; a, b, e1/r ) =

n    n i=0

i

Bi(k1 ,k2 ,...,kr ) (x; a, b, e1/r )y n−i

(68)

The above addition formula only works for the generalized multi poly-Bernoulli polynomials with two parameters a, b. However, using the method of generating function, we can derive the addition formula for the generalized multi poly-Bernoulli polynomials with three parameters a, b, c as follows ∞ 

Bn(k1 ,k2 ,...,kr ) (x + y; a, b, c)

n=0

Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) (x+y)r t tn = c n! (bt − a −t )r

Li(k1 ,k2 ,...,kr ) (1 − (ab)−t ) xr t yr t c c (bt − a −t )r ∞

∞

n n   t t = Bn(k1 ,k2 ,...,kr ) (x; a, b, c) (yr ln c)n n! n! n=0 n=0

∞ n     tn n n−i (k1 ,k2 ,...,kr ) . (yr ln c) Bi (x; a, b, c) = n! i n=0 i=0 =

Comparing the coefficients of

tn n!

yields the following addition formula.

Theorem 28 ([13]) The generalized multi poly-Bernoulli polynomials with parameters a, b, c satisfy the following addition formula Bn(k1 ,k2 ,...,kr ) (x + y; a, b, c) =

n    n (r ln c)n−i Bi(k1 ,k2 ,...,kr ) (x; a, b, c)y n−i . i i=0

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

711

Being classified as Appell polynomials, the generalized multi poly-Bernoulli polynomials Bn(k1 ,k2 ,...,kr ) (x; a, b, e1/r ) must satisfy Bn(k1 ,k2 ,...,kr ) (x; a, b, e1/r ) =

n    n i=0

Bn(k1 ,k2 ,...,kr ) (x; a, b, e1/r )

=

i

∞  ci i=0

i!

ci x n−i

D

i

xn.

for some scalar ci = 0. Using Theorem 23, ci = Bi(k1 ,k2 ,...,kr ) (a, b). This implies the following theorem. Theorem 29 The generalized multi poly-Bernoulli polynomials satisfy the following formula ∞

 B (k1 ,k2 ,...,kr ) (a, b) i (k1 ,k2 ,...,kr ) 1/r i D xn. Bn (x; a, b, e ) = i! i=0 There are other forms of multi-parameter generalizations of Bernoulli numbers. One of these is the universal Bernoulli numbers [2], which are defined as coefficients of the following generating function ∞

 tn t = Bˆ n , G(t) n! n=0 where G(t) is the compositional inverse of the function F(t) = t + c1 t 2 /2 + c2 t 3 /3 + ...

5 Multi Poly-Euler Numbers and Polynomials with Parameters a, b, c The multi poly-Euler numbers and polynomials were first introduced in the paper by Jolany et al. [16]. These numbers and polynomials were also defined by means of the multiple polylogarithm. The following definition states formally the multi poly-Euler numbers and polynomials. Definition 10 ([16]) The multi poly-Euler polynomials, denoted by E n(k1 ,k2 ,...,kr ) (x), are defined by ∞

2Li(k1 ,k2 ,...,kr ) (1 − e−t ) r xt  (k1 ,k2 ,...,kr ) t n e = En (x) . (1 + et )r n! n=0

(69)

712

R. B. Corcino

When x = 0, the numbers E n(k1 ,k2 ,...,kr ) := E n(k1 ,k2 ,...,kr ) (0) are called multi poly-Euler numbers, which are given by ∞

2Li(k1 ,k2 ,...,kr ) (1 − e−t )  (k1 ,k2 ,...,kr ) t n . = En (1 + et )r n! n=0

(70)

These numbers and polynomials have possessed the following explicit formula: Theorem 30 ([16]) The multi poly-Euler polynomials E n(k1 ,k2 ,...,kr ) (x) equal E n(k1 ,k2 ,...,kr ) (x) =

n 



mr  2(r x − j)n−i r !(−1) j+c1 +2c2 +...

i=0 0≤m 1 ≤m 2 ≤···≤m r j=0 c1 +c2 +···=r

(c1 !c2 ! . . .)(m k11 m k22 . . . m rkr )

   mr n . × (c1 + 2c2 + · · · ) j i i

When x = 0, we have n 

E n(k1 ,k2 ,...,kr ) =



mr  (2(− j)n−i r !(−1) j+c1 +2c2 +···

i=0 0≤m 1 ≤m 2 ≤···≤m r j=0 c1 +c2 +···=r

× (c1 + 2c2 + . . .)i

(c1 !c2 ! . . .)(m k11 m k22 . . . m rkr )

×

   mr n . j i

Proof Using the definition of multiple polylogarithm, we have  Li(k1 ,k2 ,...,kr ) (1 − e−t ) r xt (1 − e−t )m r r xt e = e k1 k2 kr (1 + et )r 0 0 (see Fig. 1a). Thus, the function φ(x) is an increasing function. Also, from the graphic of the function φ(x) or from the computational solution of the equation 1 2x + 1 − (x + 2)e x+1 = 0, we see that x0 = 2.4898 is a numerical root of this equation (see Fig. 1b and Eq. 1). 1 Equation 1. 2x + 1 − (x + 2)e x+1 = 0. Computational numerical solution is: x0 = 2.4898. Therefore, 2μ + 1 − (μ + 2)e1/(μ+1) ≥ 0 for every μ ≥ x0 . Thus, the proof of Corollary 2 is completed. By setting β = 1 in Theorem 1 and using the second relationship in (6), we arrive at the following corollary.

730

N. Mustafa et al.

Fig. 1 a Graphic of y = φ  (x) = 2 − 1

1 x 2 +x−1 x+1 e . (x+1)2

b Graphic of y = φ(x) = 2x + 1 −

(x + 2)e x+1

Corollary 3 The normalized Wright function W1 (λ, μ; z) belongs to the class T C(α) (α ∈ [0, 1)) if λ ≥ 1, μ ≥ μ0 = 0.462 and the following condition is satisfied:

Geometric Properties of Normalized Wright Functions

731

 1 (1 − α)(μ + 1)(2μ + 1) − (1 − α)(μ + 1)2 + (3 − α)(μ + 1) + 1 e μ+1 ≥ 0. By taking α = 0 in Corollary 3, we obtain the following corollary. Corollary 4 The normalized Wright function W1 (λ, μ; z) belongs to the class TC if λ ≥ 1 and μ ≥ x1 . Here, x1 = 4.8523 is the numerical root of the equation 1

2x 2 + 3x + 1 − (x 2 + 5x + 5)e x+1 = 0. 1

Proof Let ψ(x) = 2x 2 + 3x + 1 − (x 2 + 5x + 5)e x+1 , x > 0. By simple computation, we get x(2x 2 + 8x + 7) 1 e x+1 . ψ  (x) = 4x + 3 − (x + 1)2 From the graphic of this function, we exact can see that ψ  (x) > 0 for each x > 1.25 (see Fig. 2a). Hence, the function ψ(x) is an increasing function for x > 1.25. Also, as it is seen from the graphic of the function ψ(x) or from the computational solution of the equation 1

2x 2 + 3x + 1 − (x 2 + 5x + 5)e x+1 = 0 x1 = 4.8523 is a numerical root of this equation (see Fig. 2b and Eq. 2). 1 Equation 2. 2x 2 + 3x + 1 − (x 2 + 5x + 5)e x+1 = 0. Computational numerical solution is: x1 = 4.8523. 1 Therefore, 2μ2 + 3μ + 1 − (μ2 + 5μ + 5)e μ+1 ≥ 0 for every μ ≥ x1 . Thus, the proof of Corollary 4 is completed. Theorem 2 Let λ ≥ 1, μ > 0 and the following condition is satisfied: (1 − α)(λ + μ) + (λ + μ + 1) [(1 − (1 − β)α)(λ + μ + 2) + (1 − αβ)]  1

− (1 − (1 − β)α)(λ + μ + 1)2 + (1 − αβ)(λ + μ + 1) + β e λ+μ+1 ≥ 0. Then, the normalized Wright function W2 (λ, μ; z) belongs to the class T (α, β) (α ∈ [0, 1) , β ∈ [0, 1)). Proof Since W2 (λ, μ; z) = z −

∞  n=2

zn Γ (λ + μ) Γ (λ(n − 1) + λ + μ) n!

by virtue of Lemma 1, it suffices to show that ∞  n=2

(n − α)(β(n − 1) + 1)

1 Γ (λ + μ) ≤ 1 − α. Γ (λ(n − 1) + λ + μ) n!

(16)

732

N. Mustafa et al.

Fig. 2 a Graphic of y = ψ  (x) = 4x + 3 − 3x + 1 − (x 2 + 5x + 5)e

1 x+1

1 x(2x 2 +8x+7) x+1 e . (x+1)2

b Graphic of y = ψ(x) = 2x 2 +

Geometric Properties of Normalized Wright Functions

733

Let ∞ 

L 2 (λ, μ; α, β) =

(n − α)(β(n − 1) + 1)

n=2

1 Γ (λ + μ) . Γ (λ(n − 1) + λ + μ) n!

We can easily write: (n − α)(β(n − 1) + 1) = βn(n − 1) + (1 − αβ)n − (1 − β)α. In that case, by simple computation, we have L 2 (λ, μ; α, β) =

∞ 

Γ (λ + μ) β (n − 2)! Γ (λ(n − 1) + λ + μ) n=2  ∞  1 1 Γ (λ + μ) 1 − αβ − + n (n − 1)! Γ (λ(n − 1) + λ + μ) n=2 +

∞  1 − (1 − β)α

n!

n=2

Γ (λ + μ) . Γ (λ(n − 1) + λ + μ)

Using (14) and (15), with μ ≡ λ + μ, we obtain L 2 (λ, μ; α, β) ≤

∞ 

1 β (n − 2)! (λ + μ)(λ + μ + 1)n−2

n=2 ∞ 

+

n=2

+

∞  1 − (1 − β)α n=2

=

1 1 − αβ (n − 1)! (λ + μ)(λ + μ + 1)n−2 n!

1 (λ + μ)(λ + μ + 1)n−2

1 1 β (1 − αβ)(λ + μ + 1) λ+μ+1 − 1) e λ+μ+1 + (e λ+μ λ+μ   1 1 (1 − (1 − β)α)(λ + μ + 1)2 λ+μ+1 e −1 . − + λ+μ λ+μ+1

Thus, (16) holds true if the following condition is satisfied:

1 (1 − (1 − β)α)(λ + μ + 1)2 (1 − αβ)(λ + μ + 1) β + + e λ+μ+1 λ+μ λ+μ λ+μ λ+μ+1 − [(1 − (1 − β)α)(λ + μ + 2) + (1 − αβ)] ≤ 1 − α. λ+μ This evidently completes the proof of Theorem 2. By setting β = 0 in Theorem 2 and using the first relationship in (6), we arrive at the following corollary.

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N. Mustafa et al.

Corollary 5 The normalized Wright function W2 (λ, μ; z) belongs to the class T S ∗ (α) (α ∈ [0, 1)) if λ ≥ 1, μ > 0 and the following condition is satisfied: (1 − α) [(λ + μ + 1) (λ + μ + 2) +λ + μ] + λ + μ + 1 1

− [(1 − α) (λ + μ + 1) + 1] (λ + μ + 1)e μ+1 ≥ 0. By taking α = 0 in Corollary 5, we obtain the following corollary. Corollary 6 The normalized Wright function W2 (λ, μ; z) belongs to the class T S ∗ if λ ≥ 1 and λ + μ ≥ x2 . Here, x2 = 1.7703 is the numerical root of the equation 1

x 2 + 5x + 3 − (x 2 + 3x + 2)e x+1 = 0. 1

Proof Let h(x) = x 2 + 5x + 3 − (x 2 + 3x + 2)e x+1 , x > 0. By simple computation, we get 2x 2 + 4x + 1 1 e x+1 . h  (x) = 2x + 5 − x +1 As it is seen from the graphic of this function h  (x) > 0 (see Fig. 3a). Thus, the function h(x) is an increasing function. Also, from the graphic of the function h(x) or from the computational solution of the equation 1 x 2 + 5x + 3 − (x 2 + 3x + 2)e x+1 = 0, we see that x2 = 1.7703 is a numerical root of this equation (see Fig. 3b and Eq. 3). 1 Equation 3. x 2 + 5x + 3 − (x 2 + 3x + 2)e x+1 = 0. Computational numerical solution is: x2 = 1.7703. Therefore, 1

2(λ + μ) + (λ + μ + 1)(λ + μ + 2) + 1 − (λ + μ + 1)(λ + μ + 2)e λ+μ+1 ≥ 0 for every λ + μ ≥ x2 . Thus, the proof of Corollary 6 is completed. By setting β = 1 in Theorem 2 and using the second relationship in (6), we arrive at the following corollary. Corollary 7 The normalized Wright function W2 (λ, μ; z) belongs to the class T C(α) (α ∈ [0, 1)) if λ ≥ 1, μ > 0 and the following condition is satisfied: (1 − α) [2(λ + μ) + 1] + (λ + μ + 1)(λ + μ + 2)  1

− (1 − α)(λ + μ + 1) + (λ + μ + 1)2 + 1 e λ+μ+1 ≥ 0.

Geometric Properties of Normalized Wright Functions

Fig. 3 a Graphic of y = h  (x) = 2x + 5 − 3 − (x 2

+ 3x + 2)e

1 2x 2 +4x+1 x+1 e . x+1

735

b Graphic of y = h(x) = x 2 + 5x +

1 x+1

By taking α = 0 in Corollary 7, we obtain the following corollary. Corollary 8 The normalized Wright function W2 (λ, μ; z) belongs to the class TC if λ ≥ 1 and λ + μ ≥ x3 . Here, x3 = 2.9689 is the numerical root of the equation

736

N. Mustafa et al. 1

x 2 + 5x + 3 − (x 2 + 3x + 3)e x+1 = 0.

(17)

1

Proof Let ω(x) = x 2 + 5x + 3 − (x 2 + 3x + 3)e x+1 , x > 0. By simple computation, we get x(2x 2 + 6x + 5) 1 e x+1 . ω (x) = 2x + 5 − (x + 1)2 From the graphic of this function, we see that ω (x) > 0 (see Fig. 4a). Hence, the function ω(x) is an increasing function. Also, as it is seen from the graphic of the function ω(x) or from the computational solution of the equation 1

x 2 + 5x + 3 − (x 2 + 3x + 3)e x+1 = 0 x3 = 2.9689 is a numerical root of this equation (see Fig. 4b and Eq. 4). 1 Equation 4. x 2 + 5x + 3 − (x 2 + 3x + 3)e x+1 = 0. Computational numerical solution is: x3 = 2.9686. Therefore,

 1 (λ + μ)2 + 5(λ + μ) + 3 − (λ + μ)2 + 3(λ + μ) + 3 e λ+μ+1 ≥ 0 for every λ + μ ≥ x3 . Thus, the proof of Corollary 8 is completed.

3 Sufficient Conditions for the Integrals Involving Normalized Wright Functions In this section, some sufficient conditions for the integrals involving the normalized Wright functions W1 (λ, μ; z) and W2 (λ, μ; z) are given. Let z G 1 (λ, μ; z) = 0

W1 (λ, μ; t) dt and G 2 (λ, μ; z) = t

z

W2 (λ, μ; t) dt, z ∈ U, t

0

(18) where W1 (λ, μ; z) and W2 (λ, μ; z) are functions, defined by (7) and (8), respectively. Note that G 1 , G 2 ∈ T . In the next theorems, we give sufficient conditions so that G 1 (λ, μ; z) and G 2 (λ, μ; z) are in the class T (α, β).

Geometric Properties of Normalized Wright Functions

Fig. 4 a Graphic of y = ω (x) = 2x + 5 − 3 − (x 2 + 3x + 3)e

1 x+1

1 x(2x 2 +6x+5) x+1 e . b Graphic of y (x+1)2

737

= ω(x) = x 2 + 5x +

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N. Mustafa et al.

Theorem 3 Let λ ≥ 1, μ ≥ μ0 = 0.462 and the following condition is satisfied: (1 − α)μ + [(1 − (1 − β)α)(μ + 2) + 1 − αβ] (μ + 1)  1

− (1 − (1 − β)α)(μ + 1)2 + (1 − αβ)(μ + 1) + β e μ+1 ≥ 0.

(19)

Then, the function G 1 (λ, μ; z) belongs to the class T (α, β) (α ∈ [0, 1) , β ∈ [0, 1)). Proof Our proof of Theorem 3 is similar of that of Theorem 2. Indeed, from the definition of function G 1 (λ, μ; z), we can easily see that G 1 (λ, μ; z) = z −

∞  n=2

zn Γ (μ) = W2 (λ, μ − λ; z). Γ (λ(n − 1) + μ) n!

Therefore, the details of the proof of Theorem 3 may be omitted. By setting β = 0 in Theorem 3 and using the first relationship in (6), we arrive at the following corollary. Corollary 9 The function G 1 (λ, μ; z) belongs to the class T S ∗ (α) (α ∈ [0, 1)) if λ ≥ 1, μ ≥ μ0 = 0.462 and the following condition is satisfied:

 1 (1 − α) (μ + 1)2 + 2μ + 1 + μ + 1 − [(1 − α)(μ + 1) + 1] (μ + 1)e μ+1 ≥ 0. By taking α = 0 in Corollary 9, we obtain the following corollary. Corollary 10 The function G 1 (λ, μ; z) belongs to the class T S ∗ if λ ≥ 1 and μ ≥ x2 . Here, x2 = 1.7703 is the numerical root of the equation 1

x 2 + 5x + 3 − (x 2 + 3x + 2)e x+1 = 0. Proof The proof of Corollary 10 is very similar of the proof of Corollary 6. Therefore, the details of the proof of Corollary 9 may be omitted. By setting β = 1 in Theorem 3, and using the second relationship in (6), we arrive at the following corollary. Corollary 11 The function G 1 (λ, μ; z) belongs to the class T C(α) (α ∈ [0, 1)) if λ ≥ 1, μ ≥ μ0 = 0.462 and the following condition is satisfied: 1

(1 − α) (2μ + 1) + (μ + 1)(μ + 2)− [(μ + 2 − α) (μ + 1) + 1] e μ+1 ≥ 0. By taking α = 0 in Corollary 11, we obtain the following corollary. Corollary 12 Let λ ≥ 1 and μ ≥ x3 , where x3 = 2.9689 is the numerical root of the equation (17), then G 1 ∈ TC.

Geometric Properties of Normalized Wright Functions

739

Proof The proof of Corollary 12 is the same of the proof of Corollary 8. Theorem 4 Let λ ≥ 1, μ > 0 and the following condition is satisfied: (1 − α)(λ + μ) + (2 − (1 + β)α) (λ + μ + 1)(λ + μ + 2) + (λ + μ + 1)β 1 − [(2 − (1 + β)α) (λ + μ + 1) + β] (λ + μ + 1)e λ+μ+1 ≥ 0. (20) Then, the function G 2 (λ, μ; z) belongs to the class T (α, β) (α ∈ [0, 1) , β ∈ [0, 1)). Proof Since G 2 (λ, μ; z) = z −

∞  n=2

zn Γ (λ + μ) Γ (λ(n − 1) + λ + μ) n · n!

by virtue of Lemma 1, it suffices to show that ∞ 

(n − α)(β(n − 1) + 1)

n=2

1 Γ (λ + μ) ≤ 1 − α. Γ (λ(n − 1) + λ + μ) n · n!

(21)

Let L 3 (λ, μ; α, β) =

∞ 

(n − α)(β(n − 1) + 1)

n=2

1 Γ (λ + μ) . Γ (λ(n − 1) + λ + μ) n · n!

We can easily write: (n − α)(β(n − 1) + 1) = n 2 β + (1 − αβ)n − nβ − (1 − β)α. Hence, by simple computation, we get L 3 (λ, μ; α, β) =

 ∞   1 β Γ (λ + μ) 1− n (n − 1)! Γ (λ(n − 1) + λ + μ) n=2   ∞  1 1 − αβ Γ (λ + μ) 1− + n n! Γ (λ(n − 1) + λ + μ) n=2 +

∞  1−α n=2

Γ (λ + μ) . n · n! Γ (λ(n − 1) + λ + μ)

By using (14) and (15), with μ ≡ λ + μ, we get L 3 (λ, μ; α, β) ≤

∞ 

1 β (n − 1)! (λ + μ)(λ + μ + 1)n−2

n=2 ∞ 

+

n=2

1 1 − αβ n! (λ + μ)(λ + μ + 1)n−2

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+

∞  1−α n=2

=

1 n · n! (λ + μ)(λ + μ + 1)n−2

1 1 {{[2 − (1 + β)α] (λ + μ + 1) + β} (λ + μ + 1)e λ+μ+1 λ+μ − {[2 − (1 + β)α] (λ + μ + 2) + β} (λ + μ + 1)} .

We easily see that (21) holds true if the following condition is satisfied: 1 1 {{[2 − (1 + β)α] (λ + μ + 1) + β} (λ + μ + 1)e λ+μ+1 λ+μ

− {[2 − (1 + β)α] (λ + μ + 2) + β} (λ + μ + 1)} ≤ 1 − α, which is equivalent to (20). Thus, the proof of Theorem 4 is completed. By setting β = 0 in Theorem 4 and using the first relationship in (6), we arrive at the following corollary. Corollary 13 The function G 2 (λ, μ; z) belongs to the class T S ∗ (α) (α ∈ [0, 1)) if λ ≥ 1, μ > 0 and the following condition is satisfied: 1

(1 − α)(λ + μ) + (2 − α)(λ + μ + 1)(λ + μ + 2) − (2 − α)(λ + μ + 1)2 e λ+μ+1 ≥ 0.

By taking α = 0 in Corollary 13, we obtain the following corollary. Corollary 14 The function G 2 (λ, μ; z) belongs to the class T S ∗ if λ ≥ 1 and λ + μ ≥ x4 . Here, x4 = 1.1728 is the numerical root of the equation 1

2x 2 + 7x + 4 − 2(x + 1)2 e x+1 = 0. 1

Proof Let σ (x) = 2x 2 + 7x + 4 − 2(x + 1)2 e x+1 , x > 0. By simple computation, we get 1 σ  (x) = 4x + 7 − 2(2x + 1)e x+1 . From the graphic of this function, we see that σ  (x) > 0 (see Fig. 5a). Thus, the function σ (x) is an increasing function. Also, as it is seen from the graphic of the function σ (x) or from the computational solution of the equation 1

2x 2 + 7x + 4 − 2(x + 1)2 e x+1 = 0 x4 = 1.1728 is a numerical root of this equation (see Fig. 5b and Eq. 5).

Geometric Properties of Normalized Wright Functions

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1

Fig. 5 a Graphic of y = σ  (x) = 4x + 7 − 2(2x + 1)e x+1 . b Graphic of y = σ (x) = 2x 2 + 7x + 1 4 − 2(x + 1)2 e x+1

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Equation 5. 2x 2 + 7x + 4 − 2(x + 1)2 e x+1 = 0. Computational numerical solution is: x4 = 1.1728. Therefore, 1

(λ + μ) − 2(λ + μ + 1)2 e λ+μ+1 + 2(λ + μ + 1)(λ + μ + 2) ≥ 0 for every λ + μ ≥ x4 . Thus, the proof of Corollary 14 is completed. By setting β = 1 in Theorem 4, and using the second relationship in (6), we arrive at the following corollary. Corollary 15 The function G 2 (λ, μ; z) belongs to the class T C(α) (α ∈ [0, 1)) if λ ≥ 1, μ > 0 and the following condition is satisfied: (1 − α) [2(λ + μ + 1)(λ + μ + 2) + λ + μ] + λ + μ + 1 1

−(λ + μ + 1) [2(1 − α)(λ + μ + 1) + 1] e λ+μ+1 ≥ 0. By taking α = 0, in Corollary 15, we obtain the following corollary. Corollary 16 The function G 2 (λ, μ; z) belongs to the class TC if λ ≥ 1 and λ + μ ≥ x5 . Here, x5 = 2.2791 is the numerical root of the equation 1

2x 2 + 8x + 5 − (2x 2 + 5x + 3)e x+1 = 0. Proof The proof of Corollary 16 is very similar of the proof of the above corollaries. Therefore, the proof of this corollary may be omitted.

4 Close-to-Convexity of Normalized Wright Functions We will consider again Wright function given by the Eq. 1 as in the introduction section of our chapter. Also, as in previous sections, we let A be the class of analytic in the open unit disk U = {z ∈ C : |z| < 1} functions f (z), normalized by f (0) = 0 = f  (0) − 1 of the form f (z) = z + a2 z + a3 z + · · · + an z + · · · = z + 2

3

n

∞ 

an z n .

(22)

n=2

Also, let S ∗ (α), C(α) and K (α) denote the subclasses of A consisting of functions which are, respectively, starlike, convex and close-to-convex with respect to starlike function g(z) (need not be normalized) of order α (α ∈ [0, 1)) in the open unit disk U . Thus, we have (see for details, [13, 15], also [47])

Geometric Properties of Normalized Wright Functions

S ∗ (α) =  C(α) = 

and K (α) =

 f ∈ A : (

z f  (z) ) > α, z ∈ U f (z)

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z f  (z) f ∈ A : (1 +  ) > α, z ∈ U f (z)

, α ∈ [0, 1) , 

z f  (z) ) > α, z ∈ U, g ∈ S ∗ f ∈ A : ( g(z)

, α ∈ [0, 1)  , α ∈ [0, 1) .

In the special cases, S ∗ = S ∗ (0), C = C(0) and K = K (0), are, respectively, starlike, convex and close-to-convex functions in U . It is well known that close-toconvex functions are univalent in U , but not necessarily the converse. It is easy to verify that C ⊂ S ∗ ⊂ K . For details on these classes, one could refer to the monograph by Goodman [15]. An interesting generalization of the function classes S ∗ (α), C(α) and K (α) are provided by the classes S ∗ (α, β), C(α, β) and K (α, β) of functions f ∈ A, which satisfies the following conditions: S ∗ (α, β) =



 C(α, β) =

f ∈ A : (

z f  (z) + βz 2 f  (z) ) > α, z ∈ U f (z)

 , α, β ∈ [0, 1) ,

  z f  (z) + βz 2 f  (z) ) > α, z ∈ U , α, β ∈ [0, 1) f ∈ A : ( f  (z)

and  K (α, β) =

z f  (z) + βz 2 f  (z) ) > α, z ∈ U, g ∈ S ∗ (α,β) f ∈ A : ( g(z)

 , α, β ∈ [0, 1)

with respect to function g(z) (need not be normalized), respectively. Note 1 The class K (α, β), α, β ∈ [0, 1) is the first time introduced and examined in the paper by the first author [25]. Clearly, for β = 0, we have K (α, 0) = K (α). Recall from the introduction section that Wright function Wλ,μ (z) defined by (1) does not belong to the class A. Thus, we consider the following two kinds of normalization of the Wright function: (1) Wλ,μ (z) := Γ (μ)zWλ,μ (z) =

∞  n=0

Γ (μ) z n+1 , λ > −1, μ > 0, z ∈ U Γ (λn + μ) n!

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and (2) (z) : = Γ (λ + μ) Wλ,μ (z) − Wλ,μ ∞ 

=

n=0

1 Γ (μ)



z n+1 Γ (λ + μ) , λ > −1, λ + μ > 0, z ∈ U. Γ (λn + λ + μ) (n + 1)!

Easily, we write (1) Wλ,μ (z) = z +

∞  n=2

(2) (z) = z + Wλ,μ

∞  n=2

zn Γ (μ) , λ > −1, μ > 0, z ∈ U, Γ (λ(n − 1) + μ) (n − 1)!

(23)

zn Γ (λ + μ) , λ > −1, λ + μ > 0, z ∈ U. (24) Γ (λ(n − 1) + λ + μ) n!

(1) (2) Furthermore, we observe that Wλ,μ and Wλ,μ are satisfying the following relations: (1) (1) (1) λz(Wλ,μ (z)) = (μ − 1)Wλ,μ−1 (z) + (λ − μ + 1)Wλ,μ (z),

(25)

(2) (2) (2) λz(Wλ,μ (z)) = (λ + μ − 1)Wλ,μ−1 (z) + (1 − μ)Wλ,μ (z),

(26)

Γ (μ) Vλ,λ+μ (z), Γ (λ + μ)

(27)

√ − W1,(1)p+1 (−z) = J¯p (z) := Γ ( p + 1)z 1− p/2 J p (2 z), p > −1.

(28)

(2) (1)  z(Wλ,μ (z)) = Wλ,λ+μ (z) and Vλ,μ (z) =

where Vλ,μ (z) = Note that

(1) Wλ,μ (z) . z

Here, J¯p (z) is the normalized Bessel function. In this section, we give sufficient conditions for the parameters of the normalized Wright functions to be in the class K (α, β). The following lemma will be required. Lemma 2 ([24]) A function f ∈ A belongs to the class S ∗ (α, β) if ∞  n=2

(n + βn(n − 1) − α) |an | ≤ 1 − α.

Geometric Properties of Normalized Wright Functions

745

4.1 Sufficient Conditions for the Class K (α, β) In this section, we will give sufficient conditions for the parameters of the normalized Wright functions to be in the class K (α, β). Theorem 5 Let λ ≥ 1, μ ≥ μ0 = 0.462 and the following condition is satisfied:

 1 (1 − α)(μ + 1)μ − (μ + 1)2 + (1 + 2β)(μ + 1) + β e μ+1 + (μ + 1)2 > 0. (1) Then, normalized Wright function Wλ,μ (z) belongs to the class K (α, β).  (1) (1) (z)) + βz(Wλ,μ (z)) > Proof By the definition, we need to show that (Wλ,μ α, z ∈ U , this can be shown by proving

   (1)  (1) (Wλ,μ (z)) + βz(Wλ,μ (z)) − 1 < 1 − α, z ∈ U. By simple computation, we obtain     (1)   ∞ (1) (Wλ,μ (z)) + βz(Wλ,μ (z)) − 1 =  n=2 ∞ βn 2 +(1−β)n Γ (μ) ≤ n=2 (n−1)! Γ (λ(n−1)+μ) . Let L 1 (λ, μ; β) =

∞  βn 2 + (1 − β)n n=2

(n − 1)!



 n+n(n−1)β Γ (μ) z n−1  (n−1)! Γ (λ(n−1)+μ)

Γ (μ) . Γ (λ(n − 1) + μ)

Setting n 2 = (n − 1)(n − 2) + 3(n − 1) + 1, n = (n − 1) + 1 and by simple computation, we write  ∞ β Γ (μ) L 1 (λ, μ; β) = ∞ n=3 (n−3)! Γ (λ(n−1)+μ) + n=2 ∞ Γ (μ) 1 + n=2 (n−1)! . Γ (λ(n−1)+μ)

1+2β Γ (μ) (n−2)! Γ (λ(n−1)+μ)

Under the hypothesis λ ≥ 1, the inequality Γ (n − 1 + μ) ≤ Γ (λ(n − 1) + μ) for n ∈ N holds, which is equivalent to Γ (μ) 1 ≤ , n ∈ N. Γ (λ(n − 1) + μ) (μ)n−1

(29)

=μ(μ + 1)(μ + 2) · · · (μ + n − 1), (μ)0 = 1 is Pochhammer Here, (μ)n = ΓΓ(n+μ) (μ) (or Appell) symbol, defined in terms of Euler gamma function. Using (29), we get

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L 1 (λ, μ; β) ≤

∞  n=3

∞

∞

 1 + 2β  1 1 1 β 1 + + . (n − 3)! (μ)n−1 n=2 (n − 2)! (μ)n−1 n=2 (n − 1)! (μ)n−1

Further, the inequality (μ)n−1 = μ(μ + 1)(μ + 2) · · · (μ + n − 2) ≥ μ(μ + 1)n−2 , n ∈ N

(30)

holds, which is equivalent to 1/(μ)n−1 ≤ 1/μ(μ + 1)n−2 , n ∈ N. Using (30), we obtain  ∞ 1+2β ∞ β 1 1 1 1 L 1 (λ, μ; β) ≤ ∞ n=3 (n−3)! μ(μ+1)n−2 + n=2 (n−2)! μ(μ+1)n−2 + n=2 (n−1)! μ(μ+1)n−2   1 β μ+1 = μ(μ+1) + 1+2β e μ+1 − μ+1 μ + μ μ < 1 − α,

which is equivalent to  1

(1 − α)(μ + 1)μ − (μ + 1)2 + (1 + 2β)(μ + 1) + β e μ+1 + (μ + 1)2 > 0. Thus, the proof of Theorem 5 is complete. By setting β = 0 in Theorem 5 and using the relationship K (α, 0) = K (α), we arrive at the following corollary. (1) Corollary 17 The normalized Wright function Wλ,μ (z) belongs to the class K (α) if λ ≥ 1, μ ≥ μ0 = 0.462 and the following condition is satisfied: 1

(2 − α)μ − (μ + 2)e μ+1 + 1 > 0. By taking α = 0 in Corollary 17 and using the relationship K (0) = K , we obtain the following corollary. Corollary 18 Let λ ≥ 1 and μ > x0 , where x0 ∼ = 2.4898 is the root of the equation 1

2x − (x + 2)e x+1 + 1 = 0, (1) ∈ K. then Wλ,μ

Proof Let ϕ(x) = 2x − (x + 2)e1/(x+1) + 1, x > 0. By simple computation, we obtain x + 2 1/(x+1) e + 2 − e1/(x+1) . ϕ  (x) = (x + 1)2 Easily, we see that ϕ  (x) > 0 for each x > 0. Thus, function ϕ(x) is an increasing function. Hence, 2μ − (μ + 2)e1/(μ+1) + 1 > 0 for every μ > x0 . Here, x0 ∼ = 2.4898 is the root of the equation

Geometric Properties of Normalized Wright Functions

747

1

2x − (x + 2)e x+1 + 1 = 0. Thus, the proof of Corollary 18 is complete. Theorem 6 Let λ ≥ 1, λ + μ > 0 and the following condition is satisfied: (1 − α)(λ + μ) 1 1 −(e λ+μ+1 − 1)(λ + μ + 1) − βe λ+μ+1 > 0. (2) Then, normalized Wright function Wλ,μ (z) belongs to the class K (α, β).  (2) (2) Proof By the definition, we need to show that (Wλ,μ (z)) + βz(Wλ,μ (z)) > α, z ∈ U , this can be shown by proving

   (2)  (2) (Wλ,μ (z)) + βz(Wλ,μ (z)) − 1 < 1 − α, z ∈ U. (2) Since (Wλ,μ (z)) =

(1) Wλ,λ+μ (z) , z

by simple computation, we write

   (1) W (z)  (2)  (2) (1) − 1 + β (Wλ,λ+μ (z)) − (Wλ,μ (z)) + βz(Wλ,μ (z)) − 1 =  λ,λ+μ z     β Γ (λ+μ) 1 n−1  = ∞  n=2 (n−2)! + (n−1)! Γ (λ(n−1)+λ+μ) z ∞  β Γ (λ+μ) 1 ≤ n=2 (n−2)! + (n−1)! Γ (λ(n−1)+λ+μ) .

(1) Wλ,λ+μ (z) z

   

Under the hypothesis λ ≥ 1, using (29) and (30) with μ ≡ λ + μ, we obtain       (2) (2) ∞ β (Wλ,μ (z)) + βz(Wλ,μ (z)) − 1 ≤ n=2 (n−2)! + =

1 λ+μ+β+1 λ+μ+1 e λ+μ

Thus,

−

1 (n−1)!



1 (λ+μ)(λ+μ+1)n−2

λ+μ+1 . λ+μ

1 λ+μ+1 λ + μ + β + 1 λ+μ+1 e < 1 − α, − λ+μ λ+μ

which is equivalent to 1

1

(1 − α)(λ + μ) − (e λ+μ+1 − 1)(λ + μ + 1) − βe λ+μ+1 > 0. Thus, the proof of Theorem 6 is complete. By setting β = 0 in Theorem 6 and using the relationship K (α, 0) = K (α), we arrive at the following corollary. (2) (z) belongs to the class K (α) Corollary 19 The normalized Wright function Wλ,μ if λ ≥ 1, λ + μ > 0 and the following condition is satisfied:

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1

(2 − α − e λ+μ+1 )(λ + μ) − e λ+μ+1 + 1 > 0. By taking α = 0 in Corollary 19 and using the relationship K (0) = K , we obtain the following corollary. Corollary 20 Let λ ≥ 1, λ + μ > 0 and λ + μ > x1 , where x1 ∼ = 1.2581 is the root of the equation 1 2x − (x + 1)e x+1 + 1 = 0, (2) ∈ K. then Wλ,μ

Proof Let h(x) = 2x − (x + 1)e1/(x+1) + 1, x > 0. By simple computation, we obtain x 1 e x+1 . h  (x) = 2 − x +1 It can be seen that h  (x) > 0 for each x > 0. Hence, function h(x)is an increasing 1 function. So 2(λ + μ) − (λ + μ + 1)e λ+μ+1 + 1 > 0 for every λ + μ > x1 . Here, x1 ∼ = 1.2581 is root of the equation 1

2x − (x + 1)e x+1 + 1 = 0. Thus, the proof of Corollary 20 is complete. Theorem 7 Let λ ≥ 1 and the following condition is satisfied: 





 1 1 1 (1 − α) (2 p + 3) − ( p + 2)e p+2 − 2(e p+2 − 1)( p + 2) + (1 + 2β)e p+2 ( p + 2) 1

−βe p+2 > 0. (1) Then, normalized Wright function Wλ, p+1 (z) belongs to the class K (α, β) with respect to J¯p (z) in U .

Proof By the definition, we need to show that 

(1) (1)  2  z(Wλ, p+1 (z)) + βz (Wλ, p+1 (z)) J¯p (z)

 > α, z ∈ U,

this can be shown by proving   (1) (1)  2    z(Wλ, p+1 (z)) + βz (Wλ, p+1 (z))    < 1 − α, z ∈ U. 1 −   J¯p (z)

Geometric Properties of Normalized Wright Functions

749

For z ∈ U , by simple computation, we obtain    J¯ (z)  (1)  (1) ∞ (Wλ, p+1 (z)) − pz + βz(Wλ, p+1 (z))  ≤ n=1  Γ ( p+1) n(n+1) +β ∞ n=1 Γ (λn+ p+1) n! .

Γ ( p+1) n!

  n+1  Γ (λn+ p+1) −

(−1)n Γ (n+ p+1)



  zn 

(31)

Under the hypothesis λ ≥ 1, the inequality Γ (n + p + 1) ≤ Γ (λn + p + 1) for each n ∈ N holds, which is equivalent to Γ ( p + 1) Γ ( p + 1) 1 , n ∈ N, ≤ = Γ (λn + p + 1) Γ (n + p + 1) ( p + 1)n

(32)

p+1) = ( p + 1)( p + 2)...( p + n). where ( p + 1)n = ΓΓ(n+ ( p+1) Further, the following inequality holds

1 1 ≤ , n ∈ N. ( p + 1)n ( p + 1)( p + 2)n−1

(33)

Using (32) and (33), from (31), we obtain   ∞    J¯p (z) 1 n+2  (1) (1)    + βz(Wλ, p+1 (z))  ≤ (Wλ, p+1 (z)) −   z n! ( p + 1)( p + 2)n−1

(34)

n=1

+β

∞  n(n + 1) 1 n! ( p + 1)( p + 2)n−1 n=1

∞  1 1 =2 n! ( p + 1)( p + 2)n−1 n=1

+(1 + 2β)

∞  n=1

+β

∞  n=2

1 1 (n − 1)! ( p + 1)( p + 2)n−1

1 1 (n − 2)! ( p + 1)( p + 2)n−1

1 1 2( p + 2) p+2 1 + 2β p+2 = − 1) + (e e p+1 p+1 1 β + e p+2 ( p + 1)( p + 2) ⎧ 1 ⎪ 2 ⎪ ⎨ 2(e p+2 − 1)( p + 2) 1 1 p+2 = ⎪ +(1 +1 2β)( p + 2)e ( p + 1)( p + 2) ⎪ ⎩ +βe p+2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

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Similarly, we obtain   ∞   J¯p (z)  Γ ( p + 1) 1  ≥ 1−   z  Γ (n + p + 1) n! n=1

(35)

∞  1 1 n! ( p + 1)( p + 2)n−1 n=1  1 1  ( p + 2)e p+2 − 1 . = 2− p+1

≥ 1−

From (34) and (35), we obtain 1

2(e p+2 − 1)( p + 2)2 1 +(1 + 2β)( p + 2)e p+2   1 (1) (1)  2    z(Wλ, +βe p+2 p+1 (z)) + βz (Wλ, p+1 (z))    . 1 − ≤  1 ¯   J p (z) (2 p + 3) − ( p + 2)e p+2 ( p + 2) Thus, we get 





 1 1 1 (1 − α) (2 p + 3) − ( p + 2)e p+2 − 2(e p+2 − 1)( p + 2) + (1 + 2β)e p+2 ( p + 2) 1

−βe p+2 > 0.

(36)

This shows that 

(1) (1)  2  z(Wλ, p+1 (z)) + βz (Wλ, p+1 (z)) J¯p (z)

 >α

if condition (36) is satisfied. For the completion of the proof of Theorem 7, we need to show that function J¯p (z) belongs to the class S ∗ (α, β) if condition (36) is satisfied. From (28), we write J¯p (z) = −W1,(1)p+1 (−z) = z +

∞  Γ ( p + 1) (−1)n+1 n z . Γ (n + p) (n − 1)! n=2

According to the Lemma 2, function J¯p (z) belongs to the class S ∗ (α, β) if the following condition is satisfied: ∞ 

n=2

n 2 β + n(1 − β) − α

 Γ ( p + 1) 1 ≤ 1 − α. Γ (n + p) (n − 1)!

Geometric Properties of Normalized Wright Functions

Let L 2 ( p; α, β) =

751

∞ 

2  Γ ( p + 1) 1 . n β + n(1 − β) − α Γ (n + p) (n − 1)! n=2

Using similar calculations to get L 1 (λ, μ; β), we write L 2 ( p; α, β) =

∞  n=3

∞

∞

n=2

n=2

β Γ ( p + 1)  1 + 2β Γ ( p + 1)  1 − α Γ ( p + 1) + + . (n − 3)! Γ (n + p) (n − 2)! Γ (n + p) (n − 1)! Γ (n + p)

The following equality is clear Γ ( p + 1) 1 = , n ∈ N. Γ (n + p) ( p + 1)n−1

(37)

Using (37) and (30), with μ ≡ p + 1, we get  ∞ 1+2β β 1 1 L 2 ( p; α, β) ≤ ∞ n=3 (n−3)! ( p+1)( p+2)n−2 + n=2 (n−2)! ( p+1)( p+2)n−2  1   (1−α)( p+2) β 1+2β 1−α 1 e p+2 + ∞ = + + n−2 n=2 (n−1)! ( p+1)( p+2) ( p+1)( p+2) p+1 p+1 p+2) − (1−α)( ≤ 1 − α, p+1

which is equivalent to    1 1 1 (1 − α) (2 p + 3) − ( p + 2)e p+2 − (1 + 2β)e p+2 ( p + 2) − βe p+2 ≥ 0. (38) Thus, condition (38) is a sufficient condition for that function J¯p (z) is in the class S ∗ (α, β). Also, from (36) the following condition is satisfied: 

  1 (1 − α) (2 p + 3) − ( p + 2)e p+2 −(1 + 2β)e

1 p+2

1

1

( p + 2) − βe p+2 > 2(e p+2 − 1)( p + 2)2 .

1

But since e p+2 > 1 for p > −1, condition (38) is also satisfied. Thus, from the hypothesis of theorem, J¯p ∈ S ∗ (α, β). The proof is complete. By setting β = 0 in Theorem 7 and using the relationship K (α, 0) = K (α), we arrive at the following corollary. Corollary 21 If λ ≥ 1 and the following condition is satisfied:     1 1 1 (1 − α) (2 p + 3) − ( p + 2)e p+2 − 2(e p+2 − 1)( p + 2) + e p+2 > 0,

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(1) ¯ then Wλ, p+1 (z) is close-to-convex function order α ∈ [0, 1)with respect to J p (z) in U .

By taking α = 0 in Corollary 21 and using relationship K (0) = K , we obtain the following corollary. Corollary 22 If λ ≥ 1 and p > x2 − 2, where x2 ∼ = 5.4904 is the root of the equation 1 4x − (3x + 1)e x − 1 = 0, (1) ¯ then Wλ, p+1 (z) is close-to-convex function with respect to J p (z) in U .

Proof Let ω(x) = 4x − (3x + 1)e1/x − 1, x > 0. By simple computation, we obtain 3x 2 − 3x − 1 1 ex . ω (x) = 4 − x2 We observe that ω (x) > 0 for each x > 0. Thus, function ϕ(x) is an increasing function. Hence, 4( p + 2) − (3 p + 7)e1/( p+2) − 1 > 0 for every p > x2 − 2. Here, x2 ∼ = 5.4904 is the root of the equation 1

4x − (3x + 1)e x − 1 = 0. Thus, the proof of Corollary 22 is complete.

5 Some Integral Operators of Normalized Wright Functions In this section, we will examine some geometric properties of the integral operators of normalized Wright functions. Recalling the geometric properties of normalized Wright function from the earlier sections, easily, we can write  (1) Γ (μ) zn (z) = z + ∞ Wλ,μ n=2 Γ (λ(n−1)+μ) (n−1)! , λ > −1, μ > 0, z ∈ U,

(39)

 (2) Γ (λ+μ) zn Wλ,μ (z) = z + ∞ n=2 Γ (λ(n−1)+λ+μ) n! , λ > −1, λ + μ > 0, z ∈ U.

(40)

(1) (2) and Wλ,μ are satisfying the following relations: Furthermore, observe that Wλ,μ (1) (1) λz(Wλ,μ (z)) = (μ − 1)Wλ,μ−1 (z) (1) (z), +(λ − μ + 1)Wλ,μ

Geometric Properties of Normalized Wright Functions

753

(2) (2) λz(Wλ,μ (z)) = (λ + μ − 1)Wλ,μ−1 (z) (2) (z), +(1 − μ)Wλ,μ (2) (1) (z)) = Wλ,λ+μ (z) and z(Wλ,μ Γ (μ)  Vλ,λ+μ (z) Vλ,μ (z) = Γ (λ + μ)

where Vλ,μ (z) = Let G (1) λ,μ (z) =

(1) Wλ,μ (z)

z

.

(41)

z

(1) Wλ,μ (t) dt t (2)  z Wλ,μ (t) = 0 t dt,

0

and G (2) λ,μ (z)

(42)

z ∈ U,

(1) (2) where Wλ,μ (z) and Wλ,μ (z) are functions defined by (39) and (40). (2) From (42) it is easy to verify that G (1) λ,μ , G λ,μ ∈ A. In the following theorems, we will give sufficient conditions so that functions (2) ∗ G (1) λ,μ (z) and G λ,μ (z) are in the classes S (α, β) and C(α, β), respectively.

Theorem 8 Let λ ≥ 1 and assume the condition 1

1

1

(1 − α)μ(μ + 2 − μe μ ) − μ(e μ − 1) − βe μ ≥ 0 ∗ then function G (1) λ,μ (z) belongs to the class S (α, β).

Proof Since G (1) λ,μ (z)

=z+

∞  n=2

zn Γ (μ) Γ ((n − 1)λ + μ) n!

by virtue of Lemma 2 given in Sect. 4, it suffices to show that ∞

Γ (μ) 1 [n + βn(n − 1) − α] Γ ((n−1)λ+μ) n! ≤ 1 − α. n=2

Let

L 1 (λ, μ; α, β)  Γ (μ) 1 = ∞ n=2 [n + βn(n − 1) − α] Γ ((n−1)λ+μ) n! .

(43)

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By simple computation, we write  β Γ (μ) L 1 (λ, μ; α, β) = ∞ n=2 (n−2)! Γ ((n−1)λ+μ)  1   Γ (μ) 1 + ∞ n=2 1 − n (n−1)! Γ ((n−1)λ+μ)  Γ (μ) 1−α + ∞ n=2 n! Γ ((n−1)λ+μ) . Under the hypothesis λ ≥ 1, the inequality Γ (n − 1 + μ) ≤ Γ ((n − 1)λ + μ) for n ∈ N, holds, which is equivalent to Γ (μ) 1 ≤ , n∈N Γ ((n − 1)λ + μ) (μ)n−1

(44)

(μ)n = ΓΓ(n+μ) (μ) = μ(μ + 1)(μ + 2) · · · (μ + n − 1), (μ)0 = 1 is Pochhammer (or Appell) symbol, defined in terms of Euler gamma function. Using (44), we obtain

where

 β 1 L 1 (λ, μ; α, β) ≤ ∞ n=2 (n−2)! (μ)n−1  ∞ (1−α) 1 1 1 + ∞ n=2 (n−1)! (μ)n−1 + n=2 n! (μ)n−1 . Further, the inequality (μ)n−1 = μ(μ + 1)(μ + 2) · · · (μ + n − 2) ≥ μn−1 , n ∈ N

(45)

holds, which is equivalent to 1/(μ)n−1 ≤ 1/μn−1 , n ∈ N. Using inequality (45), we obtain  β 1 L 1 (λ, μ; α, β) ≤ ∞ n=2 (n−2)! μn−1  ∞ 1−α 1 1 1 + ∞ n=2 (n−1)! μn−1 + n=2 n! μn−1 = μβ e μ + (e μ − 1) + (1 − α)μ(e μ − ≤ 1 − α, 1

1

1

μ+1 ) μ

which is equivalent to 1

1

1

(1 − α)μ(μ + 2 − μe μ ) − μ(e μ − 1) − βe μ ≥ 0. Thus, the proof of Theorem 8 is complete. By setting β = 0 in Theorem 8 and using the relationship S ∗ (α, 0) = S ∗ (α), we arrive at the following corollary.

Geometric Properties of Normalized Wright Functions

755

∗ Corollary 23 The function G (1) λ,μ (z) belongs to the class S (α) if λ ≥ 1 and the following condition is satisfied: 1

1

(1 − α)(μ + 2 − μe μ ) − (e μ − 1) ≥ 0. By taking α = 0 in Corollary 23 and using relationship S ∗ (0) = S ∗ , we arrive at the following corollary. ∗ Corollary 24 The function G (1) λ,μ (z) belongs to the class S if λ ≥ 1 and μ > x 1 . Here, x1 ≈ 1.9133 is the root of the equation 1

x + 3 − (x + 1)e x = 0. Proof Let ϕ(x) = x + 3 − (x + 1)e1/x , x > 0. By simple computation, we obtain ϕ  (x) = 1 −

x2 − x − 1 1 e x , x > 0. x2

We observe that ϕ  (x) > 0 for each x > 0. Thus, function ϕ(x) is an increasing function. 1 Hence, μ + 3 − (μ + 1)e μ > 0 for everyμ > x1 where x1 ≈ 1.9133 is the root of the equation 1 x + 3 − (x + 1)e x = 0. Thus, the proof of Corollary 24 is complete. Theorem 9 Let λ ≥ 1 and assume the condition 1

1

(1 − α)(2 − e μ )μ2 − (μ + (2μ + 1)β)e μ ≥ 0, then the function G (1) λ,μ (z) belongs to the class C(α, β). (1)  Proof The function G (1) λ,μ (z) belongs to the class C(α, β) if and only if z · (G λ,μ (z)) ∈ S ∗ (α, β). Since (1) (z)) = Wλ,μ (z) z · (G (1) λ,μ ∞ Γ (μ) zn = z + n=2 Γ ((n−1)λ+μ) (n−1)!

by virtue of Lemma 2 given in Sect. 4, it suffices to show that ∞

Γ (μ) 1 [n + βn(n − 1) − α] Γ ((n−1)λ+μ) (n−1)! ≤ 1 − α, n=2

or

 ∞ 2 Γ (μ) 1 n=2 n β + n(1 − β) − α Γ ((n−1)λ+μ) (n−1)! ≤ 1 − α.

756

Let

N. Mustafa et al.

L 2 (λ, μ; α, β)   Γ (μ) 1 2 = ∞ n=2 n β + n(1 − β) − α Γ ((n−1)λ+μ) (n−1)! .

Setting n 2 = (n − 1)(n − 2) + 3(n − 1) + 1, n = (n − 1) + 1 and by simple computation, we get  β Γ (μ) L 2 (λ, μ; α, β) = ∞ n=3 (n−3)! Γ ((n−1)λ+μ)  1+2β Γ (μ) + ∞ n=2 (n−2)! Γ ((n−1)λ+μ)  Γ (μ) 1−α + ∞ n=2 (n−1)! Γ ((n−1)λ+μ) . Using (44) and (45), we obtain  β 1 L 2 (λ, μ; α, β) ≤ ∞ n=3 (n−3)! μn−1  ∞ 1−α 1 1+2β 1 + ∞ n=2 (n−2)! μn−1 + n=2 (n−1)! μn−1 = μβ2 e μ + ≤ 1 − α, 1

1+2β μ1 e +(1 μ

1

− α)(e μ − 1)

which is equivalent to 1

1

(1 − α)(2 − e μ )μ2 − (μ + (2μ + 1)β)e μ ≥ 0. Thus, the proof of Theorem 9 is complete. By setting β = 0 in Theorem 9 and using the relationship C(α, 0) = C(α), we arrive at the following corollary. Corollary 25 The function G (1) λ,μ (z) belongs to the class C(α) if λ ≥ 1 and the following condition is satisfied: 1

1

(1 − α)(2 − e μ )μ − e μ ≥ 0. By taking α = 0 in Corollary 25 and using relationship C(0) = C, we arrive at the following corollary. Corollary 26 The function G (1) λ,μ (z) belongs to the class C if λ ≥ 1 and μ > x 2 . Here, x2 ≈ 2.6679 is the root of the equation 1

2x − (x + 1)e x = 0. Proof Let φ(x) = 2x − (x + 1)e1/x , x > 0. By simple computation, we obtain φ  (x) = 2 −

x2 − x − 1 1 e x , x > 0. x2

Geometric Properties of Normalized Wright Functions

757

We observe that φ  (x) > 0 for each x > 0. Thus, function φ(x) is an increasing function. 1 Hence, 2μ − (μ + 1)e μ > 0 for everyμ > x2 where x2 ≈ 2.6679 is the root of the equation 1 2x − (x + 1)e x = 0. Thus, the proof of Corollary 26 is complete. Theorem 10 Let λ ≥ 1 and assume the condition   1 (1 − α) 2 − (λ + μ)(e λ+μ − 1) 1

−(λ+ μ)(e λ+μ − 1)  1 +β (λ + μ − 1)(e λ+μ − 1) − 1 + 1 ≥ 0, ∗ then function G (2) λ,μ (z) belongs to the class S (α, β).

Proof Since G (2) λ,μ (z) = z +

∞  n=2

zn Γ (λ + μ) Γ ((n − 1)λ + λ + μ) n!n

by virtue of Lemma 2, it suffices to show that ∞

Γ (λ+μ) 1 [n + βn(n − 1) − α] Γ ((n−1)λ+λ+μ) n!n ≤ 1 − α. n=2

Let

L (λ, μ; α, β) = 3∞ Γ (λ+μ) 1 . n=2 [n + βn(n − 1) − α] Γ ((n−1)λ+λ+μ) n!n

Simple computation, we write L 3 (λ, μ; α, β) =

∞

β Γ (λ+μ) n=2 (n−1)! Γ ((n−1)λ+λ+μ)

   Γ (μ) 1 1 + ∞ n=2 1 − β − n n! Γ ((n−1)λ+μ)  1−α Γ (μ) + ∞ n=2 n!n Γ ((n−1)λ+μ) . Using inequality (44) and (45), with μ ≡ λ + μ, we obtain  β 1 L 3 (λ, μ; α, β) ≤ ∞ n=2 (n−1)! (λ+μ)n−1  2−α−β 1 1 + ∞ = β(e λ+μ − 1) n=2 n! (λ+μ)n−1 1

+(2 − α − β)(λ + μ)(e λ+μ − ≤ 1 − α,

λ+μ+1 ) λ+μ

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which is equivalent to   1 (1 − α) 2 − (λ + μ)(e λ+μ − 1) 1

−(λ+ μ)(e λ+μ − 1)  1 +β (λ + μ − 1)(e λ+μ − 1) − 1 + 1 ≥ 0. Thus, the proof of Theorem 10 is complete. By setting β = 0 in Theorem 10 and using the relationship S ∗ (α, 0) = S ∗ (α), we arrive at the following corollary. ∗ Corollary 27 The function G (2) λ,μ (z) belongs to the class S (α) if λ ≥ 1 and the following condition is satisfied:

  1 (1 − α) 2 − (λ + μ)(e λ+μ − 1) 1

−(λ + μ)(e λ+μ − 1) + 1 ≥ 0. By taking α = 0 in Corollary 27 and using relationship S ∗ (0) = S ∗ , we arrive at the following corollary. ∗ Corollary 28 The function G (2) λ,μ (z) belongs to the class S if λ ≥ 1 and λ + μ > x 3 . Here, x3 ≈ 1.3112 is the root of the equation 1

3 − 2x(e x − 1) = 0. Proof Let ψ(x) = 2x + 3 − 2xe1/x , x > 0. By simple computation, we obtain ψ  (x) = 2 −

2(x − 1) 1 e x , x > 0. x

We observe that ψ  (x) > 0 for each x > 0. Thus, function ψ(x)is an increasing function. 1 Hence, 3 − 2(λ + μ)(e λ+μ − 1) > 0 for every λ + μ > x3 where x3 ≈ 1.3112 is the root of the equation 1 3 − 2x(e x − 1) = 0. Thus, the proof of Corollary 28 is complete. Theorem 11 Let λ ≥ 1 and assume the condition (1 − α)(λ + μ)(λ + μ + 2− 1 1 1 (λ + μ)e λ+μ ) − (λ + μ)(e λ+μ − 1) − βe λ+μ ≥ 0, then the function G (2) λ,μ (z) belongs to the class C(α, β).

(46)

Geometric Properties of Normalized Wright Functions

759

(2)  Proof The function G (2) λ,μ (z) belongs to the class C(α, β) if and only if z · (G λ,μ (z)) ∈ ∗ S (α, β). Since (2)  z · (G (2) λ,μ (z)) = Wλ,μ (z)  Γ (λ+μ) zn =z+ ∞ n=2 Γ (λ(n−1)+λ+μ) n!

by virtue of Lemma 2, if suffices to show that ∞

Γ (λ+μ) 1 [n + βn(n − 1) − α] Γ ((n−1)λ+λ+μ) n! ≤ 1 − α. n=2

Since

L 1 (λ, λ + μ; α, β)  Γ (λ+μ) 1 = ∞ n=2 [n + βn(n − 1) − α] Γ ((n−1)λ+λ+μ) n!

the proof of Theorem 11 is clear from the assume of Theorem 8. Indeed, we obtain condition (46) if in (43) replace μ with λ + μ. This evidently completes the proof. By setting β = 0 in Theorem 11 and using the relationship C(α, 0) = C(α), we arrive at the following corollary. Corollary 29 The function G (2) λ,μ (z) belongs to the class C(α) if λ ≥ 1 and the following condition is satisfied: 1

1

(1 − α)(λ + μ + 2 − (λ + μ)e λ+μ ) − (e λ+μ − 1) ≥ 0. By taking α = 0 in Corollary 29 and using relationship C(0) = C, we arrive at the following corollary. Corollary 30 The function G (2) λ,μ (z) belongs to the class C if λ ≥ 1 and λ + μ > x 1 . Here, x1 ≈ 1.9133 is the root of the equation 1

x + 3 − (x + 1)e x = 0.

6 Univalence of Certain Integral Operators Involving Normalized Wright Functions Let A be the class of analytic functions f (z) in the open unit disk U = {z ∈ C : |z| < 1}, normalized by f (0) = 0 = f  (0) − 1 of the form

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N. Mustafa et al.

f (z) = z + a2 z 2 + a3 z 3 + · · · + an z n + · · · = z +

∞ 

an z n ,

(47)

n=2

It is well-known that a function f : C → C is said to be univalent if the following condition is satisfied: z 1 = z 2 if f (z 1 ) = f (z 2 ). We denote by S the subclass of A consisting of functions which are also univalent in U . For some recent investigations of various subclasses of the univalent functions class S, see the works by Altinta¸s et al. [3], Gao et al. [14], and Owa et al. [34]. In recent years there have been many studies (see for example [6, 8–11, 35, 36]) on the univalence of the following integral operators:   z 1/ p p−1  t f (t)dt , G p (z) = p

(48)

0

  z t p−1 G q1 ,q2 ,...,qn , p (z) = p 0

G p,q (z) = and



n

k=1

f k (t) t

1/ p

qk dt

,

    z 1/ p f (t) q p t p−1 dt , t 0

  z 1/q  f (t) q q−1 G q (z) = q e t dt

(49)

(50)

(51)

0

where the function f (z) belong to the class A and the parameters p, q are complex numbers such that the integrals in (48)–(51) exist. Furthermore, Breaz et al. [10] have obtained various sufficient conditions for the univalence of the following integral operator:

G n,α (z) =

⎧ ⎨ ⎩

[n(α − 1) + 1]

 z! 0

⎫1/[n(α−1)+1] ⎬

"α−1

n

f k (t) k=1

dt

⎭

(52)

where n is a natural number, α is a real number and functions f k ∈ A, k = 1, ..., n. By Baricz and Frasin [6] was obtained some sufficient conditions for the univalence of the integral operators of the type (50)–(52) when the function f (z) is the normalized Bessel function with various parameters. The Wright function is defined by the following infinite series: Wλ,μ (z) =

∞  n=0

zn 1 , Γ (λn + μ) n!

(53)

Geometric Properties of Normalized Wright Functions

761

where Γ is Euler gamma function, λ > −1, μ, z ∈ C. This series is absolutely convergent in C, when λ > −1 and absolutely convergent in open unit disk for λ = −1. Furthermore, for λ > −1 the Wright function Wλ,μ (z) is an entire function. The Wright function was introduced by Wright in [49] and has appeared for the first time in the case λ > 0 in connection with his investigation in the asymptotic theory of partitions. Later on, it has found many other applications, first of all, in the Mikusinski operational calculus and in the theory of integral transforms of Hankel type. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright functions and of the integral operators involving Wright functions. As we have already noted in the previous sections, Wright function Wλ,μ (z), defined by (53) does not belong to the class A and so it is natural to consider the following two kinds of normalization of the Wright function: (1) Wλ,μ (z) := Γ (μ)zWλ,μ (z) =

∞  n=0

Γ (μ) z n+1 , λ > −1, μ > 0, z ∈ U Γ (λn + μ) n!

and

 ∞ z n+1 Γ (λ + μ) 1 = , := Γ (λ + μ) Wλ,μ (z) − Γ (μ) Γ (λn + λ + μ) (n + 1)! n=0

(2) Wλ,μ (z)

λ > −1, λ + μ > 0, z ∈ U. Easily, we write (1) Wλ,μ (z)

=z+

∞  n=2

zn Γ (μ) , λ > −1, μ > 0, z ∈ U Γ (λ(n − 1) + μ) (n − 1)!

(54)

∞  Γ (λ + μ) z n , λ > −1, λ + μ > 0, z ∈ U. Γ (λn + μ) n! n=2

(55)

and (2) Wλ,μ (z) = z +

Note that √ W1,(1)p+1 (−z) = −J p(1) (z) = Γ ( p + 1)z 1− p/2 J p (2 z) where J p (z) is the Bessel function and J p(1) (z) the normalized Bessel function. In this section, we give various sufficient conditions for integral operators of type (48)–(51) when the function f (z) is the normalized Wright functions to be univalent in the open unit disk U . We would like to show that the univalence of integral operators which involve normalized Wright functions can be derived easily via some well-known univalence criteria.

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In our investigation, we shall need the following lemmas. Lemma 3 ([8]) If f ∈ A and the following condition is satisfied: 

    z f  (z)  ≤1  1 − |z|   f (z)  2

for all z ∈ U , then the function f (z) is univalent in U . Lemma 4 ([34]) Let q ∈ C such that (q) > 0. If the function f ∈ A satisfies the inequality   1 − |z|2 (q)  z f  (z)   f  (z)  ≤ 1 (q) for all z ∈ U , then for all p ∈ C such that ( p) ≥ (q), the function defined by (48) is univalent in U . Lemma 5 ([36]) Let q ∈ C and a ∈ R  such that (q) ≥ 1, a > 1 and 2a |q| ≤ √ 3 3. If f ∈ A satisfies the inequality z f  (z) ≤ a for all z ∈ U , then the function G q : U → C defined by (51) univalent in U . U . Lemma 6 ([35]) Let p and c be complex numbers such that ( p) > 0 and |c| ≤ 1, c = −1. If the function f ∈ A satisfies the inequality      c |z|2 p + (1 − |z|2 p ) z f (z)  ≤ 1   p f (z)  for all z ∈ U , then the function G p : U → C defined by (48) is univalent in U . Lemma 7 Let λ ≥ 1 and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation 1

2μ − (μ + 1)e μ+1 + 1 = 0.

(56)

Then, the following inequalities hold for all z ∈ U       (1)  z Wλ,μ  (z) e1/(μ+1)   ≤ − 1 ,    W (1) (z)  (2μ + 1) − (μ + 1)e1/(μ+1) λ,μ         1 z W (1) (z)  ≤ 1 + 1 (μ + 2)e μ+1 − (μ + 1) . λ,μ   μ

(57)

(58)

Geometric Properties of Normalized Wright Functions

763

(1) Proof By using the definition of the normalized Wright function Wλ,μ (z), we obtain for all z ∈ U       ∞  z W (1) (z)   z W (1) (z) −W (1) (z)  Γ (μ) 1 λ,μ   λ,μ   λ,μ n=2 Γ (λ(n−1)+μ) (n−2)! ∞ ≤ .   W (1) (z) − 1 =  (1) Γ (μ) 1 1− W (z)   λ,μ   n=2 Γ (λ(n−1)+μ) (n−1)! λ,μ

Under hypothesis λ ≥ 1, the inequality Γ (n − 1 + μ) ≤ Γ (λ(n − 1) + μ), n ∈ N holds, which is equivalent to Γ (μ) 1 ≤ , n∈N Γ (λ(n − 1) + μ) (μ)n−1 where (μ)n = Γ (n + μ)/Γ (μ) = μ(μ + 1) · · · (μ + n − 1), (μ)0 = 1 Pochhammer (or Appell) symbol, defined in terms of Euler gamma function. Using (59), we obtain ∞  n=2

(59) is

∞

 1 1 Γ (μ) 1 ≤ . Γ (λ(n − 1) + μ) (n − 2)! n=2 (n − 2)! (μ)n−1

Further, the inequality (μ)n−1 = μ(μ + 1) · · · (μ + n − 2) ≥ μ(μ + 1)n−2 , n ∈ N

(60)

is true, which is equivalent to 1/(μ)n−1 ≤ 1/μ(μ + 1)n−2 , n ∈ N. Using (60), we get ∞  n=2

∞

 1 1 Γ (μ) 1 e1/(μ+1) ≤ . = Γ (λ(n − 1) + μ) (n − 2)! n=2 (n − 2)! μ(μ + 1)n−2 μ

(61)

Similarly, we have ∞  n=2

 1 μ + 1  1/(μ+1) Γ (μ) ≤ e −1 . Γ (λ(n − 1) + μ) (n − 1)! μ

(62)

Combining inequalities (61) and (62), we immediately get that first assertion (57) of Lemma 7 holds. Let’s prove second assertion of lemma. From the definition of the normalized (1) (z), we have Wright function Wλ,μ       z W (1) (z)  ≤ 1 + ∞ λ,μ n=2  

Γ (μ) 1 (n−2)! Γ (λ(n−1)+μ)

+

∞

Γ (μ) 1 n=2 (n−1)! Γ (λ(n−1)+μ) .

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Using (61) and (62), we get     1/(μ+1)   μ + 1  1/(μ+1) z W (1) (z)  ≤ 1 + e e −1 + λ,μ   μ μ  1 (μ + 2)e1/(μ+1) − (μ + 1) . = 1+ μ Thus, the proof of Lemma 7 is complete. (2) For the normalized Wright function Wλ,μ (z), we can give the following lemma.

Lemma 8 Let λ ≥ 1 and λ + μ > x0 where x0 ∼ = 1.2581 is the root of the equation 1

2x − (x + 1)e x+1 + 1 = 0.

(63)

Then the following inequalities hold for all z ∈ U         (2)  z Wλ,μ  (z) (λ + μ + 1) e1/(μ+1) − 1    , − 1 ≤   W (1) (z)  (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 λ,μ          z W (2) (z)  ≤ 1 + λ + μ + 1 e1/(μ+1) − 1 . λ,μ   λ+μ

(64)

(65)

Proof The proof of this lemma is very similar to the proof of Lemma 7, so the details of the proof may be omitted. Here our main aim is to give sufficient conditions for the integral operators of the type (48)–(51) when the function f (z) is the normalized Wright functions to be univalent in the open unit disk U . To this end, firstly we consider the following integral operator: q G λ,μ (z)

=

 z! 0

(1) Wλ,μ (t)

t

"q dt, λ > −1, μ > 0, z ∈ U.

(66)

For this integral operator, we can give the following theorem. ∼ 1.2581 is the root of the equation Theorem 12 Let λ ≥ 1 and μ > μ0 where μ0 = (56). Moreover, suppose that q is a complex number such that |q| ≤ q

(2μ + 1) − (μ + 1)e1/(μ+1) . e1/(μ+1)

Then, the function G λ,μ : U → C defined by (66) is univalent in U .

Geometric Properties of Normalized Wright Functions

765

 q  q q (1) Proof Since Wλ,μ ∈ A, clearly G λ,μ ∈ A, i. e. G λ,μ (0) = G λ,μ (0) − 1 = 0. On the other hand, it is easy to see that 

and

 q G λ,μ (z)



  q z G λ,μ (z)  q  G λ,μ (z)

! =

(1) Wλ,μ (z)

"q

z

⎡  ⎤  (1) z Wλ,μ (z) ⎢ ⎥ =q⎣ − 1⎦ . (1) Wλ,μ (z)

(67)

By using first assertion (57) of Lemma 7, we obtain            (1)  q   z W (z)  z G λ,μ (z)  λ,μ |q| e1/(μ+1)      ≤ |q|  ≤ − 1       W (1) (z)  (2μ + 1) − (μ + 1)e1/(μ+1)  G qλ,μ (z)  λ,μ   for all z ∈ U and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation 1

(2μ + 1) − (μ + 1)e μ+1 = 0. Hence, for all z ∈ U and μ > μ0 , we write the following inequality:       q   z G (z) |q| e1/(μ+1) λ,μ 2 (1 − |z|2 )   q .   ≤ (1 − |z| ) (2μ + 1) − (μ + 1)e1/(μ+1)  G λ,μ (z)  This last expression is bounded by 1 if |q| ≤

(2μ + 1) − (μ + 1)e1/(μ+1) . e1/(μ+1)

But, this is true by hypothesis of theorem. Thus, according to the Lemma 3, function q G λ,μ (z) is univalent in U . With this the proof of Theorem 12 is complete. By setting q = 1 in Theorem 12, we have the following result. Corollary 31 Let λ ≥ 1 and μ > μ1 where μ1 ∼ = 2.4898 is the root of the equation 1

(2μ + 1) − (μ + 2)e μ+1 = 0.

(68)

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Then, the function G λ,μ : U → C defined by 

z

G λ,μ (z) =

(1) Wλ,μ (t)

t

0

dt

is univalent in U . If we take λ = 1, μ = p + 1 in Theorem 12, we arrive at the following corollary. q

Corollary 32 The function G p : U → C defined by G qp (z)

=

 z!

J p(1) (−t)

"q dt

t

0

is univalent in U if p > μ0 − 1 where μ0 ∼ = 1.2581 is the root of the equation (56) and q is a complex number such that |q| ≤

(2 p + 3) − ( p + 2)e1/( p+2) e1/( p+2)

Here, function J p(1) (z) is normalized Bessel function. By taking q = 1 in Corollary 32, we obtain the following result. Corollary 33 Let p > μ1 − 1 where μ1 ∼ = 2.4898 is the root of the equation (68). Then, the function G p : U → C defined by 

z

G p (z) = 0

J p(1) (−t) t

dt

is univalent in U . Here, J p(1) (z) is normalized Bessel function. For the integral operator q Fλ,μ (z)

=

 z! 0

(2) (t) Wλ,μ

t

"q dt, λ > −1, λ + μ > 0, z ∈ U

(69)

we can give the following theorem. Theorem 13 Let λ ≥ 1 and λ + μ > x0 where x0 ∼ = 1.2581 is the root of the equation (63). Moreover, suppose that q is a complex number such that   (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1   |q| ≤ . (λ + μ + 1) e1/(μ+1) − 1 q

Then, the function Fλ,μ : U → C defined by (69) is univalent in U .

Geometric Properties of Normalized Wright Functions

767

Proof The proof of this theorem is very similar to the proof of Theorem 12, so the details of the proof may be omitted. By setting q = 1 in Theorem 13, we have the following corollary. Corollary 34 Let λ ≥ 1 and λ + μ > x1 where x1 ∼ = 2.3325 is the root of the equation 1 (70) 3x − 2(x + 1)e x+1 + 2 = 0. Then, the function Fλ,μ : U → C defined by  Fλ,μ (z) = 0

z

(2) Wλ,μ (t)

t

dt

is univalent in U . Now, we consider the following integral operator: p,q G λ,μ (z)

  ! (1) "q 1/ p z Wλ,μ (t) p−1 = p t dt , λ > −1, μ > 0, z ∈ U. t 0

(71)

p,q

On the univalence of the function G λ,μ (z), we give the following theorem. Theorem 14 Let λ ≥ 1 and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation (56). Moreover, suppose that p, q and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied: |c| ≤ 1 −

|q| e1/(μ+1) . | p| (2μ + 1) − (μ + 1)e1/(μ+1)

p,q

Then, the integral operator G λ,μ : U → C defined by (71) is univalent in U Proof We can rewrite the integral operator (71) as p,q

G λ,μ (z) =

  z 1/ p  q  p t p−1 G λ,μ (t) dt

(72)

0

q

where function G λ,μ : U → C is defined in (66). Under hypothesis of theorem, using (67) and (57), we obtain   q       z G (z) |q| e1/(μ+1) λ,μ  ≤ |c| + c |z|2 p + (1 − |z|2 p ) 

.     q | p| (2μ + 1) − (μ + 1)e1/(μ+1)  p G λ,μ (z) 

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This last expression is bounded by 1 if |c| ≤ 1 −

|q| e1/(μ+1) . | p| (2μ + 1) − (μ + 1)e1/(μ+1)

But this is true by hypothesis of theorem. Thus, according to the Lemma 6, function p,q G λ,μ (z) defined by (72) is univalent in U . With this, the proof of Theorem 14 is complete. By setting q = 1 in Theorem 14, we arrive at the following result. Corollary 35 Let λ ≥ 1 and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation (56). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied: |c| ≤ 1 −

e1/(μ+1) . | p| (2μ + 1) − (μ + 1)e1/(μ+1)

p

Then, the integral operator G λ,μ : U → C defined by p G λ,μ (z)

  z 1/ p p−2 (1) = p t Wλ,μ (t)dt .

(73)

0

is univalent in U . p

Remark 1 Note that, recently the function G λ,μ : U → C defined by (73) was investigated by Prajapat [44] and he obtained some sufficient conditions for the univalence this function. Now, on the univalence of the integral operator p,q Fλ,μ (z)

  ! (2) "q 1/ p z Wλ,μ (t) p−1 = p t dt , λ > −1, λ + μ > 0, z ∈ U (74) t 0

we can give the following theorem. ∼ 1.2581 is the root of the Theorem 15 Let λ ≥ 1 and λ + μ > x0 where x0 = equation (63). Moreover, suppose that p, q and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:   |q| (λ + μ + 1) e1/(μ+1) − 1

  . |c| ≤ 1 − | p| (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 p,q

Then, the integral operator Fλ,μ : U → C defined by (74) is univalent in U .

Geometric Properties of Normalized Wright Functions

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Proof The proof of Theorem 15 is similar to the proof of Theorem 14. Hence, the details of the proof of Theorem 15 may be omitted. By setting q = 1 in Theorem 15, we obtain the following corollary. Corollary 36 Let λ ≥ 1 and λ + μ > x0 where x0 ∼ = 1.2581 is the root of the equation (63). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:   (λ + μ + 1) e1/(μ+1) − 1

  . |c| ≤ 1 − | p| (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 p

Then, the function Fλ,μ : U → C defined by p

Fλ,μ (z) =

  z 1/ p (2) p t p−2 Wλ,μ (t)dt , z∈U

(75)

0

is univalent in U . Now, we consider integral operator of the type (51) when the function f (z) is the normalized Wright function. Let q Hλ,μ (z)

  z  (1) q 1/q q−1 e Wλ,μ (t) = q t , λ > −1, λ + μ > 0, z ∈ U.

(76)

0

On univalence of the function (76), we give the following theorem. Theorem 16 Let q ∈ C, λ ≥ 1 and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation (56). If (q) ≥ 1 and the following condition is satisfied: |q| ≤

√ 3 3μ

, 2 (μ + 2)e1/(μ+1) − 1

q

then, the function Hλ,μ : U → C defined by (76) is univalent in U . Proof From (58), we write       1 z W (1) (z)  ≤ 1 + 1 (μ + 2)e μ+1 − (μ + 1) λ,μ   μ for all z ∈ U . Taking a =1+

1 1 (μ + 2)e μ+1 − (μ + 1) , μ

(77)

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√ we easily see that 2a |q| ≤ 3 3 if provided (77). Thus, under hypothesis of theorem, all hypothesis of the Lemma 5 is provided. Hence, the proof of Theorem 16 is complete. By setting q = 1 in Theorem 16, we have the following result. Corollary 37 Let λ ≥ 1 and μ > μ1 where μ1 ∼ = 1.6692 is the root of the equation √ (78) 3 3μ − 2(μ + 2)e1/(μ+1) + 2 = 0. Then, the function Hλ,μ : U → C defined by 

z

Hλ,μ (z) =

(1)

e Wλ,μ (t) dt

0

is univalent in U . Now, let q Q λ,μ (z)

  z  (2) q 1/q q−1 e Wλ,μ (t) = q t , λ > −1, λ + μ > 0, z ∈ U.

(79)

0

For the function (79), we can give the following theorem which will be proved similarly to the Theorem 16. ∼ 1.2581 is the root of Theorem 17 Let q ∈ C, λ ≥ 1 and λ + μ > x0 where x0 = the equation (56). If (q) ≥ 1 and the following condition is satisfied: √ 3 3(λ + μ) , |q| ≤ 2 (λ + μ + 1)e1/(λ+μ+1) − 1 q

then, the function Q λ,μ : U → C defined by (79) is univalent in U . By setting q = 1 in Theorem 17, we obtain the following corollary. Corollary 38 Let λ ≥ 1 and λ + μ > x1 where x1 ∼ = 0.83232 is the root of the equation √ 3 3x − 2(x + 1)e1/(x+1) + 2 = 0. Then, the function Q λ,μ : U → C defined by  Q λ,μ (z) = 0

is univalent in U .

z

(2)

e Wλ,μ (t) dt

Geometric Properties of Normalized Wright Functions

771

7 More on Univalence of Integral Operators Involving Wright Functions In this section our main aim is to give sufficient conditions for the integral operators of the type (49)–(52) when the functions f k (z), k = 1, 2, ..., n are the normalized Wright functions with various parameters to be univalent in the open unit disk. Let p,q ,q ,...,q G λ,μ11 ,μ22 ,...,μn n (z)

  z = p t p−1

n

0

!

(1) Wλ,μ (t) k

, λ > −1,

dt

t

k=1

1/ p

"qk

μk > 0, k = 1, 2, ..., n, z ∈ U

(80)

(1) (z) is normalized Wright functions defined by (54) when μ is μk for where Wλ,μ k each k = 1, 2, ..., n. p,q ,q ,...,q On the univalence of the function G λ,μ11 ,μ22 ,...,μn n (z), we give the following theorem.

Theorem 18 Let λ ≥ 1, n be a natural number and q1 , q2 , ..., qn are all non-zero complex numbers, and μ > μ0 where μ = min {μk : k = 1, 2, ..., n}, μ0 ∼ = 1.2581 is the root of the equation (56). Moreover, suppose that p and c be complex numbers such that ( p) > 0 and the following condition is satisfied:  n    qk  e1/(μ+1) .  |c| ≤ 1 − (2μ + 1) − (μ + 1)e1/(μ+1) k=1  p  p,q ,q ,...,q

Then the integral operator G λ,μ11 ,μ22 ,...,μn n : U → C defined by (80) is univalent in U . q ,q ,...,q

1 2 n : U → C by Proof Firstly, we define the function G λ,μ 1 ,μ2 ,...,μn

q1 ,q2 ,...,qn (z) G λ,μ 1 ,μ2 ,...,μn

 =

z n

!

(1) Wλ,μ (t) k

"qk

t

0 k=1

dt.

(81)

(1) We observe that, since for all k = 1, 2, ..., n we have Wλ,μ ∈ A, clearly k ∈ A. Also, from (81) it is easy to see that

q1 ,q2 ,...,qn G λ,μ 1 ,μ2 ,...,μn



 q1 ,q2 ,...,qn (z) G λ,μ 1 ,μ2 ,...,μn

n

= k=1

!

(1) Wλ,μ (z) k

z

"qk .

From (82), by simple computation, we have     (1) (1) q1 ,q2 ,...,qn n z Wλ,μ (z) − Wλ,μ (z)  (z) G λ,μ k k 1 ,μ2 ,...,μn qk · ,  =  q1 ,q2 ,...,qn (1) zWλ,μ (z) G λ,μ1 ,μ2 ,...,μn (z) k=1 k



(82)

772

and

N. Mustafa et al.



  q1 ,q2 ,...,qn z G λ,μ (z) 1 ,μ2 ,...,μn  q1 ,q2 ,...,qn  G λ,μ1 ,μ2 ,...,μn (z)

⎛  ⎞  (1) (z) z Wλ,μ k ⎜ ⎟ − 1⎠ . = qk ⎝ (1) Wλ,μk (z) k=1 n 

Using (57) for each μk , k = 1, 2, ..., n, we obtain            (1)  q1 ,q2 ,...,qn n   z W (z)  z G λ,μ1 ,μ2 ,...,μn (z)   λ,μk      |q | ≤ − 1   k     1 ,q2 ,...,qn   W (1) (z)  G qλ,μ  (z) λ,μk k=1   1 ,μ2 ,...,μn ≤

n 

|qk |

k=1

e1/(μk +1) . (2μk + 1) − (μk + 1)e1/(μk +1)

Now, we define the function g : (1.2581, +∞) → R by g(x) =

e1/(x+1) . (2x + 1) − (x + 1)e1/(x+1)

(83)

It can be easily see that the function g : (1.2581, +∞) → R, defined by (83) is decreasing. Consequently for all μk , k = 1, 2, ..., n we have e1/(μk +1) e1/(μ+1) ≤ . (2μk + 1) − (μk + 1)e1/(μk +1) (2μ + 1) − (μ + 1)e1/(μ+1)

(84)

By using the triangle inequality, we obtain the following inequality:  n  qk   q1 ,q2 ,...,qn    1/(μ+1) e  z G (z)  k=1  p   ≤ |c| + c |z|2 p + (1 − |z|2 p )  λ,μ1 ,μ2 ,...,μn .     q1 ,q2 ,...,qn (2μ + 1) − (μ + 1)e1/(μ+1)  p G λ,μ (z)  1 ,μ2 ,...,μn This last expression is bounded by 1 if  n    qk  e1/(μ+1) .  |c| ≤ 1 − (2μ + 1) − (μ + 1)e1/(μ+1) k=1  p  But this is true by hypothesis of theorem. Thus, according to the Lemma 6, function p,q ,q ,...,q G λ,μ11 ,μ22 ,...,μn n (z) defined by (80) is univalent in U . With this the proof of Theorem 18 is complete. By setting q1 = q2 = · · · = qn = q in Theorem 18, we arrive at the following corollary. Corollary 39 Let λ ≥ 1, n be a natural number and q be a nonzero complex number, and μ > μ0 where μ = min {μk : k = 1, 2, ..., n}, μ0 ∼ = 1.2581 is the root of the

Geometric Properties of Normalized Wright Functions

773

equation (56). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:   q  ne1/(μ+1) |c| ≤ 1 −   . p (2μ + 1) − (μ + 1)e1/(μ+1) p,q

Then the integral operator G λ,μ1 ,μ2 ,...,μn : U → C defined by p,q G λ,μ1 ,μ2 ,...,μn (z)

  z = p t p−1 0

n

!

k=1

(1) Wλ,μ (t) k

1/ p

"q

t

dt

is univalent in U . Taking n = 1 in Theorem 18, we immediately obtain the following result. Corollary 40 Let λ ≥ 1 and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation (56). Moreover, suppose that p, q and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied: |c| ≤ 1 −

|q| e1/(μ+1) . | p| (2μ + 1) − (μ + 1)e1/(μ+1)

p,q

Then the integral operator G λ,μ : U → C defined p,q G λ,μ (z)

  ! (1) "q 1/ p z Wλ,μ (t) p−1 = p t dt t 0

is univalent in U . By taking q = 1 in Corollary 40, we have the following corollary. ∼ 1.2581 is the root of the equaCorollary 41 Let λ ≥ 1 and μ > μ0 where μ0 = tion (56). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied: |c| ≤ 1 −

e1/(μ+1) . | p| (2μ + 1) − (μ + 1)e1/(μ+1)

p

Then the integral operator G λ,μ : U → C defined p

G λ,μ (z) = is univalent in U .

  z 1/ p (1) p t p−2 Wλ,μ (t)dt 0

(85)

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N. Mustafa et al. p

Remark 2 Note that, recently the function G λ,μ : U → C defined by (85) was investigated by Prajapat [44] and obtained some sufficient conditions for the univalence of this function. On the univalence of the integral operator p,q ,q ,...,q Fλ,μ11,μ22 ,...,μnn (z)

  z = p t p−1 0

n

!

(2) Wλ,μ (t) k

, λ > −1,

dt

t

k=1

1/ p

"qk

λ+μk > 0, k = 1, 2, ..., n, z ∈ U

(86)

(2) (z) is normalized Wright functions defined by (54) when μ is μk for where Wλ,μ k each k = 1, 2, ..., n, we can give the following theorem.

Theorem 19 Let λ ≥ 1, n be a natural number and q1 , q2 , ..., qn are all nonzero complex numbers, and λ + μ > x0 where μ = min {μk : k = 1, 2, ..., n}, x0 ∼ = 1.2581 is the root of the equation (63). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:    n    qk  (λ + μ + 1) e1/(μ+1) − 1  .   |c| ≤ 1 − (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 k=1  p  p,q ,q ,...,q

Then the integral operator Fλ,μ11,μ22 ,...,μnn : U → C defined by (86) is univalent in U . Proof The proof of Theorem 19 is similar to the proof of Theorem 18. Hence, the details of the proof of Theorem 19 may be omitted. By setting q1 = q2 = · · · = qn = q in Theorem 19, we arrive at the following corollary. Corollary 42 Let λ ≥ 1, n be a natural number and q be a nonzero complex number, and λ + μ > x0 where μ = min {μk : k = 1, 2, ..., n}, x0 ∼ = 1.2581 is the root of the equation (63). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:     q  n(λ + μ + 1) e1/(μ+1) − 1  . |c| ≤ 1 −   p (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 p,q

Then, the integral operator Fλ,μ1 ,μ2 ,...,μn : U → C defined by p,q Fλ,μ1 ,μ2 ,...,μn (z)

is univalent in U .

  z = p t p−1 0

n

k=1

!

(2) Wλ,μ (t) k

t

1/ p

"q dt

Geometric Properties of Normalized Wright Functions

775

Taking n = 1 in Theorem 19, we immediately obtain the following result. Corollary 43 Let λ ≥ 1 and λ + μ > x0 where x0 ∼ = 1.2581 is the root of the equation (63). Moreover, suppose that p, q and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:     q  (λ + μ + 1) e1/(μ+1) − 1  . |c| ≤ 1 −   p (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 p,q

Then, the integral operator Fλ,μ : U → C defined p,q Fλ,μ (z)

  ! (2) "q 1/ p z Wλ,μ (t) = p t p−1 dt t 0

is univalent in U . By taking q = 1 in Corollary 43, we have the following corollary. Corollary 44 Let λ ≥ 1 and λ + μ > x0 where x0 ∼ = 1.2581 is the root of the equation (56). Moreover, suppose that p and c be complex numbers such that ( p) > 0, |c| < 1 and the following condition is satisfied:   (λ + μ + 1) e1/(μ+1) − 1

  . |c| ≤ 1 − | p| (λ + μ) − (λ + μ + 1) e1/(μ+1) − 1 p

Then, the integral operator Fλ,μ : U → C defined p

Fλ,μ (z) =

  z 1/ p (2) p t p−2 Wλ,μ (t)dt 0

is univalent in U . Let  n,α Hλ,μ (z) 1 ,μ2 ,...,μn

= (nα + 1)

 z! 0

n

"α (1) Wλ,μ (t) k

1/(nα+1) dt

,λ > −1,

k=1

μk > 0, k = 1, 2, ..., n, z ∈ U.

(87)

(1) where Wλ,μ (z) is normalized Wright functions defined by (54) when μ is μk for k each k = 1, 2, ..., n. The following theorem contains another sufficient conditions for integral operator (87) to be univalent in the open unit disk U .

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Theorem 20 Let λ ≥ 1, n be a natural number and μ > μ0 (n), where μ = min {μk : k = 1, 2, ..., n}, μ0 (n) is the root of the equation 2μ − (μ + n + 1)e1/(μ+1) + 1 = 0. Moreover, suppose that α be a complex number such that (α) > 0 and the following condition is satisfied: |α| ≤

(2μ + 1) − (μ + 1)e1/(μ+1) (α). ne1/(μ+1)

n,α Then the integral operator Hλ,μ : U → C defined by (87) is univalent in U . 1 ,μ2 ,...,μn α Proof Let us define the function Hλ,μ : U → C by 1 ,μ2 ,...,μn

α Hλ,μ (z) 1 ,μ2 ,...,μn

 =

z n

!

(1) Wλ,μ (t) k

"α

t

0 k=1

dt.

(88)

α We easily see that Hλ,μ ∈ A. Also, from (88) it is easy to see that 1 ,μ2 ,...,μn



α Hλ,μ (z) 1 ,μ2 ,...,μn



n

= k=1

!

(1) Wλ,μ (z) k

"α

z

.

(89)

From (89) simple computation, we have 

α (z) Hλ,μ 1 ,μ2 ,...,μn

 and

 

α Hλ,μ (z) 1 ,μ2 ,...,μn



z 

 =

α·

k=1

α (z) Hλ,μ 1 ,μ2 ,...,μn

α Hλ,μ (z) 1 ,μ2 ,...,μn

n 





  (1) (1) z Wλ,μ (z) − Wλ,μ (z) k k (1) zWλ,μ (z) k

,

⎞ ⎛   (1) (z) z Wλ,μ k ⎟ ⎜ − 1⎠ . = α·⎝ (1) Wλ,μk (z) k=1 n 

Using (57) for each μk , k = 1, 2, ..., n, we obtain            (1)  n α   z W (z)  z Hλ,μ1 ,μ2 ,...,μn (z)   λ,μk      |α| ≤ − 1       α   W (1) (z)  Hλ,μ  (z) λ,μk k=1   1 ,μ2 ,...,μn ≤

n  k=1

|α|

e1/(μk +1) . (2μk + 1) − (μk + 1)e1/(μk +1)

Geometric Properties of Normalized Wright Functions

777

On the other hand, by using (84), we obtain that for all z ∈ U 1 − |z|2 (α) (α)

      α   z Hλ,μ (z) |α| n e1/(μ+1) 1 ,μ2 ,...,μn    ≤ .     1/(μ+1) α  Hλ,μ (z)  (α) (2μ + 1) − (μ + 1)e 1 ,μ2 ,...,μn

Under hypothesis of theorem this last expression is bounded by 1. Since the function n,α Hλ,μ (z) can be rewritten in the form 1 ,μ2 ,...,μn  n,α Hλ,μ (z) 1 ,μ2 ,...,μn



n

z

= (nα + 1)

t

nα

0

!

(1) Wλ,μ (t) k

k=1

t

1/(nα+1)

"α dt

and (nα + 1) > (α), applying Lemma 4, we obtain the required result. Thus, the proof of Theorem 20 is complete. By setting n = 1 in Theorem 20, we arrive at the following corollary. Corollary 45 Let λ ≥ 1, and μ > μ0 where μ0 ∼ = 2.4898 is the root of the equation 1

2μ − (μ + 2)e μ+1 + 1 = 0.

(90)

Moreover, suppose that α be a complex number such that (α) > 0 and the fol lowing condition is satisfied: e1/(μ+1) |α| ≤ 2μ + 1 − (μ + 1)e1/(μ+1) (α). Then α the integral operator Hλ,μ : U → C defined by α Hλ,μ (z)

  z α 1/(α+1) (1) Wλ,μ (t) dt = (α + 1) 0

is univalent in U . By taking α = 1 in the Corollary 45, we have the following result. Corollary 46 Let λ ≥ 1 and μ > μ0 where μ0 ∼ = 2.4898 is the root of the equation (90). Then the integral operator Hλ,μ : U → C defined by Hλ,μ (z) =

√ 2

 0

z

(1) Wλ,μ (t)dt

1/2

is univalent in U . (1) Note 2 For the integral operator (87) when the function Wλ,μ (z) is normalized k (2) Wright function Wλ,μk (z) defined by (55) when μ is μk for each k = 1, 2, ..., n can be proved similar theorem.

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N. Mustafa et al.

Finally, we give the following theorem, which contain another sufficient conditions for integral operator   z  (1) q 1/q q Q λ,μ (z) = q t q−1 e Wλ,μ (t) dt , λ > −1, λ + μ > 0, z ∈ U

(91)

0

to be univalent in the open unit disk U . Theorem 21 Let q ∈ C, λ ≥ 1 and μ > μ0 where μ0 ∼ = 1.2581 is the root of the equation (56). If (q) ≥ 1 and the following condition is satisfied: |q| ≤

√ 3 3μ

, 2 (μ + 2)e1/(μ+1) − 1

(92)

q

then, the function Q λ,μ : U → C defined by (91) is univalent in U . Proof From (58), we write       1 z W (1) (z)  ≤ 1 + 1 (μ + 2)e μ+1 − (μ + 1) λ,μ   μ a =1+

1 1 (μ + 2)e μ+1 − (μ + 1) , μ

√ we easily see that 2a |q| ≤ 3 3 if provided (92). Thus, under hypothesis of theorem, all hypothesis of the Lemma 5 is provided. Hence, the proof of Theorem 21 is complete. By setting q = 1 in Theorem 21, we have the following result. Corollary 47 Let λ ≥ 1 and μ > μ2 where μ2 ∼ = 1.6692 is the root of the equation √ (93) 3 3μ − 2(μ + 2)e1/(μ+1) + 2 = 0. Then, the function Q λ,μ : U → C defined by  Q λ,μ (z) =

z

(1)

e Wλ,μ (t) dt

0

is univalent in U . Now, let q Dλ,μ (z)

  z  (2) q 1/q q−1 e Wλ,μ (t) dt = q t , λ > −1, λ + μ > 0, z ∈ U. 0

(94)

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For the function (94), we can give the following theorem which will be proved similarly to Theorem 21. Theorem 22 Let q ∈ C, λ ≥ 1 and λ + μ > x0 where x0 ∼ = 1.2581 is the root of the equation (63). If (q) ≥ 1 and the following condition is satisfied: √ 3 3(λ + μ) , |q| ≤ 2 (λ + μ + 1)e1/(λ+μ+1) − 1 q

then, the function Dλ,μ : U → C defined by (94) is univalent in U . By setting q = 1 in Theorem 22, we obtain the following corollary. Corollary 48 Let λ ≥ 1 and λ + μ > x1 where x1 ∼ = 0.83232 is the root of the equation √ 3 3x − 2(x + 1)e1/(x+1) + 2 = 0. Then, the function Dλ,μ : U → C defined by 

z

Dλ,μ (z) =

(2)

e Wλ,μ (t) dt

0

is univalent in U .

8 Applications of a Poisson Distribution Series on the Analytic Functions In this section, we introduce a Poisson distribution series, whose construction is alike Wright functions, and obtain necessary and sufficient conditions for this series belonging to the class S ∗ C(α, β; γ ), and necessary and sufficient conditions for those belonging to the class T S ∗ C(α, β; γ ). We also introduce two integral operators related to this series and investigate various geometric properties of these integral operators. Let A be the class of analytic functions f (z) in the open unit disk U = {z ∈ C : |z| < 1}, normalized by f (0) = 0 = f  (0) − 1, in the form f (z) = z + a2 z 2 + a3 z 3 + · · · + an z n + · · · = z +

∞ 

an z n , an ∈ C.

(95)

n=2

It is well-known that a function f : C → C is said to be univalent if the following condition is satisfied: z 1 = z 2 if f (z 1 ) = f (z 2 ) or f (z 1 ) = f (z 2 ) if z 1 = z 2 . We will denote the family of all functions in A by S which are univalent in U .

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Let T denote the subclass of all functions f (z) in A of the form f (z) = z − a2 z 2 − a3 z 3 − · · · − an z n − · · · = z −

∞ 

an z n , an ≥ 0.

(96)

n=2

Some of the important and well-investigated subclasses of the univalent functions class S include the classes S ∗ (α) and C(α), respectively, starlike and convex of order α (α ∈ [0, 1)) in the open unit disk U . By definition, we have (see for details, [13, 15, 32] also [47]) 

∗

S (α) = 

and C(α) =



z f  (z) f ∈ A: f (z) 



z f  (z) f ∈ A : 1+  f (z)

 > α, z ∈ U

, α ∈ [0, 1) ,



 > α, z ∈ U

, α ∈ [0, 1) .

Note that, we will use T S ∗ (α) = S ∗ (α) ∩ T and T C(α) = C(α) ∩ T. Interesting generalization of the functions classes S ∗ (α) and C(α), are classes ∗ S (α, β) and C(α, β), which defined by S ∗ (α, β) =



 f ∈ A:

z f  (z) βz f  (z) + (1 − β) f (z)







 > α, z ∈ U , α, β ∈ [0, 1)

and C(α, β) =

f ∈ A:

f  (z) + z f  (z) f  (z) + βz f  (z)



 > α, z ∈ U , α, β ∈ [0, 1) ,

respectively. We will denote T S ∗ (α, β) = S ∗ (α, β) ∩ T and T C(α, β) = C(α, β) ∩ T . These classes T S ∗ (α, β) and T C(α, β) were extensively studied by Altinta¸s and Owa [4], Porwal [42], and certain conditions for hypergeometric functions and generalized Bessel functions for these classes were studied by Moustafa [23] and Porwal and Dixit [43]. The coefficient problems for the subclasses T S ∗ (α, β) and T C(α, β) were investigated by Altınta¸s and Owa in [4]. They, also investigated properties like starlike and convexity of these classes. Also, the coefficient problems, representation formula and distortion theorems for these subclasses S ∗ (α, β, μ) and C ∗ (α, β, μ) of the analytic functions were given by Owa and Aouf in [33]. In [18], results of Silverman were extended by Kadio˘glu. Inspired by the studies mentioned above, we define a unification of the functions classes S ∗ (α, β) and C(α, β) as follows.

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Definition 1 A function f ∈ A given by (95) is said to be in the class S ∗ C(α, β; γ ), α, β ∈ [0, 1) , γ ∈ [0, 1] if the following condition is satisfied 

z f  (z) + γ z 2 f  (z) γ z ( f  (z) + βz f  (z)) + (1 − γ ) (βz f  (z) + (1 − β) f (z))

 > α, z ∈ U.

Also, we will denote T S ∗ C(α, β; γ ) = S ∗ C(α, β; γ ) ∩ T. In special case, we have: S ∗ C(α, β; 0) = S ∗ (α, β); S ∗ C(α, β; 1) = C(α, β); S ∗ C(α, 0; 0) = S ∗ (α); ∗ S C(α, 0; 1) = C(α); T S ∗ C(α, β; 0) = T S ∗ (α, β); T S ∗ C(α, β; 1) = T C(α, β); T S ∗ C(α, 0; 0) = T S ∗ (α); T S ∗ C(α, 0; 1) = T C(α). Suitably specializing the parameters we note that (1) (2) (3) (4) (5) (6)

S ∗ C(α, 0; 0) = S ∗ (α) [22] S ∗ C(α, 0; 1) = C(α) [22] T S ∗ C(α, β; 0) = T S ∗ (α, β) [3, 5, 6, 42] T S ∗ C(α, 0; 0) = T S ∗ (α) [22] T S ∗ C(α, β; 1) = T C(α, β) [4, 42] T S ∗ C(α, 0; 1) = T C(α) [22]

Theorem 23 ([48]) Let f ∈ A. Then, the function f (z) belongs to the class S ∗ C(α, β; γ ) (α, β ∈ [0, 1) , γ ∈ [0, 1]) if the following condition is satisfied: ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ) |an | ≤ 1 − α.

(97)

n=2

The result is sharp for the functions f n (z) = z +

1−α z n , z ∈ U, n = 2, 3, ... . (98) (1 + (n − 1)γ ) (n − α − (n − 1)αβ)

Theorem 24 ([48]) Let f ∈ T . Then, the function f (z) belongs to the class T S ∗ C(α, β; γ ) (α, β ∈ [0, 1) , γ ∈ [0, 1]) if and only if ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ) an ≤ 1 − α.

(99)

n=2

The result is sharp for the functions f n (z) = z −

1−α z n , z ∈ U, n = 2, 3, ... . (1 + (n − 1)γ ) (n − α − (n − 1)αβ) (100)

Now, we will investigate certain class of analytic functions associated with Poisson distribution series.

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A variable x is said to have Poisson distribution if it takes the values 0, 1, 2, 3, ... 2 3 with probabilities e− p , 1!p e− p , p2! e− p , p3! e− p , ..., respectively, where p is called the parameter. Thus, pn − p e , n = 0, 1, 2, 3, .... (101) P(x = n) = n! Now, we introduce a Poisson distribution series as follows: z+

∞  n=2

p n−1 − p n e z , z ∈ U. (n − 1)!

(102)

We can easily show that series (102) is convergent and the radius of convergence is infinity. We define function F : C → C by F( p, z) = z +

∞  n=2

p n−1 − p n e z , z ∈ U. (n − 1)!

(103)

We define also the function G( p, z) = 2z − F( p, z) = z −

∞  n=2

p n−1 − p n e z , z ∈ U. (n − 1)!

(104)

It is clear that F ∈ A and G ∈ T , respectively. Now, we will give sufficient condition for the function F( p, z) defined by (103), belonging to the class S ∗ C(α, β; γ ), and necessary and sufficient condition for the function G( p, z) defined by (104), belonging to the class T S ∗ C(α, β; γ ), respectively. Theorem 25 Let p > 0 and the following condition is provided: {(1 − αβ) γ p + (1 − αβ + (2 − (1 + β)α)γ )} pe p ≤ 1 − α. Then, the function F( p, z) defined by (103) is in the class S ∗ C(α, β; γ ) (α, β ∈ [0, 1) , γ ∈ [0, 1]). Proof Since F ∈ A and F( p, z) = z +

∞  n=2

p n−1 − p n e z , z ∈ U, (n − 1)!

according to Theorem 23, we must show that

(105)

Geometric Properties of Normalized Wright Functions ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

783

p n−1 − p e ≤ 1 − α. (n − 1)!

(106)

Let L 1 ( p; α, β; γ ) =

∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

p n−1 − p e . (n − 1)!

Setting (1 + (n − 1)γ ) (n − α − (n − 1)αβ) = (n − 2)(n − 1) (1 − αβ)γ + (n − 1) (1 − αβ + (2 − (1 + β)α) γ ) + 1 − α

and by simple computation, we obtain L 1 ( p; α, β; γ ) = e− p

∞   (n − 2)(n − 1) (1 − αβ) γ n=2

+(n − 1) (1 − αβ + (2 − (1 + β)α) γ ) + 1 − α



pn−1 (n − 1)!

= (1 − αβ) γ p 2 + (1 − αβ + (2 − (1 + β)α) γ ) p + (1 − α)(1 − e− p ).

Therefore, inequality (106) holds true if (1 − αβ) γ p 2 + (1 − αβ + (2 − (1 + β)α)γ ) p + (1 − α)(1 − e− p ) ≤ 1 − α, which is equivalent to (105). Thus, the proof of Theorem 25 is completed. By taking γ = 0 and γ = 1 in Theorem 25, we arrive at the following results. Corollary 49 If p > 0 and satisfied the following condition (1 − αβ) pe p ≤ 1 − α, then the function F( p, z) defined by (103) belongs to the class S ∗ (α, β) (α, β ∈ [0, 1)). Corollary 50 If p > 0 and satisfied the following condition /

0 (1 − αβ) p 2 + (3 − 2αβ − α) p e p ≤ 1 − α

then the function F( p, z) defined by (103) belongs to the class C(α, β) (α, β ∈ [0, 1)).



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Remark 3 Further consequences of the results given by Corollaries 49 and 50 can be obtained for each of the classes, by specializing the various parameters involved. Theorem 26 If p > 0, then the function G( p, z) defined by (104) belongs to the class T S ∗ C(α, β; γ )(α, β ∈ [0, 1) , γ ∈ [0, 1]) if and only if {(1 − αβ) γ p + (1 − αβ + (2 − (1 + β)α)γ )} pe p ≤ 1 − α.

(107)

Proof From (104), we have G( p, z) = z −

∞  n=2

p n−1 − p n e z , z ∈ U. (n − 1)!

Further, from the proof of Theorem 25, we write: ∞

n−1

p −p n=2 (1 + (n − 1)γ ) (n − α − (n − 1)αβ) (n−1)! e = (1 − αβ) γ p2 + (1 − αβ + (2 − (1 + β)α) γ ) p + (1 − α)(1 − e− p ).

(108) Now, suppose that condition (107) is satisfied. Then, (1 − αβ) γ p 2 + (1 − αβ + (2 − (1 + β)α)γ ) p ≤ (1 − α)e− p , which is equivalent to (1 − αβ) γ p 2 + (1 − αβ + (2 − (1 + β)α) γ ) p + (1 − α)(1 − e− p ) ≤ 1 − α. Considering (108), we obtain: ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

p n−1 − p e ≤ 1 − α. (n − 1)!

Thus, according to Theorem 24, the function G( p, z) belongs to the class T S ∗ C(α, β; γ ). Now, let we prove of the necessity. Assume that G ∈ T S ∗ C(α, β; γ ). Then, according to Theorem 24, the following condition is satisfied: ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

p n−1 − p e ≤ 1 − α. (n − 1)!

Therefore, from (108), we have (1 − αβ) γ p 2 + (1 − αβ + (2 − (1 + β)α) γ ) p + (1 − α)(1 − e− p ) ≤ 1 − α,

Geometric Properties of Normalized Wright Functions

785

which readily yields (1 − αβ) γ p 2 + (1 − αβ + (2 − (1 + β)α) γ ) p ≤ (1 − α)e− p . Last condition is equivalent to condition (107). This completes the proof of necessity, and the proof of Theorem 26. By taking γ = 0 and γ = 1 in Theorem 26, we can readily deduce the following results, respectively. Corollary 51 If p > 0, then the function G( p, z) defined by (104) is in the class T S ∗ C(α, β) (α, β ∈ [0, 1)) if and only if (1 − αβ) e p ≤ 1 − α. Corollary 52 If p > 0, then the function G( p, z) defined by (104) is in the class T C(α, β) (α, β ∈ [0, 1)) if and only if /

0 (1 − αβ) p 2 + (3 − 2αβ − α) p e p ≤ 1 − α.

Remark 4 Results obtained in Corollaries 51 and 52 verifies results given by Theorem 3 and Theorem 4 in [42], respectively.

8.1 Some Integral Operators Involving the Functions F( p, z) and G( p, z) In this subsection, we will examine some inclusion properties of integral operators associated with the functions F( p, z) and G( p, z) as follows: ˆ p, z) = F(



z 0

F( p, t) ˆ p, z) = dt and G( t

 0

z

G( p, t) dt. t

(109)

Theorem 27 Let p > 0 and the following condition is provided:   (1 − αβ) γ pe p + (1 − β)(1 − γ )α(e p − 1) − (1 − β)(1 − γ )αp −1 e p − (1 + p) ≤ 1 − α.

(110)

ˆ p, z) defined by (109) is in the class Then, the function F( S ∗ C(α, β; γ ) (α, β ∈ [0, 1) , γ ∈ [0, 1]). Proof Since ˆ p, z) = z + F(

∞  p n−1 − p n e z , z∈U n! n=2

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N. Mustafa et al.

ˆ p, z) belongs to the class S ∗ C(α, β; γ ) if according to Theorem 23, the function F( the following condition is satisfied: ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

p n−1 − p e ≤ 1 − α. n!

(111)

Let L 2 ( p; α, β; γ ) =

∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

p n−1 − p e n!

Setting (1 + (n − 1)γ ) (n − α − (n − 1)αβ) = (n − 1)n (1 − αβ) γ + n ((1 − αβ)(1 − γ ) + (1 − α)γ ) − (1 − β)(1 − γ )α

and by simple computation, we obtain ⎧ ⎫ ⎬ ∞ ⎨ (n − 1)n (1 − αβ) γ L 2 ( p; α, β; γ ) = n=2 +n(1 − β)(1 − γ )α − (1 − β)(1 − γ )α ⎩ ⎭ +n(1 − α) = (1 − αβ) γ p + (1 − β)(1 − γ )α(1 − e− p ) − + (1 − α)(1 − e− p ).

p n−1 e p n!

(1−β)(1−γ )α p

  1 − (1 + p)e− p

Therefore, inequality (111) holds true if (1 − β)(1 − γ )α(1 − e− p ) (1 − αβ) γ p +  (1−β)(1−γ )α  1 − (1 + p)e− p + (1 − α)(1 − e− p ) ≤ 1 − α, − p which is equivalent to (110). Thus, the proof of Theorem 27 is completed. By taking γ = 0 and γ = 1 in Theorem 27, we arrive at the following results. Corollary 53 If p > 0 and the following condition is satisfied   (1 − β)α (1 − p −1 )e p + p −1 ≤ 1 − α, ˆ p, z) defined by (109) belongs to the class S ∗ (α, β) (α, β ∈ then the function F( [0, 1)). Corollary 54 If p > 0 and the following condition is satisfied (1 − αβ) pe p ≤ 1 − α,

Geometric Properties of Normalized Wright Functions

787

ˆ p, z) defined by (109) belongs to the class C(α, β) (α, β ∈ then the function F( [0, 1)). Remark 5 Further consequences of the results given by Corollaries 53 and 54 can be obtained for each of the classes, by specializing the various parameters involved. ˆ p, z) defined by (109) is in the class Theorem 28 If p > 0, then the function G( ∗ T S C(α, β; γ )(α, β ∈ [0, 1) , γ ∈ [0, 1]) if and only if (1 − αβ) γ pe p + (1 − β)(1 − γ )α(e p − 1) −(1 − β)(1 − γ )αp −1 (e p − (1 + p)) ≤ 1 − α.

(112)

Proof From (109), we obtain ˆ p, z) = z − G(

∞  p n−1 − p n e z , z ∈ U. n! n=2

Further, from the proof of Theorem 27, we write: ∞ n=2

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

pn−1 − p e n!

= (1 − αβ) γ p + (1 − β)(1 − γ )α(1 − e− p )  )α  − (1−β)(1−γ 1 − (1 + p)e− p + (1 − α)(1 − e− p ). p

(113)

Now, suppose that (112) is satisfied. Then, − e−p ) (1 − αβ) γ p + (1 − β)(1  − γ−)α(1 −1 p −(1 − β)(1 − γ )αp 1 − e (1 + p) ≤ (1 − α)e− p , which is equivalent to (1 − αβ) γ p + (1 − β)(1 − γ )α(1 − e− p )    )α  − (1−β)(1−γ 1 − e− p (1 + p) + (1 − α) 1 − e− p ≤ 1 − α. p Hence, from (113), we have ∞  n=2

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

p n−1 − p e ≤ 1 − α. n!

ˆ p, z) belongs to the class Thus, according to Theorem 24, the function G( ∗ T S C(α, β; γ ). Now, let we prove of the necessity. Assume that Gˆ ∈ T S ∗ C(α, β; γ ). Then, according to Theorem 24, the following condition is satisfied:

788

N. Mustafa et al. ∞ 

(1 + (n − 1)γ ) (n − α − (n − 1)αβ)

n=2

p n−1 − p e ≤ 1 − α. n!

This follows (1 − αβ) γ p + (1 − β)(1 − γ )α(1 − e− p )  )α  − (1−β)(1−γ 1 − (1 + p)e− p + (1 − α)(1 − e− p ) ≤ 1 − α, p which readily yields (1 − αβ) γ p + (1 − β)(1 − γ )α(1 − e− p )  )α  − (1−β)(1−γ 1 − (1 + p)e− p ≤ (1 − α)e− p . p Last condition is equivalent to (112). This completes the proof of necessity. Thus, the proof of Theorem 28 is completed. By taking γ = 0 and γ = 1 in Theorem 28, we can readily deduce the following results, respectively. ˆ p, z) defined by (109) is in the class Corollary 55 If p > 0, then the function G( ∗ T S C(α, β) (α, β ∈ [0, 1)) if and only if   (1 − β)α (1 − p −1 )e p + p −1 ≤ 1 − α. ˆ p, z) defined by (109) is in the class Corollary 56 If p > 0, then the function G( T C(α, β) (α, β ∈ [0, 1)) if and only if (1 − αβ) pe p ≤ 1 − α. Remark 6 Result obtained in Corollary 56 verifies result given by Theorem 5 in [42].

References 1. Altinta¸s, O.: On a subclass of certain starlike functions with negative coefficiets. Math. Jpn. 36, 489–495 (1991) 2. Altinta¸s, O., Irmak, H., Srivastava, H.M.: Fractional calculus and certain starlike functions with negative coefficients. Comput. Math. Appl. 30(2), 9–15 (1995) 3. Altinta¸s, O., Irmak, H., Owa, S., Srivastava, H.M.: Coefficient bounds for some families of starlike and convex functions of complex order. Appl. Math. Lett. 20(12), 1218–1222 (2007) 4. Altinta¸s, O., Owa, S.: On subclasses of univalent functions with negative coefficients. Pusan Kyongnam Math. J. 4, 41–46 (1988) 5. Altinta¸s, O., Özkan, Ö., Srivastava, H.M.: Neighborhoods of a certain family of multivalent functions with negative coefficients. Comput. Math. Appl. 47(10–11), 1667–1672 (2004)

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6. Baricz, A., Frasin, B.A.: Univalence of integral operators involving Bessel functions. Appl. Math. Lett. 23(4), 371–376 (2010) 7. Baricz, A., Ponnusamy, S.: Starlikeness and convexity of generalized Bessel functions. Integr. Transform. Spec. Funct. 21(9), 641–653 (2010) 8. Becker, J.: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. J. reine angew. Math. 255, 23–43 (1972) 9. Blezu, D.: On univalence criteria. Gen. Math. 14(1), 77–84 (2006) 10. Breaz, D., Breaz, N., Srivastava, H.M.: An extension of the univalent condition for a family of integral operators. Appl. Math. Lett. 22(1), 41–44 (2009) 11. Breaz, D., Günay, H.Ö.: On the univalence criterion of a general integral operator. J. Inequalities Appl. 2008(1), 702715 (2008) 12. Buckwar, E., Luchko, Y.: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227(1), 81–97 (1998) 13. Duren, P.L.: Univalent Functions (Grundlehren der mathematischen Wissenschaften), vol. 259. Springer, New York (1983) 14. Gao, C.Y., Yuan, S.M., Srivastava, H.M.: Some functional inequalities and inclusion relationships associated with certain families of integral operators. Comput. Math. Appl. 49(11–12), 1787–1795 (2005) 15. Goodman, A.W.: Univalent Functions. Mariner Company (1983) 16. Gorenflo, R., Luchko, Y., Mainardi, F.: Analytic properties and applications of Wright functions. Fract. Calc. Appl. Anal. 2(4), 383–414 (1999) 17. Irmak, H., Lee, S.H., Cho, N.E.: Some multivalently starlike functions with negative coefficients and their subclasses defined by using a differential operator. Kyungpook Math. J. 37(1), 43 (1997) 18. Kadio˘glu, E.: On subclass of univalent functions with negative coefficients. Appl. Math. Comput. 146(2–3), 351–358 (2003) 19. Luchko, Y., Gorenflo, R.: Scale-invariant solutions of a partial differential equation of fractional order. Fract. Calc. Appl. Anal. 3(1), 63–78 (1998) 20. Maharana, S., Prajapat, J.K., Bansal, D.: Geometric properties of Wright function. Math. Bohemica 143(1), 99–111 (2018) 21. Mainardi, F.: Fractional calculus some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Berlin (1971) 22. Miller, S.S., Mocanu, P.T.: Univalence of Gaussian and confluent hypergeometric functions. Proc. Am. Math. Soc. 1, 333–342 (1990) 23. Mostafa, A.O.: A study on starlike and convex properties for hypergeometric functions. J. Inequalities Pure Appl. Math. 10(3), 1–6 (2009) 24. Murugusundaramoorthy, G., Vijaya, K., Porwal, S.: Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series. Hacet. J. Math. Stat. 45(4), 1101–1107 (2016) 25. Mustafa, N.: Close-to-convexity of normalized Wright function. J. Sci. Eng. 18(54), 290–303 (2016) 26. Mustafa, N.: Some geometric properties of the Wright functions. In: AIP Conference Proceedings 2016, vol. 1726, No. 1, p. 020080. AIP Publishing, New York (2016) 27. Mustafa, N.: Geometric properties of normalized Wright functions. Math. Comput. Appl. 21(2), 14 (2016) 28. Mustafa, N.: Integral operators of the normalized Wright functions and their some geometric properties. Gazi Univ. J. Sci. 30(1), 333–343 (2017) 29. Mustafa, N.: Univalence of certain integral operators involving normalized Wright functions. Commun. Fac. Sci. Univ. Ank.-Ser. A1 Math. Stat. 66(1), 19–28 (2017) 30. Mustafa, N., Altinta¸s, O.: Normalized Wright functions with negative coefficients and their some integral transforms. TWMS J. Pure Appl. Math. 9(2), 190–206 (2018) 31. Mustafa, N., Nezir, V.: Applications of a Poisson distribution series on the analytic functions. In: AIP Conference Proceedings 2017, vol. 1833, No. 1, p. 020012. AIP Publishing, New York (2017)

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32. Owa, S.: Some applications of the fractional calculus. Res. Notes Math. 138, 164–175 (1985) 33. Owa, S., Aouf, M.K.: On subclasses of univalent functions with negative coefficients. East Asian Math. J. 4, 57–73 (1988) 34. Owa, S., Nunokawa, M., Saitoh, H., Srivastava, H.M.: Close-to-convexity, starlikeness, and convexity of certain analytic functions. Appl. Math. Lett. 15(1), 63–69 (2002) 35. Pescar, V.: A new generalization of Ahlforss and Beckers criterion of univalence. Bull. Malays. Math. Sci. Soc. (Second. Ser.) 19, 53–54 (1996) 36. Pescar, V.: Univalence of certain integral operators. Acta Univ. Apulensis 12, 43–48 (2006) 37. Podlubny, I.: Fractional Differential Equations. Academic, San Diego 38. Ponnusamy, S., Ronning, F.: Geometric properties for convolutions of hypergeometric functions and functions with the derivative in a halfplane. Integr. Transform. Spec. Funct. 8(1–2), 121– 138 (1999) 39. Ponnusamy, S., Singh, V., Vasundhra, P.: Starlikeness and convexity of an integral transformisntegral transform. Integr. Transform. Spec. Funct. 15(3), 267–280 (2004) 40. Ponnusamy, S., Vuorinen, M.: Univalence and convexity properties for confluent hypergeometric functions. Complex Var. Elliptic Equ. 36(1), 73–97 (1998) 41. Ponnusamy, S., Vuorinen, M.: Univalence and convexity properties for Gaussian hypergeometric functions. Rocky Mt. J. Math. 1, 327–353 (2001) 42. Porwal, S.: An application of a Poisson distribution series on certain analytic functions. J. Complex Anal. Art. ID 984135, 1–3 (2014) 43. Porwal, S., Dixit, K.K.: An application of generalized Bessel functions on certain analytic functions. In: Acta Universitatis Matthiae Belii. Series Mathematics, pp. 51–57 (2013) 44. Prajapat, J.K.: Certain geometric properties of the Wright function. Integr. Transform. Spec. Funct. 26(3), 203–212 (2015) 45. Raza, M., Din, M.U., Malik, S.N.: Certain geometric properties of normalized Wright functions. J. Funct. Spaces Article ID 1896154, p. 8 (2016). http://dx.doi.org/10.1155/2016/1896154 46. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Translated from the 1987 Russian Original. Gordon and Breach, Yverdon (1993) 47. Srivastava, H.M., Owa, S.: Current Topics in Analytic Function Theory. World Scientific, Singapore (1992) 48. Topkaya, S., Mustafa, N.: The general subclasses of the analytic functions and their various properties. Asian Res. J. Math. 7(4), 1–12 (2017) 49. Wright, E.M.: On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 1(1), 71–79 (1933)

On the Spectra of Difference Operators Over Some Banach Spaces Hemen Dutta and P. Baliarsingh

Abstract The primary objective of this chapter is to provide a literature review on spectral subdivisions of difference operators and compute the spectrum and the fine spectrum of third order difference operator 3 over the Banach space c. The generalized difference operator 3 : c → c is defined by (3 x)k =

3  i=0

(−1)i

  3 xk−i = xk − 3xk−1 + 3xk−2 − xk−3 , (k ∈ N0 ) i

It is presumed that xn = 0 if n < 0. The operator 3 represents a lower forth band infinite matrix. Finally, we find the estimates for the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the above operator over the Banach spaces c, c0 and 1 . Keywords Third order difference operator 3 · Spectrum of an operator · Sequence spaces 2010 Mathematics Subject Classification 47A10 · 40A05 · 46A45

1 Introduction, Preliminaries and Definitions The study of spectrum and fine spectrum of linear bounded operators has been considered as one of the emerging areas of research in operator theory, which generalizes the notion of eigenvalues. The idea is usually used in linear algebra such as in solving H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, India e-mail: [email protected] P. Baliarsingh (B) Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, Studies in Systems, Decision and Control 177, https://doi.org/10.1007/978-3-319-99918-0_23

791

792

H. Dutta and P. Baliarsingh

system of linear equations, matrix inversions, problems related to spectral analysis in matrix theory etc. (see [1, 2]). In particular, the spectrum of difference operators has several applications in different scientific and engineering problems concerning the eigenvalues and others. Let w be the space of all sequences of real or complex numbers and any subspace of it is called a sequence space. Usually, we write ∞ , c, c0 and 1 for the spaces of all bounded, convergent, null and absolutely summable sequences, respectively. It is known that the spaces c, c0 and ∞ are Banach spaces under the usual sup norm .∞ = supk |xk |. Also, by  p , bv and bv p , we denote the spaces of all pabsolutely summable sequences, bounded variation sequences and p-bounded variation sequences, respectively. Let us summarize the definitions of certain triangle limitation matrices generated by difference operators. We write U for the set of all sequences u = (u n ) such that u n = 0 for all n ∈ N0 . For u ∈ U, let 1/u = (1/u n ). Let u, v ∈ U. Then define the Cesàro matrix of order one C1 = (cnk ), the difference matrix of order one  = (δnk ), r ) and the generalized weighted the generalized rth order difference matrix r = (δnk mean G(u, v) = (gnk ) as follows:  cnk = 

1/n, (0 ≤ k ≤ n), 0, (k > n).

(−1)n−k , (n − 1 ≤ k ≤ n), 0, otherwise.   r  (−1)n−k n−k , (max{0, n − r } ≤ k ≤ n), = , 0, otherwise.  u n vk , (0 ≤ k ≤ n), = 0, (k > n).

δnk = r δnk

gnk

for all k, n, r ∈ N0 , where u n depends only on n and vk only on k. Let A = (ank ) be an infinite matrix of real numbers ank , where n, k ∈ N0 . For the sequence spaces X and Y we write a matrix mapping A : X → Y defined by (Ax)n =



ank xk ,

(n ∈ N0 ).

(1)

k

 For every x = (xk ) ∈ X , we call Ax as the A−transform of x if the series k ank xk converges for each n ∈ N0 . By (X : Y ), we denote the class of all infinite matrices A such that A : X → Y . Thus, A ∈ (X : Y ) if and only if Ax ∈ Y for every x ∈ X . With the help of matrix transformations, define the set S(X, Y ) by S(X, Y ) = {z = (z k ) : x z = (xk z k ) ∈ Y for all x ∈ X }.

(2)

On the Spectra of Difference Operators Over Some Banach Spaces

793

Recently, the investigation on finding the spectra of operators derived from different means or difference operators with variable orders has become an active topic of research in functional analysis. Many prominent authors have contributed largely in the literature by adding different new and novel ideas in the field of spectral theory. In the present literature, there are many works concerning the spectrum and the fine spectra of an operator over different sequence spaces. For instance, the fine spectrum of the Cesàro operator on the sequence space  p for 1 < p < ∞ has been studied by Gonzalez [3]. The fine spectrum of the integer power of the Cesàro operator over c was examined by Wenger [4] and then Rhoades [5] generalized this result to the weighted mean methods. Reade [6] studied the spectrum of the Cesàro operator over the sequence space c0 . Okutoyi [7] computed the spectrum of the Cesàro operator over the sequence space bv. The fine spectra of the Cesàro operator over the sequence spaces c0 and bv p have been determined by Akhmedov and Ba¸sar [8, 9]. Akhmedov and Ba¸sar [10, 11] have studied the fine spectrum of the difference operator  over the sequence spaces  p and bv p , where 1 < p < ∞. Altay and Ba¸sar [12, 13] have determined the fine spectrum of the difference operator  over the sequence spaces c0 , c and  p , for 0 < p < 1. Furkan and Kayaduman [14] studied the fine spectrum of the generized difference operator B(r, s) over the sequence spaces 1 and bv. The fine spectrum of the difference operator  over the sequence spaces 1 and bv was investigated by Kayaduman and Furkan [15]. Srivastava and Kumar [16] have examined the fine spectrum of the generalized difference operator ν over the sequence spaces c0 . Akhmedov and Shabrawy [17] studied the fine spectrum of the operator a,b over the sequence space c, where ab was defined as a lower double sequential band matrix with the convergent sequences a = (ak ) and b = (bk ) having certain properties. Recently, Dutta and Baliarsingh [18, 19] have computed the fine spectra of the generalized difference operator rν over the sequence spaces 1 and c0 , where r ∈ N and ν = (νk ) is either a decreasing sequence or constant sequence of positive real numbers satisfying certain conditions. Kızmaz [20] introduced the idea of forward difference operator with order one by defining xk = xk − xk+1 , (k ∈ N0 ), and studied the related sequence spaces in detail. The idea was extended to the case of order m by Et and Colak [21] as   m  i m xk+i , (k ∈ N0 ), (−1)  xk = i i=0 m

and further it was extended and studied by many other prominent authors. The backward difference operator , defined by (x)k = xk − xk−1 (k ∈ N0 ), and its spectral properties have been studied by Altay and Ba¸sar [13]. Several difference operators with different orders have been studied and been applied in the diverse fields of pure and applied sciences such as linear algebra, operator theory, approximation theory, theory of calculus etc. Some of such applications are found in many research articles such as [20–31].

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Now, the third order difference operator 3 : c → c is defined by 3 x = {(3 x)k }, where (3 x)k =

3  i=0

(−1)i

  3 xk−i = xk − 3xk−1 + 3xk−2 − xk−3 , (k ∈ N0 ) i

(3)

with xk = 0 for k < 0, where x ∈ c. It is easy to verify that the operator 3 represents a lower triangular 4-th band infinite matrix ⎛

1 ⎜ −3 ⎜ ⎜ 3 = ⎜ 3 ⎜ −1 ⎝ .. .

0 1 −3 3 .. .

0 0 1 −3 .. .

0 0 0 1 .. .

⎞ ... ...⎟ ⎟ ...⎟ ⎟. ...⎟ ⎠ .. .

Now, we provide the definition of the spectrum and fine spectrum of a linear bounded operator: Let X and Y be Banach spaces and T : X → Y be a bounded linear operator. By R(T ), we denote the range of T , i.e. R(T ) = {y ∈ Y : y = T x ; x ∈ X }. By B(X ), we denote the set all bounded linear operators on X into itself. If X is any Banach space and T ∈ B(X ) then the adjoint T ∗ of T is a bounded linear operator on the dual X ∗ of X defined by (T ∗ φ)(x) = φ(T x) for all φ ∈ X ∗ and x ∈ X with T  = T ∗ . Let X = {0} be a normed linear space over the complex field and T : D(T ) → X be a linear operator, where D(T ) denotes the domain of T . With T, for a complex number α, we associate an operator Tα = (T − αI ), where I is called the identity operator on D(T ) and if Tα has an inverse, we denote it by Tα−1 i.e. Tα−1 = (T − αI )−1 and is called the resolvent operator of T. Many properties of Tα and Tα−1 depend on α and the spectral theory is concerned with those properties. We are interested in the set of all α’s in the complex plane such that Tα−1 exists/ Tα−1 is bounded/ domain of Tα−1 is dense in X . For our investigation, we need some basic concepts in spectral theory which are given as some definitions and lemmas. Definition 1 ([2], pp. 371) Let X and T be defined as above. A regular value of T is a complex number α such that (R1) Tα−1 exists; (R2) Tα−1 is bounded; (R3) Tα−1 is defined on a set which is dense in X .

On the Spectra of Difference Operators Over Some Banach Spaces

795

The resolvent set ρ(T, X ) of T is the set of all regular values of T . Its complement σ(T, X ) = C \ ρ(T, X ) in the complex plane C is called the spectrum of T . Furthermore, the spectrum σ(T, X ) is partitioned into three disjoint sets as follows. (I) Point spectrum σ p (T, X ): It is the set of all α ∈ C such that (R1) does not hold. The elements of σ p (T, X ) are called eigenvalues of T . (II) Continuous spectrum σc (T, X ): It is the set of all α ∈ C such that (R1) holds and satisfies (R3) but does not satisfy (R2). (III) Residual spectrum σr (T, X ): It is the set of all α ∈ C such that (R1) holds but does not satisfy (R3). The condition (R2) may or may not hold. Goldberg’s classification of operator Tα : ([1], pp. 58–71) Let X be a Banach space and Tα = (T − αI ) ∈ B(X ), where α is a complex number. Again, let R(Tα ) and Tα−1 denote the range and inverse of the operator Tα respectively. Then the following possibilities may occur: (A) R(Tα ) = X ; (B) R(Tα ) = R(Tα ) = X ; (C) R(Tα ) = X ; and (1) Tα is injective and Tα−1 is continuous; (2) Tα is injective and Tα−1 is discontinuous; (3) Tα is not injective. Taking the permutations (A), (B), (C) and (1), (2), (3), we get nine different states. These are labeled by A1 , A2 , A3 , B1 , B2 , B3 , C1 , C2 and C3 . If α is a complex number such that Tα ∈ A1 or Tα ∈ B1 , then α is in the resolvent set ρ(T, X ) of T on X . The other classifications give rise to the fine spectrum of T . We use α ∈ B2 σ(T, X ) means the operator Tα ∈ B2 , i.e. R(Tα ) = R(Tα ) = X and Tα is injective but Tα−1 is discontinuous. Similarly others. Lemma 1 ([1], pp. 59) A linear operator T has a dense range if and only if the adjoint T ∗ is one to one. Lemma 2 ([1], pp. 60) The adjoint operator T ∗ is onto if and and only if T has a bounded inverse. Lemma 3 ([32], pp. 126) The matrix A = (ank ) gives rise to a bounded linear operator T ∈ B(1 ) from 1 to itself if and only if the supremum of 1 norms of the columns of A is bounded. Lemma 4 ([32], pp. 6) The matrix A = (ank ) gives rise to a bounded linear operator T ∈ B(c) from c to itself if and only if (i) the rows of A are in 1 and their 1 norms are bounded, (ii) the columns of A are in c and (iii) the sequence of row sums of A is in c.

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The operator norm of T is the supremum of 1 norms of the rows. Lemma 5 ([32], pp. 129) The matrix A = (ank ) gives rise to a bounded linear operator T ∈ B(c0 ) from c0 to itself if and only if (i) The rows of A are in 1 and their 1 norms are bounded, (ii) The columns of A are in c0 , The operator norm of T is the supremum of 1 norms of the rows. If T : c → c is a bounded operator represented by the matrix A, then the adjoint operator T ∗ : c∗ → c∗ acting on C ⊕ 1 has a matrix representation of the form 

χ 0 B At

 ,

where χ is the limit of the sequence of row sums of A minus the sum of the limits of the columns of A and B is the column vector whose kth entry is the limit of the kth column of A for each k ∈ N0 . For 3 : c → c, the matrix (3 )∗ ∈ B(1 ) is of the form   0 0 . 0 (3 )t As a part of review, we quote some existing results on the spectrum of the difference operators over the spaces c0 , 1 and c in the following table. The notations used in Table 1 are as follows: L = lim νk , k→∞

D = {α ∈ C : |a − α| ≤ |b|}, E = {ak : ak ∈ / D, k ∈ N}, G = {α ∈ C : ∃k0 ∈ N such that |ak − α| = |bk |, for all k ≥ k0 }. H = σ p (a,b , c) ∪ {a + b}.

[33] [34] [34]

[16] [14, 35] [14, 35] [14, 35] [14, 35]

∅

{α ∈ C : |α − 1| < 2}

{α ∈ C : 2 < |α − 1| ≤ 3}   α ∈ C : |α − 1| ≤ 43

∅

{α ∈ C : |α − 1| < 1}   α ∈ C : 1 < |α − 1| ≤ 43   α ∈ C : |1 − αL | ≤ 1

∅  α ∈ C : |1 −

∅

{α ∈ C : |α − r| ≤ |s|}

∅

{α ∈ C : |α − r| < |s|} ∪ {r + s}

{α ∈ C : |α − r| = |s|} \ {r + s}       ≤1 √2(r−α) α ∈ C :  −s+ s 2 −4t (r−α) 

σ(2 , c0 )

σ p (2 , c0 )

σr (2 , c0 )

σc (2 , c0 )

σ(i2 , c0 /1 )

σ p (i2 , c0 /1 )

σr (i2 , c0 /1 )

σ p (ν , c0 )

σc (ν , c0 )

σ(B(r, s), c0 /c/1 )

σ p (B(r, s), c0 /c/1 )

σr (B(r, s), c0 /c/1 )

σc (B(r, s), c0 /c/1 )

σr (B(r, s, t), c0 /c/1 )

σc (B(r, s, t), c0 /c/1 )

σ(B(r, s, t), c0 /c/1 )

σr (ν , c0 )

σ(ν , c0 )

∅    α ∈ C : 

[33] [33]

{α ∈ C : |α − 1| ≤ 3}

σc (, c/c0 / p )

    < 1 ∪ {r + s + t} √2(r−α)  −s+ s 2 −4t (r−α)

 ≤1

[10, 13] [33]

{α ∈ C : |α − 1| = 1} (\{0})

σr (, c/c0 / p )

α L|

[10, 13]

{α ∈ C : |α − 1| < 1} (∪{0})

σ p (, c/c0 / p )

σc (i2 , c0 /1 )

[10, 13]

∅

σ(, c/c0 / p )

[36, 37]

[36, 37]

[36, 37]

[16]

[16]

[16]

[34]

[34]

Ref. to: [10, 13]

Defined sets

{α ∈ C : |α − 1| ≤ 1}

Spectral subdivisions

Table 1 Spectra of some difference operators

(continued)

On the Spectra of Difference Operators Over Some Banach Spaces 797

k

L−α

∅

k

σc (rν , c0 /1 )

σr (rν , c0 /1 )

∅     ⎧  α ⎪ ⎪ ⎨ α ∈ C : 1 − L  ≤ r , for an increasing  sequence of reals ⎪ ⎪ ⎩n and n 1/k  Lr  < 1 (for large k)

   ⎧  α ⎪ ⎪ ⎨ α ∈ C : 1 − L  ≤ r , for an increasing  sequence of reals ⎪ ⎪ ⎩n and n 1/k  Lr  < 1 (for large k) k k L−α

⎪ (G \ {a + b}) ⊆ σc (a,b , c), ⎪    ⎪ ⎪ ⎩ α ∈ C : inf  ak −α  < 1 ∩ E \ {a + b} ⊆ σ ( , c). k  bk  c a,b

⎧ σc (a,b , c) ⊆ ({α ∈ C : |a − α| = |b|} ∪ E) \ H, ⎪  ⎪  ⎪ ⎪ ⎨σc (a,b , c) ⊆ ((D ∪ E) ∩ α ∈ C : sup  ak −α  ≥ 1 \ H, k bk

⎧ ⎪ ⎪{α ∈ C : |a − α| < |b|} ∪ {a + b} ⊆ σr (a,b , c), ⎪ ⎪ ⎪ {a : k ∈ N} \ σ ⎪ ⎪  p (a,b  , c)⊆ σr (a,b , c), ⎨ k   α ∈ C : supk  akb−α  < 1 ⊆ σr (a,b , c),     k ⎪ ⎪   ⎪ ⎪ σ ( , c) ⊆ α ∈ C : inf k  akb−α  < 1 ∪ {a + b}, ⎪ r a,b k ⎪ ⎪ ⎩ σr (a,b , c) ⊆ ((D ∪ E) \ G) ∪ {a + b}.

D∪E  E, ∃m ∈ N : ai  = a j , ∀ j ≥ m ∅, otherwise

    = 1 \ {r + s + t} √2(r−α)  2 −s+ s −4t (r−α)

Defined sets    α ∈ C : 

σ p (rν , c0 /1 )

σ(rν , c0 /1 )

σc (a,b , c)

σr (a,b , c)

σ p (a,b , c)

σ(a,b , c)

σr (B(r, s, t), c0 /c/1 )

Spectral subdivisions

Table 1 (continued) Ref. to:

[18, 19]

[18, 19]

[18, 19]

[18, 19]

[17]

[17]

[17]

[17]

[36, 37]

798 H. Dutta and P. Baliarsingh

On the Spectra of Difference Operators Over Some Banach Spaces

799

2 Main Results In this section, we compute the point spectrum, the spectrum, the continuous spectrum, the residual spectrum and the fine spectrum of the difference operator 3 on the sequence spaces c, c0 and 1 . Theorem 1 The operator 3 : c → c is a linear and bounded operator and  3 (c:c) = 8.

(4)

Proof The proof follows from the Lemma 4 and the following result 1 + | − 3| + 3 + 1 = 8.  Theorem 2 The point spectrum of the difference operator 3 on the sequence c is given by σ p (3 , c) = ∅. Proof Suppose x ∈ c and consider 3 x = αx for x = θ in c, which provides a system of linear equations: x0 = αx0 −3x0 + x1 = αx1 3x0 − 3x1 + x2 = αx2 −x0 + 3x1 − 3x2 + x3 = αx3 ........ −xk−3 + 3xk−2 − 3xk−1 + xk = αxk ........

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(5)

On solving above system of equations, it is clear that for α = 1, we have a solution x0 = 0, x1 = 0, x2 = 0, .... which contradicts our assumption. Suppose x is an eigenvector corresponds to the eigenvalue α and xk , k = 0 is the first non zero entry of x = (xk ). Then from the above system of equations, one 2 3 x0 , similarly, x2 = 3(α+2) x , x3 = α +16α+10 x0 and so on. can obtain that x1 = 1−α (1−α)2 0 (1−α)3 Proceeding this way, we can get (3 − αI )x = 0 has a solution for α = 1 and  x0 = 0, which is a contradiction. Thus σ p (3 , c) = ∅. Theorem 3 The spectrum of the difference operator 3 over the sequence space c is given by   σ(3 , c) = α ∈ C : |1 − α| ≤ 7 . (6) Proof We divide the proof into two parts: Part 1: In the first part , we have to show that

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  σ(3 , c) ⊆ α ∈ C : |1 − α| ≤ 7 . Equivalently, we need to show that if α ∈ C with |1 − α| > 7 ⇒ α ∈ / σ(3 , c). Let α ∈ C with |1 − α| > 7. Now (3 − αI ) = (ank ) is a triangle and hence has an inverse (3 − αI )−1 = (bnk ) where ⎛

(3 − αI )−1

1 1−α 3 (1−α)2

⎜ ⎜ ⎜ ⎜ 3(α+2) ⎜ ⎜ (1−α)3 ⎜ ⎜ α2 +16α+10 (1−α)4 =⎜ ⎜ ⎜ 15α2 +51α+15 ⎜ (1−α)5 ⎜ ⎜ α3 +90α2 +106α+41 ⎜ (1−α)6 ⎜ ⎝ .. .

0

0

0

0

1 1−α

0

0

0

3 (1−α)2

1 1−α

0

0

3(α+2) (1−α)3

3 (1−α)2

1 1−α

0

α2 +16α+10 (1−α)4

3(α+2) (1−α)3

3 (1−α)2

1 1−α

15α2 +51α+15 α2 +16α+10 3(α+2) 3 (1−α)4 (1−α)3 (1−α)2 (1−α)5

.. .

.. .

.. .

.. .

0 ...

⎞

⎟ 0 ...⎟ ⎟ ⎟ 0 ...⎟ ⎟ ⎟ 0 ...⎟ ⎟, ⎟ ⎟ 0 ...⎟ ⎟ ⎟ 1 ⎟ . . . 1−α ⎟ ⎠ .. . . . .

It is noticed that the operator (3 − αI )−1 represents a banded matrix and the general expression of the elements bnk for k ≤ n can be computed as bnn = bn,n−1 = bn,n−2 = bn,n−3 = bn,n−4 = bn,n−5 = bn,n−6 = bn,n−7 = bn,n−8 = bn,n−9 =

1 , 1−α 3 , (1 − α)2 9 3 − , (1 − α)3 (1 − α)2 27 18 1 − + , (1 − α)4 (1 − α)3 (1 − α)2 81 81 15 − + , (1 − α)4 (1 − α)3 (1 − α)5 243 324 108 6 − + − (1 − α)6 (1 − α)4 (1 − α)3 (1 − α)5 729 1215 594 81 1 − + − + (1 − α)7 (1 − α)6 (1 − α)4 (1 − α)3 (1 − α)5 2187 4374 2835 648 36 − + − + (1 − α)8 (1 − α)7 (1 − α)6 (1 − α)4 (1 − α)5 6561 15309 12393 4050 459 9 − + − + − (1 − α)9 (1 − α)8 (1 − α)7 (1 − α)6 (1 − α)4 (1 − α)5 19683 52488 51030 21870 3915 216 1 − + − + − + (1 − α)10 (1 − α)9 (1 − α)8 (1 − α)7 (1 − α)6 (1 − α)4 (1 − α)5

. . . bn,k = . . .

1  3bn−1,k − 3bn−2,k + bn−3,k ; 0 ≤ k ≤ n, 1−α

On the Spectra of Difference Operators Over Some Banach Spaces

801

Now, we need to prove that (3 − αI )−1 ∈ B(c), i.e., (i) the series

∞ 

|bnk | is convergent for each n ∈ N0 and sup

k=0

n

∞ 

|bnk | < ∞,

k=0

(ii) lim bnk exists for each k ∈ N0 or the sequence (b0k , b1k , b2k , . . . ) is convergent. n→∞

(iii) the sequence of row sums of (bnk ) is convergent or lim

n→∞

∞ 

bnk exists.

k=0

In order to prove (i), we consider Sn =

n 

|bnk | = |bn0 | + |bn,1 | + |bn,2 | + · · · + |bnn | = |bnn | + |bn,n−1 | + |bn,n−2 | + . . .

k=0

            27   1   3 9 3 18 1 + + + + − − + =        1−α (1 − α)2 (1 − α)3 (1 − α)2 (1 − α)4 (1 − α)3 (1 − α)2       81  81 15   243 243 108 6    (1 − α)5 − (1 − α)4 + (1 − α)3  +  (1 − α)6 − (1 − α)5 + (1 − α)4 − (1 − α)3  +     729   2187 1215 594 90 1 4374     (1 − α)7 − (1 − α)6 + (1 − α)5 − (1 − α)4 + (1 − α)3  +  (1 − α)8 − (1 − α)7 +  648 36  2835 + ... − + (1 − α)6 (1 − α)4  (1 − α)5             !  3      3   27   18   9 1 + + + +  + + 1  + ≤ 1 +            2 3 2 |1 − α| (1 − α) (1 − α) (1 − α) (1 − α) (1 − α) (1 − α)                 81              6   + − 81  +  15  +  243  +  243  +  108  +    (1 − α)4   (1 − α)3   (1 − α)2   (1 − α)5   (1 − α)4   (1 − α)3   (1 − α)2  +                729   1215   2187   4374    594   90   1                (1 − α)6  +  (1 − α)5 + +  (1 − α)4  +  (1 − α)3  +  (1 − α)2  +  (1 − α)7  +  (1 − α)6  +       "  2835   648   36         (1 − α)5  +  (1 − α)4  +  (1 − α)3  + . . .       !  3 + 3 + 1   9 + 18 + 15 + 6 + 1   27 + 81 + 108 + 81 + 36 + 9 + 1  1 + + + = 1 +     |1 − α| (1 − α)   (1 − α)2 (1 − α)3   "  81 + 324 + 594 + 648 + 459 + 216 + 66 + 12 + 1    + ...   (1 − α)4         " ! 2 3  7   7   7   74  1 + + +  + ... = 1 +         2 3 4 |1 − α| (1 − α) (1 − α) (1 − α) (1 − α)

Now, with the condition |1 − α| > 7, we have      $ #     7   72   73   74  1   + lim Sn ≤ 1+ + +  + ... n→∞ |1 − α| (1 − α)   (1 − α)2   (1 − α)3   (1 − α)4  =

1 < ∞. |1 − α| − 7

Therefore, the sequence (Sn ) is convergent. Since (Sn ) is a sequence of positive real numbers and convergent, it is bounded. Also, under the condition |1 − α| > 7, it is observed that the sequence (bnk )∞ n=0 is convergent for each k ∈ N0 .

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Finally, considering the sequence of row sums (X k ) of (bnk ), we have for each k ≥ 1, X k = X k−1 + bk0 , 1 . As limn→∞ Sn is finite and the sequence (bnk )∞ and X 0 = 1−α n=0 is convergent, the n is also bounded and convergent. As a consequence, the sequence ( k=0 bnk )∞ n=0 sequence (X k ) is convergent. Thus, (3 − αI )−1 ∈ B(c) with |1 − α| > 7. Now, we show that domain of the operator (3 − αI )−1 is dense in c, or the range of (3 − αI ) is dense in c, which implies the operator (3 − αI )−1 is onto. Hence we have   (7) σ(3 , c) ⊆ α ∈ C : |1 − α| ≤ 7 .

Part 2:   Conversely, to show α ∈ C : |1 − α| ≤ 7 ⊆ σ(3 , c). Consider α = 1 and |1 − α| < 7, clearly (3 − αI ) is a triangle and hence (3 − αI )−1 exists. Also, Sn is sequence of positive reals and limn Sn = supn Sn is not finite, thus, Sn is unbounded. ⇒

/ (c, c) with |1 − α| < 7. (3 − αI )−1 ∈

Again 1 = α ∈ C with |1 − α| = 7 which implies limk Sk = ∞. Thus Sn is unbounded. / (c, c) with |1 − α| = 7. ⇒ (3 − αI )−1 ∈ Finally, we check the result under the assumption α = 1. We have ⎛

0 ⎜ −3 ⎜ ⎜ (3 − αI ) = ⎜ 3 ⎜ −1 ⎝ .. .

0 0 −3 −3 .. .

0 0 0 3 .. .

0 0 0 0 .. .

⎞ ... ...⎟ ⎟ ...⎟ ⎟, ...⎟ ⎠ .. .

which is not invertible. Hence   α ∈ C : |1 − α| ≤ 7 ⊆ σ(3 , c). Combining (6) and (7) we conclude the proof.

(8) 

Theorem 4 The point spectrum of the adjoint operator (3 )∗ of 3 over c∗  1 is given by   α ∈ C : |1 − α| < 7 = σ p ((3 )∗ , 1 ). (9) Proof Let us consider (3 )∗ f = α f for 0 = f ∈ 1 , where

On the Spectra of Difference Operators Over Some Banach Spaces

⎛

0  ⎜  ⎜0 χ 0 ⎜ 3 ∗ = ⎜0 ( ) = B (3 )t ⎜0 ⎝ .. .

0 1 0 0 .. .

0 −3 1 0 .. .

0 3 −3 1 .. .

0 −1 3 −3 .. .

0 0 −1 3 .. .

803

⎛ ⎞ ⎞ f0 ... ⎜ f1 ⎟ ...⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ...⎟ ⎟ and f = ⎜ f 2 ⎟ . ⎜ f3 ⎟ ⎟ ...⎠ ⎝ ⎠ .. .. . .

Then the system of linear equations, above can be rewritten as 0 = α f0 f1 − 3 f2 + 3 f3 − f4 = α f1 f2 − 3 f3 + 3 f4 − f5 = α f2 f3 − 3 f4 + 3 f5 − f6 = α f3 .. .

f k − 3 f k+1 + 3 f k+2 − f k+3 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ = α fk ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(10)

The general form of the above equations can be expressed as | fk | =

   3  f k+1 − f k+2 + 1 f k+3  , k ≥ 1.  |1 − α|  3

It is clear that for each f ∈ 1 of the form ⎛ ⎜ ⎜ ⎜ f =⎜ ⎜ ⎝

⎞ 0 f1 ⎟ ⎟ f2 ⎟ ⎟, f3 ⎟ ⎠ .. .

can be treated as an eigenvector corresponding to an eigenvalue α if and only if |1 − α| < 7. For example: let us consider a sequence f = ( f k ), defined by f k = r k for all k ∈ N and |r | < 1. Then f ∈ 1 and from above system of equations, we have for all k ≥ 1,     1 |1 − α|r k = 3 r k+1 − r k+2 + r k+3  3   2 3  ⇒ |1 − α| = 3r − 3r + r   < |3r | + |3r 2 | + r 3  < 7.   σ p ((3 )∗ , 1 ) ⊆ α ∈ C : |1 − α| < 7 .

(11)

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Now consider the general form of f k , i.e., | fk | =

  1    3 f k+1 − 3 f k+2 + f k+3  , k ≥ 1. |1 − α|

For |1 − α| < 7, ( f k ) is unbounded and hence the adjoint operator (3 − λI )∗ is unbounded and not one to one. Therefore, (3 − λI )∗ is not invertible and   α ∈ C : |1 − α| < 7 ⊆ σ p ((3 )∗ , 1 ). Combining (11) and (12), we conclude the proof.

(12) 

Theorem 5 The residual spectrum of the operator 3 over c is given by   σr (3 , c) = α ∈ C : |1 − α| < 7 . Proof For |1 − α| < 7, the operator 3 − αI has an inverse. By Theorem 4 the operator (3 )∗ − αI is not one to one for α ∈ C with |1 − α| < 7. By using Lemma 2, we have the proof.  Remark 1 {1, 0} ⊂ σr (3 , c). From the system equations (12), it is noticed that for α = 1, the system of equations has a non zero solution i.e., ⎛ ⎞ 0 ⎜1⎟ ⎜ ⎟ ⎜ ⎟ f = ⎜ 0 ⎟ ∈ 1 . ⎜0⎟ ⎝ ⎠ .. . Also, for α = 0 the system of above equations may have solution of the form ⎛ ⎞ 1 ⎜0⎟ ⎜ ⎟ ⎜ ⎟ f = ⎜ 0 ⎟ ∈ 1 , ⎜0⎟ ⎝ ⎠ .. . Therefore, α = 0, 1 are considered as two eigenvalues for the matrix (3 )∗ , which implies that {0, 1} ⊂ σ p ((3 )∗ , 1 ).

On the Spectra of Difference Operators Over Some Banach Spaces

805

Thus {0, 1} ⊂ σr (3 , c).

Theorem 6 The continuous spectrum of the operator 3 over c is given by   σc (3 , c) = α ∈ C : |1 − α| = 7 . Proof The proof follows from Theorems 2, 3, 5 and the fact that σ(3 , c) = σ p (3 , c) ∪ σr (3 , c) ∪ σc (3 , c).  Since the fine spectrum of the operator 3 over the sequence space c0 and 1 can be obtained using arguments similar to those used in the case of the space c, we omit the details and provide the following theorems without mentioning proof. Theorem 7 (i)  3 (c0 :c0 ) = 8. (ii)

  σ(3 , c0 ) = α ∈ C : |1 − α| ≤ 7 .

(iii) σ p (3 , c0 ) = ∅. (iv)

  σr (3 , c0 ) = α ∈ C : |1 − α| < 7 \ {0}.

(v)

  σc (3 , c0 ) = α ∈ C : |1 − α| = 7 ∪ {0}.

Proof To prove (iii), we need to solve the system of equations (3 )∗ f = αI , i.e., f0 − 3 f1 + 3 f2 − f1 − 3 f2 + 3 f3 − f2 − 3 f3 + 3 f4 − f3 − 3 f4 + 3 f5 − .. .

f3 f4 f5 f6

= α f0 = α f1 = α f2 = α f3

f k − 3 f k+1 + 3 f k+2 − f k+3 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ = α fk ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(13)

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Clearly, for α = 0 the above system of equations has a non zero solution of the form ⎛ ⎞ 1 ⎜1⎟ ⎜ ⎟ ⎜ ⎟ f = ⎜1⎟. ⎜1⎟ ⎝ ⎠ .. . ⎛ ⎞ 1 ⎜1⎟ ⎜ ⎟ ⎜ ⎟ Thus, α = 0 is an eigenvalue corresponds to the eigenvector f = ⎜ 1 ⎟, which is ⎜1⎟ ⎝ ⎠ .. .

not in 1 . Due the uniqueness of the eigenvector f correspond to eigenvalue α = 0, / σr (3 , c0 ).  we conclude that 0 ∈ / σ p ((3 )∗ , 1 ) and therefore, 0 ∈ Theorem 8 (i)  3 (1 :1 ) = 8. (ii)

  σ(3 , 1 ) = α ∈ C : |1 − α| ≤ 7 .

(iii) σ p (3 , 1 ) = ∅. (iv)

  σr (3 , 1 ) = α ∈ C : |1 − α| ≤ 7 .

(v) σc (3 , 1 ) = ∅. Corollary 1 If α satisfies |1 − α| > 7, then (3 − αI ) ∈ A1 . Proof Let α ∈ C with the condition that |1 − α| > 7. Then it is clear that α = 1. As the operator (3 − αI ) is a triangle, hence it has an bounded inverse (3 − αI )−1 . This implies the operator (3 − αI )−1 is continuous with |1 − α| > 7, Now, let (3 − αI )x = y ⇒ x = (3 − αI )−1 y. ⇒ xk = ((3 − αI )−1 y)∞ k=0 , k ∈ N0 . For every y ∈ c, we have a corresponding x ∈ c such that (3 − αI )x = y. This  shows that the operator (3 − αI ) is onto.

On the Spectra of Difference Operators Over Some Banach Spaces

807

Corollary 2 If α = 1 and α ∈ σr (3 , c), then α ∈ C2 . Proof Suppose we consider α = 1. Then by Theorem 4, the operator (3 )∗ − αI is not one to one which implies that the operator 3 − αI does not have dense range. As a result 3 − αI ∈ C, as desired. For the second part, since α = 1 the operator 3 − αI has an inverse. As α ∈ σr (3 , c), this implies that |1 − α| ≤ 7, hence the operator (3 − αI )−1 is discontinuous with the condition |1 − α| ≤ 7. Therefore, (3 − αI )−1 ∈ 2. This completes the proof.  Conclusion: As mentioned in the literature, the spectrum of the difference operator over the sequence space c has been calculated maximum up to order 2. The present study attempts a step to compute the spectrum of the operator 3 over the sequence spaces S, where S ∈ {c, c0 , 1 }. In this context, the results obtained are as follows: (i)

  σ(3 , S) = α ∈ C : |1 − α| ≤ 7 . Geometrically, this represents set of all complex numbers inside or boundary of a closed circular disc with radius 7 and center (1, 0) (see Fig. 1).

(ii) σ p (3 , S) = ∅. This suggests that the infinite matrix induced by the operator 3 does not have any eigenvalues. (iii)

  σr (3 , c) = α ∈ C : |1 − α| < 7 . The residual spectrum of the operator 3 consists of all complex numbers inside an open circular disc with radius 7 and center (1, 0). However, the residual spectrum of the operator 3 over the sequence c0 represents the same open circular disc excluding the point (0, 0).

(iv)

  σc (3 , c) = α ∈ C : |1 − α| = 7 . The continuous spectrum of the operator 3 over c and 1 represents the set of all complex numbers lying only on the circle (or the boundary of the circular disc) with radius 7 and center (1, 0), whereas it includes the point (0, 0) over the space c0 (see Fig. 2).

Most of works on linear algebra or functional analysis provide at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many mathematical disciplines involving systems of linear equations,

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3

σ(Δ , c) 3

σc(Δ , c)

8

3

σr(Δ , c)

6

Imaginary

4 2 0 −2 −4 −6 −6

−4

−2

0

2

4

6

8

10

Real Fig. 1 Spectral subdivisions of the operator 3 over the Banach space c

Sturm-Liouville operators, Schrdinger operators, classical harmonic analysis, graph theory etc. The spectral properties of different self-adjoint operators are extensively used in the study of many fundamental results in physics such as solid states physics, statistical physics, quantum mechanics and large particle systems etc. The present investigation may be very useful for finding the eigenvalues and eigenvectors of 3rd order difference matrix for finite dimensional case, and solving the systems of linear equations involving certain 4th order banded matrices. The results presented by this chapter may be used for finding the inverse of certain triangular matrices and other relevant properties. This chapter is devoted to find the spectral subdivisions of the 3rd order backward difference matrices over different Banach spaces, and the results may be extended to the case of 3rd order forward difference matrices. As a further scope, the present investigation may be carried out for the difference operators of any arbitrary orders(including the case of fractional orders).

On the Spectra of Difference Operators Over Some Banach Spaces

809

10 3

σc(Δ ,c0)

8

3

6

σ (Δ ,c ) r

Imaginary

4

0

(0,0)

2

0

(1,0)

−2

−4

−6 −6

−4

−2

0

2

4

6

8

10

Real Fig. 2 Spectral subdivisions of the operator 3 over the Banach space c0

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