Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives 3031348761, 9783031348761

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Synthesis Lectures on Mathematics & Statistics

Mouffak Benchohra · Erdal Karapınar · Jamal Eddine Lazreg · Abdelkrim Salim

Fractional Differential Equations New Advancements for Generalized Fractional Derivatives

Synthesis Lectures on Mathematics & Statistics Series Editor Steven G. Krantz, Department of Mathematics, Washington University, Saint Louis, MO, USA

This series includes titles in applied mathematics and statistics for cross-disciplinary STEM professionals, educators, researchers, and students. The series focuses on new and traditional techniques to develop mathematical knowledge and skills, an understanding of core mathematical reasoning, and the ability to utilize data in specific applications.

Mouffak Benchohra · Erdal Karapınar · Jamal Eddine Lazreg · Abdelkrim Salim

Fractional Differential Equations New Advancements for Generalized Fractional Derivatives

Mouffak Benchohra Laboratory of Mathematics Djillali Liabes University Sidi Bel-Abbes, Algeria

Erdal Karapınar Department of Mathematics Çankaya University Etimesgut, Turkey

Jamal Eddine Lazreg Djillali Liabes University Sidi Bel-Abbes, Algeria

Abdelkrim Salim Faculty of Technology Hassiba Benbouali University of Chlef Chlef, Algeria

ISSN 1938-1743 ISSN 1938-1751 (electronic) Synthesis Lectures on Mathematics & Statistics ISBN 978-3-031-34876-1 ISBN 978-3-031-34877-8 (eBook) https://doi.org/10.1007/978-3-031-34877-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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

We dedicate this book to our family members. In particular, Mouffak Benchohra makes his dedication to the memory of his father Yahia Benchohra and his wife Kheira Bencherif; Erdal Karapınar dedicates it to his sons, Ula¸s Ege Karapınar and Can Karapınar, his wife Senem Pınar Karapınar, his mother Elife Karapınar, and the memory of his father, Hüseyin Karapınar; Jamal E. Lazreg dedicates it to the memory of his father Mohammed Lazreg and Abdelkrim Salim makes his dedication to his mother, his brother and his sisters.

Preface

Over the past two decades, fractional calculus has made significant strides in the field of mathematical analysis, both theoretically and practically. Fundamentally, fractional calculus is a mathematical analysis tool used to study integrals and derivatives of arbitrary order, unifying and generalizing the classical concepts of differentiation and integration. Fractional and derivative integrals have been demonstrated as powerful instruments in modeling problems across various scientific fields, including fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing, and entropy theory. Fractional calculus provides a tool for modeling physical processes, which is often more useful than classical formulations, and this has made the application of fractional calculus a topic of intense research internationally. This book aims to explore the existence and uniqueness of solutions for various classes of problems under varying conditions. The problems tackled in this book involve fractional differential equations and some generalized Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. Each chapter builds on the previous one, providing a partial continuation or generalization of the results obtained earlier. The classic and novel fixed point theorems related to the concept of measure of noncompactness in Banach spaces are used as tools. Every chapter concludes with a section devoted to remarks and bibliographical suggestions, and all abstract results are illustrated. This monograph presents original content that contributes to the current literature on fractional calculus. Each chapter contains the authors’ most recent research on the subject, making it suitable for advanced graduate courses, seminars, and research projects across various applied disciplines.

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We are grateful to J. R. Graef, J. Henderson, G. N’Guérékata, J. J. Nieto, and Y. Zhou for their contributions to research on the problems covered in this book. Sidi Bel-Abbès, Algeria Ankara, Turkey Sidi Bel-Abbès, Algeria Chlef, Algeria

Mouffak Benchohra Erdal Karapınar Jamal Eddine Lazreg Abdelkrim Salim

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 9

2 Preliminary Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notations and Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Space of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Spaces of Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Spaces of Continuous Functions with Weight . . . . . . . . . . . . . . . . . . 2.2 Special Functions of the Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 k-Gamma and k-Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Elements from Fractional Calculus Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Fractional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Necessary Lemmas, Theorems and Properties . . . . . . . . . . . . . . . . . 2.4 Kuratowski Measure of Noncompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 16 16 18 18 19 20 20 21 23 27 27 29

3 Hybrid Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Boundary Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 33 33 40 42 42 43 49 ix

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3.4 Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Initial Value Problem for Hybrid ψ-Hilfer Fractional Implicit Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fractional Differential Equations with Retardation and Anticipation . . . . . . 4.1 Introduction and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 On k-Generalized ψ-Hilfer Boundary Value Problems with Retardation and Anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem with Retarded and Advanced Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Impulsive Fractional Differential Equations with Retardation and Anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 On k-Generalized ψ-Hilfer Impulsive Boundary Value Problem with Retarded and Advanced Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 On k-Generalized ψ-Hilfer Impulsive Boundary Value Problem with Retarded and Advanced Arguments in Banach Spaces . . . . . . . . . . . . 5.3.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Nonlocal k-Generalized ψ-Hilfer Impulsive Initial Value Problem with Retarded and Advanced Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 62 64 65 71 75 75 77 77 78 79 87 89 90 103 105 106 109 109 110 111 124 125 126 133 135 136 152 153 153

Contents

6 Coupled Systems for Fractional Differential Equations . . . . . . . . . . . . . . . . . . 6.1 Introduction and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 On Coupled Systems for k-Generalized ψ-Hilfer Fractional Differential Equations with Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 6.2.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Implicit Coupled k-Generalized ψ-Hilfer Fractional Differential Systems with Terminal Conditions in Banach Spaces . . . . . . . . . . . . . . . . . 6.3.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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157 157 159 159 170 171 172 181 183 183

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Introduction

Fractional calculus is an area of mathematical analysis that extends the concepts of integer differential calculus to involve real or complex order derivatives and integrals. The study of fractional calculus has developed over time to meet the scientific demands expressed by numerous prominent mathematicians. Its origins can be traced back to the work of Leibnitz, L’Hopital (1695), Bernoulli (1697), Euler (1730), and Lagrange (1772). Subsequently, Laplace (1812), Fourier (1822), Abel (1823), Liouville (1832), Riemann (1847), Gr¨unwald (1867), Letnikov (1868), Nekrasov (1888), Hadamard (1892), Heaviside (1892), Hardy (1915), Weyl (1917), Riesz (1922), P. Levy (1923), Davis (1924), Kober (1940), Zygmund (1945), Kuttner (1953), J. L. Lions (1959), Liverman (1964), and many others have made significant contributions to the field of fractional calculus. In recent years, there has been significant interest in fractional differential equations, with several works devoted to the subject, including the books of Miller and Ross [1], Podlubny [2], Kilbas et al. [3], Diethelm [4], Ortigueira [5], Abbas et al. [6], and Baleanu et al. [7]. This work focuses on a novel extension of the well-known Hilfer fractional derivative. The definition of this new derivative was developed by taking into account the publications of Diaz et al. [8], who introduced the k-gamma and k-beta functions and demonstrated a number of their properties. Similar properties can also be found in [9–12]. Additionally, Sousa’s numerous publications [13–19] inspired us to establish another type of fractional operator known as the ψ-Hilfer fractional derivative with respect to a particular function and provide several essential properties about this type of fractional operator. The work presented in this monograph can be viewed as a continuation and generalization of previous research in the field of fractional calculus. Fixed point theory is a field of mathematics concerned with the existence and properties of solutions to equations of the form x = f (x), where f is a given function. These solutions, known as fixed points, have been studied for millennia and have applications in

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Benchohra et al., Fractional Differential Equations, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-34877-8_1

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1 Introduction

nonlinear dynamics, differential equations, and functional analysis. Fixed point theory, as it is commonly known, has its origins in the system of successive approximations (or the Picard iterative technique) used to solve certain differential equations. Cauchy demonstrated the existence of fixed points for certain kinds of functions known as contraction mappings in 1821. Later, in the late nineteenth and early twentieth centuries, Brouwer, Schauder, and Tarski established the groundwork for contemporary fixed point theory. Browder created the theory of nonlinear operators in the mid-twentieth century, and Schauder founded the theory of nonlinear equations. Birkhoff and Banach developed the theory of set-valued analysis in the 1960s and 1970s by expanding fixed point theory to include multivalued mappings and set-valued mappings. Today, fixed point theory is a lively and quickly growing field with applications in a wide range of mathematics fields such as nonlinear dynamics, differential equations, and functional analysis. It also has useful applications in physics, engineering, economics, computer science, and optimization. It has also been used in the study of chaotic systems and fractals, as well as in the solution of control theory and image processing problems. It is common in the literature to propose a solution to fractional differential equations by combining several types of fractional derivatives, see e.g. [6, 20–27, 27–31, 31–41]. Differential equations with delay are a type of mathematical equation that captures the behavior of systems that respond to changes with a delay. A delay component is added into the equation to reflect the time lag between the incidence of an event and the system’s response to it in order to account for this delay. See [33, 42–52], for more information. Differential equations can represent a variety of delays, including continuous delay, statedependent delay, distributed delay, and functional delay. The former is the most fundamental sort, whereas the latter is a function of both system status and control inputs. The study of differential equations with delay is a busy research topic, with uses ranging from physics to engineering, biology, and economics. The type of delay chosen is determined by the nature of the system and the issue at hand, with each delay form having unique features, advantages, and limitations. External actions cause short-term perturbations in many physical phenomena at various stages of their development. For such events, impulsive differential equations are a suitable modeling paradigm. There are two kinds of impulses extensively addressed in the literature: instantaneous impulses with negligible duration and non-instantaneous impulses with changes that begin impulsively and stay active over a limited time period. The meaning of impulsive fractional differential equation solutions is primarily determined by the sort of fractional derivative used, and two methods are frequently used. The first method keeps the lower bound of the fractional derivative constant at the start, while the second changes the lower bound of the fractional derivative at the impulsive moments. Ordinary derivatives, such as the derivative of a constant, share some characteristics with fractional derivatives, resulting in comparable initial value problems and impulsive conditions for both instantaneous and non-instantaneous impulses. The class of problems for fractional differential equations with abrupt and instantaneous impulses has received a great deal of attention, with numerous studies focusing on the existence and qualitative properties of solutions. The dynamics of

1 Introduction

3

certain development processes in pharmacotherapy cannot be properly explained by instantaneous impulses. For example, medication administration into the bloodstream and following absorption by the body is a gradual and ongoing procedure. Both instantaneous (IIDEs) and non-instantaneous (NIIDEs) impulsive differential equations have distinct characteristics, benefits, and drawbacks. IIDEs have a clearer physical meaning and are more analytically tractable. However, in some instances, the impulse length is not insignificant, and NIIDEs are better suited for simulating system behavior. The literature investigates a wide range of initial value and boundary value problems for various fractional differential equations with both instantaneous and non-instantaneous impulses. Examples include the works [28, 40, 53–61], among others. The measure of noncompactness is a crucial tool in the field of nonlinear analysis. Its development can be traced back to the pioneering works of Alvàrez [62], Mönch [63], and subsequently, Bana´s and Goebel [64], who contributed significantly to its development. Many researchers in the literature have also made significant contributions to its advancement. The measure of noncompactness finds its applications in various branches of applied mathematics. For instance, it has a wide range of applications in the theory of differential equations, as evident in the works of Agarwal [65], Ortega [66], and other relevant references. Recently, the measure of noncompactness has been applied to some classes of differential equations in Banach spaces. The authors of [29, 64, 67] have made significant contributions in this regard. The concept of nonlocal conditions in fractional differential equations was first introduced by Byszewski in his influential work [68], where he established the existence and uniqueness of mild and classical solutions to nonlocal Cauchy problems. Nonlocal conditions can be more appropriate than the usual initial conditions for describing certain physical phenomena. The study of fractional differential equations with nonlocal conditions has since been a popular area of research, with several important contributions in the literature [69, 70] and beyond. In particular, these types of equations have found applications in a variety of fields, including physics, engineering, and mathematical biology, where they are used to model phenomena such as anomalous diffusion, viscoelasticity, and population dynamics. Overall, the use of nonlocal conditions has proven to be a powerful and versatile tool in the analysis of fractional differential equations. Hybrid fractional differential equations have been an area of interest for several researchers. These equations are perturbed either linearly, quadratically, or through the combination of both. When the terms in an equation are perturbed in the form of a sum or difference, it is called a linear perturbation. Conversely, when the equation is perturbed through the product or quotient of the terms, it is called a quadratic perturbation. Therefore, the study of hybrid fractional differential equations is more general and encompasses several dynamic systems as particular cases. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. The study of hybrid fractional differential equations is important in understanding complex physical phenomena where both linear and quadratic perturbations play a role. Recent research has focused on the

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qualitative analysis of solutions to hybrid fractional differential equations. Some of the recent results on hybrid fractional differential equations can be found in a series of papers, including [71–77]. These works provide insights into the existence and uniqueness of solutions, stability properties, and numerical methods for solving hybrid fractional differential equations. Many papers and monographs have lately been published in which the authors studied a wide range of outcomes for systems with various forms of fractional differential equations, inclusions, and conditions. One may see the papers [78–82], and the references therein. In the following we give an outline of this monograph organization, which consists of six chapters defining the contributed work. Chapter 2 provides the notation and preliminary results, descriptions, theorems and other auxiliary results that will be needed for this study. In the first section we give some notations and definitions of the functional spaces used in this book. In the second section, we give the definitions of the elements from fractional calculus theory, then we present some necessary lemmas, theorems and properties. In the third section, we give some properties to the Measure of noncompactness. We finish the chapter in the last section by giving all the fixed point theorems that are used throughout the book. Chapter 3 is devoted to proving some results concerning the existence of solutions for a class of initial and boundary value problems for nonlinear fractional Hybrid differential equations and generalized Hilfer fractional derivative. The obtained results in this chapter are based on fixed point theorems due to Dhage. Further, examples are provided in each section to illustrate our results. Section 3.2 deals with some existence results for the following problem: ⎧   x(t) ⎪ ρ D α,β ⎪ = ϕ(t, x(t)), t ∈ J , ⎪ ⎨ a+ f (t, x(t))    ⎪ x(τ ) ⎪ ρ J 1−γ ⎪ (a + ) = c0 , ⎩ a+ f (τ , x(τ )) α,β

where ρ Da + , ρ Ja + are the generalized Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and generalized fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, c0 ∈ R, f ∈ C([a, b] × R, R\{0}) and ϕ ∈ C([a, b] × R, R). In Sect. 3.3, we consider the following problem with hybrid fractional differential equations: ⎧   x(t) ⎪ ρ D α,β ⎪ = ϕ(t, x(t)), t ∈ J , ⎪ ⎨ a+ f (t, x(t))       ⎪ x(τ ) x(τ ) ⎪ 1−γ 1−γ ⎪ (a + ) + c2 ρ Ja + (b) = c3 , ⎩c1 ρ Ja + f (τ , x(τ )) f (τ , x(τ )) α,β

1−γ

where ρ Da + ,ρ Ja + are the generalized Hilfer operator of order α ∈ (0, 1) and type β ∈ [0, 1] and generalized fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, c1 , c2 , c3 ∈ R, c1 + c2  = 0, f ∈ C([a, b] × R, R\{0}) and ϕ ∈ C([a, b] × R, R). In Sect. 3.4, we establish existence results to the nonlocal initial value problem with nonlinear implicit hybrid generalized Hilfer-type fractional differential equation: 1−γ

1 Introduction

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⎧    α D α,β x(t)−χ(t,x(t)) = ϕ t, x(t), α D α,β x(t)−χ(t,x(t)) ⎪ , t ∈ (a, b], ⎪ + f (t,x(t)) f (t,x(t)) a+ ⎪ ⎨ a      m

x( i ) − χ( i , x( i )) ⎪ α J 1−γ x(τ ) − χ(t, x(t)) + ⎪ (a ) = , ci ⎪ a+ ⎩ f (τ , x(τ )) f ( i , x( i )) i=1

α,β

where α Da + ,α Ja + are the generalized Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and generalized fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, ci , i = 1, . . . , m, are real numbers, i , i = 1, . . . , m, are pre-fixed points satisfying a < 1 ≤ . . . ≤ m < b, f ∈ C([a, b] × R, R\{0}), χ ∈ C([a, b] × R, R), ϕ ∈ m ¯ γ ( i , a)  = 1, for further details see the definitions in the ci  C([a, b] × R2 , R) and i=1 chapter. Then, in Sect. 3.5, we consider the initial value problem with nonlinear implicit hybrid ψ-Hilfer type fractional differential equation: ⎧      x(t) x(t) ⎪ H D α,β;ψ H D α,β;ψ ⎪ = f t, x(t), , t ∈ (a, b], ⎪ ⎨ a+ a+ g(t, x(t)) g(t, x(t))    ⎪ x(τ ) ⎪ 1−γ;ψ ⎪ (a + ) = x0 , ⎩ Ja + g(τ , x(τ )) 1−γ

α,β;ψ

1−γ;ψ

where H Da + , Ja + are the ψ-Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and ψ-Riemann-Liouville fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, x0 ∈ R, g ∈ C([a, b] × R, R\{0}) and f ∈ C([a, b] × R2 , R). In Chap. 4, we prove some existence and uniqueness results for a class of boundary and terminal value problems for implicit nonlinear k-generalized ψ-Hilfer fractional differential equations involving both retarded and advanced arguments. Further, examples are given to illustrate the viability of our results in each section. In Sect. 4.2, we consider the boundary value problem for the nonlinear implicit k-generalized ψ-Hilfer type fractional differential equation involving both retarded and advanced arguments:   ⎧ H D α,β;ψ x (t) = f t, x (·), H D α,β;ψ x (t) , t ∈ (a, b], ⎪ t ⎪ a+ a+ k k ⎪ ⎪   ⎪ ⎪ k(1−ξ),k;ψ k(1−ξ),k;ψ ⎪ ⎨ϑ1 Ja+ x (a + ) + ϑ2 Ja+ x (b) = ϑ3 , ⎪ ⎪ x(t) = (t), t ∈ [a − λ, a], λ > 0, ⎪ ⎪ ⎪

⎪ ⎪ ⎩x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0, α,β;ψ

k(1−ξ),k;ψ

are, respectively, the k-generalized ψ-Hilfer fractional where kH Da+ and Ja+ derivative of order α ∈ (0, k) and type β ∈ [0, 1], and k-generalized ψ-fractional 

inte 1 ˜ gral of order k(1 − ξ), where ξ = k (β(k − α) + α), k > 0, f : [a, b] × C −λ, λ , R × ˜ R −→ R is a given function, and ϑ1 , ϑ2 , ϑ3 ∈ R such that ϑ 1 + ϑ2  =

0, and (t) and (t) are, respectively, continuous functions on [a − λ, a] and b, b + λ˜ . For each function x

6

1 Introduction



defined on a − λ, b + λ˜ and for any t ∈ (a, b], we denote by xt the element defined

by xt (τ ) = x(t + τ ), τ ∈ −λ, λ˜ . The results are founded on the Banach contraction principle and Schauder’s fixed point theorem. Further, examples are given to illustrate the viability of our results. In Sect. 4.3, we consider the terminal value problem with nonlinear implicit k-generalize ψ-Hilfer type fractional differential equation involving both retarded and advanced arguments:   ⎧ H D α,β;ψ x (t) = f t, x (·), H D α,β;ψ x (t) , t ∈ (a, b], ⎪ t a+ ⎪ k a+ k ⎪ ⎪ ⎪ ⎪ m ⎪

⎪ ⎪ ⎨x(b) = ϑi x( i ), i=1 ⎪ ⎪ ⎪ ⎪ x(t) = (t), t ∈ [a − λ, a], λ > 0, ⎪ ⎪ ⎪

⎪ ⎪ ⎩x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0, α,β;ψ

k(1−ξ),k;ψ

are the k-generalize ψ-Hilfer fractional derivative of order α ∈ where kH Da+ , Ja+ (0, k) and type β ∈ [0, 1], and k-generalize ψ-fractional of order k(1 − ξ), where 

integral 1 ˜ ξ = k (β(k − α) + α), k > 0, f : [a, b] × C −λ, λ , R × R −→ R is a given appropriate function specified latter, ϑi , i = 1, . . . , m, are real numbers and i , i = 1, . . . , m, are pre-fixed points satisfying a < 1 ≤ . . . ≤ m < b. The results are founded on the Banach contraction principle, Schauder and Krasnoselskii fixed point theorems. Further, illustrative examples are provided in support of the obtained results. Chapter 5 deals with the existence and uniqueness results for a class of impulsive initial and boundary value problems for implicit nonlinear fractional differential equations and kGeneralized ψ-Hilfer fractional derivative involving both retarded and advanced arguments. Our results are based on some necessary fixed point theorems. Suitable illustrative examples are provided for each section. In Sect. 5.2, we establish existence and uniqueness results to the following k-generalized ψ-Hilfer problem with nonlinear implicit fractional differential equation with impulses involving both retarded and advanced arguments:      ⎧ ⎪ H D α,β;ψ x (t) = f t, x t (·), H D α,β;ψ x (t) , t ∈ J , i = 0, . . . , m, ⎪ ⎪ i k k ⎪ ti+ ti+ ⎪ ⎪ ⎪    ⎪ ⎪ ⎪ k(1−ξ),k;ψ k(1−ξ),k;ψ ⎪ J x (ti+ ) = Jt + x (ti− ) + L i (x(ti− )); i = 1, . . . , m, ⎪ ⎪ ⎨ ti+ i−1   k(1−ξ),k;ψ k(1−ξ),k;ψ ⎪ ϑ1 Ja+ x (a + ) + ϑ2 Jt + x (b) = ϑ3 , ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ x(t) = (t), t ∈ [a − λ, a], λ > 0, ⎪ ⎪ ⎪

⎪ ⎪ ⎩x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

1 Introduction α,β;ψ

7 k(1−ξ),k;ψ

where kH Da+ , Ja+ are the k-generalized ψ-Hilfer fractional derivative of order α ∈ (0, k) and type β ∈ [0, 1], and k-generalized ψ-fractional integral  of order

k(1 − ξ), where 1 ˜ ˜ ∈ C b, b + λ , R , f : [a, b] × ξ = k (β(k − α) + α), k > 0, ∈ C ([a − λ, a], R), 

PCξ,k;ψ −λ, λ˜ × R −→ R is a given appropriate function specified latter, ϑ1 , ϑ2 , ϑ3 ∈ R such that ϑ1 + ϑ2  = 0, Ji := (ti , ti+1 ]; i = 0, . . . , m, a = t0 < t1 < . . . < tm < tm+1 = b < ∞, u(ti+ ) = lim u(ti + ) and u(ti− ) = lim u(ti + ) represent the right and left hand

→0+

→0−

limits of u(t) at t = ti and L i : R → R; i =

1, . . . , m are given continuous functions. For ˜ each function x defined on a − λ, b + λ and for any t ∈ (a, b], we denote by x t the

element defined by x t (τ ) = x(t + τ ), τ ∈ −λ, λ˜ . Our results are based on the Banach contraction principle and Schauder’s fixed point theorem. Suitable illustrative examples are provided. Section 5.3 presents some existence result to the following k-generalized ψ-Hilfer problem with nonlinear implicit fractional differential equation with impulses involving both retarded and advanced arguments:      ⎧ H D α,β;ψ x (t) = f t, x t (·), H D α,β;ψ x (t) , t ∈ J , i = 0, . . . , m, ⎪ ⎪ i ⎪ k k ti+ ti+ ⎪ ⎪ ⎪     ⎪ ⎪ ⎪ k(1−ξ),k;ψ k(1−ξ),k;ψ ⎪ ⎪ x (ti+ ) = Jt + x (ti− ) + L i (x(ti− )); i = 1, . . . , m, ⎪ ⎨ Jti+ i−1   k(1−ξ),k;ψ k(1−ξ),k;ψ ⎪ α1 Ja+ x (a + ) + α2 Jt + x (b) = α3 , ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ x(t) =

(t), t ∈ [a − λ, a], λ > 0, ⎪ ⎪ ⎪

⎪ ⎪ ⎩x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0, α,β;ψ

k(1−ξ),k;ψ

where kH Da+ , Ja+ are the k-generalized ψ-Hilfer fractional derivative of order α ∈ (0, k) and type β ∈ [0, 1], and k-generalized ψ-fractional integral  of order k(1 − ξ), 1 where ξ = k (β(k − α) + α), k > 0, ∈ C ([a − λ, a], E), ˜ ∈ C b, b + λ˜ , E , f : 

[a, b] × PCξ,k;ψ −λ, λ˜ × E −→ E is a given appropriate function specified latter, α1 , α2 ∈ R such that α1 + α2  = 0, α3 ∈ E, Ji := (ti , ti+1 ]; i = 0, . . . , m, a = t0 < t1 < . . . < tm < tm+1 = b < ∞, x(ti+ ) = lim x(ti + ) and x(ti− ) = lim x(ti + ) represent

→0+

→0−

the right and left hand limits of x(t) at t = ti and L i : E → E; i = 1, . . . , m are given continuous functions, where (E,  · ) is a Banach space. The result are based on Mönch fixed point theorem associated with the technique of measure of noncompactness. An illustrative example is provided to indicate the applicability of our results. Section 5.4 provides some existence and uniqueness results to the following k-generalized ψ-Hilfer nonlocal initial value problem with nonlinear implicit fractional differential equation with non-instantaneous impulses involving both retarded and advanced arguments:

8

1 Introduction

     ⎧ α,β;ψ α,β;ψ H H ⎪ t ⎪ k D + x (t) = f t, x (·), k D + x (t) , t ∈ Ji , i = 0, . . . , m, ⎪ si si ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ⎪ x(t) = σi (t, x(t)); t ∈ Ji , i = 1, . . . , m, ⎪ ⎪ ⎨ m

k(1−ξ),k;ψ +) = J x (a ω j x( j ), a+ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ x(t) = (t), t ∈ [a − λ, a], λ > 0, ⎪

⎪ ⎪ ⎩x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0, α,β;ψ

k(1−ξ),k;ψ

where kH Da+ , Ja+ are the k-generalized ψ-Hilfer fractional derivative of order α ∈ (0, k) and type β ∈ [0, 1], and k-generalized ψ-fractional integral of order k(1 − 1 ˜ ξ), where ξ = k (β(k − α) + α), k > 0, ∈ C ([a − λ, a], R), ˜ ∈ C b, b + λ , R , 

f : [a, b] × PCξ;ψ −λ, λ˜ × R −→ R is a given appropriate function specified later, Ji := (si , ti+1 ]; i = 0, . . . , m, J˜i := (ti , si ]; i = 1, . . . , m, a = t0 = s0 < t1 ≤ s1 < t2 ≤ s2 < . . . ≤ sm−1 < tm ≤ sm < tm+1 = b < ∞, x(ti+ ) = lim x(ti + ) and x(ti− ) =

→0+

lim x(ti + ) represent the right and left hand limits of x(t) at t = ti , ω j , j = 1, . . . , m,

→0−

are real numbers and j ; j = 1, . . . , m, are pre-fixed points satisfying a < 1 ≤ . . . ≤

m < b and σi : J˜i × R → R; i = 1, . . . , m are given continuous functions such that  k(1−ξ),k;ψ Js + σi (t, x(t)) t=si = ci ∈ R . Our reasoning is based on some relevant fixed i

point theorems. In addition, an example is provided to demonstrate the effectiveness of the main results. In Chap. 6, we prove some existence and uniqueness results for a class of coupled systems for nonlinear k-generalized ψ-Hilfer fractional differential equations with boundary and terminal conditions. Our results are based on some necessary fixed point theorems. Furthermore, an illustration is presented for each section to demonstrate the plausibility of our results. In Sect. 6.2, we study the existence and uniqueness results for a class of coupled systems of nonlinear k-generalized ψ-Hilfer type fractional differential equation and boundary conditions as follows: ⎧ α1 ,β1 ;ψ ⎪ ⎨ kH Da+ x (t) = f 1 (t, x(t), y(t)) , t ∈ (a, b],  α2 ,β2 ;ψ ⎪ ⎩ kH Da+ y (t) = f 2 (t, x(t), y(t)) , t ∈ (a, b], ⎧   k(1−ξ1 ),k;ψ k(1−ξ ),k;ψ ⎪ ⎨c1 Ja+ x (a + ) + c2 Ja+ 1 x (b) = c3 ,   k(1−ξ2 ),k;ψ k(1−ξ ),k;ψ ⎪ ⎩d1 Ja+ y (a + ) + d2 Ja+ 2 y (b) = d3 , α ,β ;ψ

k(1−ξ ),k;ψ

are, respectively, the k-generalized ψ-Hilfer where for i = 1, 2, kH Da+i i and Ja+ i fractional derivative of order αi ∈ (0, k) and type βi ∈ [0, 1], and k-generalized ψ-fractional

References

9

integral of order k(1 − ξi ), where ξi = k1 (βi (k − αi ) + αi ), k > 0, f i : [a, b] × R × R −→ R are given functions, and c1 , c2 , c3 , d1 , d2 , d3 ∈ R such that c1 + c2  = 0 and d1 + d2  = 0. The results are founded on the Banach contraction principle and Schauder’s fixed point theorem. Lastly, we illustrate the viability of our results with an example. In Sect. 6.3, we investigate the existence results for a class of coupled systems of nonlinear implicit kgeneralized ψ-Hilfer type fractional differential equation and terminal conditions as follows: ⎧    α1 ,β1 ;ψ α ,β ;ψ α ,β ;ψ ⎪ ⎨ kH Da+ x (t) = f 1 t, x(t), y(t), kH Da+1 1 x (t), kH Da+2 2 y (t) ,     α2 ,β2 ;ψ α ,β ;ψ α ,β ;ψ ⎪ ⎩ kH Da+ y (t) = f 2 t, x(t), y(t), kH Da+1 1 x (t), kH Da+2 2 y (t) , 

x(b) = c1 , y(b) = c2 ,

α ,β ;ψ

where t ∈ (a, b] and for i = 1, 2, kH Da+i i is the k-generalized ψ-Hilfer fractional derivative of order αi ∈ (0, k) and type βi ∈ [0, 1], where ξi = k1 (βi (k − αi ) + αi ), k > 0, c1 , c2 ∈ E, and f i : [a, b] × E 4 −→ E are given functions, where (E,  · ) is a Banach space. The results are founded on the fixed point theorem Mönch combined with the technique of measure of noncompactness. In the last part, we present some illustrations to demonstrate the practicability of our results.

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12

1 Introduction

52. S. Krim, A. Salim, M. Benchohra, On implicit Caputo tempered fractional boundary value problems with delay. Lett. Nonlinear Anal. Appl. 1, 12–29 (2023) 53. J. Wang, M. Feckan, Periodic solutions and stability of linear evolution equations with noninstantaneous impulses. Miskolc Math. Notes 20(2), 1299–1313 (2019) 54. A. Salim, M. Benchohra, J.R. Graef, J.E. Lazreg, Boundary value problem for fractional generalized Hilfer-type fractional derivative with non-instantaneous impulses. Fractal Fract. 5, 1–21 (2021). https://doi.org/10.3390/fractalfract5010001 55. F. Vaadrager, J. Van Schuppen, Hybrid Systems, Computation and Control, Lecture Notes in Computer Sciences, vol. 1569 (Springer, New York, 1999) 56. M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions (Hindawi Publishing Corporation, New York, 2006) 57. R.P. Agarwal, S. Hristova, D. O’Regan, Non-Instantaneous Impulses in Differential Equations (Springer, New York, 2017) 58. I. Stamova, G. Stamov, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications (CRC Press, 2017) 59. L. Bai, J.J. Nieto, J.M. Uzal, On a delayed epidemic model with non-instantaneous impulses. Commun. Pure Appl. Anal. 19(4), 1915–1930 (2020) 60. E. Hernández, K.A.G. Azevedo, M.C. Gadotti, Existence and uniqueness of solution for abstract differential equations with state-dependent delayed impulses. J. Fixed Point Theory Appl. 21(1) (2019), Paper No. 36, 17 pp 61. F. Kong, J.J. Nieto, Control of bounded solutions for first-order singular differential equations with impulses. IMA J. Math. Control Inform. 37(3), 877–893 (2020) 62. J.C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces. Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid. 79, 53–66 (1985) 63. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980) 64. J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces (Marcel Dekker, New York, 1980) 65. R. Agarwal, Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365– 370 (1969) 66. D. O’Regan, Fixed point theory for weakly sequentially continuous mapping. Math. Comput. Model. 27, 1–14 (1998) 67. J.M. Ayerbee Toledano, T. Dominguez Benavides, G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications (Berlin, 1997) 68. L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1991) 69. S.A. Abd-Salam, A.M.A. El-Sayed, On the stability of a fractional-order differential equation with nonlocal initial condition. Electron. J. Qual. Theory Differ. Equ. 29, 1–8 (2008) 70. G.M. N’Guérékata, A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70, 1873–1876 (2009) 71. B. Ahmad, S.K. Ntouyas, Initial value problems for hybrid Hadamard fractional differential equations. Electron. J. Differ. Equ. 2014, 161 (2014) 72. Z. Baitiche, K. Guerbati, M. Benchohra, Y. Zhou, Boundary value problems for hybrid Caputo fractional differential equations. Mathematics 2019, 282 (2019) 73. K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order. Adv. Differ. Equ. 2015, 183 (2015) 74. Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations. Comput. Math. Appl. 62, 1312–1324 (2011)

References

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75. C. Derbazi, H. Hammouche, M. Benchohra et al., Fractional hybrid differential equations with three-point boundary hybrid conditions. Adv. Differ. Equ. 2019, 125 (2019) 76. C. Derbazi, H. Hammouche, A. Salim, M. Benchohra, Measure of noncompactness and fractional hybrid differential equations with hybrid conditions. Differ. Equ. Appl. 14, 145–161 (2022). https://doi.org/10.7153/dea-2022-14-09 77. C. Derbazi, H. Hammouche, A. Salim, M. Benchohra, Weak solutions for fractional Langevin equations involving two fractional orders in banach spaces. Afr. Mat. 34 (2023), 10 p. https:// doi.org/10.1007/s13370-022-01035-3 78. K.S. Samina, R.A. Khan, Stability theory to a coupled system of nonlinear fractional hybrid differential equations. Indian J. Pure Appl. Math. 51, 669–687 (2020) 79. A. Ali, K. Shah, R.A. Khan, Existence of solution to a coupled system of hybrid fractional differential equations. Bull. Math. Anal. Appl. 9, 9–18 (2017) 80. K. Guida, K. Hilal, L. Ibnelazyz, Existence of mild solutions for a class of impulsive Hilfer fractional coupled systems. Adv. Math. Phys. 2020 (2020), 12pp 81. M.A. Almalahi, O. Bazighifan, S.K. Panchal, S.S. Askar, G.I. Oros, Analytical study of two nonlinear coupled hybrid systems involving generalized Hilfer fractional operators. Fractal Fract. 5 (2021), 22pp 82. L. Lin, Y. Liu, D. Zhao, Study on implicit-type fractional coupled system with integral boundary conditions. Math. 9 (2021), 15pp

2

Preliminary Background

This chapter discusses the mathematical tools, notations, and concepts that will be required in subsequent chapters. We will look at some of the most important properties of fractional differential operators. We also go through some of the fundamental properties of noncompactness measures and fixed point theorems, which are important in our findings for fractional differential equations.

2.1

Notations and Functional Spaces

In this part, we will offer all of the notations and definitions of functional spaces that have been regarded essential and constant throughout all of the next chapters. Let 0 < a < b, J = (a, b] where J¯ = [a, b]. Consider the following parameters α, β, γ satisfying γ = α + β − αβ and 0 < α, β, γ < 1. Let ξ = k1 (β(k − α) + α) where k > 0. Let ψ be an increasing and positive function on J¯ such that ψ  is continuous on J¯.

2.1.1

Space of Continuous Functions

By C( J¯, R) we denote the Banach space of all continuous functions from J¯ into R with the norm u∞ = sup{|u(t)| : t ∈ J¯}. Let (E,  · ) be a Banach space. By C( J¯, E) we denote the Banach space of all continuous functions from J¯ into E with the norm u E = sup{u(t) : t ∈ J¯}. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Benchohra et al., Fractional Differential Equations, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-34877-8_2

15

16

2 Preliminary Background

AC n (J , R), C n (J , R) be the spaces of n-times absolutely continuous and n-times continuously differentiable functions on J , respectively.

2.1.2

Spaces of Integrable Functions p

Consider the space X c (a, b), (c ∈ R, 1 ≤ p ≤ ∞) of those real-valued Lebesgue measurable functions f on J¯ for which  f  X cp < ∞, where the norm is defined by   f  X cp = In particular, when c =

b

|t f (t)| c

p dt

 1p

t

a

, (1 ≤ p < ∞, c ∈ R).

p p 1 p p , the space X c (a, b) coincides with the L (a, b) space: X 1 (a, b)

=

p

L p (a, b). By L 1 (J ), we denote the space of Bochner–integrable functions f : J −→ E with the norm  b  f 1 =  f (t)dt. a p

Consider the space X ψ (a, b), (1 ≤ p ≤ ∞) of those real-valued Lebesgue measurable functions g on J¯ for which g X p < ∞, where the norm is defined by ψ

 g X p = ψ

b



ψ (t)|g(t)| dt p

 1p

,

a

where ψ is an increasing and positive function on [a, b] such that ψ  is continuous on J¯. In p particular, when ψ(x) = x, the space X ψ (a, b) coincides with the L p (a, b) space.

2.1.3

Spaces of Continuous Functions with Weight

We consider the weighted spaces of continuous functions     ρ   t − a ρ 1−γ u(t) ∈ C J¯, E , Cγ,ρ (J ) = u : J → E : ρ and

n (J ) = u ∈ C n−1 : u (n) ∈ Cγ,ρ (J ) , n ∈ N, Cγ,ρ 0 Cγ,ρ (J ) = Cγ,ρ (J ),

2.1

Notations and Functional Spaces

with the norms uCγ,ρ

17

    t ρ − a ρ 1−γ   = sup  u(t) ,   ρ t∈ J¯

and n = uCγ,ρ

n−1

u (k) ∞ + u (n) Cγ,ρ .

k=0

We define the spaces

α,β α,β (J ) = u ∈ Cγ,ρ (J ), ρ Da + u ∈ Cγ,ρ (J ) , Cγ,ρ and

 γ γ Cγ,ρ (J ) = u ∈ Cγ,ρ (J ), ρ Da + u ∈ Cγ,ρ (J ) , α,β

γ

where ρ Da + and ρ Da + are factional derivatives defined in the following sections. Consider the weighted Banach space

 Cγ;ψ (J ) = u : J → R : t → (ψ(t) − ψ(a))1−γ u(t) ∈ C( J¯, R) , with the norm   uCγ;ψ = sup (ψ(t) − ψ(a))1−γ u(t) , t∈ J¯

and

n Cγ;ψ (J ) = u ∈ C n−1 (J ) : u (n) ∈ Cγ;ψ (J ) , n ∈ N, 0 Cγ;ψ (J ) = Cγ;ψ (J ),

with the norm n uCγ;ψ =

n−1

u (i) ∞ + u (n) Cγ;ψ .

i=0 α,β

The weighted space Cγ;ψ (J ) is defined by α,β Cγ;ψ (J ) = u ∈ Cγ;ψ (J ), α,β;ψ

where H Da +

H

α,β;ψ

Da +

u ∈ Cγ;ψ (J ) .

is a factional derivatives defined in the following sections.

18

2 Preliminary Background

Consider the weighted Banach space

Cξ;ψ (J ) = x : J → E : t → (ψ(t) − ψ(a))1−ξ x(t) ∈ C( J¯, E) , with the norm     xCξ;ψ = sup (ψ(t) − ψ(a))1−ξ x(t) , t∈ J¯

and

n Cξ;ψ (J ) = x ∈ C n−1 (J ) : x (n) ∈ Cξ;ψ (J ) , n ∈ N, 0 Cξ;ψ (J ) = Cξ;ψ (J ),

with the norm n xCξ;ψ =

n−1

x (i) ∞ + x (n) Cξ;ψ .

i=0 α,β

The weighted space Cξ,k;ψ (J ) is defined by α,β Cξ;ψ (J ) = x ∈ Cξ;ψ (J ), α,β;ψ

where kH Da+

H α,β;ψ k Da+ x

∈ Cξ;ψ (J ) ,

is defined in the sequel.

2.2

Special Functions of the Fractional Calculus

2.2.1

Gamma Function

Euler’s gamma function (z) is a fundamental function in fractional calculus. It generalizes the factorial n! to non-integer and even complex values of n. Leonhard Euler, the Swiss mathematician, physicist, astronomer, geographer, logician, and engineer, is credited with pioneering and inspiring breakthroughs in several fields, including analytic number theory, complex analysis, and infinitesimal calculus. He introduced many of today’s mathematical language and notation, such as the concept of a mathematical function. Euler also made significant contributions to mechanics, fluid dynamics, optics, astronomy, and music theory. The gamma function was extended to complex numbers with a positive real part by Daniel Bernoulli, a Swiss mathematician and physicist from the famous Bernoulli family in Basel. Bernoulli is most recognized for his contributions to mechanics, particularly fluid mechanics, and his seminal work in probability and statistics.

2.2

Special Functions of the Fractional Calculus

19

Definition 2.1 ([1]) The gamma function is defined via a convergent improper integral:  +∞ t z−1 e−t dt, (z) = 0

where Re(z) > 0. One of the basic properties of the gamma function is that it satisfies the following functional equation: (z + 1) = z(z), so, for positive integer values n, the gamma function becomes (n) = (n − 1)! and thus can be seen as an extension of the factorial function to real values. A useful particular value √ of the function: ( 21 ) = π, is used throughout many examples in this monograph.

2.2.2

k-Gamma and k-Beta Functions

In 2005, Diaz and Petruel [2] have defined new functions called k-gamma and k-beta functions given by:  ∞ tk t α−1 e− k dt, α > 0, k (α) = 0

and Bk (α, β) =

1 k



1

α

β

t k −1 (1 − t) k −1 dt.

0

It is noteworthy that if k → 1 then k (α) → (α) and Bk (α, β) → B(α, β). We have also the following useful relations: α α , k (α + k) = αk (α), k (k) = (1) = 1, k (α) = k k −1  k

Bk (α, β) =

1 B k



α β , k k

 , Bk (α, β) =

k (α)k (β) . k (α + β)

20

2.3

2 Preliminary Background

Elements from Fractional Calculus Theory

In this part, we will review some definitions of fractional integral and fractional differential operators, which are used throughout this book. We wrap it off with some required lemmas, theorems, and properties.

2.3.1

Fractional Integrals

Definition 2.2 (Generalized Fractional Integral [3]) Let α ∈ R+ and g ∈ L 1 (J ). The generalized fractional integral of order α is defined by ρ

 Jaα+ g (t) =



t

s ρ−1



a

t ρ − sρ ρ

α−1

g(s) ds, t > a, ρ > 0. (α)

Definition 2.3 (ψ-Riemann-Liouville Fractional Integral [3]) Let (a, b) (−∞ ≤ a < b ≤ p ∞) be a finite or infinite interval of the real line R, α > 0, c ∈ R and h ∈ X c (a, b). Also let ψ(t) be an increasing and positive monotone function on J , having a continuous derivative ψ  (t) on (a, b). The left and right-sided fractional integrals of a function h of order α with respect to another function ψ on J are defined by  t   h(τ ) α;ψ Ja + h (t) = ψ  (τ ) (ψ(t) − ψ(τ ))α−1 dτ , (α) a and    α;ψ Jb− h (t) =

b t

ψ  (τ ) (ψ(τ ) − ψ(t))α−1

h(τ ) dτ , (α) p

Definition 2.4 (k-Generalized ψ-fractional Integral [4]) Let g ∈ X ψ (a, b), ψ(t) > 0 be an increasing function on J and ψ  (t) > 0 be continuous on (a, b) and α > 0. The generalized k-fractional integral operators of a function g (left-sided and right-sided) of order α are defined by  t ψ  (s)g(s)ds 1 α,k;ψ Ja+ g(t) = α , kk (α) a (ψ(t) − ψ(s))1− k  b ψ  (s)g(s)ds 1 α,k;ψ Jb− g(t) = α , kk (α) t (ψ(s) − ψ(t))1− k with k > 0.

2.3

Elements from Fractional Calculus Theory

2.3.2

21

Fractional Derivatives

Definition 2.5 (Generalized Fractional Derivative [3]) Let α ∈ R+ \ N and ρ > 0. The generalized fractional derivative ρ Daα+ of order α is defined by: ρ

 Daα+ g (t) = δρn (ρ Jan−α + g)(t)  ρ  n  t ρ n−α−1 g(s) 1−ρ d ρ−1 t − s s = t ds, t > a, ρ > 0, dt ρ (n − α) a   d n n 1−ρ . where n = [α] + 1 and δρ = t dt Definition 2.6 (Generalized Hilfer type Fractional Derivative [5]) Let order α and type β satisfy n − 1 < α < n and 0 ≤ β ≤ 1, with n ∈ N. The generalized Hilfer type fractional derivative to t, with ρ > 0 of a function g, is defined by       d n ρ (1−β)(n−α) β(n−α) ρ α,β t ρ−1 Da + g (t) = ρ Ja + Ja + g (t) dt   ρ β(n−α) n ρ (1−β)(n−α) = Ja + δρ Ja + g (t). Definition 2.7 (ψ-Riemann-Liouville fractional derivative [3]) Let ψ  (t) = 0 (−∞ ≤ a < t < b ≤ ∞), α > 0 and n ∈ N. The Riemann-Liouville derivatives of a function h of order α with respect to another function ψ on J¯ are defined by   α;ψ n−α;ψ Da + h (t) = δ n (Ja + h)(t)  t h(τ ) ψ  (τ ) (ψ(t) − ψ(τ ))n−α−1 dτ , = δn (n − α) a and   α;ψ n−α;ψ Db− h (t) = (−1)n δ n (Ja + h)(t)  b = (−1)n δ n ψ  (τ ) (ψ(τ ) − ψ(t))n−α−1 t

 where n = [α] + 1 and δ n =

1 d  ψ (t) dt

h(τ ) dτ , (n − α)

n .

Definition 2.8 (ψ-Hilfer Fractional Derivative [6]) Let order α and type β satisfy n − 1 < α < n and 0 ≤ β ≤ 1, with n ∈ N, let h, ψ ∈ C n ( J¯, R) be two functions such that ψ is increasing and ψ  (t) = 0. The ψ-Hilfer fractional derivatives to t of a function h, are defined by

22

2 Preliminary Background



H

α,β;ψ

Da +

     1 d n (1−β)(n−α);ψ β(n−α);ψ h (t) = Ja + J h (t) + a ψ  (t) dt

and 

H

α,β;ψ

Db −

     1 d n (1−β)(n−α);ψ β(n−α);ψ −  h (t) = Jb− Jb− h (t). ψ (t) dt

In this monograph we consider the case n = 1 only, because 0 < α < 1. We are now able to define the k-generalized ψ-Hilfer derivative as follows. α Definition 2.9 (k-Generalized ψ-Hilfer derivative) Let n − 1 < ≤ n with n ∈ N, −∞ ≤ k a < b ≤ ∞ and g, ψ ∈ C n ( J¯, R) two functions such that ψ is increasing and ψ  (t) = 0, for all t ∈ J . The k-generalized ψ-Hilfer fractional derivatives (left-sided and right-sided) H α,β;ψ H α,β;ψ k Da+ (·) and k Db− (·) of a function g of order α and type 0 ≤ β ≤ 1, with k > 0 are defined by     1 d n  n (1−β)(kn−α),k;ψ  β(kn−α),k;ψ H α,β;ψ k D g = J J g (t) (t) a+ a+ a+ k ψ  (t) dt    β(kn−α),k;ψ n (1−β)(kn−α),k;ψ = Ja+ δψ k n Ja+ g (t) , and H α,β;ψ k Db− g (t)

 where δψn =

    1 d n  n (1−β)(kn−α),k;ψ  β(kn−α),k;ψ −  k Jb− = Jb− g (t) ψ (t) dt    β(kn−α),k;ψ (1−β)(kn−α),k;ψ = Jb− (−1)n δψn k n Jb− g (t) ,

1 d  ψ (t) dt

n .

Property 2.10 It is worth noting that the k-generalized ψ-Hilfer fractional derivative is thought to be an expansion to many fractional operators defined over the years; indeed, in the following part, we will give a list of some of the most commonly used fractional derivatives α,β;ψ that are considered to be a particular case of our operator. The fractional derivative kH Da+ interpolate the following fractional derivatives: • • • • •

The ψ-Hilfer fractional derivative (k = 1); The ψ-Riemann–Liouville fractional derivative (k = 1, β = 0); The ψ-Caputo fractional derivative (k = 1, β = 1); The Hilfer fractional derivative (k = 1, ψ(t) = t); The Riemann–Liouville fractional derivative (k = 1, ψ(t) = t, β = 0);

2.3

• • • • • • • •

Elements from Fractional Calculus Theory

23

The Caputo fractional derivative (k = 1, ψ(t) = t, β = 1); The Hilfer–Hadamard fractional derivative (k = 1, ψ(t) = ln(t)); The Caputo–Hadamard fractional derivative (k = 1, ψ(t) = ln(t), β = 1); The Hadamard fractional derivative (k = 1, ψ(t) = ln(t), β = 0); The Hilfer–generalized fractional derivative (k = 1, ψ(t) = t ρ ); The Caputo–generalized fractional derivative (k = 1, ψ(t) = t ρ , β = 1); The generalized fractional derivative (k = 1, ψ(t) = t ρ , β = 0); The Weyl fractional derivative (k = 1, ψ(t) = t ρ , β = 0, a = −∞).

2.3.3

Necessary Lemmas, Theorems and Properties

Theorem 2.11 ([3]) Let α > 0, β > 0, 1 ≤ p ≤ ∞, 0 < a < b < ∞. Then, for g ∈ L 1 (J ) we have     α+β ρ α ρ β Ja + Ja + g (t) = ρ Ja + g (t). α;ψ

Lemma 2.12 ([7]) Let α > 0, 0 ≤ γ < 1. Then, Ja + Cγ;ψ (J ). In addition, if γ ≤ α, then

α;ψ Ja +

is bounded from Cγ;ψ (J ) into is bounded from Cγ;ψ (J ) into C( J¯, R).

Theorem 2.13 ([8]) Let g : J¯ → R be an integrable function, and take α > 0 and k > 0. α,k;ψ Then Ja+ g exists for all t ∈ J¯. α,k;ψ

Theorem 2.14 ([8]) Let g ∈ X ψ (a, b) and take α > 0 and k > 0. Then JG,a+ g ∈ C( J¯, R). p

Lemma 2.15 ([9, 10]) Let α > 0, β > 0 and k > 0. Then, we have the following semigroup property given by α,k;ψ

Ja+

α,k;ψ

Jb−

Ja+

β,k;ψ

f (t) = Ja+

α+β,k;ψ

f (t) = Ja+

β,k;ψ

f (t) = Jb−

β,k;ψ

Ja+

α+β,k;ψ

f (t) = Jb−

α,k;ψ

f (t),

β,k;ψ

Jb−

α,k;ψ

f (t).

and

Jb−

Lemma 2.16 ([11]) Let t > a. Then, for α ≥ 0 and β > 0, we have     ρ   ρ t − a ρ α+β−1 s − a ρ β−1 (β) ρ α Ja + , (t) = ρ (α + β) ρ    ρ  s − a ρ α−1 ρ α (t) = 0, 0 < α < 1. Da + ρ

24

2 Preliminary Background

Lemma 2.17 ([3, 6]) Let t > a. Then, for α ≥ 0 and β > 0, we have 

 α;ψ Ja + (ψ(τ ) − ψ(a))β−1 (t) =

(β) (ψ(t) − ψ(a))α+β−1 . (α + β)

Lemma 2.18 ([9, 10]) Let α, β > 0 and k > 0. Then, we have β

α,k;ψ

[ψ(t) − ψ(a)] k −1 =

α,k;ψ

[ψ(b) − ψ(t)] k −1 =

Ja+

α+β k (β) (ψ(t) − ψ(a)) k −1 k (α + β)

and

Jb−

β

α+β k (β) (ψ(b) − ψ(t)) k −1 . k (α + β)

α,β

Property 2.19 ([5]) The operator ρ Da + can be written as ρ

α,β

β(1−α)

Da + = ρ Ja +

δρ ρ Ja +

1−γ

β(1−α) ρ

= ρ Ja +

γ

Da + , γ = α + β − αβ.

Lemma 2.20 ([3, 5]) Let α > 0, and 0 ≤ γ < 1. Then, ρ Jaα+ is bounded from Cγ,ρ (J ) into α,β

β(1−α) ρ γ Da + u,

Cγ,ρ (J ). Since ρ Da + u = ρ Ja + γ

it follows that

α,β

C1−γ,ρ (J ) ⊂ C1−γ,ρ (J ) ⊂ C1−γ,ρ (J ). Lemma 2.21 ([5]) Let 0 < a < b < ∞, α > 0, 0 ≤ γ < 1 and u ∈ Cγ,ρ (J ). If α > 1 − γ, then ρ Jaα+ u is continuous on J and ρ

   Jaα+ u (a) = lim ρ Jaα+ u (t) = 0. t→a +

Lemma 2.22 ([6]) Let 0 < a < b < ∞, α > 0, 0 ≤ γ < 1, u ∈ Cγ;ψ (J ). If α > 1 − γ, α;ψ then Ja + u ∈ C( J¯, R) and     α;ψ α;ψ Ja + u (a) = lim Ja + u (t) = 0. t→a +

Theorem 2.23 ([9, 10]) Let 0 < a < b < ∞, α > 0, 0 ≤ ξ < 1, k > 0 and u ∈ Cξ;ψ (J ). α If > 1 − ξ, then k     α,k;ψ α,k;ψ Ja+ u (a) = lim Ja+ u (t) = 0. t→a +

Lemma 2.24 ([5]) Let α > 0, 0 ≤ γ < 1 and g ∈ Cγ,ρ (J ). Then, ρ

 Daα+ ρ Jaα+ g (t) = g(t),

for all t ∈ J .

2.3

Elements from Fractional Calculus Theory

25

1 (J ). Then, Lemma 2.25 ([6, 6]) Let α > 0, 0 ≤ β ≤ 1, and h ∈ Cγ;ψ



H

α,β;ψ

Da +

 α;ψ Ja + h (t) = h(t),

for all t ∈ J .

1 (J ), where k > 0, then for t ∈ Lemma 2.26 ([9, 10]) Let α > 0, 0 ≤ β ≤ 1, and u ∈ Cξ;ψ J , we have   α,k;ψ H α,β;ψ D J u (t) = u(t). a+ a+ k

Lemma 2.27 ([6, 12]) Let t > a, α > 0, 0 ≤ β ≤ 1. Then for 0 < γ < 1; γ = α + β − αβ, we have   γ;ψ Da + (ψ(τ ) − ψ(a))γ−1 (t) = 0, and 

H

α,β;ψ

Da +

 (ψ(τ ) − ψ(a))γ−1 (t) = 0.

Lemma 2.28 ([9, 10]) Let t > a, α > 0, 0 ≤ β ≤ 1, k > 0. Then for 0 < ξ < 1; ξ = 1 k (β(k − α) + α), we have 

H α,β;ψ k Da+

 (ψ(s) − ψ(a))ξ−1 (t) = 0.

1 Lemma 2.29 ([5]) Let 0 < α < 1, 0 ≤ γ < 1. If g ∈ Cγ,ρ (J ) and ρ Ja1−α + g ∈ C γ,ρ (J ), then   ρ J 1−α g (a)  ρ  + ρ α ρ α  t − a ρ α−1 a Ja + Da + g (t) = g(t) − , for all t ∈ J . (α) ρ γ

Lemma 2.30 ([5]) Let 0 < α < 1, 0 ≤ β ≤ 1 and γ = α + β − αβ. If u ∈ Cγ,ρ (J ), then ρ

γ

γ

α,β

Ja + ρ Da + u = ρ Jaα+ ρ Da + u,

and ρ

γ

β(1−α)

Da + ρ Jaα+ u = ρ Da +

u.

1 (J ). Then, Lemma 2.31 ([6, 12]) Let α > 0, 0 ≤ β ≤ 1, and h ∈ Cγ;ψ

 α;ψ Ja +

H

α,β;ψ

Da +



h (t) = h(t) −



1−γ;ψ

Ja +

 h (a)

(γ)

(ψ(t) − ψ(a))γ−1 , for all t ∈ J .

26

2 Preliminary Background

n [a, b], n − 1 < α < n, 0 ≤ β ≤ 1, where n ∈ N and Theorem 2.32 ([9, 10]) If f ∈ Cξ;ψ k > 0, then



α,k;ψ H α,β;ψ k Da+

Ja+

n  

− (ψ(t) − ψ(a))ξ−i n−i  k(n−ξ),k;ψ δψ f (t) = Ja+ f (a) i−n k k (k(ξ − i + 1)) i=1

+ f (t), where ξ=

1 (β(kn − α) + α) . k

In particular, if n = 1, we have  α,k;ψ Ja+

H α,β;ψ k Da+

 (ψ(t) − ψ(a))ξ−1 (1−β)(k−α),k;ψ f (t) = f (t) − J f (a). k (β(k − α) + α) a+ α,β;ψ

Property 2.33 ([6]) The operator H Da + H

α,β;ψ

Da +

β(1−α);ψ

= Ja +

can be written as γ;ψ

Da + , γ = α + β − αβ.

p

Lemma 2.34 ([3, 6]) Let α > 0, β > 0, 0 < a < b < ∞. Then, for h ∈ X c (a, b) the semigroup property is valid, i.e.     α;ψ β;ψ α+β;ψ Ja + Ja + h (t) = Ja + h (t). γ

Lemma 2.35 ([5]) Let f be a function such that f ∈ Cγ,ρ (J ). Then u ∈ Cγ,ρ (J ) is a solution of the differential equation:   ρ α,β Da + u (t) = f (t), for each , t ∈ J , 0 < α < 1, 0 ≤ β ≤ 1, if and only if u satisfies the following Volterra integral equation:   ρ J 1−γ u (a + )  ρ    t ρ t − s ρ α−1 ρ−1 1 t − a ρ γ−1 a+ u(t) = + s f (s)ds, (γ) ρ (α) a ρ where γ = α + β − αβ.

2.5

Fixed Point Theorems

2.4

27

Kuratowski Measure of Noncompactness

As stated in the introduction, one of the key tools in the theory of nonlinear analysis is the measure of noncompactness. In this part, we will review several core elements related to the concept of measure of noncompactness. Throughout this work, we use the Kuratowski measure of noncompactness in particular. Let  X be the class of all bounded subsets of a metric space X . Definition 2.36 ([13]) A function μ :  X → [0, ∞) is said to be a measure of noncompactness on X if the following conditions are verified for all B, B1 , B2 ∈  X . (a) Regularity, i.e., μ(B) = 0 if and only if B is precompact, (b) invariance under closure, i.e., μ(B) = μ(B), (c) semi-additivity, i.e., μ(B1 ∪ B2 ) = max{μ(B1 ), μ(B2 )}. Definition 2.37 ([13]) Let X be a Banach space. The Kuratowski measure of noncompactness is the map μ :  X −→ [0, ∞) defined by μ(M) = in f { > 0 : M ⊂

m 

M j , diam(M j ) ≤ },

j=1

where M ∈  X . The map μ satisfies the following properties : • • • • • •

μ(M) = 0 ⇔ M is compact (M is relatively compact). μ(M) = μ(M). M1 ⊂ M2 ⇒ μ(M1 ) ≤ μ(M2 ). μ(M1 + M2 ) ≤ μ(B1 ) + μ(B2 ). μ(cM) = |c|μ(M), c ∈ R. μ(conv M) = μ(M).

2.5

Fixed Point Theorems

In this part, we will go through all of the fixed point theorems that are employed in the various studies throughout the monograph. Fixed point theory has been one of the most intensely researched study subjects in recent decades. The fixed point notion dates back to the middle of the 18th century. Although fixed point theory appears to be a separate academic field nowadays, it first emerged in articles dealing with the solution of certain differential equations, see e.g. Liouville [14], Picard [15], Poincaré [16]. One of the first independent

28

2 Preliminary Background

fixed point results was obtained by Banach [17] by abstracting the successive approximation method of Picard. Theorem 2.38 (Banach’s fixed point theorem [18]) Let D be a non-empty closed subset of a Banach space E, then any contraction mapping N of D into itself has a unique fixed point. The most crucial difference between Banach’s fixed point theorem and others is that it ensures not only the existence but also the uniqueness of the fixed point. More crucially, it not only informs you the existence and uniqueness of a fixed point, but also how to obtain it. Following that, we will mention several more fixed point theorems that have shown to be useful in solving differential equations. Theorem 2.39 (Schauder’s fixed point theorem [18]) Let X be a Banach space, D be a bounded closed convex subset of X and T : D → D be a compact and continuous map. Then T has at least one fixed point in D. Theorem 2.40 (Schaefer’s fixed point theorem [18]) Let E be a Banach space and N : E → E be a completely continuous operator. If the set D = {u ∈ E : u = λN u, for some λ ∈ (0, 1)} is bounded, then N has a fixed point. Theorem 2.41 (Darbo’s fixed point Theorem [19]) Let D be a non-empty, closed, bounded and convex subset of a Banach space X , and let T be a continuous mapping of D into itself such that for any non-empty subset C of D, μ(T (C)) ≤ kμ(C),

(2.1)

where 0 ≤ k < 1, and μ is the Kuratowski measure of noncompactness. Then T has a fixed point in D. Theorem 2.42 (Mönch’s fixed point Theorem [20]) Let D be closed, bounded and convex subset of a Banach space X such that 0 ∈ D, and let T be a continuous mapping of D into itself. If the implication V = convT (V ), or V = T (V ) ∪ {0} ⇒ μ(V ) = 0, holds for every subset V of D, then T has a fixed point.

(2.2)

References

29

Theorem 2.43 (Krasnoselskii’s fixed point theorem [18]) Let D be a closed, convex, and nonempty subset of a Banach space E, and A, B the operators such that (1) Au + Bv ∈ D for all u, v ∈ D; (2) A is compact and continuous; (3) B is a contraction mapping. Then there exists w ∈ D such that w = Aw + Bw. Theorem 2.44 (Dhage fixed point theorem [21]) Let  be a closed, convex, bounded and nonempty subset of a Banach algebra (X ,  · ), and let T1 : E → E and T2 :  → E be two operators such that (1) (2) (3) (4)

T1 is Lipschitzian with Lipschitz constant λ, T2 is completely continuous, v = T1 y T2 w ⇒ v ∈  for all w ∈ , λM < 1, where M = B() = sup{B(w) : w ∈ }.

Then the operator equation T1 y T2 y = v has a solution in . Theorem 2.45 (Dhage fixed point theorem with three operators [22]) Let B be a closed, convex, bounded and nonempty subset of a Banach algebra (X ,  · ), and let P , R : X → X and Q : B → X be three operators such that (1) (2) (3) (4)

P and R are Lipschitzian with Lipschitz constants η1 and η2 , respectively, Q is compact and continuous, u = P u Qv + Ru ⇒ u ∈ B for all v ∈ B η1 β + η2 < 1, where β = Q(B) = sup{Q(v) : v ∈ B}.

Then the operator equation P u Qu + Ru = u has a solution in B.

References 1. I. Podlubny, Fractional Differential Equations (Academic, San Diego, 1999) 2. R. Diaz, C. Teruel, q, k-Generalized gamma and beta functions. J. Nonlinear Math. Phys 12, 118–134 (2005) 3. A.A. Kilbas, H.M. Srivastava, J. Juan Trujillo, Theory and Applications of Fractional Differential Equations (North-Holland Mathematics Studies, Amsterdam, 2006) 4. S. Rashid, M. Aslam Noor, K. Inayat Noor, Y.M. Chu, Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions. AIMS Math. 5, 2629–2645 (2020)

30

2 Preliminary Background

5. D.S. Oliveira, E. Capelas de Oliveira, Hilfer-Katugampola fractional derivatives. Comput. Appl. Math. 37, 3672–3690 (2018) 6. J.V. da C. Sousa, E.C. de Oliveira, On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018) 7. J.V. da C. Sousa, E.C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator. Differ. Equ. Appl. 11, 87–106 (2019) 8. J.E. Nápoles Valdés, Generalized fractional Hilfer integral and derivative. Contr. Math. 2, 55–60 (2020) 9. A. Salim, M. Benchohra, J.E. Lazreg, G. N’Guérékata, Existence and k-Mittag-Leffler-UlamHyers stability results of k-generalized ψ-Hilfer boundary value problem. Nonlinear Stud. 29, 359–379 (2022) 10. A. Salim, J.E. Lazreg, B. Ahmad, M. Benchohra, J.J. Nieto, A study on k-generalized ψ-Hilfer derivative operator. Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00561-8 11. R. Almeida, A.B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Caputo-Katugampola derivative. J. Comput. Nonlinear Dyn. 11, 1–11 (2016) 12. J.V. Sousa, E.C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. J. Fixed Point Theory Appl. 20(3), 96 (2018) 13. J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces (Marcel Dekker, New York, 1980) 14. J. Liouville, Second mémoire sur le développement des fonctions ou parties de fonctions en séries dont divers termes sont assujettis á satisfaire à une même équation différentielle du second ordre contenant un paramétre variable. J. Math. Pure et Appl. 2, 16–35 (1837) 15. E. Picard, Mémoire sur la théorie des équations aux derivées partielles et la méthode des approximations successives. J. Math. Pures Appl. 6, 145–210 (1890) 16. H. Poincaré, Sur les courbes definies par les équations différentielles. J. Math. 2, 54–65 (1886) 17. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922) 18. A. Granas, J. Dugundji, Fixed Point Theory (Springer, New York, 2003) 19. K. Goebel, Concise Course on Fixed Point Theorems (Yokohama Publishers, Japan, 2002) 20. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980) 21. B.C. Dhage, On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 18, 273–280 (2005) 22. B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations. Kyungpook Math. J. 44, 145–155 (2004)

3

Hybrid Fractional Differential Equations

3.1

Introduction and Motivations

This chapter is devoted to proving some results concerning the existence of solutions for a class of initial and boundary value problems for nonlinear fractional Hybrid differential equations and Generalized Hilfer fractional derivative. The results obtained in this chapter are based on fixed point theorems due to Dhage. Further, examples are provided in each section to illustrate our results. We took as motivation the following works: • The papers of Delouei et al. [1–5], Baleanu et al. [6–11], Qureshi et al. [9–11], where the authors studied exact and analytical solutions for several classes of functional equations. • The papers of Ahmad et al. [12] and Baitiche et al. [13], which deal with an interesting class of problems that involves hybrid fractional differential equations appeared recently and has achieved a great deal of interest and attention of several researchers. • The paper of Benchohra et al. [14], in it, the authors discussed the following terminal value problem for fractional differential equations with generalized Hilfer fractional derivative:       ρ D α,β x (t) = f t, x(t), ρ D α,β x (t) , t ∈ I := [a, T ], a > 0, + + a a x(T ) = c ∈ R, α,β

where ρ Da + is the generalized Hilfer type fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and f : (a, T ] × R × R → R is a given function. • The paper of Wang and Zhang [15], where they proved some existence results for the following nonlocal initial value problem for differential equations involving Hilfer’s fractional derivative:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Benchohra et al., Fractional Differential Equations, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-34877-8_3

31

32

3 Hybrid Fractional Differential Equations

⎧ α,β ⎪ ⎨ Da + u(t) = f (t, u(t)), t ∈ (a, b], m   1−γ +) = J u (a λi u(τi ), ⎪ ⎩ a+ i=1 α,β

1−γ

where Da + , Ja + are the left-sided Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and the left-sided Riemann-Liouville fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, f : (a, b] × R → R is a given function, λi , i = 1, . . . , m, are real numbers and τi , i = 1, . . . , m, are pre-fixed points satisfying a < τ1 ≤ . . . ≤ τm < b. • The paper of Zhao et al. [16], where the authors discussed the following hybrid differential equations involving Riemann-Liouville fractional derivative: 

⎧ u(t) ⎨ R L Dβ = h(t, u(t)), t ∈ I := [0, T ], g(t, u(t)) ⎩ u(0) = 0, where 0 < β < 1, g ∈ C(I × R, R\{0}) and h ∈ C(I × R, R). • The paper of Derbazi et al. [17], which deals with the existence and uniqueness of solutions of the following three-point boundary value problem for fractional hybrid differential equations with Caputo’s fractional derivative:   ⎧ u(t)− f (t,u(t)) c Dα ⎪ = h(t, u(t)), t ∈ J := [0, T ], ⎪ + g(t,u(t))  ⎪   ⎨ 0 u(0)− f (0,u(0)) )− f (T ,u(T )) + b1 u(Tg(T = c1 , a1 g(0,u(0)) ,u(T ))     ⎪ ⎪ β u(t)− f (t,u(t)) ⎪ ⎩ a c Dβ u(t)− f (t,u(t)) + b cD =c , 2

0+

g(t,u(t))

t=η

2

0+

g(t,u(t))

t=T

2

where 1 < α ≤ 2, 0 < β ≤ 1, 0 < η < T , g ∈ C(I × R, R\{0}), f , h ∈ C(I × R, R) and ai , bi , ci ∈ R, with i = 1, 2 such that a1 + b1  = 0, a2 η 1−β + b2 T 1−β  = 0. • The paper of Sousa and Oliveira [18], in which they proved some existence, uniqueness and stability results for following initial value problem for fractional differential equations involving ψ-Hilfer derivative:    H D α,β;ψ y(t) = f t, y(t), H D α,β;ψ y(t) a+ a+ 1−γ;ψ

Ja + α,β;ψ

1−γ;ψ

y(a) = ya ,

where H Da + , Ja + are the ψ-Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and ψ-Riemann-Liouville fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, ya ∈ R and f ∈ C([a, T ] × R2 , R). The existence result is based on Banach’s contraction principle. • The paper of Zhao et al. [16], in it, they studied the following hybrid differential equations:

3.2

Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

⎧ ⎨ Dβ

u(t) f (t, u(t)) ⎩ u(0) = 0,

33

 = g(t, u(t)), t ∈ I := [0, T ],

where 0 < β < 1, f ∈ C(I × R, R\{0}), g ∈ C(I × R, R), and Dβ is RiemannLiouville fractional derivative. • The paper of Hilal and Kajouni [19], where the authors discussed the following problem ⎧ 

u(t) ⎪ c Dα ⎪ = g(t, u(t)), t ∈ I := [0, T ], ⎨ 0+ f (t, u(t))  



u(0) u(T ) ⎪ ⎪ + c2 = c3 , ⎩ c1 f (0, u(0)) f (T , u(T )) where 0 < α < 1, f ∈ C(I × R, R\{0}), g ∈ C(I × R, R), c1 , c2 , c3 ∈ R, such that c1 + c2  = 0, and c Dα is the Caputo fractional derivative.

3.2

Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

3.2.1

Introduction

In this section, we consider the problem: 

x(t) ρ α,β = ϕ(t, x(t)), t ∈ J , Da + f (t, x(t))

ρ

α,β

1−γ Ja +

x(τ ) f (τ , x(τ ))



(a + ) = c0 ,

(3.1)

(3.2)

where ρ Da + , ρ Ja + are the generalized Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and generalized fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, c0 ∈ R, f ∈ C([a, b] × R, R\{0}) and ϕ ∈ C([a, b] × R, R).

3.2.2

1−γ

Existence Results

We consider the following fractional differential equation 

x(t) ρ α,β = v(t), t ∈ J , Da + f (t, x(t)) where 0 < α < 1, 0 ≤ β ≤ 1, ρ > 0, with the condition

(3.3)

34

3 Hybrid Fractional Differential Equations



ρ

1−γ Ja +

x(τ ) f (τ , x(τ ))



(a + ) = φ0 ,

(3.4)

and γ = α + β − αβ, φ0 ∈ R, f ∈ C([a, b] × R, R\{0}) and v ∈ Cγ,ρ (J ). Theorem 3.1 Let γ = α + β − αβ, where 0 < α < 1 and 0 ≤ β ≤ 1. If v : [a, b] → R is a function such that v(·) ∈ Cγ,ρ (J ), and the function f ∈ C([a, b] × R, R\{0}) then x satisfies (3.3)–(3.4) if and only if it satisfies 

ρ   t − a ρ γ−1 ρ α φ0 + Ja + v(τ ) (t) , (3.5) x(t) = f (t, x(t)) (γ) ρ Proof Assume  x satisfies the problem (3.3)–(3.4) and such that the function g : t −→

x(t) γ ∈ Cγ,ρ (J ). We prove that x is a solution to the Eq. (3.5). We have f (t, x(t))   ρ 1−γ Ja + g(τ ) (t) ∈ Cγ,ρ (J ), and

ρ

  γ 1−γ Da + g(t) = δρ ρ Ja + g(τ ) (t) ∈ Cγ,ρ (J ). 

We have

ρ

 1−γ 1 Ja + g(τ ) (t) ∈ Cγ,ρ (J ).

Thus, we deduce 

ρ

 γ γ Ja + ρ Da + g(τ ) (t) = g(t) −

ρ J 1−γ g(τ ) a+



(a) t ρ − a ρ γ−1

(γ)

ρ

, for all t ∈ J .

Then

ρ

   γ γ α,β Ja + ρ Da + g(τ ) (t) = ρ Jaα+ ρ Da + g(τ ) (t) 

= ρ Jaα+ v(τ ) (t).

Then,

 x(t) = f (t, x(t))

which implies that

ρ J 1−γ g(τ ) a+

(γ)



(a) t ρ − a ρ γ−1 ρ

+

ρ

 Jaα+ v(τ ) (t),

3.2

Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

⎡

⎢ x(t) = ⎢ ⎣

ρ J 1−γ a+



x(τ ) f (τ , x(τ )) (γ)

 (a)

tρ − a ρ

ρ γ−1

35

⎤ +

ρ

 ⎥ Jaα+ v(τ ) (t)⎥ ⎦

× f (t, x(t)), where t ∈ J , that is x satisfies the Eq. (3.5). assume x satisfies the Eq. (3.5) such that the function g : t −→

Reciprocally,  x(t) γ ∈ Cγ,ρ (J ). We prove that x is a solution to the Eqs. (3.3) and (3.4). Apply f (t, x(t)) γ operator ρ Da + on both sides of (3.5). And since f (t, x(t))  = 0 for all t ∈ [a, b], then γ

(ρ Da + g(τ ))(t) = γ



ρ

β(1−α)

Da +

 v(τ ) (t).

(3.6)

γ

Since g ∈ Cγ,ρ (J ) we have ρ Da + g ∈ Cγ,ρ (J ), then (3.6) implies that     γ 1−β(1−α) β(1−α) (ρ Da + g(τ ))(t) = δρ ρ Ja + v(τ ) (t) = ρ Da + v(τ ) (t) ∈ Cγ,ρ (J ). As v(·) ∈ Cγ,ρ (J ), it follows that 

ρ

1−β(1−α)

Ja +

 v ∈ Cγ,ρ (J ).

(3.7)

(3.8)

n (J ), we deduce From (3.7), (3.8) and definition of Cγ,ρ



ρ

β(1−α)

Applying operator ρ Ja + 

ρ

1−β(1−α)

Ja +

 1 v ∈ Cγ,ρ (J ).

on both sides of (3.7), we have

  α,β β(1−α) ρ γ Da + g(τ ) (t) = ρ Ja + Da + g(τ ) (t)   ρ J 1−β(1−α) v(τ ) (a) ρ  t − a ρ β(1−α)−1 a+ = v(t) − (β(1 − α)) ρ = v(t).

Now, applying ρ Ja +

1−γ



on both sides of (3.5), we get

ρ

   1−γ 1−γ+α Ja + g(τ ) (t) = φ0 + ρ Ja + v(τ ) (t).

By passing to the limit we obtain



ρ J 1−γ a+

x(τ ) f (τ , x(τ ))



(a + ) = φ0 .

(3.9)

(3.10) 

36

3 Hybrid Fractional Differential Equations

We can now introduce our existence result which is based on a fixed point theorem due to Dhage. Theorem 3.2 Assume that the following conditions hold. (3.2.1) Function ϕ ∈ C([a, b] × R, R) and β(1−α) (J ), for any x ∈ Cγ,ρ (J ). ϕ(·, x(·)) ∈ Cγ,ρ

(3.2.2) Function f ∈ C([a, b] × R, R\{0}) is continuous and there exists function p ∈ C([a, b], [0, ∞)) that

| f (t, x) − f (t, x)| ¯ ≤ p(t)

t ρ − aρ ρ

1−γ |x − x| ¯

for any x, x¯ ∈ R and t ∈ J . (3.2.3) There exists a function λ ∈ C([a, b], [0, ∞)) such that |ϕ(t, x)| ≤ λ(t)|x| for t ∈ J , and x ∈ R. (3.2.4) There exists R > 0 such that 

  |c0 | f∗ λ∗ R(γ) bρ − a ρ α R≥ , + 1 −  (γ) (α + γ) ρ and = p

∗



|c0 | λ∗ R(γ) + (γ) (α + γ)



bρ − a ρ ρ

α  < 1,

where p ∗ = sup p(t), λ∗ = sup λ(t), and f ∗ = sup | f (t, 0)|. t∈[a,b]

If

t∈[a,b]



bρ − a ρ ρ

t∈[a,b]

1−γ  < 1,

then the problem (3.1)–(3.2) has at least one solution in Cγ,ρ (J ). Proof Define a subset  of Cγ,ρ (J ) by  = {x ∈ Cγ,ρ (J ) : x γ,ρ ≤ R}.

(3.11)

3.2

Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

37

Let the operators T1 : Cγ,ρ (J ) → Cγ,ρ (J ) by (T1 x)(t) = f (t, x(t)), t ∈ J ,

(3.12)

and T2 :  → Cγ,ρ (J ) by c0 (T2 x)(t) = (γ)



t ρ − aρ ρ

γ−1 +

ρ

 Jaα+ ϕ(τ , x(τ )) (t), t ∈ J .

(3.13)

Set x = T1 x T2 x. Step 1: T1 is a Lipschitz on Cγ,ρ (J ). Let x, y ∈ Cγ,ρ (J ), t ∈ J . Then in view of condition (3.2.2), we get    ρ 

ρ  t − a ρ 1−γ t − a ρ 1−γ   | f (t, x(t)) − f (t, y(t))|, ((T1 x)(t) − (T1 y)(t)) ≤   ρ ρ

ρ  t − a ρ 1−γ x(t) − y(t) γ,ρ , ≤ p(t) ρ

ρ  b − a ρ 1−γ x(t) − y(t) γ,ρ . ≤ p∗ ρ Then, for each t ∈ J we obtain T1 x − T1 y γ,ρ ≤ p ∗



bρ − a ρ ρ

1−γ x(t) − y(t) γ,ρ .

Step 2: T2 is completely continuous on . We firstly show that T2 is continuous on . Let {xn } be a sequence in  such that xn → x in . Then for each t ∈ J , we have 

ρ     t − a ρ 1−γ  ρ α   ≤ Ja + |ϕ(τ , xn (τ )) − ϕ(τ , x(τ ))| (t) (T2 xn )(t) − (T2 x)(t))   ρ 

ρ t − a ρ 1−γ × , ρ Since xn → x and by the continuity of ϕ on J we get ϕ(τ , xn (τ )) → ϕ(τ , x(τ )) as n → ∞ for each t ∈ J , so by Lebesgue’s dominated convergence theorem, we deduce T2 xn − T2 x Cγ,ρ → 0 as n → ∞. Then T2 is continuous. Next we prove that T2 () is uniformly bounded on Cγ,ρ (J ). Let any x ∈ . For t ∈ J , by (3.13), we have

38

3 Hybrid Fractional Differential Equations

 



ρ   t ρ − a ρ 1−γ  |c0 | t − a ρ 1−γ ρ α   (T2 x)(t) ≤ Ja + |ϕ(τ , x(τ ))| (t) +   (γ)  ρ ρ 

ρ t − a ρ 1−γ |c0 | ∗ ≤ + λ x Cγ,ρ (γ) ρ  γ−1 

ρ ρ τ −a (t) × ρ Jaα+ ρ

 λ∗ x Cγ,ρ (γ) t ρ − a ρ α |c0 | ≤ + (γ) (α + γ) ρ

ρ α ∗ ρ |c0 | λ R(γ) b − a ≤ . + (γ) (α + γ) ρ Then, we obtain T2 x Cγ,ρ ≤

|c0 | λ∗ R(γ) + (γ) (α + γ)



bρ − a ρ ρ

α .

Next we prove that the operator T2  equicontinuous. We take x ∈  and a < ε1 < ε2 ≤ b. Then,  

1−γ

ρ   ερ − a ρ 1−γ ε2 − a ρ   1 (T2 x)(ε1 ) − (T2 x)(ε2 )    ρ ρ 



ρ 1−γ  ερ − a ρ 1−γ 

ρ α   ε2 − a ρ  1  ρ α ≤ Ja + ϕ(τ , x(τ )) (ε1 ) − Ja + ϕ(τ , x(τ )) (ε2 )   ρ ρ

ρ   1−γ   ε1   ε2 − a ρ 1 ρ α τ ρ−1 ψ(τ )ϕ(τ , x(τ )) dτ , ≤ Jε+ |ϕ(τ , x(τ ))| (ε2 ) + 1 ρ (α) a where

ψ(τ ) =

ρ

ε1 − a ρ ρ

1−γ

ρ

ε1 − τ ρ ρ

α−1

−

ρ

ε2 − a ρ ρ

1−γ

ρ

ε2 − τ ρ ρ

For each t ∈ J , we have  

1−γ

ρ   ερ − a ρ 1−γ ε2 − a ρ   1 (T2 x)(ε1 ) − (T2 x)(ε2 )    ρ ρ

ρ 1−γ ρ ρ α+γ−1 ρ ∗ ε2 − ε1 Rλ (γ) ε2 − a ≤ (α + γ)  ρ ρ   ε1 ρ−1  τ ρ − a ρ γ−1  τ ∗  ψ(τ ) +Rλ dτ ,  (α)  ρ a note that

α−1  .

3.2

Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

39

 

1−γ

ρ   ερ − a ρ 1−γ ε2 − a ρ   1 (T2 x)(ε1 ) − (T2 x)(ε2 ) → 0 as ε1 → ε2 .    ρ ρ This proves that T2  is equicontinuous on J . Therefore by the Arzela-Ascoli Theorem, T2 is completely continuous on . Step 3: Now we show that the third hypothesis of Lemma 2.44 is satisfied. Let x ∈ Cγ,ρ (J ) and y ∈  be arbitrary such that x = T1 x T2 y. Then, for t ∈ J we have  

 

  t ρ − a ρ 1−γ   t ρ − a ρ 1−γ     x(t) =  (T1 x T2 y)(t)      ρ ρ

ρ  t − a ρ 1−γ |(T1 x)(t)| |(T2 y)(t)| = ρ   

ρ  c   t − a ρ 1−γ ρ α  0 = | f (t, x(t))|  Ja + |ϕ(τ , y(τ ))| (t) +  (γ)  ρ ≤ (| f (t, x(t)) − f (t, 0)| + | f (t, 0)|)

   |c0 | λ∗ R(γ) bρ − a ρ α × + (γ) (α + γ) ρ

    |c0 |

λ∗ R(γ) bρ − a ρ α . ≤ p ∗ x Cγ,ρ + f ∗ + (γ) (α + γ) ρ Thus, we obtain

  |c0 | λ∗ R(γ) bρ − a ρ α + (γ) (α + γ) ρ  = ∗ R(γ) bρ − a ρ α  |c | λ 0 1 − p∗ + (γ) (α + γ) ρ ≤ R. f∗

x Cγ,ρ



Then x ∈ , thus the third hypothesis of Lemma 2.44 is satisfied. 

ρ b − a ρ 1−γ ∗ L < 1, where Step 4: Now, we show that p ρ L = T2 () Cγ,ρ = sup{ T2 y Cγ,ρ : y ∈ }. Since L≤

|c0 | λ∗ R(γ) + (γ) (α + γ)



bρ − a ρ ρ

α ,

40

3 Hybrid Fractional Differential Equations

 

ρ bρ − a ρ 1−γ b − a ρ 1−γ L≤  < 1. That is, the last hypothesis of ρ ρ Lemma 2.44 is satisfied. Thus, the operator equation x = T1 x T2 x = x has at least one solution x ∗ ∈ Cγ,ρ , which is a point fixed for the operator . x ∗ (t) Step 5: We prove that for such fixed point x ∗ ∈ Cγ,ρ (J ), the function g : t → f (t, x ∗ (t)) γ is in Cγ,ρ (J ). We have 

ρ   c0 t − a ρ γ−1 ρ α ∗ ∗ ∗ + Ja + ϕ(τ , x (τ )) (t) .

x (t) = f (t, x (t)) (γ) ρ then p ∗



γ

Applying ρ Da + to both sides, we have ρ Dγ a+



x ∗ (t) f (t, x ∗ (t))



ρ

 γ Da + ρ Jaα+ ϕ(τ , x ∗ (τ )) (t)   β(1−α) ϕ(τ , x ∗ (τ )) (t). = ρ Da + =

γ

Thus ρ Da + g ∈ Cγ,ρ (J ). Its clear that g ∈ Cγ,ρ (J ), since f ∈ C([a, b] × R → R\{0}), then γ  g ∈ Cγ,ρ (J ). We deduce that (3.1)–(3.2) admit at least a solution in Cγ,ρ (J ).

3.2.3

Examples

Example 3.3 Consider the following initial problem of hybrid generalized Hilfer fractional differential equation √ 

1 x(t) t − 1|sin(t)x(t)| 1 2 ,0 = , for each t ∈ (1, 2], (3.14) D1+ √ −t+3 f (t, x(t)) 43e (1 + t − 1|x(t)|)

1

1



J12+

x(τ ) f (t, x(τ ))



(1+ ) = 1,

(3.15)

where J = (1, 2], a = 1, b = 2 and f (t, x(t)) =

√  1 t − 1|x(t)| + |sin(t)| + 1 , t ∈ [1, 2], x ∈ C 1 ,1 (J ). 2 72e−t+2

Set √ t − 1|sin(t)||x| ϕ(t, x) = , t ∈ [1, 2], x ∈ R. √ 43e−t+3 (1 + t − 1|x|) We have

3.2

Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

41

  √ β(1−α) Cγ,ρ (J ) = C 01 ,1 (J ) = u : (1, 2] → R : ( t − 1)u ∈ C([1, 2], R) , 2

with γ = α = 21 , ρ = 1, β = 0. Clearly, the continuous function ϕ ∈ C 01 (J ). Hence con2 ,1

dition (3.2.1) is satisfied. We have √

| f (t, x) − f (t, x)| ¯ ≤

t −1 |x − x| ¯ . 72e−t+2

Hence condition (3.2.2) is satisfied with p(t) =

1 1 , and p ∗ = . 72e−t+2 72

For x ∈ R, we have √ |ϕ(t, x)| ≤

t − 1|sin(t)||x| , t ∈ [1, 2], 43e−t+3

and so condition (3.2.3) is satisfied with √ e t − 1|sin(t)| , and λ∗ ≤ . λ(t) = −t+3 43e 43 Also, condition (3.2.4) and the condition (3.11) of Theorem 3.2 are satisfied if we take √ 

3096 1548 π − 43 1 316 ≈ ≤ R < 1− √ √ ≈ 637. eπ 72 π e π Then the problem (3.14)–(3.15) has at least one solution in C 1 ,1 ([1, 2]). 2

Example 3.4 Consider now the following initial problem of hybrid generalized Hilfer fractional differential equation 

1 t| ln(t)| x(t) 1 2 ,0 = 1−t D0 + , for each t ∈ (0, 1], (3.16) f (t, x(t)) e (1 + |x(t)|)

1

1 2 0+

J



x(τ ) f (t, x(τ ))



(0+ ) = 0,

(3.17)

where f (t, x(t)) =

1 (|x(t)| + t| ln(t + 1)| + 1) , t ∈ [0, 1], x ∈ C 1 ,1 ([0, 1]). 2 e1−t

Simple computations show that conditions of Theorem 3.2 are satisfied. Then the problem (3.16)–(3.17) has at least one solution in C 1 ,1 ([0, 1]). 2

42

3 Hybrid Fractional Differential Equations

Example 3.5 Now, by taking α = 21 , ρ = 1, β = 1 and γ = 1, we obtain an initial problem of hybrid Caputo fractional differential equation which is a particular case of problem (3.1)– (3.2), given by  



1 1 | ln(t)x(t)| x(t) x(t) 1 2 ,1 = C D12+ = D1+ , t ∈ (1, e], −t+e f (t, x(t)) f (t, x(t)) 103e (1 + |x(t)|) (3.18) 



x(τ ) x(1) 1 0 (1+ ) = J1+ = 0, (3.19) f (t, x(τ )) f (1, x(1)) where J = (1, e], a = 1, b = e and f (t, x(t)) =

t (|x(t)| + |cos(t)| + 1) , t ∈ J , x ∈ C1,1 (J ). 102e2

Then, the conditions of Theorem 3.2 are satisfied with p∗ =

1 1 , λ∗ = , 102e 103

and √ √ 1548 π − 43 10506e π ≈ 19307. 9650 ≈ ≤R< √ eπ 2 e−1 Then the problem (3.18)–(3.19) has at least one solution in C1,1 (J ).

3.3

Boundary Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations

3.3.1

Introduction

In this section, we consider the following problem with hybrid fractional differential equations: 

x(t) ρ α,β = ϕ(t, x(t)), t ∈ J , (3.20) Da + f (t, x(t))

c1

ρ

1−γ

Ja + α,β

x(τ ) f (τ , x(τ ))



(a + ) + c2



ρ

1−γ

Ja +

x(τ ) f (τ , x(τ ))

 (b) = c3 ,

(3.21)

where ρ Da + ,ρ Ja + are the generalized Hilfer operator of order α ∈ (0, 1) and type β ∈ [0, 1] and generalized fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, c1 , c2 , c3 ∈ R, c1 + c2  = 0, f ∈ C([a, b] × R, R\{0}) and ϕ ∈ C([a, b] × R, R). 1−γ

3.3

Boundary Value Problem for Hybrid Generalized Hilfer Fractional …

3.3.2

43

Existence Results

For the purpose of simplification, we will assume the following: ¯ α (t, a) = 

α−1 ρ1−α ρ , t − aρ (α)

1−γ . γ (t, a) = ργ−1 t ρ − a ρ We consider the following fractional differential equation 

x(t) ρ α,β = v(t), t ∈ J , Da + f (t, x(t)) where 0 < α < 1, 0 ≤ β ≤ 1, ρ > 0, with the condition





  x(τ ) x(τ ) 1−γ 1−γ c1 ρ Ja + (a + ) + c2 ρ Ja + (b) = c3 , f (τ , x(τ )) f (τ , x(τ ))

(3.22)

(3.23)

The following theorem shows that the problem (3.22)–(3.23) have a unique solution given by Theorem 3.6 If v(·) ∈ Cγ,ρ (J ), and f ∈ C([a, b] × R, R\{0}), then x satisfies (3.22)– (3.23) if and only if it satisfies

     c3 c2 ρ 1−γ+α ρ α ¯ − Ja + v(τ ) (b) + Ja + v(τ ) (t) x(t) = γ (t, a) c1 + c2 c1 + c2 × f (t, x(t)).

(3.24)

Proof Assume  x satisfies the Eqs. (3.22) and (3.23) and such that the function g : t −→

x(t) γ ∈ Cγ,ρ (J ). We have f (t, x(t))   ρ 1−γ Ja + g(τ ) (t) ∈ Cγ,ρ (J ), and

ρ



Thus Hence,

  γ 1−γ Da + g(t) = δρ ρ Ja + g(τ ) (t) ∈ Cγ,ρ (J ). ρ

ρ

 1−γ 1 Ja + g(τ ) (t) ∈ Cγ,ρ (J ).

   γ γ ¯ γ (t, a) ρ J 1−γ Ja + ρ Da + g(τ ) (t) = g(t) −  g(τ ) (a). a+

44

3 Hybrid Fractional Differential Equations

Hence

ρ

Then,

   γ γ α,β Ja + ρ Da + g(τ ) (t) = ρ Jaα+ ρ Da + g(τ ) (t) 

= ρ Jaα+ v(τ ) (t).

 

 x(t) ¯ γ (t, a) ρ J 1−γ g(τ ) (a) + ρ Jaα+ v(τ ) (t), = + a f (t, x(t))

which implies that



 ¯ γ (t, a) ρ J 1−γ x(t) = f (t, x(t))  a+ Thus

ρ

1−γ

Ja +

x(τ ) f (τ , x(τ ))



(b) =

ρ

x(τ ) f (τ , x(τ ))

1−γ

Ja +

 (a) +

x(τ ) f (τ , x(τ ))

ρ

  Jaα+ v(τ ) (t) .

 (a) +



ρ

1−γ+α

Ja +

 v(τ ) (b).

Then by using condition (3.23), we get 



 c3 x(τ ) c2 ρ 1−γ+α ρ 1−γ (a) = Ja + − Ja + v(τ ) (b). f (τ , x(τ )) c1 + c2 c1 + c2 Substituting (3.26) into (3.25), we obtain (3.24).



x(t) f (t, x(t))   γ β(1−α) v(τ ) (t). (ρ Da + g(τ ))(t) = ρ Da +

Reciprocally, assume x satisfies (3.24) and g : t −→

γ

γ



(3.25)

(3.26)

γ

∈ Cγ,ρ (J ). We obtain (3.27)

γ

Since g ∈ Cγ,ρ (J ) and by definition of Cγ,ρ (J ), we have ρ Da + g ∈ Cγ,ρ (J ), then (3.27) implies that     γ 1−β(1−α) β(1−α) (ρ Da + g(τ ))(t) = δρ ρ Ja + v(τ ) (t) = ρ Da + v(τ ) (t) ∈ Cγ,ρ (J ). (3.28) As v(·) ∈ Cγ,ρ (J ) it follows 

ρ

1−β(1−α)

Ja +

 v ∈ Cγ,ρ (J ).

From (3.28), (3.29) we obtain 

ρ

β(1−α)

Applying operator ρ Ja +

1−β(1−α)

Ja +

we get

 1 v ∈ Cγ,ρ (J ).

(3.29)

3.3

Boundary Value Problem for Hybrid Generalized Hilfer Fractional …



ρ

45

  α,β β(1−α) ρ γ Da + g(τ ) (t) = ρ Ja + Da + g(τ ) (t)   ¯ β(1−α) (t, a) ρ J 1−β(1−α) = v(t) −  v(τ ) (a) + a = v(t),

that is, (3.22) holds. Now, applying ρ Ja +

1−γ



ρ

1−γ



Ja + g(τ ) (t) =

c3 c2 − c1 + c2 c1 + c2

on both sides of (3.24) we obtain 

ρ

1−γ+α

Ja +

   1−γ+α v(τ ) (b) + ρ Ja + v(τ ) (t). (3.30)

Passing to the limit we obtain



  x(τ ) c3 c2 ρ 1−γ+α ρ J 1−γ +) = − J v(τ ) (b). (a + + a a f (τ , x(τ )) c1 + c2 c1 + c2 Next, taking t = b in (3.30), we obtain



 c3 x(τ ) c2 ρ 1−γ+α ρ 1−γ (b) = Ja + − Ja + v(τ ) (b) f (τ , x(τ )) c1 + c2 c1 + c2   1−γ+α + ρ Ja + v(τ ) (b).

(3.31)

(3.32)

From (3.31) and (3.32), we find that  





x(τ ) x(τ ) 1−γ 1−γ (a + ) + c2 ρ Ja + (b) = c3 . c1 ρ Ja + f (τ , x(τ )) f (τ , x(τ ))  Now, we will provide and demonstrate the existence result of our problem. Theorem 3.7 Suppose that the following assumptions are met. (3.7.1) ϕ : [a, b] × R → R is continuous on [a, b] and β(1−α) ϕ(·, x(·)) ∈ Cγ,ρ (J ), for any x ∈ Cγ,ρ (J ).

(3.7.2) f : J × R → R\{0} is continuous and there exists function p ∈ C([a, b], [0, ∞)) that | f (t, x) − f (t, x)| ¯ ≤ p(t)γ (t, a)|x − x| ¯ for any x, x¯ ∈ R and t ∈ [a, b]. (3.7.3) There exists a function λ ∈ C([a, b], [0, ∞)) such that |ϕ(t, x)| ≤ λ(t)|x| for t ∈ [a, b], and x ∈ R.

46

3 Hybrid Fractional Differential Equations

(3.7.4) There exists a number R > 0 such that   ρ    |φ1 | b − aρ α f∗ |φ2 | (γ) ∗ R≥ , +λ R + 1 −  (γ) (1 + α) (α + γ) ρ

= p

∗



  ρ   |φ1 | |φ2 | (γ) b − aρ α ∗ < 1, +λ R + (γ) (1 + α) (α + γ) ρ

where p ∗ = sup p(t), λ∗ = sup λ(t), f ∗ = sup | f (t, 0)|. t∈[a,b]

t∈[a,b]

t∈[a,b]

If γ (b, a) < 1,

(3.33)

then (3.20)–(3.21) admits at least one solution in Cγ,ρ (J ). Proof We define a subset  of Cγ,ρ (J ) by  = {x ∈ Cγ,ρ (J ) : x γ,ρ ≤ R}. We consider the operators S : Cγ,ρ (J ) → Cγ,ρ (J ) by: (S x)(t) = f (t, x(t)), t ∈ J ,

(3.34)

    ¯ γ (t, a) φ1 − φ2 ρ J 1−γ+α (T x)(t) =  ϕ(τ , x(τ )) (b) a+ 

+ ρ Jaα+ ϕ(τ , x(τ )) (t), t ∈ J .

(3.35)

and T :  → Cγ,ρ (J ) by:

Set x = S x T x. Step 1: The operator S is a Lipschitz on Cγ,ρ (J ). Let x, y ∈ Cγ,ρ (J ) and t ∈ J . Then by condition (3.7.2), we obtain   ((S x)(t) − (S y)(t)) γ (t, a) ≤ γ (t, a)| f (t, x(t)) − f (t, y(t))|, ≤ p(t)γ (t, a) x(t) − y(t) γ,ρ , ≤ p ∗ γ (b, a) x(t) − y(t) γ,ρ , then for each t ∈ J we obtain S x − S y γ,ρ ≤ p ∗ γ (b, a) x(t) − y(t) γ,ρ ,

3.3

Boundary Value Problem for Hybrid Generalized Hilfer Fractional …

47

Step 2: The operator T is completely continuous on . We firstly show that the operator T is continuous on . Let {xn } be a sequence in  such that xn → x in . Then for each t ∈ J , we have     (T xn )(t) − (T x)(t)) γ (t, a) ≤ |φ2 | ρ J 1−γ+α |ϕ(τ , x (τ )) − ϕ(τ , x(τ ))| (b) n + a (γ)

ρ α  + γ (t, a) Ja + |ϕ(τ , xn (τ )) − ϕ(τ , x(τ ))| (t), Since xn → x and ϕ is continuous function on [a, b] then we get ϕ(τ , xn (τ )) → ϕ(τ , x(τ )) as n → ∞ for each t ∈ J , so by Lebesgue’s dominated convergence theorem, we have T xn − T x Cγ,ρ → 0 as n → ∞. Then T is continuous. Let any x ∈ . Then   γ (t, a)(T x)(t) 

 |φ1 | |φ2 | ρ 1−γ+α ≤ Ja + |ϕ(τ , x(τ ))| (b) + γ (t, a) ρ Jaα+ |ϕ(τ , x(τ ))| (t) + (γ) (γ)    |φ1 | 1−γ+α ¯ γ (τ , a) (b) ≤ + λ∗ x Cγ,ρ |φ2 | ρ Ja + (γ)

  ¯ γ (τ , a) (t) +γ (t, a)(γ) ρ Jaα+   |φ1 | ¯ 1+α (b, a) + γ (t, a)(γ) ¯ α+γ (t, a) ≤ + λ∗ R |φ2 | (γ)  ρ   b − aρ α |φ1 | |φ2 | (γ) ≤ . + λ∗ R + (γ) (1 + α) (α + γ) ρ Thus T x Cγ,ρ

 ρ   b − aρ α |φ1 | |φ2 | (γ) ∗ ≤ . +λ R + (γ) (1 + α) (α + γ) ρ

Next we prove that the operator T  equicontinuous. We take x ∈  and a < ε1 < ε2 ≤ b. Then,   γ (ε1 , a)(T x)(ε1 ) − γ (ε2 , a)(T x)(ε2 )  

ρ α 

 ≤ γ (ε1 , a) Ja + ϕ(τ , x(τ )) (ε1 ) − γ (ε2 , a) ρ Jaα+ ϕ(τ , x(τ )) (ε2 ) ε1   ¯ α (ε1 , τ ) − γ (ε2 , a) ¯ α (ε2 , τ )| τ ρ−1 ϕ(τ , x(τ )) dτ , ≤ |γ (ε1 , a) a

 α ρ + γ (ε2 , a) Jε+ |ϕ(τ , x(τ ))| (ε2 ). 1

Then we have   γ (ε1 , a)(T x)(ε1 ) − γ (ε2 , a)(T x)(ε2 )  +

ε1

¯ α (ε1 , τ ) − γ (ε2 , a) ¯ α (ε2 , τ )| ¯ γ (τ , a)dτ , τ ρ−1 |γ (ε1 , a) a ¯ α+γ (ε2 , ε1 ). Rλ∗ (γ)γ (ε2 , a)

≤ Rλ∗ (γ)

48

3 Hybrid Fractional Differential Equations

Note that   γ (ε1 , a)(T x)(ε1 ) − γ (ε2 , a)(T x)(ε2 ) → 0 as ε1 → ε2 . This proves that T  is equicontinuous on J . Therefore by the Arzela-Ascoli Theorem, T is completely continuous on . Step 3: Now we show that the third hypothesis of Lemma 2.44 is satisfied. Let x ∈ Cγ,ρ (J ) and y ∈  be arbitrary such that x = S x T y. Then, for t ∈ J we have     γ (t, a)x(t) = γ (t, a)(S x T y)(t) = γ (t, a) |(S x)(t)| |(T y)(t)|   |φ1 | |φ2 | ρ 1−γ+α = | f (t, x(t))| Ja + |ϕ(τ , y(τ ))| (b) + (γ) (γ) 

ρ α  + γ (t, a) Ja + |ϕ(τ , y(τ ))| (t) ≤ (| f (t, x(t)) − f (t, 0)| + | f (t, 0)|)  ρ     b − aρ α |φ2 | (γ) |φ1 | + λ∗ R + × (γ) (1 + α) (α + γ) ρ  ρ     b − aρ α |φ1 | |φ2 | (γ) ≤ + λ∗ R + (γ) (1 + α) (α + γ) ρ 

∗ ∗ × p x Cγ,ρ + f , then,  ρ    b − aρ α |φ1 | |φ2 | (γ) + λ∗ R + (γ) (1 + α) (α + γ) ρ    ρ   = |φ |φ | | − aρ α (γ) b 1 2 1 − p∗ + λ∗ R + (γ) (1 + α) (α + γ) ρ ≤ R. f∗

x Cγ,ρ



Then x ∈ , thus the third hypothesis of Lemma 2.44 is satisfied. Step 4: Now, we show that p ∗ γ (b, a)L < 1, where L = T () Cγ,ρ = sup{ T y Cγ,ρ : y ∈ }. Since  ρ   b − aρ α |φ2 | (γ) |φ1 | , + λ∗ R + L≤ (γ) (1 + α) (α + γ) ρ That is, the last hypothesis of then p ∗ γ (b, a)L ≤ γ (b, a) < 1. Lemma 2.44 is satisfied. Thus, the operator equation x = S x T x = x has at least one solution x ∗ ∈ Cγ,ρ , which is a point fixe for the operator .

3.3

Boundary Value Problem for Hybrid Generalized Hilfer Fractional …

Step 5: We prove that for such fixed point x ∗ ∈ Cγ,ρ (J ), the function g : t → γ

49

x ∗ (t) f (t, x ∗ (t))

is in Cγ,ρ (J ). Since x ∗ is a fixed point of operator in Cγ,ρ (J ), then      ∗ ∗ ∗ ¯ γ (t, a) φ1 − φ2 ρ J 1−γ+α ϕ(τ , x (τ )) (b)

x (t) = f (t, x (t))  + a 

 + ρ Jaα+ ϕ(τ , x ∗ (τ )) (t) . γ

Applying ρ Da + to both sides we get ρ Dγ a+



x ∗ (t) f (t, x ∗ (t))



ρ

 γ Da + ρ Jaα+ ϕ(τ , x ∗ (τ )) (t)   β(1−α) ϕ(τ , x ∗ (τ )) (t). = ρ Da + =

γ

Thus ρ Da + g ∈ Cγ,ρ (J ). Its clear that g ∈ Cγ,ρ (J ), since f ∈ C([a, b] × R → R\{0}), γ then g ∈ Cγ,ρ (J ). Theorem 3.7 implies that (3.20)–(3.21) has at least one solution in  Cγ,ρ (J ).

3.3.3

Examples

Example 3.8 Consider the terminal problem 1 2

D

1 2 ,0 e+



x(t) f (t, x(t))



√ √ 1 ( t − e) 2 x(t) = , t ∈ (e, π], √ √ 1 105e−t+π (1 + |cos(t)|( t − e) 2 |x(t)|) 



1 1 1 x(τ ) 2J 2 (π) = , e+ f (t, x(τ )) 2

(3.36)

(3.37)

where J = (e, π], a = e, b = π and √ √  t− e |sin(t)|x(t) + tan −1 (t) + π , t ∈ [e, π], x ∈ C 1 , 1 (J ). f (t, x(t)) = π−t 2 2 52e Set √ √ 1 ( t − e) 2 x ϕ(t, x) = , t ∈ [e, π], x ∈ R. √ √ 1 105e−t+π (1 + |cos(t)|( t − e) 2 |x|) We have

! " √ √ √ 1 β(1−α) Cγ,ρ (J ) = C 01 , 1 (J ) = u : (e, π] → R : 2( t − e) 2 u ∈ C([e, π], R) , 2 2

50

3 Hybrid Fractional Differential Equations

with γ = α = 21 , ρ = 21 , β = 0. Clearly, the continuous function ϕ ∈ C 01 1 (J ). Hence, con2,2

dition (3.7.1) is satisfied. We have √ √ ( t − e)|sin(t)| |x − x| | f (t, x) − f (t, x)| ¯ ≤ ¯ . 52eπ−t Thus, condition (3.7.2) is satisfied with √ √ 1 1 ( t − e) 2 |sin(t)| and p ∗ ≤ √ . p(t) = √ π−t 52 2e 52 2 Let x ∈ R. Then we have √ √ 1 ( t − e) 2 |x| , t ∈ J, |ϕ(t, x)| ≤ 105e−t+π and so condition (3.7.3) is satisfied with √ √ 1 1 ( t − e) 2 , and λ∗ ≤ . λ(t) = 105e−t+π 105

Also, condition (3.7.4) and the condition (3.33) of Theorem 3.7 are satisfied if we take

 √ 1 √ 1− 5460 π √ 283920 2π − 5460 104 2π ≤R< ≈ 5330. 2655 ≈ √ √ √ 1 √ √ 21 ( π − e) 2 (2 + π) 104 2( π − e) (2 + π) Then the problem (3.36)−(3.37) has at least one solution in C 1 , 1 (J ). 2 2

Example 3.9 Consider the anti-periodic problem √ 

1 et−2 t − 1ln(t)x(t) x(t) 1 2 ,0 = D1+ , for each t ∈ (1, 2], f (t, x(t)) 333(1 + x C 1 )

(3.38)

2 ,1

1

1

J12+



x(τ ) f (t, x(τ ))



(1+ ) = −

1

1

J12+



x(τ ) f (t, x(τ ))

 (2),

(3.39)

where J = (1, 2], a = 1, b = 2 and √ f (t, x(t)) = Set

t − 1|tan −1 (t)|x(t) ln(|cos(t)| + + √ 111e−t+3 e3 t

√ t)

, t ∈ [1, 2], x ∈ C 1 ,1 (J ). 2

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

ϕ(t, x(t)) =

51

√ et−2 t − 1ln(t)x(t) , t ∈ [1, 2], x ∈ C 1 ,1 (J ). 2 333(1 + x C 1 ) 2 ,1

We have   √ β(1−α) (J ) = C 01 ,1 (J ) = u : J → R : ( t − 1)u ∈ C([1, 2], R) , Cγ,ρ 2

with γ = α = 21 , ρ = 1, β = 0. Clearly, ϕ ∈ C 01 (J ). Hence, condition (3.7.1) is satisfied. 2 ,1

We have √ | f (t, x) − f (t, x)| ¯ ≤

t − 1|tan −1 (t)| |x − x| ¯ . 111e−t+3

Hence condition (3.7.2) is satisfied with p(t) =

1 |tan −1 (t)| , and p ∗ = . 111e−t+3 222e

Let x ∈ R. Then we have

√ et−2 t − 1ln(t)|x| |ϕ(t, x)| ≤ , t ∈ J, 333

and so condition (3.7.3) is satisfied with √ ln(2) et−2 t − 1ln(t) , and λ∗ = . λ(t) = 333 333 Also, condition (3.7.4) and the condition (3.33) of Theorem 3.7 are satisfied if we take √ √ 111e × 333 π 222e × 333 π 62036 ≈ ≤R< ≈ 124072. (1 + π)ln(2) (1 + π)ln(2) Then the problem (3.38)−(3.39) has at least one solution in C 1 ,1 (J ). 2

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

Motivated by the works of the papers mentioned in the introduction of the chapter, in this section, we establish existence results to the nonlocal initial value problem with nonlinear implicit hybrid generalized Hilfer-type fractional differential equation: 





α,β x(t) − χ(t, x(t)) ρ α,β x(t) − χ(t, x(t)) = ϕ t, x(t), ρ Da + , t ∈ (a, b], Da + f (t, x(t)) f (t, x(t)) (3.40)

52

3 Hybrid Fractional Differential Equations



ρ

1−γ Ja + α,β

x(τ ) − χ(t, x(t)) f (τ , x(τ ))



+

(a ) =

m

ci

i=1

 x(i ) − χ(i , x(i )) , f (i , x(i ))

(3.41)

where ρ Da + , ρ Ja + are the generalized Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and generalized fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, ci , i = 1, . . . , m, are real numbers, i , i = 1, . . . , m, are pre-fixed points satisfying a < 1 ≤ . . . ≤ m < b, f ∈ C([a, b] × R, R\{0}), χ ∈ C([a, b] × R, R), ϕ ∈ #m ¯ γ (i , a)  = 1, for further details see the definitions in the ci  C([a, b] × R2 , R) and i=1 following subsection.

3.4.1

1−γ

Existence Results

For the purpose of simplification, we will assume the following: ¯ α (t, a) = 

α−1 ρ1−α ρ , t − aρ (α)

1−γ . γ (t, a) = ργ−1 t ρ − a ρ We consider the following fractional differential equation 

ρ α,β x(t) − χ(t, x(t)) = v(t), t ∈ J , Da + f (t, x(t))

(3.42)

where 0 < α < 1, 0 ≤ β ≤ 1, ρ > 0, with the nonlocal condition

ρ

1−γ

Ja +

x(τ ) − χ(t, x(t)) f (τ , x(τ ))



(a + ) =

m i=1

ci

 x(i ) − χ(i , x(i )) , f (i , x(i ))

(3.43)

where γ = α + β − αβ, ci , i = 1, . . . , m, are real numbers, f ∈ C([a, b] × R, R\{0}), χ ∈ C([a, b] × R, R), i , i = 1, . . . , m, are pre-fixed points satisfying a < 1 ≤ . . . ≤ m < b, m ¯ γ (i , a)  = 1. The following theorem shows that the problem (3.42)– ci  v ∈ Cγ,ρ (J ) and i=1

(3.43) have a solution given by ⎡

m

ρ

Jaα+ v(τ )



⎤

ci (i ) ⎥ ⎢ ⎢  ⎥

ρ α i=1 ⎥ + χ(t, x(t)). ¯ γ (t, a)  J v(τ ) (t) + x(t) = f (t, x(t)) ⎢ + a ⎥ ⎢ m ⎦ ⎣ ¯ γ (i , a) 1− ci  i=1

(3.44)

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

53

Theorem 3.10 Let γ = α + β − αβ, where 0 < α < 1 and 0 ≤ β ≤ 1. If v : J → R is a function such that v(·) ∈ Cγ,ρ (J ), f ∈ C([a, b] × R, R\{0}), and the function χ ∈ C([a, b] × R, R), then x satisfies Eqs. (3.42) and (3.43) if and only if it satisfies (3.44). Proof Assume  x satisfies the Eqs. (3.42) and (3.43) and such that the function σ : t −→  γ x(t)−χ(t,x(t)) ∈ Cγ,ρ (J ). We prove that x is a solution to the Eq. (3.44). From the definition f (t,x(t)) γ

of the space Cγ,ρ (J ) and by using Lemma 2.20 and Definition 2.5, we have   ρ 1−γ Ja + σ(τ ) (t) ∈ Cγ,ρ (J ), and

ρ

  γ 1−γ Da + σ(t) = δρ ρ Ja + σ(τ ) (t) ∈ Cγ,ρ (J ).

n (J ), we have By the definition of the space Cγ,ρ



ρ

 1−γ 1 Ja + σ(τ ) (t) ∈ Cγ,ρ (J ).

Hence, Lemma 2.29 implies that

ρ

   γ γ ¯ γ (t, a) ρ J 1−γ Ja + ρ Da + σ(τ ) (t) = σ(t) −  σ(τ ) (a), for all t ∈ (a, b]. + a

Using Lemma 2.30 we have

ρ

Then,

   γ γ α,β Ja + ρ Da + σ(τ ) (t) = ρ Jaα+ ρ Da + σ(τ ) (t) 

= ρ Jaα+ v(τ ) (t).

 

 x(t) − χ(t, x(t)) ¯ γ (t, a) ρ J 1−γ σ(τ ) (a) + ρ Jaα+ v(τ ) (t), = + a f (t, x(t))

which implies that 





ρ α  x(τ ) − χ(τ , x(τ )) ¯ γ (t, a) ρ J 1−γ (a) + x(t) =  J v(τ ) (t) a+ a+ f (τ , x(τ )) × f (t, x(t)) + χ(t, x(t)),

(3.45)

where t ∈ J . Next, we substitute t = i into (3.45), then we multiply ci to both sides, we obtain 

 

ρ α  x(i ) − χ(i , x(i )) ¯ γ (i , a) ρ J 1−γ = ci  Ja + v(τ ) (i ). ci + σ(τ ) (a) + ci a f (i , x(i ))

54

3 Hybrid Fractional Differential Equations

Then by using condition (3.43), we have 

ρ

 m  x(i ) − χ(i , x(i )) 1−γ Ja + σ(τ ) (a + ) = ci f (i , x(i )) i=1

=



ρ

m m 

 1−γ ¯ γ (i , a) + Ja + σ(τ ) (a) ci  ci ρ Jaα+ v(τ ) (i ), i=1

which implies

m



ρ



Ja + σ(τ ) (a + ) = 1−γ

i=1

ci

ρ

 Jaα+ v(τ ) (i )

i=1

1−

m

.

(3.46)

¯ γ (i , a) ci 

i=1

Substituting (3.46) into (3.45), we obtain (3.44).  assume x satisfies the Eq. (3.44) such that the function σ : t −→  Reciprocally, γ x(t)−χ(t,x(t)) ∈ Cγ,ρ (J ). We prove that x is a solution to the problem (3.42)–(3.43). Apply f (t,x(t)) γ

operator ρ Da + on both sides of (3.44). And since f (t, x(t))  = 0 for all t ∈ J , then, from Lemmas 2.16 and 2.30 we obtain   γ β(1−α) (ρ Da + σ(τ ))(t) = ρ Da + v(τ ) (t). (3.47) γ

γ

γ

Since σ ∈ Cγ,ρ (J ) and by definition of Cγ,ρ (J ), we have ρ Da + σ ∈ Cγ,ρ (J ), then (3.47) implies that     γ 1−β(1−α) β(1−α) v(τ ) (t) = ρ Da + v(τ ) (t) ∈ Cγ,ρ (J ). (3.48) (ρ Da + σ(τ ))(t) = δρ ρ Ja + As v(·) ∈ Cγ,ρ (J ) and from Lemma 2.20, follows   ρ 1−β(1−α) Ja + v ∈ Cγ,ρ (J ).

(3.49)

n (J ), we obtain From (3.48), (3.49) and by the definition of the space Cγ,ρ



ρ

1−β(1−α)

Ja +

β(1−α)

 1 v ∈ Cγ,ρ (J ).

Applying operator ρ Ja + on both sides of (3.48) and using Lemma 2.29, Lemma 2.21 and Property 2.19, we have    β(1−α) ρ γ ρ α,β Da + σ(τ ) (t) = ρ Ja + Da + σ(τ ) (t)   ¯ β(1−α) (t, a) ρ J 1−β(1−α) v(τ ) (a) = v(t) −  + a = v(t),

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

that is, (3.42) holds. Now, applying ρ Ja + and Theorem 2.11, we get

1−γ

m



ρ



1−γ

Ja + σ(τ ) (t) =

ci

ρ

on both sides of (3.44) and using Lemma 2.16

 Jaα+ v(τ ) (i )

i=1

1−

m

55

+



ρ

¯ γ (i , a) ci 

1−γ+α

Ja +

 v(τ ) (t).

(3.50)

i=1

Taking the limit t → a + of (3.50) and using Lemma 2.21, with 1 − γ < 1 − γ + α, we obtain m



ρ J 1−γ a+



x(τ ) − χ(τ , x(τ )) f (τ , x(τ ))



(a + ) =

ρ

ci

 Jaα+ v(τ ) (i )

i=1

1−

m

.

(3.51)

¯ γ (i , a) ci 

i=1

Substituting t = i into (3.44), we have m

ρ

ci

 Jaα+ v(τ ) (i )



x(i ) − χ(i , x(i )) ¯ γ (i , a) i=1 + ρ Jaα+ v(τ ) (i ). = m f (i , x(i )) ¯ γ (i , a) 1− ci  i=1

Then, we have m m i=1

ci

x(i ) − χ(i , x(i )) f (i , x(i ))

 =

m

ci

¯ γ (i , a) i=1 ci  1−

i=1

ρ

 Jaα+ v(τ ) (i )

m

¯ γ (i , a) ci 

i=1

+

m

ci

ρ



Jaα+ v(τ ) (i ),

i=1

thus,

m m i=1

ci

x(i ) − χ(i , x(i )) f (i , x(i ))

 =

ci

ρ

 Jaα+ v(τ ) (i )

i=1

1−

m i=1

. ¯ γ (i , a) ci 

(3.52)

56

3 Hybrid Fractional Differential Equations

From (3.51) and (3.52), we find that

ρ

1−γ

Ja +

x(τ ) − χ(τ , x(τ )) f (τ , x(τ ))



(a + ) =

m

ci

i=1

 x(i ) − χ(i , x(i )) , f (i , x(i ))

which shows that the initial condition (3.43) is satisfied. This completes the proof.



As a consequence of Theorem 3.10, we have the following result. Lemma 3.11 Let γ = α + β − αβ where 0 < α < 1 and 0 ≤ β ≤ 1, let f ∈ C([a, b] × R, R\{0}), χ ∈ C([a, b] × R, R) and ϕ : J × R2 → R, be a function such that  ϕ(·, x(·),  γ y(·)) ∈ Cγ,ρ (J ), for any x, y ∈ Cγ,ρ (J ). If the function t −→ x(t)−χ(t,x(t)) ∈ Cγ,ρ (J ), f (t,x(t)) then x satisfies the problem (3.40)–(3.41) if and only if x is the fixed point of the operator

: Cγ,ρ (J ) → Cγ,ρ (J ) defined by  m

ρ α 

ρ α  ¯ ci Ja + v(τ ) (i ) + Ja + v(τ ) (t)

x(t) = K γ (t, a) i=1

× f (t, x(t)) + χ(t, x(t)), where K = 1 −

m

(3.53)

−1 ¯ γ (i , a) ci 

and v : J → R be a function satisfying the functional

i=1

equation v(t) = ϕ(t, x(t), v(t)). Since the functions f and χ are continuous and ϕ(·, x(·), y(·)) ∈ Cγ,ρ (J ), then, by Lemma 2.20, we have x ∈ Cγ,ρ (J ). We are now in a position to state and prove our existence result for the problem (3.40)– (3.41) based on based on Lemma 2.45. Theorem 3.12 Assume that the following hypotheses hold. (3.12.1) The function ϕ : J × R2 → R be continuous on J and β(1−α) ϕ(·, x(·), y(·)) ∈ Cγ,ρ (J ), for any x, y ∈ Cγ,ρ (J ).

(3.12.2) The functions f : [a, b] × R → R\{0} and χ : [a, b] × R → R are continuous and there exist two functions p, q ∈ C([a, b], [0, ∞)) such that | f (t, x) − f (t, x)| ¯ ≤ p(t)γ (t, a)|x − x| ¯

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

57

and |χ(t, x) − χ(t, x)| ¯ ≤ q(t)|x − x| ¯ for any x, x¯ ∈ R and t ∈ J . (3.12.3) There exists functions λ1 , λ2 , λ3 ∈ C([a, b], [0, ∞)) such that |ϕ(t, x, y)| ≤ λ1 (t) + λ2 (t)|x| + λ3 (t)|y| for t ∈ J , and x, y ∈ R. (3.12.4) There exists a number  > 0 such that ≥

f ∗ M + χ∗ , 1 − p∗ M − q ∗

where p ∗ = sup p(t), q ∗ = sup q(t), t∈[a,b]

t∈[a,b]

λi∗ = sup λi (t), i = 1, 2, λ∗3 = sup λ3 (t) < 1, t∈[a,b]

t∈[a,b]

f ∗ = sup | f (t, 0)|, χ∗ = sup γ (t, a)|χ(t, 0)|,  := t∈[a,b]

t∈[a,b]

γ (b, a)λ∗1 + λ∗2  1 − λ∗3

and  m  

ρ

ρ ρ−α 1−γ ρ α+γ−1 ρ α |K |ρ . |ci | i − a + (γ) b − a M= (α + γ) i=1

If

max{ p ∗ M, p ∗ γ (b, a)M} + q ∗ < 1,

(3.54)

then the problem (3.40)−(3.41) has at least one solution in Cγ,ρ (J ). Proof We define a subset  of Cγ,ρ (J ) by  = {x ∈ Cγ,ρ (J ) : x γ,ρ ≤ }. We consider the operator defined in (3.53), and define three operators S , N : Cγ,ρ (J ) → Cγ,ρ (J ) by (3.55) (S x)(t) = f (t, x(t)), t ∈ J ,

58

3 Hybrid Fractional Differential Equations

(N x)(t) = χ(t, x(t)), t ∈ J ,

(3.56)

and T :  → Cγ,ρ (J ) by ¯ γ (t, a) (T x)(t) = K 

m

ci

ρ



 Jaα+ v(τ ) (i ) + ρ Jaα+ v(τ ) (t), t ∈ J .

(3.57)

i=1

Then we get x = S x T x + N x. We shall show that the operators S , T and N satisfies all the conditions of Lemma 2.45. The proof will be given in several steps. Step 1: The operators S and N are Lipschitzian on Cγ,ρ (J ). Let x, y ∈ Cγ,ρ (J ) and t ∈ J . Then by (3.12.2) we have   ((S x)(t) − (S y)(t)) γ (t, a) ≤ γ (t, a)| f (t, x(t)) − f (t, y(t))|, ≤ p(t)γ (t, a) x − y γ,ρ , ≤ p ∗ γ (b, a) x − y γ,ρ ,

then for each t ∈ J we obtain S x − S y γ,ρ ≤ p ∗ γ (b, a) x − y γ,ρ .

Also, for each t ∈ J we have   ((N x)(t) − (N y)(t)) γ (t, a) ≤ γ (t, a)|χ(t, x(t)) − χ(t, y(t))|, ≤ q(t) x − y γ,ρ , ≤ q ∗ x − y γ,ρ , then, N x − N y γ,ρ ≤ q ∗ x − y γ,ρ . Step 2: The operator T is completely continuous on . We firstly show that the operator T is continuous on . Let {xn } be a sequence in  such that xn → x in . Then for each t ∈ J , we have m  

 (T xn )(t) − (T x)(t)) γ (t, a) ≤ |K | |ci | ρ Jaα+ |vn (τ ) − v(τ )| (i ) (γ) i=1

 + γ (t, a) ρ Jaα+ |vn (τ ) − v(τ )| (t),

where vn , v ∈ Cγ,ρ (J ) such that

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

59

vn (t) = ϕ(t, xn (t), vn (t)), v(t) = ϕ(t, x(t), v(t)). Since xn → x and ϕ is continuous function on J then we get vn (t) → v(t) as n → ∞ for each t ∈ J . So by Lebesgue’s dominated convergence theorem, we have T xn − T x Cγ,ρ → 0 as n → ∞. Then T is continuous. Next we prove that T () is uniformly bounded on Cγ,ρ (J ). Let any x ∈ . By (3.12.3), we have for each t ∈ J     γ (t, a)v(t) = γ (t, a)ϕ(t, x(t), v(t)) ≤ γ (t, a)(λ1 (t) + λ2 (t)|x(t)| + λ3 (t)|v(t)|) ≤ γ (b, a)λ∗1 + λ∗2  + λ∗3 |γ (t, a)v(t)|. Which implies that ∗ ∗   γ (t, a)v(t) ≤ γ (b, a)λ1 + λ2  . 1 − λ∗3

Then, we have  γ (b, a)λ∗1 + λ∗2   := . sup γ (t, a)v(t) ≤ 1 − λ∗3 t∈(a,b] For t ∈ J , by (3.57) and Lemma 2.16, we have   γ (t, a)(T x)(t) m



 |K | |ci | ρ Jaα+ |v(τ )| (i ) + γ (t, a) ρ Jaα+ |v(τ )| (t) ≤ (γ) i=1 m



 ¯ γ (τ , a) (i ) + γ (t, a)(γ) ρ J α+  ¯ γ (τ , a) (t) |ci | ρ J α+  ≤ |K | a

≤ |K |

i=1 m

a

¯ α+γ (i , a) + γ (t, a)(γ) ¯ α+γ (t, a) |ci |

i=1  ρ α+γ−1

ρ  m i − a ρ b − aρ α (γ) |K | |ci | + . ≤ (α + γ) ρ (α + γ) ρ i=1

60

3 Hybrid Fractional Differential Equations

Then for t ∈ J , we obtain T x Cγ,ρ ≤ M. This prove that the operator T is uniformly bounded on . Next we prove that the operator T  equicontinuous. We take x ∈  and a < ε1 < ε2 ≤ b. Then,   γ (ε1 , a)(T x)(ε1 ) − γ (ε2 , a)(T x)(ε2 )  



 ≤ γ (ε1 , a) ρ Jaα+ v(τ ) (ε1 ) − γ (ε2 , a) ρ Jaα+ v(τ ) (ε2 ) ε1   ¯ α (ε1 , τ ) − γ (ε2 , a) ¯ α (ε2 , τ )| τ ρ−1 v(τ ) dτ , |γ (ε1 , a) ≤ a

 + γ (ε2 , a) ρ Jεα+ |v(τ )| (ε2 ). 1

Then by Lemma 2.16, we have for each t ∈ (a, b]   γ (ε1 , a)(T x)(ε1 ) − γ (ε2 , a)(T x)(ε2 )  ε1 ¯ α (ε1 , τ ) − γ (ε2 , a) ¯ α (ε2 , τ )| ¯ γ (τ , a)dτ , ≤ (γ) τ ρ−1 |γ (ε1 , a) a

¯ α+γ (ε2 , ε1 ). +(γ)γ (ε2 , a) Note that   γ (ε1 , a)(T x)(ε1 ) − γ (ε2 , a)(T x)(ε2 ) → 0 as ε1 → ε2 . This proves that T  is equicontinuous on J . Therefore by the Arzela-Ascoli Theorem, T is completely continuous on . Step 3: Now we show that the third hypothesis of Lemma 2.45 is satisfied. Let x ∈ Cγ,ρ (J ) and y ∈  be arbitrary such that x = S x T y + N x and v˜ ∈ Cγ,ρ (J ) with v(t) ˜ = ϕ(t, y(t), v(t)). ˜ Then, for t ∈ J we have   γ (t, a)x(t)   = γ (t, a)(S x T y)(t) + γ (t, a)(N x)(t) ≤ γ (t, a) |(S x)(t)| |(T y)(t)| + |γ (t, a)(N x)(t)|  m

ρ α 

ρ α  |K | |ci | Ja + |v(τ ˜ )| (i ) + γ (t, a) Ja + |v(τ ˜ )| (t) ≤ | f (t, x(t))| (γ) i=1

+γ (t, a)|χ(t, x(t))| ≤ M (| f (t, x(t)) − f (t, 0)| + | f (t, 0)|) +γ (t, a) (|χ(t, x(t)) − χ(t, 0)| + |χ(t, 0)|) 

≤ M p ∗ x Cγ,ρ + f ∗ + q ∗ x Cγ,ρ + χ∗ .

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

61

Thus, x Cγ,ρ =

f ∗ M + χ∗ ≤ . 1 − p∗ M − q ∗

Consequently, x ∈  and the third hypothesis of Lemma 2.45 is satisfied. Step 4: Now, we show that p ∗ γ (b, a)L + q ∗ < 1, where L = T () Cγ,ρ = sup{ T y Cγ,ρ : y ∈ }. Since L ≤ M, we have p ∗ γ (b, a)L + q ∗ ≤ p ∗ γ (b, a)M + q ∗ < 1. That is, the last hypothesis of Lemma 2.45 is satisfied. Thus, the operator equation

x = S x T x + N x = x has at least one solution x ∗ ∈ Cγ,ρ , which is a point fixed for the operator . Step 5: We prove that for such fixed point x ∗ ∈ Cγ,ρ (J ), the function σ : t → ∗ x (t) − χ(t, x ∗ (t)) γ is in Cγ,ρ (J ). f (t, x ∗ (t)) Since x ∗ is a fixed point of operator in Cγ,ρ (J ), then for each t ∈ J , we have  m

ρ α 

ρ α  ∗ ¯ ci Ja + v(τ ) (i ) + Ja + v(τ ) (t)

x (t) = K γ (t, a) i=1

× f (t, x ∗ (t)) + χ(t, x ∗ (t)). where v ∈ Cγ,ρ (J ) such that

(3.58)

v(t) = ϕ(t, x ∗ (t), v(t)).

γ

Applying ρ Da + to both sides of 3.58, and by Lemmas 2.16 and 2.30, we have ρ Dγ a+



x ∗ (t) − χ(t, x ∗ (t)) f (t, x ∗ (t))



ρ

 γ Da + ρ Jaα+ v(τ ) (t)   β(1−α) v(τ ) (t). = ρ Da + =

γ

Since γ ≥ α, by (3.12.1), the right hand side is in Cγ,ρ (J ) and thus ρ Da + σ ∈ Cγ,ρ (J ). Its clear that σ ∈ Cγ,ρ (J ), since f ∈ C([a, b] × R → R\{0}) and χ ∈ C([a, b] × R → R), γ then σ ∈ Cγ,ρ (J ). As a consequence of Steps 1 and 5 with Theorem 3.12, we can conclude  that the problem (3.40)–(3.41) has at least a solution in Cγ,ρ (J ).

62

3 Hybrid Fractional Differential Equations

3.4.2

Example

Example 3.13 Consider the nonlocal initial value problem of hybrid generalized type Hilfer Fractional differential equation 

  1 √ x(t)−χ(t,x(t)) 2 ,0 1 +1 t − 1 x(t) + D1+   f (t,x(t)) 1 1 2 ,0 x(t)−χ(t,x(t)) , t ∈ (1, 2], = D1+ √ f (t,x(t)) 111e−t+2 (1 + t − 1|x(t)|) (3.59)  

1

 x( 23 ) − χ( 23 , x( 23 )) x(τ ) − χ(τ , x(τ )) 1 2 + (1 ) = 2 J1+ , (3.60) f (τ , x(τ )) f ( 23 , x( 23 )) where I = (1, 2], a = 1, b = 2 and

f (t, x(t)) =

|sin(πt)|(t − 1)|x(t)| + 1 , t ∈ [1, 2], x ∈ C 1 ,1 ([1, 2]), 2 41e−t+4

and √ 1 t − 1 ln(|cos(t)| + 1)x(t) + χ(t, x(t)) = , t ∈ [1, 2], x ∈ C 1 ,1 ([1, 2]). √ 3 2 55e−t+2 33e 6 − t Set √ t − 1 (x + y + 1) , t ∈ I , x, y ∈ R. ϕ(t, x, y) = √ −t+2 111e (1 + |x| t − 1) We have   √ β(1−α) Cγ,ρ (I ) = C 01 ,1 (I ) = v : I → R : t → ( t − 1)v(t) ∈ C([1, 2], R) , 2

with γ = α = 21 , ρ = 1, β = 0. Clearly, the continuous function ϕ ∈ C 01 (I ). Hence the 2 ,1

condition (3.12.1) is satisfied. For each x, x¯ ∈ R and t ∈ I , we have | f (t, x) − f (t, x)| ¯ ≤

|sin(πt)|(t − 1) |x − x| ¯ , 41e−t+4

and √ t − 1 ln(|cos(t)| + 1) |x − x| |χ(t, x) − χ(t, x)| ¯ ≤ ¯ . √ 33e3 6 − t Hence condition (3.12.2) is satisfied with

3.4

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type …

p(t) =

63

√ √ |sin(πt)| t − 1 t − 1 ln(|cos(t)| + 1) , , and q(t) = √ 41e−t+4 33e3 6 − t

so we have p∗ ≤

1 ln(2) , and q ∗ ≤ . 2 41e 66e3

Let x, y ∈ R. Then we have √ t −1 |ϕ(t, x, y)| ≤ (|x| + |y| + 1) , t ∈ I , 111e−t+2 and so the condition (3.12.3) is satisfied with √

λ1 (t) = λ2 (t) = λ3 (t) =

t −1 , 111e−t+2

and λ∗1 = λ∗2 = λ∗3 =

1 . 111

Also, the condition (3.12.4) and the condition (3.54) of Theorem 3.12 is satisfied if we take √ √ 4510e2 (2 2 − π) 2956 ≈ 400e2 ≤  < √ − 1 ≈ 6496, √ (2 2π + 2 π − π) where f∗ =

1 , 41e2

χ∗ =

1 55 ,

k=

√ π √ √ π−2 2

and =

1+ . 110

Then the problem (3.61)–(3.62) has at least one solution in C 1 ,1 (I ). 2

Example 3.14 In this example, we change the boundary condition (3.62) which give us the following nonlocal initial value problem of hybrid generalized Hilfer fractional differential equation √   1 t − 1 (x(t) + 1) 1 2 ,0 x(t)−χ(t,x(t)) = D1+ √ f (t,x(t)) 111e−t+2 (1 + t − 1|x(t)|)  √ 1 1 D 2 ,0 x(t)−χ(t,x(t)) t −1 + f (t,x(t)) 1 + , for each t ∈ (1, 2], (3.61) √ 111e−t+2 (1 + t − 1|x(t)|)

64

3 Hybrid Fractional Differential Equations

1

1 2 1+

J



x(τ ) − χ(τ , x(τ )) f (τ , x(τ ))





+

(1 ) = 3  +2

x( 45 ) − χ( 45 , x( 45 )) f ( 45 , x( 45 )) x( 43 ) − χ( 43 , x( 43 )) f ( 43 , x( 43 ))

  .

(3.62)

All the hypothesis of Theorem 3.12 are satisfied, indeed, we have √ √ 4510e2 (66e3 − ln(2))(2 3 − π + 6) 2 2956 ≈ 400e ≤  < − 1 ≈ 11387, √ √ 66e3 (11 π + 2 3π − π) where m = 2, f ∗ =

1 , 41e2

χ∗ =

1 55 ,

K =

√

√ π √ , π−2 3−6

and =

1+ . 110

Then the problem (3.61)–(3.62) has at least one solution in C 1 ,1 (I ). 2

3.5

Initial Value Problem for Hybrid ψ-Hilfer Fractional Implicit Differential Equations

In this section, we consider the initial value problem with nonlinear implicit hybrid ψ-Hilfer type fractional differential equation:





 x(t) x(t) α,β;ψ H α,β;ψ Da + = f t, x(t), H Da + , t ∈ (a, b], (3.63) g(t, x(t)) g(t, x(t)) 



x(τ ) 1−γ;ψ (a + ) = x0 , Ja + (3.64) g(τ , x(τ )) α,β;ψ

1−γ;ψ

where H Da + , Ja + are the ψ-Hilfer fractional derivative of order α ∈ (0, 1) and type β ∈ [0, 1] and ψ-Riemann-Liouville fractional integral of order 1 − γ, (γ = α + β − αβ) respectively, x0 ∈ R, g ∈ C([a, b] × R, R\{0}) and f ∈ C([a, b] × R2 , R).

3.5

Initial Value Problem for Hybrid ψ-Hilfer Fractional …

3.5.1

65

Existence Results

We consider the following fractional differential equation 

x(t) H α,β;ψ = v(t), t ∈ (a, b], Da + g(t, x(t)) where 0 < α < 1, 0 ≤ β ≤ 1, with the condition



x(τ ) 1−γ;ψ (a + ) = φ0 , Ja + g(τ , x(τ ))

(3.65)

(3.66)

where φ0 ∈ R, g ∈ C([a, b] × R, R\{0}). The following theorem shows that the Eqs. (3.65) and (3.66) have a unique solution given by   φ0 (ψ(t) − ψ(a))γ−1  α;ψ (3.67) + Ja + v(τ ) (t) . x(t) = g(t, x(t)) (γ) Theorem 3.15 Let γ = α + β − αβ, where 0 < α < 1 and 0 ≤ β ≤ 1. If v : J → R is 1 (J ) and the function g ∈ C([a, b] × R, R\{0}) then x a given function such that v ∈ Cγ,ψ satisfies problem (3.65)–(3.66) if and only if it satisfies (3.67). Proof Assume x satisfies the Eqs. (3.65) and (3.66) such that the function h : t −→   x(t) 1 (J ). We prove that x is a solution to the Eq. (3.67). Applying the frac∈ C γ,ψ g(t,x(t)) α;ψ

tional integral Ja + to both sides of Eq. (3.65) and using Lemma 2.31, we get h(t) −

   (ψ(t) − ψ(a))γ−1  1−γ;ψ α;ψ Ja + h(τ ) (a) = Ja + v(τ ) (t). (γ)

Thus,

   (ψ(t) − ψ(a))γ−1  1−γ;ψ  x(τ )  α;ψ (a) + Ja + v(τ ) (t) . x(t) = g(t, x(t)) Ja + g(τ ,x(τ )) (γ) By using the condition (3.66), we obtain Eq. (3.67). Reciprocally, assume x satisfies the Eq. (3.67). We prove that x is a solution to the α,β;ψ equations (3.65) and (3.66). Applying operator H Da + to both sides of (3.67) and using Lemmas 2.27 and 2.25, we have

   γ−1 x(t) α,β;ψ φ0 (ψ(t) − ψ(a)) α,β;ψ α;ψ H α,β;ψ Da + Ja + v(τ ) (t) = H Da + + H Da + g(t, x(t)) (γ) = v(t),

66

3 Hybrid Fractional Differential Equations

that is, (3.65) holds. 1−γ Now, applying Ja + to both sides of (3.67) and using Lemmas 2.34 and 2.17, we get     1−γ;ψ 1−γ+α;ψ Ja + h(τ ) (t) = φ0 + Ja + v(τ ) (t).

(3.68)

Next, taking the limit t → a + of (3.68) and using Lemma 2.22, with 1 − γ < 1 − γ + α, we obtain 



x(τ ) 1−γ;ψ (a + ) = φ0 . Ja + (3.69) g(τ , x(τ )) which shows that the initial condition (3.66) is satisfied. This completes the proof.



As a consequence of Theorem 3.15, we have the following result Lemma 3.16 Let γ = α + β − αβ where 0 < α < 1 and 0 ≤ β ≤ 1, let g ∈ C([a, b] × 1 (J ), for any R, R\{0}) and f : J × R2 → R, be a function such that f (·, x(·), y(·)) ∈ Cγ,ψ   x(t) 1 (J ), then x satisfies the problem ∈ Cγ,ψ x, y ∈ Cγ,ψ (J ). If the function t −→ g(t,x(t)) (3.63)–(3.64) if and only if x is the fixed point of the operator : Cγ,ψ (J ) → Cγ,ψ (J ) defined by   x0 (ψ(t) − ψ(a))γ−1  α;ψ (3.70)

x(t) = g(t, x(t)) + Ja + v(τ ) (t) , (γ) where v : J −→ R be function satisfying the functional equation v(t) = f (t, x(t), v(t)). Since the function g is continuous and f (·, x(·), y(·)) ∈ Cγ,ψ (J ), then, by Lemma 2.12, we have x ∈ Cγ,ψ (J ). We are now in a position to state and prove our existence result for the problem (3.63)−(3.64) based on based on Lemma 2.44. Theorem 3.17 Assume that the requirement that follow are met. (3.17.1) The function f : J × R → R be continuous on J and 1 (J ), for any x, y ∈ Cγ;ψ (J ). f (·, x(·), y(·)) ∈ Cγ;ψ

(3.17.2) The function g : [a, b] × R → R\{0} is continuous and there exists function p ∈ C([a, b], [0, ∞)) that

3.5

Initial Value Problem for Hybrid ψ-Hilfer Fractional …

67

|g(t, x) − g(t, x)| ¯ ≤ p(t) (ψ(t) − ψ(a))1−γ |x − x| ¯ for any x, x¯ ∈ R and t ∈ (a, b]. (3.17.3) There exist functions η1 , η2 , η3 ∈ C([a, b], [0, ∞)) such that | f (t, x, y)| ≤ η1 (t) + η2 (t)|x| + η3 (t) for t ∈ (a, b], and x, y ∈ R. (3.17.4) There exists a number R > 0 such that   |x0 | g∗ η(γ) R≥ + (ψ(b) − ψ(a))α , 1 −  (γ) (α + γ) where p ∗ = sup p(t), ηi∗ = sup ηi (t), i = 1, 2, η3∗ = sup η3 (t) < 1, t∈[a,b]

t∈[a,b]

g ∗ = sup |g(t, 0)|, η = t∈[a,b]

t∈[a,b]

(ψ(b) − ψ(a))1−γ η1∗ + η2∗ R , 1 − η3∗

and = p

∗



 |x0 | η(γ) α + (ψ(b) − ψ(a)) < 1. (γ) (α + γ)

If (ψ(b) − ψ(a))1−γ  < 1,

(3.71)

then the problem (3.63)−(3.64) has at least one solution in Cγ;ψ (J ). Proof We define a subset D of Cγ;ψ (J ) by D = {x ∈ Cγ;ψ (J ) : x γ;ψ ≤ R}. We consider the operator defined in (3.70), and define two operators N1 : Cγ;ψ (J ) → Cγ;ψ (J ) by (3.72) (N1 x)(t) = g(t, x(t)), t ∈ (a, b], and N2 : D → Cγ;ψ (J ) by (N2 x)(t) =

 x0 (ψ(t) − ψ(a))γ−1  α;ψ + Ja + v(τ ) (t), t ∈ (a, b]. (γ)

(3.73)

Then we get x = N1 x N2 x. We shall show that the operators N1 and N2 satisfies all the conditions of Lemma 2.44. The proof will be given in several steps.

68

3 Hybrid Fractional Differential Equations

Step 1: The operator N1 is a Lipschitz on Cγ;ψ (J ). Let x, y ∈ Cγ;ψ (J ) and t ∈ (a, b]. Then by condition (3.17.2) we have   (ψ(t) − ψ(a))1−γ ((N1 x)(t) − (N1 y)(t)) ≤ (ψ(t) − ψ(a))1−γ |g(t, x(t)) − g(t, y(t))| ≤ p(t) (ψ(t) − ψ(a))1−γ x − y γ,ψ ≤ p ∗ (ψ(b) − ψ(a))1−γ x − y γ,ψ , then for each t ∈ (a, b] we obtain N1 x − N1 y γ,ψ ≤ p ∗ (ψ(b) − ψ(a))1−γ x − y γ,ψ . Step 2: The operator N2 is completely continuous on D. We firstly show that the operator N2 is continuous on D. Let {xn } be a sequence in D such that xn → x in D. Then for each t ∈ (a, b], we have   (ψ(t) − ψ(a))1−γ (N2 xn )(t) − (N2 x)(t)) ≤ (ψ(t) − ψ(a))1−γ   α;ψ × Ja + |vn (τ ) − v(τ )| (t), where vn , v ∈ Cγ,ψ (J ) such that vn (t) = f (t, xn (t), vn (t)), v(t) = f (t, x(t), v(t)). Since xn → x and f is continuous function on J then we get vn (t) → v(t) as n → ∞ for each t ∈ (a, b], so by Lebesgue’s dominated convergence theorem, we have N2 xn − N2 x Cγ;ψ → 0 as n → ∞. Then N2 is continuous. Next we prove that N2 (D) is uniformly bounded on Cγ;ψ (J ). Let x ∈ D. By hypothesis (3.17.3), we have for each t ∈ (a, b]     (ψ(t) − ψ(a))1−γ v(t) = (ψ(t) − ψ(a))1−γ f (t, x(t), v(t)) ≤ (ψ(t) − ψ(a))1−γ (η1 (t) + η2 (t)|x(t)| + η3 (t)|v(t)|) ≤ (ψ(b) − ψ(a))1−γ η1∗ + η2∗ R + η3∗ | (ψ(t) − ψ(a))1−γ v(t)|. Witch implies that

3.5

Initial Value Problem for Hybrid ψ-Hilfer Fractional …

69

1−γ ∗   η1 + η2∗ R (ψ(t) − ψ(a))1−γ v(t) ≤ (ψ(b) − ψ(a)) . ∗ 1 − η3

Then, we have   (ψ(b) − ψ(a))1−γ η1∗ + η2∗ R sup (ψ(t) − ψ(a))1−γ v(t) ≤ := η. 1 − η3∗ t∈(a,b] For t ∈ (a, b], by (3.73), condition (3.17.3) and Lemma 2.17, we have   (ψ(t) − ψ(a))1−γ (N2 x)(t)   |x0 | α;ψ ≤ + (ψ(t) − ψ(a))1−γ Ja + |v(τ )| (t) (γ)   |x0 | α;ψ + η (ψ(t) − ψ(a))1−γ Ja + (ψ(τ ) − ψ(a))γ−1 (t) ≤ (γ) |x0 | η(γ) ≤ + (ψ(t) − ψ(a))α (γ) (α + γ) |x0 | η(γ) ≤ + (ψ(b) − ψ(a))α . (γ) (α + γ) Then, for t ∈ (a, b] we obtain N2 x Cγ;ψ ≤

|x0 | η(γ) + (ψ(b) − ψ(a))α . (γ) (α + γ)

This prove that the operator N2 is uniformly bounded on D. Next we prove that the operator N2 D equicontinuous. We take x ∈ D and a < ε1 < ε2 ≤ b. Then,   (ψ(ε1 ) − ψ(a))1−γ (N2 x)(ε1 ) − (ψ(ε2 ) − ψ(a))1−γ (N2 x)(ε2 )         α;ψ α;ψ ≤ (ψ(ε1 ) − ψ(a))1−γ Ja + v(τ ) (ε1 ) − (ψ(ε2 ) − ψ(a))1−γ Ja + v(τ ) (ε2 )

  ε1   1 α;ψ 1−γ ψ (τ )(τ )v(τ ) dτ , ≤ (ψ(ε2 ) − ψ(a)) Jε+ |v(τ )| (ε2 ) + (α) a 1 and (τ ) = (ψ(ε1 ) − ψ(a))1−γ (ψ(ε1 ) − ψ(τ ))α−1

 − (ψ(ε2 ) − ψ(a))1−γ (ψ(ε2 ) − ψ(τ ))α−1 .

Then, by Lemma 2.17, we have for each t ∈ (a, b]

70

3 Hybrid Fractional Differential Equations

  (ψ(ε1 ) − ψ(a))1−γ (N2 x)(ε1 ) − (ψ(ε2 ) − ψ(a))1−γ (N2 x)(ε2 ) η(γ) ≤ (ψ(ε2 ) − ψ(a))1−γ (ψ(ε2 ) − ψ(ε1 ))α+γ−1 (α + γ)   ε1   ψ (τ )  γ−1  +η dτ . (τ ) (α)  (ψ(τ ) − ψ(a)) a

Note that   (ψ(ε1 ) − ψ(a))1−γ (N2 x)(ε1 ) − (ψ(ε2 ) − ψ(a))1−γ (N2 x)(ε2 ) → 0 as ε1 → ε2 . This proves that N2 D is equicontinuous on J . Therefore by the Arzela-Ascoli Theorem, N2 is completely continuous. Step 3: Now we show that the third hypothesis of Lemma 2.44 is satisfied. Let x ∈ Cγ;ψ (J ) and y ∈ D be arbitrary such that x = N1 x N2 y. Then, for t ∈ (a, b] we have   (ψ(t) − ψ(a))1−γ x(t)   = (ψ(t) − ψ(a))1−γ (N1 x N2 y)(t) = (ψ(t) − ψ(a))1−γ |(N1 x)(t)| |(N2 y)(t)|         x0  α;ψ  α,β;ψ y(τ ) =  + (ψ(t) − ψ(a))1−γ Ja +  f τ , y(τ ), H Da +  (t) g(τ ,y(τ )) (γ) × |g(t, x(t))|

  |x0 | η(γ) ≤ (|g(t, x(t)) − g(t, 0)| + |g(t, 0)|) + (ψ(b) − ψ(a))α (γ) (α + γ)  

∗  |x0 | η(γ) ≤ p x Cγ;ψ + g ∗ + (ψ(b) − ψ(a))α , (γ) (α + γ) then,  |x0 | η(γ) α + (ψ(b) − ψ(a)) (γ) (α + γ)   = |x0 | η(γ) 1 − p∗ + (ψ(b) − ψ(a))α (γ) (α + γ) g∗

x Cγ;ψ



≤ R. Then x ∈ D, thus the third hypothesis of Lemma 2.44 is satisfied. Step 4: Now, we show that p ∗ (ψ(b) − ψ(a))1−γ L < 1, where L = N2 (D) Cγ;ψ = sup{ N2 y Cγ;ψ : y ∈ D}. Since L≤

|x0 | η(γ) + (ψ(b) − ψ(a))α , (γ) (α + γ)

3.5

Initial Value Problem for Hybrid ψ-Hilfer Fractional …

71

then p ∗ (ψ(b) − ψ(a))1−γ L ≤ (ψ(b) − ψ(a))1−γ  < 1. That is, the last hypothesis of Lemma 2.44 is satisfied. Thus, the operator equation x = N1 x N2 x = x has at least one solution x ∈ Cγ;ψ , witch is a point fixe for the operator . 

x(t) 1 (J ). Then, as a It is clear that by hypothesis (3.17.1) we have t −→ g(t,x(t)) ∈ Cγ,ψ consequence of Steps 1–4 with Theorem 3.17, we can conclude that the problem (3.63)–  (3.64) has at least a solution in Cγ;ψ (J ).

3.5.2

Examples

Example 3.18 Taking β → 0, α = 21 , ψ(t) = t, a = 1, b = 2 and x0 = 0, we obtain a particular case of problem (3.63)–(3.64) with Riemann-Liouville fractional derivative, given by 





1 1 x(t) x(t) ,0;t R L 2 ,0;t = f t, x(t), R L D12+ , t ∈ (1, 2], (3.74) D1+ g(t, x(t)) g(t, x(t))

1 2 ;t 1+

J



x(τ ) g(τ , x(τ ))



(1+ ) = 0,

(3.75)

where J = (1, 2]. Set √

g(t, x(t)) =

t −1 (|x(t)sin(t)| + 3) , t ∈ [1, 2], x ∈ C 1 ,t (J ). 2 33e−t+2

and f (t, x, y) =

√ t − 1|cos(t)|(1 + x + y) , t ∈ J , x, y ∈ R. 55e−t+4 (2 + |x|)

We have   √ Cγ,ψ (J ) = C 1 ,t (J ) = u : J → R : ( t − 1)u ∈ C([a, b], R) , 2

and   1 Cγ,ψ (J ) = C 11 ,t (J ) = u ∈ C 1 ,t (J ) : u ∈ C 1 ,t (J ) , 2

2

2

with γ = α = 21 , ψ(t) = t, β = 0. Clearly, the function f ∈ C 11 (J ). Hence the condition 2 ,t

(3.17.1) is satisfied. For each x, x¯ ∈ R and t ∈ J , we have √ |g(t, x) − g(t, x)| ¯ ≤

t − 1|sin(t)| |x − x| ¯ . 33e−t+2

72

3 Hybrid Fractional Differential Equations

Hence condition (3.17.2) is satisfied with p(t) =

|sin(t)| 1 and p ∗ = . 33e−t+2 33

Let x, y ∈ R. Then we have √ t − 1|cos(t)|(1 + |x| + |y|) , t ∈ J, | f (t, x, y)| ≤ 55e−t+4 and so condition (3.17.3) is satisfied with √ t − 1|cos(t)| , η1 (t) = η2 (t) = η3 (t) = 55e−t+4 and η1∗ = η2∗ = η3∗ =

1 . 55e2

Also, condition (3.17.4) and the condition (3.71) of Theorem 3.17 is satisfied if we take √ √ 605e2 − π − 11 1815e2 − π − 33 2515 ≈ ≤R< ≈ 7547. √ √ π π Then the problem (3.74)–(3.75) has at least one solution in C 1 ,t (J ). 2

Example 3.19 Taking β → 1, α = 21 , ψ(t) = t, a = 1, b = π and x0 = 0, we obtain a particular case of problem (3.63)–(3.64) involving Caputo fractional derivative, given by 





1 1 x(t) x(t) ,1;t C 2 ,1;t = f t, x(t),C D12+ , t ∈ (1, π], (3.76) D1+ g(t, x(t)) g(t, x(t))



J11;t +

x(τ ) g(τ , x(τ ))



(1+ ) = 0,

(3.77)

where J = (1, π]. Set g(t, x(t)) =

 1 |x(t)| + |tan −1 (t)| , t ∈ [1, π], x ∈ C1,t (J ). 115

and |sin 2 (t)| f (t, x, y) = 77t We have



x y + 1 + |x| 2 + |y|



+ |tan −1 (t)| + 5π, t ∈ J , x, y ∈ R.

3.5

Initial Value Problem for Hybrid ψ-Hilfer Fractional …

73

Cγ,ψ (J ) = C1,t (J ) = {u : J → R : u ∈ C([1, π], R)} , and $ % 1 1 Cγ,ψ (J ) = C1,t (J ) = u ∈ C1,t (J ) : u ∈ C1,t (J ) , 1 (J ). Hence, condition with α = 21 , γ = 1, ψ(t) = t, β = 1. Clearly, the function f ∈ C1,t (3.17.1) is satisfied. For each x, x¯ ∈ R and t ∈ J , we have

|g(t, x) − g(t, x)| ¯ ≤

1 |x − x| ¯ . 115

1 Thus, condition (3.17.2) is satisfied with p(t) = . 115 Let x, y ∈ R. Then we have | f (t, x, y)| ≤

|sin 2 (t)|(|x| + |y|) + |tan −1 (t)| + 5π, t ∈ J , 77t

and so condition (3.17.3) is satisfied with η1 (t) = |tan −1 (t)| + 5π, η2 (t) = η3 (t) =

|sin 2 (t)| , 77t

and η1∗ =

11π 1 , η2∗ = η3∗ = . 2 77

Same as the last example, conditions (3.17.4) and (3.71) of Theorem 3.17 is satisfied if we choose a convenient constant R. Then the problem (3.76)–(3.77) has at least one solution in C1,t ([1, π]). Example 3.20 Taking β → 0, α = 21 , ψ(t) = ln t, a = 1, b = e and x0 = e, we get a particular case of problem (3.63)–(3.64) using the Hadamard fractional derivative, given by 





1 1 x(t) x(t) ,0;ln t H D 2 ,0;ln t = f t, x(t), H D D12+ , t ∈ (1, e], (3.78) D1+ g(t, x(t)) g(t, x(t))

1

2 ;ln t J1+

x(τ ) g(τ , x(τ ))



(1+ ) = e,

(3.79)

where J = (1, e]. Set g(t, x(t)) =

 √ e−t+1 |cos(πt)|  √ −t+e ( + e π, t ∈ [1, e], x ∈ C 1 ,ln t (J ). ln t)|x(t)| + e 2 12π + 111e2t

74

3 Hybrid Fractional Differential Equations

and f (t, x, y) =

e+x+y , t ∈ J , x, y ∈ R. 22et

We have   √ Cγ,ψ (J ) = C 1 ,ln t (J ) = u : J → R : ( ln t)u ∈ C([1, e], R) , 2

and   1 Cγ,ψ (J ) = C 11 ,ln t (J ) = u ∈ C 1 ,ln t (J ) : u ∈ C 1 ,ln t (J ) , 2

2

2

with α = γ = 1, ψ(t) = ln t, β = 0. Clearly, the continuous function f ∈ C 11

2 ,ln t

condition (3.17.1) is satisfied. For each x, x¯ ∈ R and t ∈ J , we have

(J ). Hence,

√ e−t+1 |cos(πt)| ln t |x − x| |g(t, x) − g(t, x)| ¯ ≤ ¯ . 12π + 111e2t Hence condition (3.17.2) is satisfied with p(t) =

1 e−t+1 |cos(πt)| , and p ∗ = . 12π + 111e2t 12π + 111e2

Let x, y ∈ R. Then we have | f (t, x, y)| ≤

e + |x| + |y| , t ∈ J, 22et

and so condition (3.17.3) is satisfied with η1 (t) =

e 1 , η2 (t) = η3 (t) = , t 22e 22et

and η1∗ =

e 1 , η2∗ = η3∗ = . 22e 22e

Same as before, we choose a suitable constant R so that conditions (3.17.4) and (3.71) of Theorem 3.17 be satisfied. Then the problem (3.78)–(3.79) has at least one solution in C 1 ,ln t (J ). 2

References

3.6

75

Notes and Remarks

This chapter’s results are based on the articles of Salim et al. [20–23]. The monographs [24– 31] and the papers [6–11, 13, 14] include additional pertinent results and investigations.

References 1. A.A. Delouei , A. Emamian , S. Karimnejad, H. Sajjadi, A closed-form solution for axisymmetric conduction in a finite functionally graded cylinder. Int. J. Heat Mass Transf. 108 (2019) 2. A.A. Delouei. A. Emamian, S. Karimnejad, H. Sajjad, D. Jing, Two-dimensional analytical solution for temperature distribution in FG hollow spheres: General thermal boundary conditions. Int. J. Heat Mass Transf. 113 (2020) 3. A.A. Delouei, A. Emamian, S. Karimnejad, H. Sajjadi, D. Jing, Asymmetric conduction in an infinite functionally graded cylinder: two-dimensional exact analytical solution under general boundary conditions. Int. J. Heat Mass Transf. 142 (2020) 4. A.A. Delouei, A. Emamian, S. Karimnejad, H. Sajjadi, D. Jing, Two-dimensional temperature distribution in FGM sectors with the power-law variation in radial and circumferential directions. J. Thermal Anal. Calorimetry 144, 611–621 (2021) 5. A.A. Delouei, A. Emamian, S. Karimnejad, H. Sajjadi, A. Tarokh, On 2D asymmetric heat conduction in functionally graded cylindrical segments: a general exact solution. Int. J. Heat Mass Transf. 143 (2019) 6. D. Baleanu, A. Jajarmi, E. Bonyah, M. Hajipour, New aspects of poor nutrition in the life cycle within the fractional calculus. Adv. Differ. Equ. 1, 1–14 (2018) 7. D. Baleanu, S.S. Sajjadi, J.H. Asad, A. Jajarmi, E. Estiri, Hyperchaotic behaviors, optimal control, and synchronization of a nonautonomous cardiac conduction system. Adv. Differ. Equ. 1, 1–24 (2021) 8. A. Jajarmi, D. Baleanu, On the fractional optimal control problems with a general derivative operator. Asian J. Control 23, 1062–1071 (2021) 9. U.T. Mustapha, S. Qureshi, A. Yusuf, E. Hincal, Fractional modeling for the spread of Hookworm infection under Caputo operator. Chaos Solitons Fractals. 137, 109878 (2020), 16 pp 10. S. Qureshi, E. Bonyah, A.A. Shaikh, Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data. Phys. A: Stat. Mech. Appl. 535 (2019) 11. S. Qureshi, A. Yusuf., A.A. Shaikh, M. Inc, D. Baleanu, Fractional modeling of blood ethanol concentration system with real data application. Chaos 29(1), 013143 (2019), 8 pp 12. B. Ahmad, S.K. Ntouyas, Initial value problems for hybrid Hadamard fractional differential equations. Electron. J. Differ. Equ. 2014, 161 (2014) 13. Z. Baitiche, K. Guerbati, M. Benchohra, Y. Zhou, Boundary value problems for hybrid Caputo fractional differential equations. Mathematics 2019, 282 (2019) 14. M. Benchohra, S. Bouriah, J.J. Nieto, Terminal value problem for differential equations with Hilfer-Katugampola fractional derivative. Symmetry. 2019, 672 (2019) 15. J.R. Wang, Y.R. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative. Appl. Math. Comput. 266, 850–859 (2015) 16. Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations. Comput. Math. Appl. 62, 1312–1324 (2011)

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17. C. Derbazi, H. Hammouche, M. Benchohra et al., Fractional hybrid differential equations with three-point boundary hybrid conditions. Adv. Differ. Equ. 2019, 125 (2019) 18. J.V. Sousa, E.C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. J. Fixed Point Theory Appl. 20(3), 96 (2018) 19. K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order. Adv. Differ. Equ. 2015, 183 (2015) 20. A. Salim, M. Benchohra, J.R. Graef, J.E. Lazreg, Initial value problem for hybrid ψ-Hilfer fractional implicit differential equations. J. Fixed Point Theory Appl. 24 (2022), 14 pp. https:// doi.org/10.1007/s11784-021-00920-x 21. A. Salim, M. Benchohra, J.E. Lazreg, J.J. Nieto, Y. Zhou, Nonlocal initial value problem for hybrid generalized Hilfer-type fractional implicit differential equations. Nonauton. Dyn. Syst. 8, 87–100 (2021). https://doi.org/10.1515/msds-2020-0127 22. A. Salim, M. Benchohra, J.E. Lazreg, Initial value problem for hybrid generalized Hilfer fractional differential equations. Discontinuity, Nonlinearity, and Complexity. (Accepted) 23. A. Salim, B. Ahmad, M. Benchohra, J.E. Lazreg, Boundary value problem for hybrid generalized Hilfer fractional differential equations. Differ. Equ. Appl. 14, 379–391 (2022). https://doi.org/ 10.7153/dea-2022-14-27 24. E. Capelas de Oliveira, Solved Exercises in Fractional Calculus (Springer International Publishing, Switzerland, 2019) 25. V. Daftardar-Gejji, Fractional Calculus and Fractional Differential Equations (Birkhäuser, Basel, 2019) 26. H. Dutta, A.O. Akdemir, A. Atangana, Fractional Order Analysis: Theory, Methods and Applications, Hoboken (Wiley, NJ, 2020) 27. M. Francesco, Fractional Calculus: Theory and Applications (MDPI, 2018) 28. B. Jin, Fractional Differential Equations: An Approach via Fractional Derivatives (Springer International Publishing, Switzerland, 2021) 29. G. Karniadakis, Handbook of Fractional Calculus with Applications. Volume 3: Numerical Methods (Berlin, Boston, De Gruyter, 2019) 30. C. Li, M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives. Society for Industrial and Applied Mathematics (2019) 31. E. Shishkina, S.M. Sitnik, Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics (Academic, 2020)

Fractional Differential Equations with Retardation and Anticipation

4.1

Introduction and Motivations

In this chapter, we prove some existence and uniqueness results for a class of boundary and terminal value problems for implicit nonlinear k-generalized ψ-Hilfer fractional differential equations involving both retarded and advanced arguments. Further, examples are given to illustrate the viability of our results in each section. The results of our analysis in this chapter can be viewed as a conditional extension of the problems discussed fairly recently in the following: • The monographs of Abbas et al. [1–3], Benchohra et al. [4], and the papers of Abbas et al. [5–9], Benchohra et al. [10, 11], Liu et al. [12], Kharade et al. [13], Sousa et al. [14, 15], in it, considerable attention has been given to the study of the existence results and Ulam stability of a class of initial and boundary value problems for fractional differential equations. • The paper of Krim et al. [6], where the authors discussed the following terminal value problem for fractional differential equations with Katugampola fractional derivative:      ρ r  D0+ x (τ ) = κ τ , x(τ ), ρ D0r + x (τ ) , τ ∈ I := [0, T ], x(T ) = x T ∈ R, with T > 0 and the function κ : I × R × R → R is continuous. Here, ρ D0r + is the Katugampola fractional derivative of order r ∈ (0, 1]. Their reasoning is mainly based upon on the α-φ-Geraghty type contraction and the fixed point theory. • The paper of Almalahi et al. [16], in it, using Krasnoselskii fixed point theorem, Picard operator method and Grönwall’s lemma, the authors proved some existence and stability results for the following nonlocal initial value problem for differential equations involving ψ-Hilfer fractional derivative: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Benchohra et al., Fractional Differential Equations, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-34877-8_4

77

4

78

4 Fractional Differential Equations with Retardation …

⎧ H α,β;ψ ⎪ = f (t, x(t), x(g(t)), t ∈ (0, b] ⎨ D0+ x(t) k 1−γ;ψ  +  J0+ x 0 = i=1 ci x (τi ) , τi ∈ (0, b) ⎪ ⎩ x(t) = ϕ(t), t ∈ [−r , 0] α,β;ψ

where H D0+

(·) is ψ-Hilfer fractional derivative operator of order α ∈ (0, 1) and type

1−γ,ψ β ∈ [0, 1], J0+ (·) is ψ-fractional integral in Riemann-Liouville sense of order 1 − γ, where γ = α + β(1 − α), 0 < γ < 1, τi , i = 1, 2, . . . , k, are prefixed points satisfy-

ing 0 < τ1 ≤ τ2 ≤ · · · ≤ τi < b, and ci ∈ R, ϕ ∈ C[−r , 0], f : (0, b] × R × R → R is a given continuous function, and g ∈ C(0, b] → [−r , b] with g(t) ≤ t, r > 0. Some of their interesting results can be found in the paper [17].

4.2

On k-Generalized ψ-Hilfer Boundary Value Problems with Retardation and Anticipation

Motivated by the above papers, we consider the boundary value problem for the nonlinear implicit k-generalized ψ-Hilfer type fractional differential equation involving both retarded and advanced arguments:

H α,β;ψ k Da+ x

α,β;ψ





  α,β;ψ (t) = f t, xt (·), kH Da+ x (t) , t ∈ (a, b],

(4.1)



  k(1−ξ),k;ψ k(1−ξ),k;ψ x (a + ) + ϑ2 Ja+ x (b) = ϑ3 , ϑ1 Ja+

(4.2)

x(t) = (t), t ∈ [a − λ, a], λ > 0,

(4.3)



x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

(4.4)

k(1−ξ),k;ψ

are, respectively, the k-generalized ψ-Hilfer fractional where kH Da+ and Ja+ derivative of order α ∈ (0, k) and type β ∈ [0, 1] defined in Sect. 2.1, and k-generalized ψ-fractional integral of order k(1 − ξ) defined in [18], where ξ = k1 (β(k − α) + α), k > 0,



 f : [a, b] × C −λ, λ˜ , R × R −→ R is a given function, and ϑ1 , ϑ2 , ϑ3 ∈ R such that ˜ are, respectively, continuous functions on [a − λ, a] and ϑ1 + ϑ2  =

0, and (t) and (t)

b, b + λ˜ . For each function x defined on a − λ, b + λ˜ and for any t ∈ (a, b], we denote by xt the element defined by

xt (τ ) = x(t + τ ), τ ∈ −λ, λ˜ .

On k-Generalized ψ-Hilfer Boundary Value …

4.2

4.2.1

79

Existence Results

For the purpose of simplification, we will assume the following: ψ

ξ (t, a) = (ψ(t) − ψ(a))1−ξ , α

− ψ(s)) k −1 ¯ αk,ψ (t, s) = (ψ(t)  . kk (α)







 Let C −λ, λ˜ , R , C = C([a − λ, a] , R) and C˜ = C b, b + λ˜ , R be the spaces endowed, respectively, with the norms 

 x ˜ = sup |x(t)| : t ∈ −λ, λ˜ , −λ,λ

xC = sup{|x(t)| : t ∈ [a − λ, a]}, and



 xC˜ = sup |x(t)| : t ∈ b, b + λ˜ .

Next, we consider the Banach space 

F = x : a − λ, b + λ˜ → R : x|[a−λ,a] ∈ C , x| with the norm

b,b+λ˜

∈ C˜ and x|(a,b] ∈ Cξ;ψ (J )



  xF = max xC , xC˜ , xCξ;ψ .

We consider the following fractional differential equation

H α,β;ψ k Da+ x



(t) = ϕ(t), t ∈ (a, b],

where 0 < α < k, 0 ≤ β ≤ 1, with the conditions



  k(1−ξ),k;ψ k(1−ξ),k;ψ ϑ1 Ja+ x (a + ) + ϑ2 Ja+ x (b) = ϑ3 ,

(4.5)

(4.6)

x(t) = (t), t ∈ [a − λ, a], λ > 0,

(4.7)



x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

(4.8)

β(k − α) + α , k > 0, ϑ1 , ϑ2 , ϑ3 ∈ R such that ϑ1 + ϑ2  = 0 and ϕ(·) ∈ C( J¯, R), k ˜ ∈ C˜.

(·) ∈ C and (·)

where ξ=

80

4 Fractional Differential Equations with Retardation …

The following theorem will be used in a result for the existence of a unique solution for the problem (4.5)–(4.8). Theorem 4.1 The function x satisfies (4.5)–(4.8) if and only if it satisfies

 ⎧ k(1−ξ)+α,k;ψ ⎪  ϑ − ϑ J ϕ (b)

3 2 ⎪ a+ ⎪ α,k;ψ ⎪ ⎪ + J ϕ (t), t ∈ (a, b], a+ ⎪ ⎪ ⎨ (ϑ1 + ϑ2 )k (kξ)ξψ (t, a) x(t) = ⎪

(t), t ∈ [a − λ, a], ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ (t), ˜ t ∈ b, b + λ˜ .

(4.9)

Proof For both directions of the proof, (4.7) and (4.8) are trivially satisfied. Assume that x satisfies the Eqs. (4.5)–(4.8). By applying the fractional integral operator α,k;ψ Ja+ (·) on both sides of the fractional Eq. (4.5), we have 



α,k;ψ α,β;ψ α,k;ψ Ja+ kH Da+ x (t) = Ja+ ϕ (t), and using Theorem 2.32, we get k(1−ξ),k;ψ

x(t) =

Ja+ ψ

x(a)

ξ (t, a)k (kξ)

 α,k;ψ + Ja+ ϕ (t).

(4.10)

k(1−ξ),k;ψ

Applying Ja+ (·) on both sides of (4.10), using Lemma 2.15, Lemma 2.18 and taking t = b, we have



 k(1−ξ),k;ψ k(1−ξ),k;ψ k(1−ξ)+α,k;ψ Ja+ x (b) = Ja+ x(a) + Ja+ ϕ (b). (4.11) Multiplying both sides of (4.11) by ϑ2 , we get



  k(1−ξ),k;ψ k(1−ξ),k;ψ k(1−ξ)+α,k;ψ x (b) = ϑ2 Ja+ x(a) + ϑ2 Ja+ ϕ (b). ϑ2 Ja+ Using condition (4.6), we obtain



  k(1−ξ),k;ψ k(1−ξ),k;ψ x (b) = ϑ3 − ϑ1 Ja+ x (a + ). ϑ2 Ja+ Thus



  k(1−ξ),k;ψ k(1−ξ),k;ψ k(1−ξ)+α,k;ψ ϑ3 − ϑ1 Ja+ x (a + ) = ϑ2 Ja+ x(a) + ϑ2 Ja+ ϕ (b).

4.2

On k-Generalized ψ-Hilfer Boundary Value …

81

Then

 k(1−ξ),k;ψ Ja+ x (a + ) =

ϑ3 ϑ2 k(1−ξ)+α,k;ψ  − Ja+ ϕ (b). ϑ1 + ϑ2 ϑ1 + ϑ2

(4.12)

Substituting (4.12) into (4.10), we obtain (4.9). For the converse, let us now prove that if x satisfies Eq. (4.9), then it satisfies (4.5)– α,β;ψ (4.8). Applying the fractional derivative operator kH Da+ (·) on both sides of the fractional Eq. (4.9), then we get

 ⎛ ⎞ k(1−ξ)+α,k;ψ 

ϕ (b) ϑ3 − ϑ2 Ja+ H α,β;ψ H α,β;ψ ⎝ ⎠ k Da+ x (t) = k Da+ ψ (ϑ1 + ϑ2 )k (kξ)ξ (t, a) 

α,β;ψ α,k;ψ + kH Da+ Ja+ ϕ (t). Using Lemma 2.28 and Lemma 2.26, we obtain Eq. (4.5). Now we apply the operator k(1−ξ),k;ψ Ja+ (·) to Eq. (4.9), to obtain

 ⎛ ⎞ k(1−ξ)+α,k;ψ

 ϕ (b) ϑ3 − ϑ2 Ja+ k(1−ξ),k;ψ k(1−ξ),k;ψ ⎝ ⎠ Ja+ x (t) = Ja+ ψ (ϑ1 + ϑ2 )k (kξ)ξ (t, a) 

k(1−ξ),k;ψ α,k;ψ Ja+ ϕ (t). + Ja+ Now, using Lemmas 2.15 and 2.18, we get ϑ3 ϑ2 k(1−ξ)+α,k;ψ  − Ja+ ϕ (b) ϑ1 + ϑ2 ϑ1 + ϑ2 

k(1−ξ)+α,k;ψ ϕ (t). + Ja+



k(1−ξ),k;ψ Ja+ x (t) =

(4.13)

Using Theorem 2.23 with t → a, we obtain

 k(1−ξ),k;ψ Ja+ x (a + ) =

ϑ3 ϑ2 k(1−ξ)+α,k;ψ  − Ja+ ϕ (b). ϑ1 + ϑ2 ϑ1 + ϑ2

(4.14)

Next, taking t = b in (4.13), we have

k(1−ξ),k;ψ

Ja+

ϑ3 ϑ2 k(1−ξ)+α,k;ψ  − Ja+ ϕ (b) ϑ1 + ϑ2 ϑ1 + ϑ2 

k(1−ξ)+α,k;ψ ϕ (b). + Ja+

 x (b) =

From (4.14) and (4.15), we obtain (4.6). This completes the proof. As a consequence of Theorem 4.1, we have the following result.

(4.15) 

82

4 Fractional Differential Equations with Retardation …

β(k − α) + α Lemma 4.2 Let ξ = where 0 < α < k and 0 ≤ β ≤ 1, let f : J¯ × k



 ˜ ∈ C˜. Then C −λ, λ˜ , R × R → R be a continuous function, and (·) ∈ C and (·) x ∈ F satisfies the problem (4.1)–(4.4) if and only if x is the fixed point of the operator T : F → F defined by

 ⎧ k(1−ξ)+α,k;ψ ⎪  ϑ − ϑ J ϕ (b)

3 2 ⎪ a+ ⎪ α,k;ψ ⎪ ⎪ + J ϕ (t), t ∈ (a, b], a+ ⎪ ⎪ ⎨ (ϑ1 + ϑ2 )k (kξ)ξψ (t, a) (4.16) (T x) (t) = ⎪

(t), t ∈ [a − λ, a], ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ (t), ˜ t ∈ b, b + λ˜ , where ϕ be a function satisfying the functional equation ϕ(t) = f (t, xt (·), ϕ(t)). By Theorem 2.14, we have T u ∈ F. We are now in a position to state and prove our existence result for the problem (4.1)–(4.4) based on Banach’s fixed point theorem [19]. Theorem 4.3 Assume that the following conditions are met.



 (4.3.1) The function f : J¯ × C −λ, λ˜ , R × R → R is continuous. (4.3.2) There exist constants ζ1 > 0 and 0 < ζ2 < 1 such that | f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ ζ1 x1 − x2 

−λ,λ˜

+ ζ2 |y1 − y2 |

 −λ, λ˜ , R , y1 , y2 ∈ R and t ∈ (a, b]. (4.3.3) There exist functions q1 , q2 , q3 ∈ C( J¯, R+ ) with for any x1 , x2 ∈ C



q1∗ = sup q1 (t), q2∗ = sup q2 (t), q3∗ = sup q3 (t) < 1, t∈J

t∈J

t∈J

such that | f (t, x, y)| ≤ q1 (t) + q2 (t)x

−λ,λ˜

for any x ∈ C



 −λ, λ˜ , R , y ∈ R and t ∈ (a, b].

+ q3 (t)|y|,

On k-Generalized ψ-Hilfer Boundary Value …

4.2

83

If 

α

L=

ζ1 (ψ(b) − ψ(a))1−ξ+ k 1 − ζ2

|ϑ2 | 1 + |ϑ1 + ϑ2 |k (kξ)k (2k − kξ + α) k (α + k)

< 1,



(4.17)

then the problem (4.1)–(4.4) has a unique solution in F. Proof We show that the operator T defined in (4.16) has a unique fixed point in F. Let x, y ∈ F. Then for any t ∈ [a − λ, a] ∪ b, b + λ˜ , we have |T x(t) − T y(t)| = 0. Thus T x − T yC = T x − T yC˜ = 0.

(4.18)

Further, for t ∈ (a, b] we have |T x(t) − T y(t)| ≤



k(1−ξ)+α,k;ψ |ϕ1 (s) − ϕ2 (s)| (b) |ϑ2 | Ja+ ψ

|ϑ1 + ϑ2 |k (kξ)ξ (t, a) 

α,k;ψ + Ja+ |ϕ1 (s) − ϕ2 (s)| (t),

where ϕ1 and ϕ1 be functions satisfying the functional equations ϕ1 (t) = f (t, xt (·), ϕ1 (t)), ϕ2 (t) = f (t, yt (·), ϕ2 (t)). By condition (4.3.2), we have |ϕ1 (t) − ϕ2 (t)| = | f (t, xt , ϕ1 (t)) − f (t, yt , ϕ2 (t))| ≤ ζ1 xt − yt  ˜ + ζ2 |ϕ1 (t) − ϕ2 (t)|. −λ,λ

Then, |ϕ1 (t) − ϕ2 (t)| ≤

ζ1 xt − yt  ˜ . −λ,λ 1 − ζ2

Therefore, for each t ∈ (a, b]  k(1−ξ)+α,k;ψ xs − ys  ζ1 |ϑ2 | Ja+ |T x(t) − T y(t)| ≤

ψ

−λ,λ˜

(1 − ζ2 )|ϑ1 + ϑ2 |k (kξ)ξ (t, a)

 (b)

84

4 Fractional Differential Equations with Retardation …

  ζ1 α,k;ψ

(t) + Ja+ xs − ys  −λ,λ˜ (1 − ζ2 )

 ⎤ ⎡ k(1−ξ)+α,k;ψ  (1) (b)

ζ1 x − yF ⎣ |ϑ2 | Ja+ α,k;ψ ≤ + Ja+ (1) (t)⎦ . ψ 1 − ζ2 |ϑ1 + ϑ2 |k (kξ) (t, a) ξ

By Lemma 2.18, we have 

α

ζ1 |ϑ2 | (ψ(b) − ψ(a))1−ξ+ k

|T x(t) − T y(t)| ≤

ψ

(1 − ζ2 )|ϑ1 + ϑ2 |k (kξ)k (k(1 − ξ) + α + k)ξ (t, a) α  ζ1 (ψ(t) − ψ(a)) k x − yF . + (1 − ζ2 )k (α + k)

Hence     ψ  (t, a) T x(t) − T y(t)) ( ξ ≤

α

ζ1 |ϑ2 | (ψ(b) − ψ(a))1−ξ+ k (1 − ζ2 )|ϑ1 + ϑ2 |k (kξ)k (2k − kξ + α) α  ζ1 (ψ(t) − ψ(a))1−ξ+ k x − yF , + (1 − ζ2 )k (α + k)

which implies that α  |ϑ2 | ζ1 (ψ(b) − ψ(a))1−ξ+ k 1 − ζ2 |ϑ1 + ϑ2 |k (kξ)k (2k − kξ + α)  1 x − yF . + k (α + k)

T x − T yCξ;ψ ≤

Thus T x − T yCξ;ψ ≤ Lx − yF .

(4.19)

By (4.18) and (4.19), we obtain   T x − T yF = max T x − T yC , T x − T yC˜ , T x − T yCξ;ψ ≤ Lx − yF . By (4.17), the operator T is a contraction on F. Hence, by Banach’s contraction principle, T has a unique fixed point x ∈ F, which is a solution to our problem (4.1)–(4.4).  Our next existence result for the problem (4.1)–(4.4) is based on Schauder’s fixed point theorem [19].

4.2

On k-Generalized ψ-Hilfer Boundary Value …

85

Theorem 4.4 Assume that the hypotheses (4.3.1) and (4.3.3) hold. If   q2∗ 1 |ϑ2 | = + 1 − q3∗ k (α + k) |ϑ1 + ϑ2 |k (kξ)k (2k − kξ + α) α

× (ψ(b) − ψ(a))1−ξ+ k < 1,

(4.20)

then the problem (4.1)–(4.4) has at least one solution in F. Proof In several steps, we will use Schauder’s fixed point theorem to prove that the operator T defined in (4.16) has a fixed point. Step 1: The operator T is continuous.

Let {xn } be a sequence such that xn −→ x in F. For each t ∈ [a − λ, a] ∪ b, b + λ˜ , we have |T xn (t) − T x(t)| = 0. And for t ∈ (a, b], we have  |ϑ2 |(ψ(t) − ψ(a))ξ−1 k(1−ξ)+α,k;ψ |ϕn (s) − ϕ(s)| (b) Ja+ |ϑ1 + ϑ2 |k (kξ) 

α,k;ψ + Ja+ |ϕn (s) − ϕ(s)| (t),

|T xn (t) − T x(t)| ≤

where ϕ1 and ϕ1 be functions satisfying the functional equations ϕ(t) = f (t, xt (·), ϕ(t)), ϕn (t) = f (t, xnt (·), ϕn (t)). Since xn → x, then we get ϕn (t) → ϕ(t) as n → ∞ for each t ∈ (a, b], and since f is continuous, then we have T xn − T xF → 0 as n → ∞. Step 2: T (B M ) ⊂ B M . Let M a positive constant such that   q1∗  |ϑ3 | ˜ C˜ . M ≥ max ,  C ,   + |ϑ1 + ϑ2 |k (kξ) (1 − ) q2∗ (1 − ) We define the following bounded closed set B M = {x ∈ F : xF ≤ M} .

86

4 Fractional Differential Equations with Retardation …

For each t ∈ [a − λ, a], we have |T x(t)| ≤  C ,

and for each t ∈ b, b + λ˜ , we have ˜ C˜ . |T x(t)| ≤   Further, for each t ∈ (a, b], (4.16) implies that 

(ψ(t) − ψ(a))ξ−1 k(1−ξ)+α,k;ψ | f (s, xs , ϕ(s))| (b) |ϑ3 | + |ϑ2 | Ja+ |ϑ + ϑ2 |k (kξ) 

1 α,k;ψ (4.21) + Ja+ | f (s, xs , ϕ(s))| (t).

|T x(t)| ≤

By the hypothesis (4.3.3), for t ∈ (a, b], we have |ϕ(t)| = | f (t, xt , ϕ(t))| ≤ q1 (t) + q2 (t)xt 

−λ,λ˜

+ q3 (t)|ϕ(t)|,

which implies that |ϕ(t)| ≤ q1∗ + q2∗ M + q3∗ |ϕ(t)|, then |ϕ(t)| ≤

q1∗ + q2∗ M := . 1 − q3∗

Thus for t ∈ (a, b], from (4.21) we get

 |ϑ3 | |ϑ2 | k(1−ξ)+α,k;ψ Ja+ (1) (b) + |ϑ1 + ϑ2 |k (kξ) |ϑ1 + ϑ2 |k (kξ)

 ψ α,k;ψ + ξ (t, a) Ja+ (1) (t).

ψ

|ξ (t, a)T x(t)| ≤

By Lemma 2.18, we have α

ψ

|ϑ2 | (ψ(b) − ψ(a))1−ξ+ k |ϑ3 | + |ϑ1 + ϑ2 |k (kξ) |ϑ1 + ϑ2 |k (kξ)k (2k − kξ + α) α ! (ψ(t) − ψ(a))1−ξ+ k . + k (α + k)

|ξ (t, a)T x(t)| ≤

4.2

On k-Generalized ψ-Hilfer Boundary Value …

Thus α |ϑ3 | +  (ψ(b) − ψ(a))1−ξ+ k |ϑ1 + ϑ2 |k (kξ)  |ϑ2 | ≤ M. + |ϑ1 + ϑ2 |k (kξ)k (2k − kξ + α)

Then, for each t ∈ a − λ, b + λ˜ we obtain

ψ

|ξ (t, a)T x(t)| ≤

87



1 k (α + k)

T xF ≤ M. Step 3: T (B M ) is relatively compact. Let τ1 , τ2 ∈ (a, b], τ1 < τ2 and let x ∈ B M . Then     ψ ψ ξ (τ1 , a)T x(τ1 ) − ξ (τ2 , a)T x(τ2 )



 ψ α,k;ψ ψ α,k;ψ ≤ ξ (τ1 , a) Ja+ ϕ(s) (τ1 ) − ξ (τ2 , a) Ja+ ϕ(s) (τ2 )





ψ α,k;ψ ψ α,k;ψ ≤  ξ (τ1 , a) Ja+ (1) (τ1 ) − ξ (τ2 , a) Ja+ (1) (τ2 ) . By Lemma 2.18, we get    ψ  ψ ξ (τ1 , a)T x(τ1 ) − ξ (τ2 , a)T x(τ2 )

α α  ≤ (ψ(τ1 ) − ψ(a))1−ξ+ k − (ψ(τ2 ) − ψ(a))1−ξ+ k . k (α + k) As τ1 → τ2 , the right-hand side of the above inequality tends to zero. The equicontinuity for the other cases is obvious, thus we omit the details. From Step 1 to Step 3, along with the Arzela-Ascoli theorem, we conclude that T : F → F continuous and compact. As a consequence of Schauder’s fixed point theorem, we deduce that T has a fixed point which is a solution of the problem (4.1)–(4.4). 

4.2.2

Examples

In this section, we investigate specific cases of our problem (4.1)–(4.4), with J = [1, 3], ξ = k1 (β(k − α) + α) and   x1 1 x2 1 − + , 33 + 31e3−t 2 + x1 1 + |x2 |



 where t ∈ J , x1 ∈ C −λ, λ˜ , R and x2 ∈ R. f (t, x1 , x2 ) =

88

4 Fractional Differential Equations with Retardation …

Example 4.5 Taking β → 0, α = 21 , k = 1, ψ(t) = ln(t), ϑ1 = 1, ϑ2 = 2, ϑ3 = 3, λ = λ˜ = 21 and ξ = 21 , we obtain a boundary value problem which is a particular case of problem (4.1)–(4.4) with Hadamard fractional derivative, given by         1 1 1 H 2 ,0;ψ HD 2 HD 2 D x (t) = D x (t) = f t, x (·), D x (t) , t ∈ (1, 3], t 1 1+ 1+ 1+ (4.22)  1   1  2 ,1;ψ 2 ,1;ψ J1+ x (1) + 2 J1+ x (3) = 3.

(4.23)

 1 ,1 , 2   7 . x(t) = (t), ˜ t ∈ 3, 2 

x(t) = (t), t ∈

We have

(4.24)

(4.25)

  " Cξ;ψ (J ) = C 1 ;ψ (J ) = x : (1, 3] → R : ln(t)x ∈ C( J¯, R) . 2

Then

  1 7

˜ → R : x| 1 ∈ C , x| 7 ∈ C and x|(1,3] ∈ C 1 ;ψ (J ) . F= x : , 3, 2 2 2 2 2 ,1 



Since the function f is continuous, then condition (4.3.1) is satisfied. $  # For each x1 ∈ C − 21 , 21 , R , x2 ∈ R and t ∈ J , we have | f (t, x1 , x2 )| ≤

1 33 + 31e3−t



2 + x1 

−λ,λ˜

 + |x2 | .

Then, condition (4.3.3) is satisfied q1 (t) =

2 1 , q2 (t) = q3 (t) = , 33 + 31e3−t 33 + 31e3−t

and q1∗ = We have =

2 1 , q ∗ = q3∗ = . 64 2 64

  2 ln(3) 2 ≈ 0.0262360046401739 < 1. √ + √ 63 π 3 π

Then, by Theorem 4.4, we deduce that the problem (4.22)–(4.25) has at least one solution in F. $  # Further, for each x1 , y1 ∈ C − 21 , 21 , R , x2 , y2 ∈ R and t ∈ J , we have 1 | f (t, x1 , x2 ) − f (t, y1 , y2 )| ≤ 33 + 31e3−t

 x1 −

y1  ˜ −λ,λ

 + |x2 − y2 | ,

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

89

1 and then, condition (4.3.2) is satisfied with ζ1 = ζ2 = . And since L = , then all the 64 assumptions of Theorem 4.3 are satisfied. Consequently, the problem (4.22)–(4.25) has a unique solution in F. Example 4.6 Taking β → 21 , α = 21 , k = 1, ψ(t) = t, ϑ1 = 1, ϑ2 = 0, ϑ3 = 0, λ = λ˜ = 15 and ξ = 43 , we obtain an initial value problem which is a particular case of problem (4.1)– (4.4) with Hilfer fractional derivative, given by         1 1 1 1 1 1 H 2 , 2 ;ψ H 2,2 H 2,2 x (t) = D1+ x (t) = f t, xt (·), D1+ x (t) , t ∈ (1, 3], (4.26) 1 D1+ 

1

,1;ψ

4 J1+

 x (1) = 0.

 4 ,1 , 5   16 . x(t) = 0, t ∈ 3, 5

(4.27)



x(t) = 0, t ∈

(4.28)

(4.29)

We have   1 Cξ;ψ (J ) = C 3 ;ψ (J ) = x : (1, 3] → R : (t − 1) 4 x ∈ C( J¯, R) , 4

and then

    4 16 → R : x| 4 ∈ C , x| 16 ∈ C˜ and x|(1,3] ∈ C 3 ;ψ (J ) . F= x : , 3, 5 4 5 5 5 ,1

Also

7

24 L = √ ≈ 0.0301222221161139 < 1. 63 π As all the conditions of Theorem 4.3 are satisfied, then the problem (4.26)–(4.29) has a unique solution in F.

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem with Retarded and Advanced Arguments

In this section, we consider the terminal value problem with nonlinear implicit k-generalize ψ-Hilfer type fractional differential equation involving both retarded and advanced arguments:

H α,β;ψ k Da+ x





  α,β;ψ (t) = f t, xt (·), kH Da+ x (t) , t ∈ (a, b],

(4.30)

90

4 Fractional Differential Equations with Retardation …

x(b) =

m %

ϑi x(i ),

(4.31)

i=1

α,β;ψ

x(t) = (t), t ∈ [a − λ, a], λ > 0,

(4.32)



x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

(4.33)

k(1−ξ),k;ψ

are the k-generalize ψ-Hilfer fractional derivative of order where kH Da+ , Ja+ α ∈ (0, k) and type β ∈ [0, 1] defined in Sect. 2.1, and k-generalize ψ-fractional integral of order k(1 − ξ) defined in [18] respectively, where ξ = k1 (β(k − α) + α), k > 0, f :



 [a, b] × C −λ, λ˜ , R × R −→ R is a given appropriate function specified latter, ϑi , i = 1, . . . , m, are real numbers and i , i = 1, . . . , m, are pre-fixed

points satisfying a < 1 ≤ . . . ≤ m < b. For each function x defined on a − λ, b + λ˜ and for any t ∈ (a, b], we denote by xt the element defined by

xt (τ ) = x(t + τ ), τ ∈ −λ, λ˜ .

4.3.1

Existence Results

We consider the following fractional differential equation:

 H α,β;ψ k Da+ x (t) = ϕ(t), t ∈ (a, b],

(4.34)

where 0 < α < k, 0 ≤ β ≤ 1, with the conditions x(b) =

m %

ϑi x(i ),

(4.35)

i=1

x(t) = (t), t ∈ [a − λ, a], λ > 0,

(4.36)



x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

(4.37)

β(k − α) + α where ξ = , k > 0, ϑi , i = 1, . . . , m, are real numbers, ϑm+1 = −1 and k i , i = 1, . . . , m + 1, are pre-fixed points satisfying a < 1 ≤ . . . ≤ m < b = m+1 such m+1 % ϑi  = 0 and where ϕ(·) ∈ C( J¯, R), (·) ∈ C and (·) ˜ ∈ C˜. that ψ i=1 ξ (i , a) The following theorem shows that the problem (4.34)–(4.37) have a unique solution.

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

Theorem 4.7 x satisfies (4.34)–(4.37) if and only if it satisfies ⎧ m+1 % α,k;ψ  ⎪ ⎪ ⎪ ⎪ − ϑi Ja+ ϕ (i ) ⎪ ⎪ 

⎪ ⎪ i=1 α,k;ψ ⎪ J ϕ (t), t ∈ (a, b], + ⎪ a+ ⎪ m+1 ⎪ % ϑi ξψ (t, a) ⎨ x(t) = ψ ⎪ ⎪ i=1 ξ (i , a) ⎪ ⎪ ⎪ ⎪

(t), t ∈ [a − λ, a], ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ (t), ˜ t ∈ b, b + λ˜ .

91

(4.38)

Proof Assume that x satisfies the Eqs. (4.34)–(4.37), by applying the fractional integral α,k;ψ operator Ja+ (·) on both sides of the fractional Eq. (4.34), we have 



α,k;ψ α,β;ψ α,k;ψ Ja+ kH Da+ x (t) = Ja+ ϕ (t), using Theorem 2.32, we get k(1−ξ),k;ψ

x(t) =

Ja+

x(a)

ψ

ξ (t, a)k (kξ)

 α,k;ψ + Ja+ ϕ (t).

(4.39)

Next, we substitute t = i into (4.39), then we multiply ϑi to both sides, we obtain k(1−ξ),k;ψ

ϑi x(i ) =

ϑi Ja+

x(a)

ψ ξ (i , a)k (kξ)

 α,k;ψ + ϑi Ja+ ϕ (i ).

Then by using condition (4.35) and (4.39) with t = b, we have k(1−ξ),k;ψ

Ja+

x(a)

m %

ϑi

i=1

ξ (i , a)k (kξ)

ψ

+

k(1−ξ),k;ψ

 J x(a) α,k;ψ ϑi Ja+ ϕ (i ) = a+ ψ ξ (b, a)k (kξ) i=1

 α,k;ψ + Ja+ ϕ (b),

m %

which implies m

 

% α,k;ψ α,k;ψ ϑi Ja+ ϕ (i ) − Ja+ ϕ (b) + k(1−ξ),k;ψ

Ja+

x(a) =

i=1

1 ψ ξ (b, a)k (kξ)

−

m %

ϑi ψ i=1 ξ (i , a)k (kξ)

92

4 Fractional Differential Equations with Retardation … m+1 %

=−

 α,k;ψ ϑi Ja+ ϕ (i )

i=1 m+1 %

ϑi

i=1

ξ (i , a)k (kξ)

.

(4.40)

ψ

Substituting (4.40) into (4.39), we obtain (4.38). Let us now prove that if x satisfies Eq. (4.38), then it satisfies (4.34)–(4.37). Applying α,β;ψ the fractional derivative operator kH Da+ (·) on both sides of the fractional Eq. (4.38), then we get ⎡ ⎤ m+1 % α,k;ψ  ⎢− ϑi Ja+ ϕ (i ) ⎥ ⎢ ⎥

  ⎢ ⎥ i=1 H α,β;ψ H α,β;ψ ⎢ ⎥ + H Dα,β;ψ J α,k;ψ ϕ (t). D x (t) = D a+ a+ a+ a+ k k k ⎢ ⎥ m+1 % ϑi ξψ (t, a) ⎢ ⎥ ⎣ ⎦ ψ i=1 ξ (i , a) Using the Lemma 2.28 and Lemma 2.26, we obtain Eq. (4.34). Now, taking t = b of Eq. (4.38), we have −

m+1 %

 α,k;ψ ϑi Ja+ ϕ (i )

i=1 m+1 %

x(b) =

ψ ϑi ξ (b, a) ψ i=1 ξ (i , a)



α,k;ψ + Ja+ ϕ (b).

(4.41)

Substituting t = i into (4.38), we get −

m+1 %

 α,k;ψ ϑi Ja+ ϕ (i )

i=1

x(i ) =

ψ

ξ (i , a)

m+1 %

ϑi ψ i=1 ξ (i , a)



α,k;ψ + Ja+ ϕ (i ).

Then, we have

m % i=1

− ϑi x(i ) =

m+1 %

 α,k;ψ ϑi Ja+ ϕ (i )

i=1 m+1 %

ϑi ψ

i=1

ξ (i , a)

m %

ϑi

i=1

ξ (i , a)

ψ

+

m % i=1

 α,k;ψ ϑi Ja+ ϕ (i ),

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

4.3

93

thus,

m %

m



  % α,k;ψ α,k;ψ Ja+ ϕ (b) − ϑi Ja+ ϕ (i ) i=1

ϑi x(i ) =

−

i=1

ψ ξ (b, a)

1 m %

ϑi

i=1

ξ (i , a)

+1

 α,k;ψ ϑi Ja+ ϕ (i )

i=1

ψ

m %

 α,k;ψ Ja+ ϕ (b) −

 α,k;ψ ϑi Ja+ ϕ (i )

i=1 ψ

ξ (b, a) =

+

m %

− ψ ξ (b, a)

m %

ϑi ψ i=1 ξ (i , a)

1 m %

ϑi

i=1

ξ (i , a)

+1

ψ

⎞

 α,k;ψ ϑi Ja+ ϕ (i ) ⎟ ⎜ ⎟ ⎜

 ⎟ ⎜ α,k;ψ i=1 = ⎜ Ja+ ϕ (b) − ⎟ m ⎟ ⎜ % ϑ i ψ ⎠ ⎝ ξ (b, a) ψ  ( , a) i=1 ξ i ⎛ ⎞ m % ϑi ⎜  ψ (b, a) ⎟ ⎜ ξ ⎟ ψ ξ (i , a) ⎟ ⎜ i=1 ⎜ ⎟ ×⎜ ⎟ m+1 % ϑi ξψ (b, a) ⎜ ⎟ ⎝ ⎠ ψ  ( , a) i i=1 ξ ⎛

m %

m

 % α,k;ψ ψ Ja+ ϕ (b)ξ (b, a) i=1

=

ϑi ψ

ξ (i , a)

−

m %

 α,k;ψ ϑi Ja+ ϕ (i )

i=1

ψ

m+1 %

ϑi ξ (b, a)

i=1

ξ (i , a)

.

ψ

Then,

m % i=1

− ϑi x(i ) =

m+1 %

 α,k;ψ ϑi Ja+ ϕ (i )

i=1 m+1 %

ψ

ϑi ξ (b, a) ψ

i=1

ξ (i , a)



α,k;ψ + Ja+ ϕ (b).

(4.42)

94

4 Fractional Differential Equations with Retardation …

From (4.41) and (4.42), we find that x(b) =

m %

ϑi x(i ),

i=1



which shows that the condition (4.35) is satisfied. As a consequence of Theorem 4.7, we have the following result

β(k − α) + α Lemma 4.8 Let ξ = where 0 < α < k and 0 ≤ β ≤ 1, f : J¯ × k



 ˜ ∈ C˜. Then, C −λ, λ˜ , R × R → R be a continuous function, (·) ∈ C and (·) x ∈ F satisfies the problem (4.30)–(4.33) if and only if x is the fixed point of the operator T : F → F defined by ⎧ m+1 % α,k;ψ  ⎪ ⎪ ⎪ ⎪ − ϑi Ja+ ϕ (i ) ⎪ ⎪ 

⎪ ⎪ i=1 α,k;ψ ⎪ J ϕ (t), t ∈ (a, b], + ⎪ a+ ⎪ m+1 ⎪ % ϑi ξψ (t, a) ⎨ (4.43) (T x) (t) = ψ ⎪ ⎪ i=1 ξ (i , a) ⎪ ⎪ ⎪ ⎪

(t), t ∈ [a − λ, a], ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ (t), ˜ t ∈ b, b + λ˜ , where ϕ be a function satisfying the functional equation ϕ(t) = f (t, xt (·), ϕ(t)), and ϑm+1 = −1, m+1 = b. By Theorem 2.14, we have T x ∈ F. We are now in a position to state and prove our existence result for the problem (4.30)– (4.33) based on Banach fixed point theorem [19]. Theorem 4.9 Assume that the hypothesis that follow hold.



 (4.9.1) The function f : J¯ × C −λ, λ˜ , R × R → R is continuous. (4.9.2) There exist constants ζ1 > 0 and 0 < ζ2 < 1 such that | f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ ζ1 x1 − x2 

−λ,λ˜

+ ζ2 |y1 − y2 |

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

95

 −λ, λ˜ , R , y1 , y2 ∈ R and t ∈ (a, b]. (4.9.3) There exist functions q1 , q2 , q3 ∈ C( J¯, R+ ) with for any x1 , x2 ∈ C



q1∗ = sup q1 (t), q2∗ = sup q2 (t), q3∗ = sup q3 (t) < 1, t∈J

t∈J

t∈J

such that | f (t, x, y)| ≤ q1 (t) + q2 (t)x

−λ,λ˜

for any x ∈ C



+ q3 (t)|y|

 −λ, λ˜ , R , y ∈ R and t ∈ (a, b].

If

α

2ζ1 (ψ(b) − ψ(a))1−ξ+ k L= < 1, k (α + k)(1 − ζ2 )

(4.44)

then the problem (4.30)–(4.33) has a unique solution in F. Proof We show that the operator T defined in (4.43) has a unique fixed point in F. Let x, y ∈ F. Then for any t ∈ [a − λ, a] ∪ b, b + λ˜ , we have |T x(t) − T y(t)| = 0. Thus T x − T yC = T x − T yC˜ = 0. Further, for t ∈ (a, b] we have m+1 %

|T x(t) − T y(t)| ≤



α,k;ψ |ϑi | Ja+ |ϕ1 (s) − ϕ2 (s)| (i )

i=1 m+1 %

ψ

|ϑi |ξ (t, a) ψ

ξ (i , a) 

α,k;ψ + Ja+ |ϕ1 (s) − ϕ2 (s)| (t), i=1

where ϕ1 and ϕ1 be functions satisfying the functional equations ϕ1 (t) = f (t, xt (·), ϕ1 (t)), ϕ2 (t) = f (t, yt (·), ϕ2 (t)).

(4.45)

96

4 Fractional Differential Equations with Retardation …

By condition (4.9.2), we have |ϕ1 (t) − ϕ2 (t)| = | f (t, xt , ϕ1 (t)) − f (t, yt , ϕ2 (t))| ≤ ζ1 xt − yt  ˜ + ζ2 |ϕ1 (t) − ϕ2 (t)|. −λ,λ

Then, |ϕ1 (t) − ϕ2 (t)| ≤

ζ1 xt − yt  ˜ . −λ,λ 1 − ζ2

Therefore, for each t ∈ (a, b] ζ1 |T x(t) − T y(t)| ≤

m+1 %

 α,k;ψ |ϑi | Ja+ xs − ys 

−λ,λ˜

i=1

(1 − ζ2 )

 (i )

ψ

m+1 %

|ϑi |ξ (t, a)

i=1

ξ (i , a)

ψ

  ζ1 α,k;ψ Ja+ xs − ys  ˜ (t) −λ,λ (1 − ζ2 ) ⎡ ⎤ m+1

 % α,k;ψ ⎢ ⎥ |ϑi | Ja+ (1) (i ) ⎢  ⎥

⎥ ζ1 x − yF ⎢ α,k;ψ ⎢ i=1 ≤ + Ja+ (1) (t)⎥ ⎢ ⎥. ψ m+1 1 − ζ2 ⎢ % |ϑi |ξ (t, a) ⎥ ⎣ ⎦ ψ  ( , a) i=1 ξ i +

By Lemma 2.18, we have |T x(t) − T y(t)| ⎡ ζ1 x − yF ≤ 1 − ζ2

m+1 %

⎤ α k

⎢ ⎥ |ϑi | (ψ(i ) − ψ(a)) ⎢ α ⎥ ⎢ i=1 k ⎥ − ψ(a)) (ψ(t) ⎢ ⎥. + ⎢ m+1 k (α + k) ⎥ % |ϑi |ξψ (t, a) ⎢ ⎥ ⎣ k (α + k) ⎦ ψ  ( , a) i i=1 ξ

Hence    ψ  ξ (t, a) (T x(t) − T y(t)) ⎡ m+1 ⎤ % α |ϑi | (ψ(i ) − ψ(a)) k ⎢ ⎥ α ζ1 x − yF ⎢ (ψ(t) − ψ(a))1−ξ+ k ⎥ ⎢ i=1 ⎥ ≤ + ⎢ ⎥, m+1 ⎥ 1 − ζ2 ⎢  (α + k) % k |ϑi | ⎣  (α + k) ⎦ k ψ i=1 ξ (i , a)

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

97

which implies that α

T x − T yCξ;ψ ≤

2ζ1 (ψ(b) − ψ(a))1−ξ+ k x − yF . k (α + k)(1 − ζ2 )

Thus T x − T yCξ;ψ ≤ Lx − yF .

(4.46)

By (4.45) and (4.46), we obtain T x − T yF ≤ Lx − yF . By (4.44), the operator T is a contraction on F. Hence, by Banach contraction principle, T has a unique fixed point x ∈ F, which is a solution to our problem (4.30)–(4.33).  Our next existence result for the problem (4.30)–(4.33) is based on Schauder fixed point theorem [19]. Theorem 4.10 Assume that the conditions (4.9.1) and (4.9.3) hold. If α

=

2q2∗ (ψ(b) − ψ(a))1−ξ+ k < 1, (1 − q3∗ )k (α + k)

(4.47)

then the problem (4.30)–(4.33) has at least one solution in F. Proof In several steps, we will use Schauder’s fixed point theorem to prove that the operator T defined in (4.43) has a fixed point. Step 1: The operator T is continuous.

Let {xn } be a sequence such that xn −→ x in F. For each t ∈ [a − λ, a] ∪ b, b + λ˜ , we have |T xn (t) − T x(t)| = 0. And for t ∈ (a, b], we have m+1 %

|T x(t) − T y(t)| ≤



α,k;ψ |ϑi | Ja+ |ϕn (s) − ϕ(s)| (i )

i=1 m+1 %

ψ

|ϑi |ξ (t, a) ψ

i=1 ξ (i , a) 

α,k;ψ + Ja+ |ϕn (s) − ϕ(s)| (t)

98

4 Fractional Differential Equations with Retardation …

where ϕ1 and ϕ1 be functions satisfying the functional equations ϕ(t) = f (t, xt (·), ϕ(t)), ϕn (t) = f (t, xnt (·), ϕn (t)). Since xn → x, then we get ϕn (t) → ϕ(t) as n → ∞ for each t ∈ (a, b], and since f is continuous, then we have T xn − T xF → 0 as n → ∞. Step 2: T (B M ) ⊂ B M . Let M a positive constant such that  M ≥ max

 q1∗  , 

 ˜ ,   C C˜ . q2∗ (1 − )

We define the following bounded closed set B M = {x ∈ F : xF ≤ M} . For each t ∈ [a − λ, a], we have |T x(t)| ≤  C ,

and for each t ∈ b, b + λ˜ , we have ˜ C˜ . |T x(t)| ≤   Further, for each t ∈ (a, b], (4.43) implies that m+1 %

|T x(t)| ≤



α,k;ψ |ϑi | Ja+ | f (s, xs , ϕ(s))| (i )

i=1 m+1 % |ϑi |ξψ (t, a) ψ i=1 ξ (i , a)



α,k;ψ + Ja+ | f (s, xs , ϕ(s))| (t).

(4.48) By the hypothesis (4.9.3), for t ∈ (a, b], we have |ϕ(t)| = | f (t, xt , ϕ(t))| ≤ q1 (t) + q2 (t)xt 

−λ,λ˜

+ q3 (t)|ϕ(t)|,

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

99

which implies that |ϕ(t)| ≤ q1∗ + q2∗ M + q3∗ |ϕ(t)|, then |ϕ(t)| ≤

q1∗ + q2∗ M := . 1 − q3∗

Thus for t ∈ (a, b], from (4.48) we get  ψ

|ξ (t, a)T x(t)| ≤

m+1 %

 α,k;ψ |ϑi | Ja+ (1) (i )

i=1 m+1 %

|ϑi | ψ i=1 ξ (i , a)

 ψ α,k;ψ + ξ (t, a) Ja+ (1) (t).

By Lemma 2.18, we have ⎡ m+1 ⎤ % α |ϑi | (ψ(i ) − ψ(a)) k ⎢ ⎥ α ⎢ (ψ(t) − ψ(a))1−ξ+ k ⎥ ⎢ i=1 ⎥ ψ |ξ (t, a)T x(t)| ≤  ⎢ + ⎥. m+1 ⎢ ⎥  (α + k) % k |ϑ | i ⎣  (α + k) ⎦ k ψ  ( , a) i=1 ξ i Thus α

2 (ψ(b) − ψ(a))1−ξ+ k x(t)| ≤ ≤ M. k (α + k)

Then, for each t ∈ a − λ, b + λ˜ we obtain ψ |ξ (t, a)T

T xF ≤ M. Step 3: T (B M ) is relatively compact. Let τ1 , τ2 ∈ (a, b], τ1 < τ2 and let x ∈ B M . Then     ψ ψ ξ (τ1 , a)T x(τ1 ) − ξ (τ2 , a)T x(τ2 )



 ψ α,k;ψ ψ α,k;ψ ≤ ξ (τ1 , a) Ja+ ϕ(s) (τ1 ) − ξ (τ2 , a) Ja+ ϕ(s) (τ2 )





ψ α,k;ψ ψ α,k;ψ ≤  ξ (τ1 , a) Ja+ (1) (τ1 ) − ξ (τ2 , a) Ja+ (1) (τ2 ) .

100

4 Fractional Differential Equations with Retardation …

By Lemma 2.18, we get     ψ ψ ξ (τ1 , a)T x(τ1 ) − ξ (τ2 , a)T x(τ2 )

α α  ≤ (ψ(τ1 ) − ψ(a))1−ξ+ k − (ψ(τ2 ) − ψ(a))1−ξ+ k . k (α + k) As τ1 → τ2 , the right-hand side of the above inequality tends to zero. The equicontinuity for the other cases is obvious, thus we omit the details. From Step 1 to Step 3 with ArzelaAscoli theorem, we conclude that T : F → F continuous and compact. As a consequence of Schauder fixed point theorem, we deduce that T has a fixed point which is a solution of the problem (4.30)–(4.33).  Our third is based on Krasnoselskii fixed point theorem [19]. Theorem 4.11 Assume that the conditions (4.9.1) and (4.9.2) hold. If α

ζ1 (ψ(b) − ψ(a))1−ξ+ k < 1, k (α + k)(1 − ζ2 )

(4.49)

then the problem (4.30)–(4.33) has at least one solution in F. Proof Consider the set Bω = {x ∈ F : ||x||F ≤ ω}, where ω ≥ r1 + r2 , with α

(q ∗ + q2∗ ω) (ψ(b) − ψ(a))1−ξ+ k r1 := 1 , (1 − q3∗ )k (α + k) * α + (q1∗ + q2∗ ω) (ψ(b) − ψ(a))1−ξ+ k . ˜ C˜ , r2 := max  C ,   (1 − q3∗ )k (α + k) We define the operators N1 and N2 on Bω by ⎧ m+1 % α,k;ψ  ⎪ ⎪ ⎪ ⎪ − ϑi Ja+ ϕ (i ) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ , t ∈ (a, b], ⎪ ⎪ m+1 ⎪ % ϑi ξψ (t, a) ⎨ N1 x(t) = ψ ⎪ ⎪ i=1 ξ (i , a) ⎪ ⎪ ⎪ ⎪ 0, t ∈ [a − λ, a], ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ 0, t ∈ b, b + λ˜ ,

(4.50)

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

and

101

⎧ α,k;ψ  ⎪ Ja+ ϕ (t), t ∈ (a, b], ⎪ ⎪ ⎪ ⎪ ⎨ N2 x(t) = (t), t ∈ [a − λ, a], ⎪ ⎪ ⎪

⎪ ⎪ ⎩ (t), ˜ t ∈ b, b + λ˜ ,

(4.51)

where ϕ be a function satisfying the functional equation ϕ(t) = f (t, xt (·), ϕ(t)). Then the fractional integral Equation (4.43) can be written as operator equation

T x(t) = N1 x(t) + N2 x(t), x ∈ F. We shall use Krasnoselskii’s fixed point theorem to prove in several steps that the operator T defined in (4.43) has a fixed point. Step 1: We prove that N1 x + N2 y ∈ Bω for any x, y ∈ Bω . By condition (4.9.3) and from (4.43), for t ∈ (a, b], we have |ϕ(t)| = | f (t, xt , ϕ(t))| ≤ q1 (t) + q2 (t)xt 

−λ,λ˜

+ q3 (t)|ϕ(t)|,

which implies that |ϕ(t)| ≤ q1∗ + q2∗ ω + q3∗ |ϕ(t)|, then |ϕ(t)| ≤

q1∗ + q2∗ ω := A. 1 − q3∗

Thus, for t ∈ (a, b] and by (4.50), we have α

ψ

|ξ (t, a)N1 x(t)| ≤

A (ψ(b) − ψ(a))1−ξ+ k . k (α + k)



Then, for each t ∈ a − λ, b + λ˜ we obtain α

A (ψ(b) − ψ(a))1−ξ+ k . N1 xF ≤ k (α + k) for t ∈ (a, b] and by (4.51), we have

(4.52)

102

4 Fractional Differential Equations with Retardation … α

ψ

|ξ (t, a)N2 y(t)| ≤

A (ψ(b) − ψ(a))1−ξ+ k . k (α + k)

For each t ∈ [a − λ, a], we have |N2 y(t)| ≤  C ,

and for each t ∈ b, b + λ˜ , we have ˜ C˜ . |N2 y(t)| ≤  



Then, for each t ∈ a − λ, b + λ˜ we get *

α

A (ψ(b) − ψ(a))1−ξ+ k ˜ C˜ , N2 yF ≤ max  C ,   k (α + k)

+ .

(4.53)



From (4.52) and (4.53), for each t ∈ a − λ, b + λ˜ we have, N1 x + N2 yF ≤ N1 xF + N2 yF ≤ r1 + r2 ≤ ω, which infers that N1 x + N2 y ∈ Bω . Step 2: N1 is a contraction. According to Theorem 4.9 and the condition (4.49), the operator N1 is a contraction mapping on F with respect to the norm  · F . Step 3: N2 is continuous and compact.

Let {xn } be a sequence such that xn −→ x in F. For each t ∈ [a − λ, a] ∪ b, b + λ˜ , we have |N2 xn (t) − N2 x(t)| = 0. And for t ∈ (a, b], we have 

α,k;ψ |N2 x(t) − N2 y(t)| ≤ Ja+ |ϕn (s) − ϕ(s)| (t) where ϕ1 and ϕ1 be functions satisfying the functional equations ϕ(t) = f (t, xt (·), ϕ(t)), ϕn (t) = f (t, xnt (·), ϕn (t)). Since xn → x, then we get ϕn (t) → ϕ(t) as n → ∞ for each t ∈ (a, b], and since f is continuous, then we have

4.3

Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem …

103

N2 xn − N2 xF → 0 as n → ∞. Then N2 is continuous. Next we prove

that N2 is uniformly bounded on Bω . Let any y ∈ Bω . Then, for each t ∈ a − λ, b + λ˜ we get *

α

A (ψ(b) − ψ(a))1−ξ+ k N2 yF ≤ max  C ,   ˜ C˜ , k (α + k)

+ .

This prove that the operator N2 is uniformly bounded on Bω . To prove the compactness of N2 , we take y ∈ Bω and τ1 , τ2 ∈ (a, b] such that τ1 < τ2 . Then     ψ ψ ξ (τ1 , a)N2 y(τ1 ) − ξ (τ2 , a)N2 y(τ2 )



 ψ α,k;ψ ψ α,k;ψ ≤ ξ (τ1 , a) Ja+ ϕ(s) (τ1 ) − ξ (τ2 , a) Ja+ ϕ(s) (τ2 )





ψ α,k;ψ ψ α,k;ψ ≤ A ξ (τ1 , a) Ja+ (1) (τ1 ) − ξ (τ2 , a) Ja+ (1) (τ2 ) . By Lemma 2.18, we get    ψ  ψ ξ (τ1 , a)N2 y(τ1 ) − ξ (τ2 , a)N2 y(τ2 )

α α A ≤ (ψ(τ1 ) − ψ(a))1−ξ+ k − (ψ(τ2 ) − ψ(a))1−ξ+ k . k (α + k) Note that

   ψ  ψ ξ (τ1 , a)N2 y(τ1 ) − ξ (τ2 , a)N2 y(τ2 ) → 0 as τ1 → τ2 .

This proves that N2 Bω is equicontinuous on (a, b]. The equicontinuity for the other cases is obvious, thus we omit the details. Therefore N2 Bω is relatively compact. By Arzela-Ascoli theorem N2 is compact. As a consequence of Krasnoselskii fixed point theorem, we deduce that T has a fixed point which is a solution of the problem (4.30)–(4.33). 

4.3.2

Examples

In this section, we give various examples by varying the parameters in problem (4.30)–(4.33), with J = [1, π], ξ = k1 (β(k − α) + α) and   y 1 x , 1+ f (t, x, y) = − 105 + 125eπ−t 3 + |y| 1 + x



 where t ∈ J , x ∈ C −λ, λ˜ , R and y ∈ R.

104

4 Fractional Differential Equations with Retardation …

Example 4.12 Taking β → 21 , α = 21 , k = 1, ψ(t) = π t , ϑ1 = 1, ϑ2 = 2, ϑ3 = 3, 1 = 45 , 2 = 43 , 3 = 23 , m = 3, λ = λ˜ = 13 and ξ = 34 , we obtain the following problem 

1 1 H 2 , 2 ;ψ x 1 D1+

 (t) = f

    1 1 2 , 2 ;ψ t, xt (·), 1H D1+ x (t) , t ∈ (1, π],

x(π) = x( 45 ) + 2x( 43 ) + 3x( 23 ),   2 x(t) = (t), t ∈ ,1 , 3   1 . x(t) = (t), ˜ t ∈ π, π + 3

(4.54)

(4.55) (4.56)

(4.57)

We have   1 Cξ;ψ (J ) = C 3 ;ψ (J ) = x : (1, π] → R : (π t − π) 4 x ∈ C( J¯, R) , 4

then

    2 1

∈ C˜ and x| → R : x| 2 ∈ C , x| F= x : ∈ C (J ) . ,π + 3 (1,π] π,π+ 31 4 ;ψ 3 3 3 ,1

Since the function f is continuous, then condition (4.9.1) is satisfied. # $  For each x ∈ C − 13 , 13 , R , y ∈ R and t ∈ J , we have 1 | f (t, x, y)| ≤ 105 + 125eπ−t



1 + x ˜ −λ,λ

 + |y| ,

then, condition (4.9.1) is satisfied with q1 (t) = q2 (t) = q3 (t) =

1 , 105 + 125eπ−t

and q1∗ = q2∗ = q3∗ = We have

1 . 230

3

=

4 (π π − π) 4 ≈ 0.136673356256352 < 1. √ 229 π

Then, by Theorem 4.10, we deduce that the problem (4.54)–(4.57) has at least one solution in F. Example 4.13 Taking β → 0, α = 21 , k = 1, ψ(t) = t ρ , ϑ1 = 1, ϑ2 = 1, ϑ3 = 5, 1 = 23 , 2 = 2, 3 = 25 , m = 3, λ = λ˜ = 21 , ρ = 21 and ξ = 21 , we obtain a problem which is a particular case of problem (4.30)–(4.33) with the Generalized fractional derivative, given

4.4

Notes and Remarks

105

by 

1 H 2 ,0;ψ 1 D1+ x



 (t) =

ρ

D

1 2 ,0 1+



  1   ,0 x (t) = f t, xt (·), ρ D12+ x (t) , t ∈ (1, 3],

x(3) = x( 23 ) + x(2) + 5x( 25 ),   1 t x(t) = e , t ∈ ,1 , 2   7 . x(t) = et , t ∈ 3, 2 We have

(4.58)

(4.59) (4.60)

(4.61)

  , √ Cξ;ψ (J ) = C 1 ;ψ (J ) = x : (1, 3] → R : ( t − 1)x ∈ C( J¯, R) , 2

then

    1 7 → R : x| 1 ∈ C , x| 7 ∈ C˜ and x|(1,3] ∈ C 3 ;ψ (J ) . F= x : , 3, 2 4 2 2 2 ,1

Further, for each x1 , y1 ∈ C

#

$  − 21 , 21 , R , x2 , y2 ∈ R and t ∈ J , we have

| f (t, x1 , x2 ) − f (t, y1 , y2 )| ≤

1 105 + 125eπ−t



x1 − y1 

−λ,λ˜

 + |x2 − y2 | ,

1 then, condition (4.9.2) is satisfied with ζ1 = ζ2 = . 230 And since √ 4 3−4 L= ≈ 0.00721424349795779 < 1. √ 229 π Then all the assumptions of Theorem 4.9 are satisfied. Consequently, the problem (4.58)– (4.61) has a unique solution in F.

4.4

Notes and Remarks

The current chapter’s conclusions are based on the papers of Salim et al. [20, 21]. One can see the monographs [1–4, 22–27] and the papers [10, 11, 14, 15, 28–35], for additional details and results on the subject.

106

4 Fractional Differential Equations with Retardation …

References 1. S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit Differential and Integral Equations: Existence and Stability (Walter de Gruyter, London, 2018) 2. S. Abbas, M. Benchohra, G.M. N’Guérékata, Advanced Fractional Differential and Integral Equations (Nova Science Publishers, New York, 2014) 3. S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in Fractional Differential Equations (Springer-Verlag, New York, 2012) 4. M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions (Hindawi Publishing Corporation, New York, 2006) 5. S. Abbas, M. Benchohra, J.E. Lazreg, A. Alsaedi, Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type. Adv. Differ. Equ. 2017, 180 (2017) 6. S. Krim, S. Abbas, M. Benchohra, E. Karapınar, Terminal value problem for implicit Katugampola fractional differential equations in b-metric spaces. J. Funct. Spaces. 2021, 7 (2021) 7. J.E. Lazreg, S. Abbas, M. Benchohra, E. Karapınar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces. Open Math. 19, 363–372 (2021) 8. S. Krim, A. Salim, S. Abbas, M. Benchohra, On implicit impulsive conformable fractional differential equations with infinite delay in b-metric spaces. Rend. Circ. Mat. Palermo (2)., 1–14 (2022). https://doi.org/10.1007/s12215-022-00818-8 9. S. Krim, A. Salim, M. Benchohra, On implicit Caputo tempered fractional boundary value problems with delay. Lett. Nonlinear Anal. Appl. 1, 12–29 (2023) 10. M. Benchohra, J.E. Lazreg, On stability for nonlinear implicit fractional differential equations. Matematiche (Catania) 70, 49–61 (2015) 11. M. Benchohra, S. Bouriah, J. Henderson, Nonlinear implicit Hadamard’s fractional differential equations with retarded and advanced arguments. Azerbaijan J. Math. 8, 72–85 (2018) 12. K. Liu, J. Wang, D. O’Regan, Ulam-Hyers-Mittag-Leffler stability for ψ-Hilfer fractional-order delay differential equations. Adv. Differ. Equ 2019, 50 (2019) 13. J.P. Kharade, K.D. Kucche, On the impulsive implicit ψ-Hilfer fractional differential equations with delay. Math. Methods Appl. Sci. 43, 1938–1952 (2020) 14. J.V. da C. Sousa, E.C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator. Differ. Equ. Appl. 11, 87–106 (2019) 15. J.V. da C. Sousa, E.C. de Oliveira, Fractional order pseudoparabolic partial differential equation: Ulam-Hyers stability. Bull. Braz. Math. Soc. 50, 481–496 (2019) 16. A. Almalahi, K. Panchal, On the Theory of ψ-Hilfer Nonlocal Cauchy Problem. J. Sib. Fed. Univ. Math. Phys. 14(2), 161–177 (2021) 17. A. Almalahi, K. Panchal, Existence results of ψ-Hilfer integro-differential equations with fractional order in Banach space. Ann. Univ. Paedagog. Crac. Stud. Math. 19, 171–192 (2020) 18. S. Rashid, M. Aslam Noor, K. Inayat Noor, Y.M. Chu, Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions. AIMS Math. 5, 2629–2645 (2020) 19. A. Granas, J. Dugundji, Fixed Point Theory (Springer-Verlag, New York, 2003) 20. A. Salim, M. Benchohra, J.E. Lazreg, J. Henderson, On k-generalized ψ-Hilfer boundary value problems with retardation and anticipation. ATNAA. 6, 173–190 (2022). https://doi.org/10. 31197/atnaa.973992 21. A. Salim, M. Benchohra, J.E. Lazreg, Nonlocal k-generalized ψ-Hilfer terminal value problem with retarded and advanced arguments. (Submitted) 22. P. Agarwal, D. Baleanu, Y. Chen, S. Momani, J.A.T. Machado, Fractional Calculus: ICFDA 2018, Amman, Jordan, July 16–18 (Springer, Singapore, 2019)

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23. R. Almeida, D. Tavares, D.F.M. Torres, The Variable-Order Fractional Calculus of Variations (Springer International Publishing, 2019) 24. G.A. Anastassiou, Generalized Fractional Calculus: New Advancements and Applications (Springer International Publishing, Switzerland, 2021) 25. V.E. Tarasov, Handbook of Fractional Calculus with Applications. Volume 4: Applications in Physics, Part B (De Gruyter, Berlin, Boston, 2019) 26. D. Kumar, Fractional Calculus in Medical and Health Science (CRC Press, Boca Raton, 2021) 27. Z.Z. Sun, G. Gao, Fractional Differential Equations: Finite Difference Methods (De Gruyter, 2020) 28. A. Salim, J.E. Lazreg, B. Ahmad, M. Benchohra, J.J. Nieto, A study on k-generalized ψ-Hilfer derivative operator. Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00561-8 29. J.E. Lazreg, M. Benchohra, A. Salim, Existence and Ulam stability of k-generalized ψ-Hilfer fractional problem. J. Innov. Appl. Math. Comput. Sci. 2, 01–13 (2022) 30. N. Laledj, A. Salim, J.E. Lazreg, S. Abbas, B. Ahmad, M. Benchohra, On implicit fractional q-difference equations: analysis and stability. Math. Methods Appl. Sci. 45(17), 10775–10797 (2022). https://doi.org/10.1002/mma.8417 31. A. Heris, A. Salim, M. Benchohra, E. Karapınar, Fractional partial random differential equations with infinite delay. Results Physics. (2022). https://doi.org/10.1016/j.rinp.2022.105557 32. S. Bouriah, A. Salim, M. Benchohra, On nonlinear implicit neutral generalized Hilfer fractional differential equations with terminal conditions and delay. Topol. Algebra. Appl. 10, 77–93 (2022). https://doi.org/10.1515/taa-2022-0115 33. M. Chohri, S. Bouriah, A. Salim, M. Benchohra, On nonlinear periodic problems with Caputo’s exponential fractional derivative. ATNAA. 7, 103–120 (2023). https://doi.org/10.31197/atnaa. 1130743 34. M. Benchohra, F. Bouazzaoui, E. Karapınar, A. Salim, Controllability of second order functional random differential equations with delay. Mathematics. 10, 16 (2022). https://doi.org/10.3390/ math10071120 35. A. Salim, F. Mesri, M. Benchohra, C. Tunç, Controllability of second order semilinear random differential equations in Fréchet spaces. Mediterr. J. Math. 20(84), 1–12 (2023). https://doi.org/ 10.1007/s00009-023-02299-0

5

Impulsive Fractional Differential Equations with Retardation and Anticipation

5.1

Introduction and Motivations

This chapter deals with the existence and uniqueness results for a class of impulsive initial and boundary value problems for implicit nonlinear fractional differential equations and kGeneralized ψ-Hilfer fractional derivative involving both retarded and advanced arguments. Our results are based on some necessary fixed point theorems. Suitable illustrative examples are provided for each section. We explored and demonstrated the results obtained in this chapter by taking into consideration the previously stated publications in the preceding chapters and the works that follow: • The papers [1–14], in it, the authors provided many recent interesting and insightful results for diverse fractional differential problems with varied conditions and based on multiple approaches. • The paper of Wang and Zhang [15], where they proved some existence results using Krasnoselskii, Schaefer and Schauder fixed point theorems for the following nonlocal initial value problem for differential equations involving Hilfer fractional derivative: ⎧ α,β ⎪ ⎨ Da + u(t) = f (t, u(t)), t ∈ (a, b], m    1−γ + λi u(τi ). ⎪ ⎩ Ja + u (a ) = i=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Benchohra et al., Fractional Differential Equations, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-34877-8_5

109

110

5 Impulsive Fractional Differential Equations with Retardation and Anticipation

On k-Generalized ψ-Hilfer Impulsive Boundary Value Problem with Retarded and Advanced Arguments

5.2

In keeping with the spirit of generalizing the previous results, in this section, we establish existence and uniqueness results to the following k-generalized ψ-Hilfer problem with nonlinear implicit fractional differential equation with impulses involving both retarded and advanced arguments:

H α,β;ψ x k Dt +



i

k(1−ξ),k;ψ

Jt + i

(t) = f

t, x t (·),

H α,β;ψ x k Dt + i



(t) , t ∈ Ji , i = 0, . . . , m,



k(1−ξ),k;ψ x (ti+ ) = Jt + x (ti− ) + L i (x(ti− )); i = 1, . . . , m,

(5.2)

i−1

    k(1−ξ),k;ψ k(1−ξ),k;ψ ϑ1 Ja+ x (a + ) + ϑ2 Jt + x (b) = ϑ3 ,

(5.3)

x(t) = (t), t ∈ [a − λ, a], λ > 0,

(5.4)

 x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

(5.5)

m

α,β;ψ

(5.1)

k(1−ξ),k;ψ

are the k-generalized ψ-Hilfer fractional derivative of order α ∈ where kH Da+ , Ja+ (0, k) and type β ∈ [0, 1], and k-generalized ψ-fractional integral of order k(1 − ξ) respec  1 tively, where ξ = k (β(k − α) + α), k > 0, ∈ C ([a − λ, a], R),

˜ ∈ C b, b + λ˜ , R ,   f : [a, b] × PCξ;ψ −λ, λ˜ × R −→ R is a given appropriate function specified latter, ϑ1 , ϑ2 , ϑ3 ∈ R such that ϑ1 + ϑ2  = 0, Ji := (ti , ti+1 ]; i = 0, . . . , m, a = t0 < t1 < . . . < tm < tm+1 = b < ∞, u(ti+ ) = lim u(ti + ) and u(ti− ) = lim u(ti + ) represent →0+

→0−

the right and left hand limits of u(t) at t = ti and L i : R → R; i = 1, . . . , m are given continuous functions. For each function x defined on a − λ, b + λ˜ and for any t ∈ (a, b], we denote by x t the element defined by  x t (τ ) = x(t + τ ), τ ∈ −λ, λ˜ .

On k-Generalized ψ-Hilfer Impulsive Boundary Value …

5.2

5.2.1

111

Existence Results

First, let us consider the following weighted Banach space:

PCξ;ψ (J ) = x : (a, b] → R : x ∈ Cξ;ψ (Ji ); i = 0, . . . , m, and there exist x(ti− )

 k(1−ξ),k;ψ + − and Jt + x (ti ); i = 1, . . . , m, with x(ti ) = x(ti ) , i



with the norm x PCξ,k;ψ = max

i=0,...,m

where

sup

t∈[ti ,ti+1 ]

     ψ ξ (t, ti )x(t) ,

ψ

ξ (t, a) = (ψ(t) − ψ(a))1−ξ , and

α

− ψ(s)) k −1 ¯ αk,ψ (t, s) = (ψ(t)  . kk (α)

Consider the weighted Banach space   PCξ;ψ −λ, λ˜

 ψ = x : −λ, λ˜ → R : τ → ξ (t, ti )x(τ ) ∈ C([τi , τi+1 ], R); i = 0, . . . , m,

k(1−ξ),k;ψ x (τi+ ); i = 1, . . . , m, and there exist x(τi− ) and Jt + i  − with x(τi ) = x(τi ) and τi = ti − t, for each t ∈ Ji , with the norm x t 

−λ,λ˜

= max

⎧ ⎪ ⎨

max ⎪ ⎩i=0,...,m

 sup

τ ∈[τi ,τi+1 ]

     ψ t ξ (t, ti )x (τ ) ,

⎫ ⎬ ⎪   a   b  sup x (τ ) , sup (τ ) .  x ⎪ τ ∈[−λ,0] ⎭ τ ∈ 0,λ˜

Next, we consider the Banach space

 F = x : a − λ, b + λ˜ → R : x|[a−λ,a] ∈ C , x|

b,b+λ˜

with the norm



 ∈ C˜, x|(a,b] ∈ PCξ;ψ (J ) ,

112

5 Impulsive Fractional Differential Equations with Retardation and Anticipation

  xF = max xC , xC˜ , x PCξ;ψ . Theorem 5.1 Let the function ϕ(·) ∈ C( J¯, R). Then x ∈ Cξ;ψ (Ji ) is a solution of the differential equation:

H α,β;ψ x (t) = ϕ(t), t ∈ Ji , i = 0, . . . , m, 0 < α < k, 0 ≤ β ≤ 1, (5.6) k Dt + i

if and only if x satisfies the following Volterra integral equation: k(1−ξ),k;ψ

Jt +

x(t) =

i

ψ

x(ti )

ξ (t, ti )k (kξ)



α,k;ψ + Jt + ϕ (t),

(5.7)

i

α,k;ψ

Proof By applying the fractional integral operator Jt + i

(·) on both sides of the fractional

Eq. (5.6) and using Theorem 2.32, we obtain the Eq. (5.7). α,β;ψ Now, applying the fractional derivative operator kH Dt + (·) on both sides of the fractional i

Eq. (5.7), then we get

H α,β;ψ x k Dt + i

⎛

(t) =

H α,β;ψ k Dt +

⎝

i

k(1−ξ),k;ψ

Jt + i

x(ti )

ψ

ξ (t, ti )k (kξ)

⎞ ⎠+



H α,β;ψ α,k;ψ Jt + ϕ k Dt + i



i

Using the Lemmas 2.28 and 2.26, we obtain Eq. (5.6). We consider the following fractional differential equation

H α,β;ψ D x (t) = ϕ(t), t ∈ Ji , i = 0, . . . , m, k t+

(t). 

(5.8)

i

where 0 < α < k, 0 ≤ β ≤ 1, with the conditions



k(1−ξ),k;ψ k(1−ξ),k;ψ Jt + x (ti+ ) = Jt + x (ti− ) + L i (x(ti− )); i = 1, . . . , m, i

(5.9)

i−1

    k(1−ξ),k;ψ k(1−ξ),k;ψ x (a + ) + ϑ2 Jt + x (b) = ϑ3 , ϑ1 Ja+

(5.10)

x(t) = (t), t ∈ [a − λ, a], λ > 0,

(5.11)

 x(t) = (t), ˜ t ∈ b, b + λ˜ , λ˜ > 0,

(5.12)

m

5.2

On k-Generalized ψ-Hilfer Impulsive Boundary Value …

113

β(k − α) + α , k > 0, ϑ1 , ϑ2 , ϑ3 ∈ R such that ϑ1 + ϑ2  = 0 and where ϕ(·) ∈ k ˜ ∈ C˜. C( J¯, R), (·) ∈ C and (·) The following theorem shows that the problem (5.8)–(5.12) have a unique solution. where ξ =

Theorem 5.2 The function x(·) satisfies (5.8)–(5.12) if and only if it satisfies ⎧  m ⎪ 1 ϑ3 ϑ2  ⎪ ⎪ − L j (x(t − ⎪ j )) ⎪ ψ ϑ + ϑ ϑ + ϑ ⎪ 1 2 1 2  (kξ) (t, a) ⎪ k j=1 ξ ⎪ ⎪ ⎪   m+1 ⎪   k(1−ξ)+α,k;ψ

⎪ ϑ2 ⎪ α,k;ψ ⎪ J ϕ (t ) + J ϕ (t), if t ∈ J0 , − ⎪ j a+ ⎪ t+ ⎪ ϑ1 + ϑ2 j−1 ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ m ⎪ ⎪ ϑ3 1 ϑ2  ⎪ ⎪ − L j (x(t − ⎪ j )) ⎪  ψ (t, t ) (kξ) ϑ1 + ϑ2 ϑ + ϑ ⎪ 1 2 j=1 ⎨ ξ i k

m+1 i x(t) =  k(1−ξ)+α,k;ψ

 ϑ2 ⎪ k(1−ξ)+α,k;ψ ⎪ Jt + ϕ (t j ) + Jt + ϕ (t j ) ⎪ ⎪ − ϑ1 + ϑ2 ⎪ j−1 j−1 ⎪ j=1 j=1 ⎪ ⎪ ⎪ 

i ⎪ ⎪ ⎪  α,k;ψ − ⎪ ⎪ + L (x(t )) + J ϕ (t), t ∈ Ji ; i = 1, . . . , m, j ⎪ j ti+ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

(t), t ∈ [a − λ, a], ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ (t), ˜ t ∈ b, b + λ˜ .

(5.13)

Proof Assume x satisfies (5.8)–(5.12). If t ∈ J0 , then   H α,β;ψ x (t) = ϕ(t), k Da + Theorem 5.1 implies that the solution can be written as k(1−ξ),k;ψ

x(t) =

Ja +

x(a)

ψ

ξ (t, a)k (kξ)

  α,k;ψ + Ja + ϕ (t).

If t ∈ J1 , then Theorem 5.1 implies k(1−ξ),k;ψ

x(t) =

Jt + 1

ψ

x(t1 )

ξ (t, t1 )k (kξ)



α,k;ψ + Jt + ϕ (t) 1

(5.14)

114

5 Impulsive Fractional Differential Equations with Retardation and Anticipation

 =

k(1−ξ),k;ψ

Ja +

 x (t1− ) + L 1 (x(t1− ))

ψ

 =



α,k;ψ + Jt + ϕ (t)

1 ξ (t, t1 )k (kξ)    k(1−ξ),k;ψ k(1−ξ)+α,k;ψ Ja + x (a) + Ja + ϕ (t1 ) + L 1 (x(t1− ))

ψ



α,k;ψ + Jt + ϕ (t).

ξ (t, t1 )k (kξ)

1

If t ∈ J2 , then Theorem 5.1 implies k(1−ξ),k;ψ

Jt +

x(t) =

2

x(t2 )

ψ



α,k;ψ + Jt + ϕ (t)

2 ξ (t, t2 )k (kξ)

k(1−ξ),k;ψ Jt + x (t2− ) + L 2 (x(t2− ))



1

α,k;ψ Jt + ϕ 2



+ (t) ψ ξ (t, t2 )k (kξ) 

 1 k(1−ξ),k;ψ k(1−ξ)+α,k;ψ = ψ Ja + x (a) + Jt + ϕ (t2 ) 1 ξ (t, t2 )k (kξ) 

  k(1−ξ)+α,k;ψ α,k;ψ +L 1 (x(t1− )) + Ja + ϕ (t1 ) + L 2 (x(t2− )) + Jt + ϕ (t).

=

2

Repeating the process in this way, the solution x(t) for t ∈ Ji ,i = 1, . . . , m, can be written as 1

x(t) =

ψ

 i   k(1−ξ),k;ψ Ja + x (a) + L j (x(t − j ))

ξ (t, ti )k (kξ) j=1



i  k(1−ξ)+α,k;ψ α,k;ψ + Jt + ϕ (t j ) + Jt + ϕ (t). j−1

j=1 k(1−ξ),k;ψ

Applying Jt + m

(5.15)

i

on both sides of (5.15), using Lemma 2.18 and taking t = b, we obtain

m      k(1−ξ),k;ψ k(1−ξ),k;ψ Jt + x (b) = Ja + x (a) + L j (x(t − j )) m

+

m  j=1

j=1



k(1−ξ)+α,k;ψ Jt + ϕ j−1

  k(1−ξ)+α,k;ψ (t j ) + Jt + ϕ (b). m

Multiplying both sides of (5.16) by ϑ2 and using condition (5.10), we obtain

(5.16)

5.2

On k-Generalized ψ-Hilfer Impulsive Boundary Value …

115

m      k(1−ξ),k;ψ k(1−ξ),k;ψ ϑ3 − ϑ1 Ja + x (a) = ϑ2 Ja + x (a) + ϑ2 L j (x(t − j ))

+ϑ2

m+1 

k(1−ξ)+α,k;ψ

Jt +

j−1

j=1

j=1

ϕ (t j ),

which implies that   k(1−ξ),k;ψ Ja + x (a) =

m ϑ3 ϑ2  − L j (x(t − j )) ϑ1 + ϑ2 ϑ1 + ϑ2 j=1

ϑ2 − ϑ1 + ϑ2

m+1  j=1

k(1−ξ)+α,k;ψ Jt + ϕ j−1

(t j ).

(5.17)

Substituting (5.17) into (5.15) and (5.14) we obtain (5.13). k(1−ξ),k;ψ on both sides of (5.13) and using Lemmas 2.18 and Reciprocally, applying Jt + i

2.15, we get

⎧ m+1  k(1−ξ)+α,k;ψ

⎪ ϑ3 ϑ2 ⎪ ⎪ ⎪ − Jt + ϕ (t j ) ⎪ ⎪ ϑ1 + ϑ2 ϑ1 + ϑ2 j−1 ⎪ ⎪ j=1 ⎪ ⎪ m ⎪   ⎪ ϑ2  ⎪ k(1−ξ)+α,k;ψ − ⎪ − L (x(t )) + J ϕ (t), t ∈ J0 , ⎪ j a+ j ⎪ ⎪ ϑ1 + ϑ2 ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪

⎨ k(1−ξ),k;ψ

m i Jt + x (t) =  ϑ2  ⎪ k(1−ξ)+α,k;ψ i − ⎪ L j (x(t j )) + Jt + ϕ (t j ) ⎪− ⎪ ⎪ ϑ1 + ϑ2 j−1 ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎪ m+1 i ⎪  k(1−ξ)+α,k;ψ

 ϑ2 ⎪ ⎪ ⎪ Jt + ϕ (t j ) + L j (x(t − − ⎪ j )) ⎪ ϑ + ϑ j−1 ⎪ 1 2 ⎪ j=1 j=1 ⎪

⎪ ⎪ ϑ3 ⎪ k(1−ξ)+α,k;ψ ⎪ ⎩+ J + ϕ (t) + , t ∈ Ji ; i = 1, . . . , m. ti ϑ1 + ϑ2 (5.18) Next, taking the limit t → a + of (5.18) and using Theorem 2.23, with k(1 − ξ) < k(1 − ξ) + α, we obtain 

k(1−ξ),k;ψ Ja + x



(a + )

m+1  k(1−ξ)+α,k;ψ

ϑ3 ϑ2 = − Jt + ϕ (t j ) ϑ1 + ϑ2 ϑ1 + ϑ2 j−1 j=1

m ϑ2  − L j (x(t − j )). ϑ1 + ϑ2 j=1

(5.19)

116

5 Impulsive Fractional Differential Equations with Retardation and Anticipation

Now, taking t = b in (5.18), we get ⎛ ⎞ m m+1   k(1−ξ)+α,k;ψ

ϑ3 +⎝ L j (x(t − Jt + ϕ (t j )⎠ j )) + ϑ1 + ϑ2 j−1 j=1 j=1

ϑ2 . (5.20) × 1− ϑ1 + ϑ2

  k(1−ξ),k;ψ Jt + x (b) = m

From (5.19) and (5.20), we find that     k(1−ξ),k;ψ k(1−ξ),k;ψ ϑ1 Ja+ x (a + ) + ϑ2 Jt + x (b) = ϑ3 , m

α,β;ψ

which shows that the boundary condition (5.10) is satisfied. Next, apply operator kH Dt + i

(·)

on both sides of (5.13), where i = 0, . . . , m. Then, from Lemmas 2.28 and 2.26 we obtain Eq. (5.8). Also, we can easily show that x satisfies the Eqs. (5.9), (5.11) and (5.12). This completes the proof.  As a consequence of Theorem 5.2, we have the following result β(k − α) + α Lemma 5.3 Let ξ = where 0 < α < k and 0 ≤ β ≤ 1, let f : J¯ × PCξ;ψ k   −λ, λ˜ × R → R be a continuous function, (·) ∈ C and (·) ˜ ∈ C˜. Then, x ∈ F satisfies the problem (5.1)–(5.5) if and only if x is the fixed point of the operator T : F → F defined by ⎧  m 1 ϑ3 ϑ2  ⎪ ⎪ ⎪ − L j (x(t − ⎪ j )) ψ ⎪ ϑ + ϑ ϑ + ϑ 1 2 1 2 ⎪  (t, t ) (kξ) i k ⎪ ξ j=1 ⎪ ⎪ ⎪ m+1 ⎪  k(1−ξ)+α,k;ψ

⎪ ⎪ ⎪ ϑ2 Jt + ϕ (t j ) ⎪ ⎪ j−1 ⎪  k(1−ξ)+α,k;ψ

⎪ j=1 ⎪ ⎪ + Jt + ϕ (ti ) − ⎪ ⎪ ⎨ ϑ1 + ϑ2 i−1 a