Toward General Theory of Differential-Operator and Kinetic Models (World Scientific Nonlinear Science Series a) 9811213747, 9789811213748

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Toward General Theory of Differential-Operator and Kinetic Models

WORLD  SCIENTIFIC  SERIES ON  NONLINEAR  SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

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Topology and Dynamics of Chaos: In Celebration of Robert Gilmore’s 70th Birthday C. Letellier & R. Gilmore

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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science (Volume VI) L. O. Chua

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Elements of Mathematical Theory of Evolutionary Equations in Banach Spaces A. M. Samoilenko & Y. V. Teplinsky

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Integral Dynamical Models: Singularities, Signals and Control D. Sidorov

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Wave Momentum and Quasi-Particles in Physical Acoustics G. A. Maugin & M. Rousseau

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Modeling Love Dynamics S. Rinaldi, F. D. Rossa, F. Dercole, A. Gragnani & P. Landi

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Deterministic Chaos in One-Dimensional Continuous Systems J. Awrejcewicz, V. A. Krysko, I. V. Papkova & A. V. Krysko

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Control of Imperfect Nonlinear Electromechanical Large Scale Systems: From Dynamics to Hardware Implementation L. Fortuna, A. Buscarino, M. Frasca & C. Famoso

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Coupled Phase Locked Loops: Stability, Synchronization, Chaos and Communication with Chaos V. V. Matrosov & V. D. Shalfeev

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Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures V. Avrutin, L. Gardini, I. Sushko & F. Tramontana

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Frequency-Domain Approach to Hopf Bifurcation Analysis: Continuous Time-Delayed Systems F. S. Gentile, J. L. Moiola & Guanrong Chen

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Toward General Theory of Differential-Operator and Kinetic Models N. Sidorov, D. Sidorov & A. Sinitsyn

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NONLINEAR SCIENCE WORLD SCIENTIFIC SERIES ON

Series A

Vol. 97

Series Editor: Leon O. Chua

Toward General Theory of Differential-Operator and Kinetic Models Nikolay Sidorov Irkutsk State University, Russia

Denis Sidorov Russian Academy of Sciences, Russia

Alexander Sinitsyn Universidad Nacional de Colombia, Bogotá, Colombia

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World Scientific Series on Nonlinear Science Series A — Vol. 97 TOWARD GENERAL THEORY OF DIFFERENTIAL-OPERATOR AND KINETIC MODELS Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Dedicated to the 100th anniversary of the Department of Mathematics and Physics, Irkutsk State University, Russia

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Foreword Differential-operator and kinetic models play an exceptional role in natural sciences and technologies. Design of the state-of-theart models needs an in-depth knowledge of the various operator equations’ solution behavior. Stability and structural change are widely modeled in terms of the bifurcation theory based on differential-operator equations. Branching solutions of differential-operator equations with parameters help one to understand complex nonlinear phenomena in mechanical and chemical processes, energy systems, material science and many other applications. Classic Vlasov equation derived in 1938 became a cornerstone for plasma physics. Various concepts ranging from the Earth’s planetary environment to the Solar system and beyond can be formulated in terms of kinetic plasma physics, represented by the Vlasov equation. Lyapunov–Schmidt reduction technique and singularity theory help one to tackle challenging bifurcation problems in hydrodynamics and non-equilibrium thermodynamics. This volume provides a comprehensive introduction to the modern theory of differential-operator and kinetic models, including Fredholm equations, Lyapunov–Schmidt branching equations and Vlasov–Maxwell equations, to name a few. This book aims to fill the gap in the considerable body of existing academic literature on the analytical methods of studies of complex behavior of differentialoperator equations and kinetic models.

vii

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Toward General Theory of Differential-Operator and Kinetic Models

This monograph will be of interest to mathematicians, physicists and engineers interested in the theory of such nonstandard systems. Vladislav V. Pukhnachev Corresponding Member of the Russian Academy of Sciences Lavrentyev Institute of Hydrodynamics SB RAS, Russia

Contents Foreword

vii

Introduction

xv

Acknowledgments

xxi

Part I: Operator and Differential-Operator Equations

1

1.

3

Auxiliary Information on the Theory of Linear Operators 1.1 1.2 1.3 1.4 1.5

2.

Generalized Jordan Chains, Sets and Root Numbers of Linear Operators . . . . . . . . . . . . . . . . . Regularization of Linear Equations with Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . Principal Theorem of Regularization of Linear Equations by the Perturbation Method . . . . . . . Regularization of Linear Equations Based on the Perturbation Theory in Hilbert Spaces . . . . . . . Regularization with Vector Regularizing Parameter for the First Kind Equations . . . . . . . . . . . .

Volterra Operator Equations with Piecewise Continuous Kernels: Solvability and Regularized Approximate Methods 2.1 2.2

.

3

.

9

. 16 . 22 . 31

39

Theory of the Volterra Operator Equations with Piecewise Continuous Kernels . . . . . . . . . . . . . 39 Numerical Methods . . . . . . . . . . . . . . . . . . . 62 ix

Toward General Theory of Differential-Operator and Kinetic Models

x

3.

Nonlinear Differential Equations Near Branching Points 3.1 3.2 3.3

4.

5.

5.2

5.3

6.3

7.

105

Reduction of a Nonlinear System in the Neighborhood of an Equilibrium Point to a Single Differential Equation . . . . . . . . . . . . . . 108 The Construction of a Solution of a Nonlinear System by the Successive Approximations Method . . . . . . . . . . . . . . . . . . . . . . . . . 111 Open Problems . . . . . . . . . . . . . . . . . . . . . 114

Nonclassic Boundary Value Problems in the Theory of Irregular Systems of Equations with Partial Derivatives 6.1 6.2

97

Nonlinear Operator Equations . . . . . . . . . . . . . 97 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 103

Nonlinear Systems’ Equilibrium Points: Stability, Branching, Blow-Up 5.1

6.

Problem Statement . . . . . . . . . . . . . . . . . . . 84 Open Problems and Generalizations . . . . . . . . . 93 Magnetic Insulation Model Example . . . . . . . . . 95

Nonlinear Operator Equations with a Functional Perturbation of the Argument 4.1 4.2

83

117

Skeleton Chains of Linear Operators . . . . . . . . . 120 Abstract Irregular Equation Reduction to the Sequence of Regular Equations . . . . . . . . . . . . 122 Skeleton Decomposition in the Theory of Irregular Ordinary Differential Equation in Banach Space . . . . . . . . . . . . . . . . . . . . 129

Epilogue for Part I

131

Contents

Part II: Lyapunov Methods in Theory of Nonlinear Equations with Parameters 8.

Lyapunov Convex Majorants in the Existence Theorems 8.1 8.2 8.3

9.

9.2

11.2

12.2 12.3

183

205

Construction of the Regularization Equation in the Problem at a Branch Point . . . . . . . . . . . 206 Definition and Properties of Simple Solutions . . . . 207 Construction of Regularization Equation of Simple Solutions . . . . . . . . . . . . . . . . . . . . . . . . 210

13. Iteration Methods, Analytical Initial Approximations, Interlaced Equations 13.1

169

Analytic Lyapunov–Schmidt Method in the Study of Branching Equations . . . . . . . . . . . . . . . . 183 Variational Methods in the Study of Branching Equations . . . . . . . . . . . . . . . . . . . . . . . . 187

12. Regularization of Computation of Solutions in a Branch Point Neighborhood 12.1

159

Open Problem . . . . . . . . . . . . . . . . . . . . . 179

11. Construction of Asymptotics in a Neighborhood of a Bifurcation Point 11.1

135

Lyapunov–Schmidt Method in the Problem of a Bifurcation Point . . . . . . . . . . . . . . . . . 161 Open Problems . . . . . . . . . . . . . . . . . . . . . 164

10. General Existence Theorems for the Bifurcation Points 10.1

133

Parameter-Independent Majorants . . . . . . . . . . 137 Majorants Depending on a Parameter . . . . . . . . 147 Solution Existence Domain . . . . . . . . . . . . . . 153

Investigation of Bifurcation Points of Nonlinear Equations 9.1

xi

213

Iterations and Uniformization of Branching Solutions . . . . . . . . . . . . . . . . . . . . . . . . 213

xii

Toward General Theory of Differential-Operator and Kinetic Models

13.2

Branching Equation and the Selection of Initial Approximation . . . . . . . . . . . . . . . . . . . . . 214

14. Iterative Methods Using Newton Diagrams 14.1 14.2 14.3

One-Step Iteration Method . . . . . . . . . . . . . . 224 N -step Iteration Method . . . . . . . . . . . . . . . . 228 Iteration Method for Nonlinear Equation Invariant Under Transformation Groups . . . . . . . 234

15. Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods 15.1 15.2

17.2 17.3 17.4

257

Existence Theorem and Successive Approximations . . . . . . . . . . . . . . . . . . . . . 258

17. Interlaced and Potential Branching Equation 17.1

241

Construction of the Minimal Branch of Solutions of Equation with Fredholm Operator . . . . . . . . . 243 Sufficient Conditions of the Minimal Branch Existence . . . . . . . . . . . . . . . . . . . . . . . . 249

16. Successive Approximations to the Solutions to Nonlinear Equations with a Vector Parameter 16.1

221

Property of (S, K)-interlacing of an Equation and Its Inheritance by Branching Equation . (T, M )-interlaced and (T 2 , M )-interlaced Branching Equation . . . . . . . . . . . . . . α-Parametric Interlaced Branching Equation Interlaced Branching Equation of Potential Type . . . . . . . . . . . . . . . .

18. Epilogue for Part-II

265 . . . . 266 . . . . 271 . . . . 274 . . . . 277 285

Contents

xiii

Part III: Kinetic Models

287

19. The Family of Steady-State Solutions of Vlasov–Maxwell System 19.1

289

Ansatz of the Distribution Function and Reduction of Stationary Vlasov–Maxwell Equations to Elliptic System . . . . . . . . . . . . . 289

20. Boundary Value Problems for the Vlasov–Maxwell System 20.1 20.2 20.3 20.4

Introduction . . . . . . . . . . . . . . . . . . Collisionless Kinetic Models (Classical and Relativistic Vlasov–Maxwell Systems) . . . Quantum Models: Wigner–Poisson and Schr¨ odinger–Poisson Systems . . . . . . Mixed Quantum-Classical Kinetic Systems .

313 . . . . . 313 . . . . . 319 . . . . . 320 . . . . . 320

21. Stationary Solutions of Vlasov–Maxwell System 21.1 21.2

Problem Reduction to the System of Nonlinear Elliptic Equations . . . . . . . . . . . . . . . . . . . 324 System Reductions . . . . . . . . . . . . . . . . . . . 329

22. Existence of Solutions for the Boundary Value Problem 22.1

23.2

335

Existence of Solution for Nonlocal Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . 342

23. Nonstationary Solutions of the Vlasov–Maxwell System 23.1

323

349

Reduction of the Vlasov–Maxwell System to Nonlinear Wave Equation . . . . . . . . . . . . . . 349 Existence of Nonstationary Solutions of the Vlasov–Maxwell System in the Bounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . 357

24. Linear Stability of the Stationary Solutions of the Vlasov–Maxwell System

363

Toward General Theory of Differential-Operator and Kinetic Models

xiv

25. Bifurcation of Stationary Solutions of the Vlasov–Maxwell System 25.1 25.2

373

Bifurcation of Solutions of Nonlinear Equations in Banach Spaces . . . . . . . . . . . . . . . . . . . . 377 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 386

26. Statement of the Boundary Value Problem and the Bifurcation Problem

387

27. Resolving Branching Equation

401

27.1

The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions . . . . 404

28. Numerical Modeling of the Limit Problem for the Magnetically Noninsulated Diode 28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8

Introduction . . . . . . . . . . . . . . . . . . . . . Description of Vacuum Diode . . . . . . . . . . . Shot Noise in a Diode . . . . . . . . . . . . . . . Description of the Mathematical Model . . . . . Solution Trajectory, Upper and Lower Solutions . Second Lower Solution Hypothesis . . . . . . . . Numerical Methods . . . . . . . . . . . . . . . . . Numerical Modeling . . . . . . . . . . . . . . . .

415 . . . . . . . .

. . . . . . . .

415 416 417 419 424 434 438 448

29. Open Problems

453

Bibliography

459

Index

471

Introduction Nonlinear parameter-dependent mathematical models are in the heart of natural sciences and are essential in various engineering fields. Many critical processes in fluid dynamics, thermodynamics and space plasma physics are modeled using the theory of nonlinear differential–operator equations. The Vlasov–Maxwell systems and other kinetic models play an important role in contemporary mathematical physics. The problems discussed in this book have an immediate mathematical appeal and are of increasing importance in plasma physics, storage modeling and control, mechanics and material science, but are not as widely known as they should be to researchers interested in these fields. Lyapunov–Schmidt methods form the basis of the bifurcation theory for the functional equations initiated by Lyapunov and Schmidt in the beginning of the 20th century and developed by other famous mathematicians. For excellent reviews, readers may refer to the seminal publications of Nekrasov [121], Lusternik [104], Vainberg and Trenogin [169], Krasnosel’skii [92], Buffoni, Dancer and Toland [29], Chow and Hale [34] and other authors. These studies were initiated in the modern nonlinear analysis construction [177] with sound applications in fluid dynamics and mathematical modeling [5, 6, 18, 80, 90, 110, 127, 128, 161, 164]. The aim of this book is to survey the relations between the various kinds of differential-operator, kinetic equations and bifurcation theory from a constructive point of view. Therefore, apart from such a qualitative theory, much attention is paid to numerical analysis and approximate methods.

xv

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Toward General Theory of Differential-Operator and Kinetic Models

We revisit classical monographs and authors’ monographs [141, 143,155]. After the publication of these monographs, the new results were obtained to further expand the differential-operator and kinetic equations theory. This book also reviews some recent development in these fields. For many other interesting results, readers may also refer to the bibliography of this book.

Scope of the Book This monograph includes three main parts. Part I focuses on the operator and differential-operator equations with parameters from normed spaces. It starts with Chapter 1, where the theory of linear operators, generalized Jordan chains and regularizing equation are introduced. The introduction to the theory of the first kind equation regularization using the perturbation method is given in Section 1.2. In Chapter 2, the first kind Volterra operator equations with piecewise continuous kernels are considered. The existence theorems for the parametric families of solutions are proved, solution’s asymptotics are constructed, and the regularized numerical methods are presented. The algorithm for the construction of branching solutions of nonlinear higher order differential equations in the neighborhood of branching points is presented in Chapter 3. Nonlinear operator equations with a functional perturbation of the argument were considered in Chapter 4. It is to be noted that the solution of the Volterra integral equations with piecewise continuous kernels can be reduced to the solution of these equations, as shown in Chapter 2 and in [141]. The nonlinear systems of differential and operator equations are studied in Chapter 5. Finally, the novel theory of skeleton chains of operators is applied to reduce the problem of linear systems of PDEs and ODEs with noninvertible operators to the well-posed problems in Chapter 6. Part II deals with abstract nonlinear parameter-dependent systems using topological, algebraic and variational methods. It starts with Chapter 8, where the Lyapunov–Kantorovich method of convex majorants is described and employed for the estimation of the

Introduction

xvii

domain of existence and possible extension of solution branches. Then, Chapter 9 starts with the definition of the bifurcation point (or branch point) of nonlinear operator equation. In Section 9.1, the branching equation (BEq) is constructed. In Chapter 10, the general existence theorems on the bifurcation points are obtained on the basis of the application of the singular points theory of finite-dimensional vector fields to the BEq. We study the BEq with the help of analytic Lyapunov–Schmidt method complemented by a certain information from the singular points theory and by the variational arguments. Under certain conditions, the asymptotics of continuous branches in a neighborhood of a branch point is constructed in Chapter 11. In the computation of asymptotics by using our method, it is necessary to solve a simple problem: find the points of conditional extremum for certain functions on the sphere. Chapter 12 deals with problem of nonlinear equation solutions in a neighborhood of the branch point. Here, we employed the power geometry method proposed by Bruno to make uniform solution’s branches. This method enables the construction of the N -step method of sequential approximations in the neighborhood of branching points. The theory of branching for interlaced equations is described in Chapter 13. Proposed theory makes it possible to investigate different branches of solutions dependent on free parameters. The supporting lines and the Newton diagrams are applied for the construction of the initial approximation in Chapter 14. Chapters 15 and 16 tackle the nonlinear equations with vector parameter, and the efficient method for minimumnorm solution is presented and applied for the fluid mechanics models and for satellite oscillations modeling. Chapter 17 deals with the interlaced and potential branching equations in the operator theory framework. The occurrence of free parameters in the branching solutions of general nonlinear equations in Banach spaces is studied. Part III deals with kinetic models, including Vlasov–Maxwell (VM) systems. The Vlasov equation describes time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g., Coulomb. The equation was first

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Toward General Theory of Differential-Operator and Kinetic Models

suggested for the description of plasma by Anatoly Vlasova in 1938 and later discussed by him in detail in his monograph.b VM systems have been studied by many scientists: readers may refer to review by Skubachevskii.c For bibliography concerning numerical analysis of one- and two-dimensional VM system modeling, including ionized plasma of ionosphere, readers may refer to B. Eliasson.d Key mathematical problems for the VM system are connected with the existence of global classical solutions in dimension three and Landau damping, which has a collisionless nature. The French mathematician Villani was awarded the 2010 Fields medal for the mathematical results of this phenomenon. The physics applications of the Vlasov equation range from magnetically confined plasmas for thermonuclear research to space plasmas in planetary magnetospheres and in stellar winds, to relativistic electromagnetic plasmas either produced in the interaction of ultraintense laser pulses with matter or present in the high energy density environment around compact astrophysical objects.e Vlasov–Maxwell (VM) equations are introduced in Chapter 19. Section 19.1 has a reference character and is devoted to students, engineers and postgraduate students. Here, we introduce an ansatz of distribution function for twocomponent plasma. Simple problem statements are introduced for nonlinear elliptic equations both for Cauchy and bifurcation cases. In Chapter 20, we introduce kinetic equations, notably Boltzmann equation, the Vlasov–Poisson (VP), VM systems, and describe their mathematical structure. To make the derivation technique comprehensible, we first start with particle shift along the traiectories of an arbitrary system of differential equations and second with a A.

A. Vlasov (1938). On Vibration Properties of Electron Gas. J. Exp. Theor. Phys. 8(3): 291 (in Russian). b A. A. Vlasov (1961). Many-Particle Theory and Its Application to Plasma. Gordon and Breach Pub. 414p. (transl. from Russian) c A. L. Skubachevskii (2014). Vlasov–Poisson equations for a two-component plasma in a homogeneous magnetic field, Russian Math. Surveys, 69:2 (2014), 291–330. d B. Eliasson (2002). “Numerical Vlasov–Maxwell Modelling of Space Plasma,” Doctoral dissertation, Acta Universitatis Upsaliensis. Uppsala Dissertations from the Faculty of Science and Technology. e F. Pegoraro, F. Califano, G. Manfredi, P.J. Morrison, Theory and applications of the Vlasov equation. Eur. Phys. J. D, 69: 68, 2015.

Introduction

xix

particle system movement in metric spaces. In Chapters 21–24, we study special cases of stationary and nonstationary solutions of the VM system. These solutions introduce the systems of nonlocal semilinear elliptic equations with boundary conditions. Applying the lower–upper solution method, we establish the existence theorems for solutions of the semilinear nonlocal elliptic boundary value problem under corresponding restrictions upon a distribution function. The bifurcation problem for the stationary VM system is considered in Chapters 25–27. It is translated into the bifurcation problem of the semilinear elliptic system and is studied as an operator equation in Banach space. Using the classical approach by Lyapunov–Shmidt, the BEq is derived and the asymptotics of nontrivial branches of solutions is studied. Here, the principal idea is to study a potential BEq, since the system of elliptic equations is potential. Further investigation established the existence theorem for the bifurcation points and revealed the asymptotic properties of nontrivial branches of the solutions of VM system. Chapter 28 is aimed at studying the stationary self-consistent problem of magnetic insulation under space charge limitation via the VM system. In a dimension form of the VM system, the ratio of the typical particle velocity at the cathode to that reached at the anode appears as a small parameter. The associated perturbation analysis provides a mathematical framework to the results of Langmuir and Compton. We study the extension of this approach, based on the Child–Langmuir asymptotics to magnetized flows. Chapter 29 presents several fundamental open problems existing for the VP and VM systems. This book is written in a reader-friendly style, built on already familiar concepts from functional analysis, operator theory, theory of integral and differential equations. This book is useful not only for experts in the fields of nonlinear analysis but also for postgraduate students who wish to gain an applied or interdisciplinary perspective.

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Acknowledgments The authors would like to express their gratitude to colleagues and experts for the fruitful discussions, numerical experiments, constructive criticism as well as for their long-standing cooperation in the production of this monograph. The authors are also deeply grateful for the useful advices of the fellow professors V.A. Trenogin, M.M. Lavrentyev, L.V. Ovsyannikov, V.V. Pukhnachev, A. Lorenzi, A.G. Yagola, V.K. Ivanov, L. Arkeryd, J. Batt, J. Toland, D. Degond, P.A. Markowich, Yong Li, H.-J. Reihardt, M. Brown, G.A. Sviridyuk, Tong Yang, G. Wolansky, B.V. Loginov and Feimin Huang. The authors delivered the invited lectures, attended seminars, and held symposia on the problems discussed in this monograph in many universities and mathematical institutes, including Banach International Mathematical Center, Hunan University, Isaac Newton Institute of Mathematical Sciences, Institute of Mathematics of Henri Poincare, Moscow State University, Sobolev Institute of Mathematics, Bath Institute for Mathematical Innovation, Chalmers University of Technology, City University of Hong Kong. Results of this monograph were used to prepare special courses and seminars on nonlinear and singular problems for postgraduate studies in Irkutsk State University, Irkutsk National Research Technical University, Universidad. Nacional de Colombia (Bogota, Colombia) and Hunan University (Changsha, China). This book benefited from the support of the Russian Foundation of Basic Research (RFBR) and the Natural Science Foundation China (NSFC) Exchange Program under NSFC Grant No. 61911530132

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and RFBR Grant No. 19-58-53011. This work was partly performed under the Russian Science Foundation Grant No. 17-77-20005 and benefited from the support of the Ministry of Education of China, State Bureau of Foreign Experts under Project 111, Grant no. B17016 (Optimization and Control of Smart Grids). This work is carried out within the framework of the research projects III.17.3.1, III.17.4 of the program of fundamental research of the Siberian Branch of the Russian Academy of Sciences, reg No. AAAA-A17-117030310442-8, No. AAAAA17-117030310438-1. We thank World Scientific for the fruitful collaboration.

Part I

Operator and Differential-Operator Equations

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Chapter 1

Auxiliary Information on the Theory of Linear Operators 1.1.

Generalized Jordan Chains, Sets and Root Numbers of Linear Operators

This section provides an introduction to the theory of Jordan chains and Jordan sets of linear operators. Suitable techniques of this theory have been used in this book to study and develop the Lyapunov– Schmidt methods with a uniform point of view. Let us consider the operator function A(λ) ∈ L(X, Y ), λ ∈ I ⊂ R; let X, Y be Banach spaces. It means that for any point λ0 ∈ I, there is a neighborhood in which the following representation holds: A(λ) = A0 −

∞ 

Ak (λ − λ0 )k ,

(1.1)

1

where 1 (k) A (λ0 ), k! Let us introduce the following definitions. A0 = A(λ0 ),

Ak = −

k = 1, 2, . . . .

Definition 1.1. If the operator A(λ) has a bounded inverse A−1 (λ) ∈ L(Y, X), then the point λ ∈ I is called a regular point of operator function A(λ). Definition 1.2. If the operator A(λ) is not continuously invertible, then the point λ ∈ I is a singular point of A(λ).

3

4

Toward General Theory of Differential-Operator and Kinetic Models

Definition 1.3. If there exists a neighborhood of the point λ0 where all points are regular, then the singular point λ0 is called an isolated singular point of A(λ). Definition 1.4. The point λ ∈ I is called a Fredholm point if the operator A(λ) is a Fredholm (see [169]), i.e., the operator A(λ) is normally solvable and has a finite-dimensional zero and defect spaces N (A(λ)) and N ∗ (A(λ)), moreover, dim N (A(λ)) = dim N ∗ (A(λ)) = n(λ). If the point λ ∈ I is a singular and Fredholm, then n(λ) ≥ 1. Let λ0 be a singular Fredholm point of operator A(λ), ϕi |n1 a basis for N (A0 ), and ψi |n1 a basis in defect subspace N ∗ (A0 ). Lemma 1.1 (Generalized Schmidt’s lemma). The operator Aˆ0 = A0 +

n  γi , ·zi , i=1

where γi ∈

X ∗,

zi ∈ Y,

detγi , ϕk |ni,k=1 = 0,

detψi , zk |ni,k=1 = 0,

has a bounded inverse Γ = Aˆ−1 0 . Having reserved the notation A˜0 for the approximate operator, we used the notation Aˆ0 for the Schmidt operator. Remark 1.1. For any y ∈ Y , we have the following identity: γl , Γy =

n 

bji aik γl , ϕi ψk , y,

i,j,k=1

where bji and aik are the elements of inverse matrices [γj , ϕi ]−1 and [ψi , zk ]−1 , respectively. There are always the functionals γi ∈ X ∗ and the elements zi ∈ Y , such that γi , ϕk  = δik , ψi , zk  = δik , i, k = 1, . . . , n. Moreover, the formulas Aˆ0 ϕi = zi , Γzi = ϕi and the dual formulas Aˆ∗0 ψi = γi , Γ∗ γi = ψi occur.

Auxiliary Information on the Theory of Linear Operators

5

Definition 1.5. Let us say that the elements (1) def

ϕi

(2)

(pi )

= ϕi , ϕi , . . . , ϕi

generate a generalized A(λ0 )-Jordan chain (or simply generalized Jordan chain [GJC]) of length pi corresponding to ϕi if the following identities hold: (k) A0 ϕi

=

k−1 

(k−j)

Aj ϕi

j=1

,

k = 2, . . . , pi ,

(1.2)

and at least for one functional ψk ∈ N ∗ (A0 ), we get   pi  (p +1−j) Aj ϕi i = 0. ψk , j=1

In this case, it is said that the GJC of operator A(λ0 ) consists of the (1) (p ) (2) (p ) elements ϕi , . . . , ϕi i . The elements ϕi , . . . , ϕi i will be called the A-adjoined elements. If there exists an infinite sequence of the (k) elements {ϕi }, k = 1, 2, 3, . . . , such that (k) A0 ϕi

=

k−1  j=1

(k−j)

Aj ϕi

,

k = 2, 3, . . . ,

then a length of the corresponding GJC will be equal to +∞. Since the solutions of equation (1.2) are not uniquely defined in space X, (k) ϕi

=

n  j=1

(k,i) cj ϕj

+Γ

k−1  j=1

(k−j)

Aj ϕi

,

(k,i)

cj

are constants,

(k)

then we require that γj , ϕi  = 0, γj , ϕi  = δji , j, i = 1, . . . , n, (k) k = 2, . . . , pi , i.e., we consider the elements ϕi , k = 2, . . . , pi in the (k,i) = 0 and the elements of GJC are defined subspace X ∞−n . Then cj in a unique way by the recurrence relations (k)

ϕi

(k)

= Γui ,

(1.3)

Toward General Theory of Differential-Operator and Kinetic Models

6

where (k) ui

=

k−1  j=1

=

(k−j)

Aj ϕi



Ai1 ΓAi2 , . . . , ΓAik−1 Γzi ,

Γzi = ϕi .

i1 +···+ik−1 =k−1 (k)

Collection of the elements {ϕi }, i = 1, . . . , n, k = 1, . . . , pi is called a generalized A(λ0 )-Jordan set (generalized Jordan set [GJS]). Definition 1.6. It is stated that the operator function A(λ) has a total generalized Jordan set (TGJS) at the point λ0 if detψi , A(λ)[I + Γ(A(λ) − A(λ0 ))]−1 ϕk |ni,k=1 = c(λ − λ0 )κ + o(|λ − λ0 |κ ), where c = 0. Moreover, the number κ is called a root number of the operator (k) A(λ) related to the point λ0 . Note that the elements {ϕi }ni=1 , k = 1, . . . , pi generate a TGJS and κ := p1 + · · · + pn if all pi < +∞ and n  pk  (pk +1−j) Aj ϕk = 0. (1.4) det ψi , j=1

i,k=1

Without loss of generality, one may assume   m  0, m = 1, . . . , pk−1 , (m+1−j) Aj ϕk = ψi , δik , m = pk , j=1



i, k = 1, . . . , n. (1.5)

Theorem 1.1. In order that in the singular Fredholm point λ0 there is a total GJS of the operator A(λ0 ), it is necessary and sufficient that λ0 be the isolated Fredholm point. The proof is similar to the proof of [169, Theorem 30.1], where it was carried out in a special case A(λ) = A0 − A1 (λ − λ0 ). Then we assume that TGJS satisfies inequality (1.4), i.e., TGJS is a completely canonical GJS by terminology proposed in [100].

Auxiliary Information on the Theory of Linear Operators

7

Example 1.1. If detψi , A1 ϕk |ni,k=1 = 0, then the elements ϕ1 , . . . , ϕn generate a TGJS, the adjoined vectors are absent, and k = dim N (A(λ0 )) = n. In particular, if A(λ)u = u+λu, X = Y = ˚ 1 (Ω) is a completion of the space of finite functions ˚ 1 (Ω), and W W 2 2 ∞ ˚ 1 (Ω), then k = dim N (A(λ0 )). C0 (Ω) in the norm W 2 Note a well-studied case: X = Y = H, A(λ) = I − λA, where A is a completely continuous operator. In this case, there exists at most a denumerable set of singular points with the only possible limit point ∞; all other singular points are Fredholm and isolated. 1.1.1.

Construction of the left and right regularizers

Our objective here is to consider the application of TGJS for the construction of the left-hand and right-hand side regularizers of singular transformations. Let A(λ) be not analytic at the point λ0 , and it is only the sufficiently smooth operator q  1 (k) A (λ0 )(λ − λ0 )k + o(|λ − λ0 |q ). A(λ) = k!

(1.6)

k=0

Let q be sufficiently large, ϕi |n1 , ψi |n1 make previous sense, but the operator A0 = A(λ0 ) is not normally solvable. Then there exists the  operator Γ = (A0 + n1 γi , ·zi )−1 defined on R(Aˆ0 ) and one may introduce the notions of GJC and TGJS considering equation (1.2). In this case, the analytic operator A(λ) possesses a TGJS if it is continuously invertible for 0 < |λ| < , Γ(A(λ) − A0 ) ≤ q < 1 and the functions [I + Γ(A(λ) − A0 )]−1 ϕi , i = 1, . . . , n, are analytic at |λ| < . The last condition is always satisfied if λ0 is an isolated singular Fredholm point of the operator A(λ). GJS will be applied for the solutions of various problems in this book. As the first example, we show how it is possible to apply TGJS to the construction of the left and right regularizers of singular transformations. Let us introduce the sets Xt , Yt of abstract functions of argument t ∈ Rm with values in Banach spaces X and Y . Consider the linear operator Lt A0 − A1 acting from Xt to Yt . The operator Lt

8

Toward General Theory of Differential-Operator and Kinetic Models

(e.g., Lt = ∂/∂ti ) is commutative with linear operators A0 and A1 . If A0 is continuously invertible, then transformation Lt A0 − A1 is reduced to the form, solved over Lt , and it will be regular in this sense. Let A0 be Fredholm, dim N (A0 ) = n ≥ 1, and elements (j)

ϕi

= (ΓA1 )j−1 ϕi , (1)

i = 1, . . . , n,

(j)

ψi

= (Γ∗ A∗1 )j−1 ψi , (1)

j = 1, . . . , pi ,

generate TGJS of operators A0 and A∗0 , respectively. Then, we verify by a simple calculation that the following operators, respectively, pi n   (p +1−j) (1) Ljt ·, ψi i ϕi , Rl = Γ − i=1 j=1

Rr = Γ −

pi n   i=1 j=1

Lpt i +1−j ·, ψi ϕi (1)

(j)

satisfy identities Rl (Lt A0 − A1 ) = Lt − ΓA1 , (Lt A0 − A1 )Rr = Lt − A1 Γ. Since from the right we have obtained operators solved over Lt , then Rl and Rr are the left and right regularizers of the operator Lt A0 −A1 . 1.1.2.

GJC and GJS for unbounded linear operator

Let A(λ) be a closed linear operator defined for λ ∈ I with dense domain D(A) ⊂ X independent of λ, D(A) = X. Let, for every x ∈ D(A), the following representation hold:   q  k Ak (λ − λ0 ) + R(λ) x (1.7) A(λ)x = A0 − 1

in a neighborhood of the point λ0 , limλ→λ0 R(λ)x |λ − λ0 |−q = 0, and the operator A0 be Fredholm (see [169]). Then D(Aˆ−1 0 ) = Y, −1 ˆ the operator A(λ) · A0 : Y → Y is everywhere defined, closed and consequently, bounded, supλ A(λ)Aˆ−1 0 < ∞. Using this property, it is easy to note that the above statement concerning GJC and GJS remains correct.

Auxiliary Information on the Theory of Linear Operators

1.2.

9

Regularization of Linear Equations with Fredholm Operators

Let A ∈ L(X, Y ) be a Fredholm operator, dim N (A) = n ≥ 1, ϕi |n1 a basis for N (A), ψi |n1 a basis for N ∗ (A), and γi |n1 and zi |n1 the systems of functionals from X ∗ and of elements from Y such that γi , ϕj  = δij , ψi , zj  = δij , i, j = 1, . . . , n. ˜ f˜, ϕ˜i , ψ˜i , γ˜i , z˜i be δ-approximations of A, f, ϕi , ψi , γi , zi , Let A, respectively, i.e., A˜ − A ≤ δ, f˜ − f ≤ δ, etc.; δ is a maximal absolute calculation error and we need to calculate a solution x of the equation Ax = f

(1.8)

˜ = f˜. Ax

(1.9)

by the approximate equation

˜ then equation (1.9) may have no Since it is possible that f˜ ∈ / R(A), solutions. Even if a solution of (1.9) exists and is unique, δ → 0, we are not able to guarantee that x ˜ is convergent to at least one solution of equation (1.8). Thus, in the general case, the problem of construction of any solution of equation (1.8) is unstable. Under these specific conditions, this problem might be regularized either by the Tikhonov regularization method [162] or by the other methods indicated in [84, 99, 108, 117]. As a basis of the offered approach, similar to the Lavrentiev method [99], one sets a perturbation of equation (1.8) by the linear continuous operator function B(α), α ∈ [0, α0 ), B : [0, α0 ) → L(X, Y ). Let us introduce the auxiliary equation (A + B(α))x = f. Lemma 1.2. Let B(α) ≡

n  γi , ·zi . i=1

(1.10)

10

Toward General Theory of Differential-Operator and Kinetic Models

Then equation (1.10) has a unique solution x = Γf, where

−1 n  Γ= A+ γi , ·zi . i=1

If f ∈ R(A), then the element Γf satisfies equation (1.8). Lemma 1.3. Let B(α) = αB, where B is a continuous operator from X to Y, such that detψi , Bϕk |ni,k=1 = 0. Then equation (1.10) has a unique solution x(α) = (A + αB)−1 f

(1.11)

for 0 < α < α0 , moreover (A + αB)−1 = O(1/α). If f ∈ R(A), then x(α) → x∗ = Γf +

n 

ci ϕi

(1.12)

i=1

at α → +0, where (c1 , . . . , cn )T = −[ψi , Bϕk ]−1 d, d = (d1 , . . . , dn )T , di = ψi , BΓf , i = 1, . . . , n. The different approaches to the selection of the operator B, permitting one to regularize the formulated problem, correspond to Lemmas 1.2 and 1.3. 1. Let the elements z˜i and the functionals γ˜i be known. Therefore, if

A˜ − A

Γ < 1, then by Lemma 1.2 and the theorem on inverse operators, we obtain 

−1    n      A˜ + γi , ·zi ≤ Γ (1 − A˜ − A

Γ )−1 .     i=1 L(X,Y )

˜ We assume the calculation errors γi , zi to be included in A.

Auxiliary Information on the Theory of Linear Operators

11

Remark 1.2. Let X and Y be Hilbert spaces. Then, one may seek a solution Γf by the method of successive approximations xn = (I − αA∗ A)xn−1 + αA∗ f + δn , x0 = 0, n = 1, . . . , N (δ), supn δn < δ, for the number of iterations N to adjust with the error of calculations δ and 0 < α < 2/( A 2 ). 2. Now, let the elements z˜i and the functionals γ˜i be unknown, but it is known that detψi , Bϕk |ni,k=1 = 0,

(1.13)

where B is a fixed continuous operator from X to Y . If | detψ˜i , B ϕ˜k | ≥ c > 0, c is a constant independent of δ, which is sufficiently small, then inequality (1.13) is satisfied. Therefore, by Lemma 1.3 and theorem of inverse operator, the formula x ˜ = (A˜ + αB)−1 f˜ determines regulating algorithm (RA) for computation of solution (1.12) of equation (1.8). Here, we need to adjust the parameter α with the value of error δ(δ/α(δ) → 0 at δ → 0) in a reasonable way. More precisely, the following assertion holds. Theorem 1.2. (1) Let A be a Fredholm operator and B be an arbitrary linear continuous operator such that ˜k | ≥ c > 0. | detψ˜i , B ϕ (2) Let α0 (δ) and β(δ) be the continuous positive functions monotonically tending to zero at δ → 0 such that δ/β(δ) ≤ α0 (δ), and α be an arbitrary number satisfying inequalities δ ≤ α ≤ α0 (δ), β(δ)

α > 0.

Then, the equation (A˜ + αB)x = f˜

(1.14)

Toward General Theory of Differential-Operator and Kinetic Models

12

possesses a unique solution x ˜ for sufficiently small δ. If, moreover, f ∈ R(A), then the following estimation holds:

˜ x − x∗ ≤ c1 δ + c2 α0 + c3 β + r(δ, α0 , β).

(1.15)

Here, x∗ is a solution (1.12) of equation (1.8), c1 , c2 , c3 the constants, and r(δ, α0 , β) = o[(δ + α0 + β)2 ]. Proof. By the conditions (1) and (2), the existence and uniqueness solution x ˜ will follow from Lemma 1.3 and the theorem on inverse operator if δ is sufficiently small. The fulfillment of estimation (1.15) remains to be controlled. Putting x ˜ = x∗ + u in equation (1.14), we obtain u = u ˆ(α) + u ˇ(α), where u ˆ(α) = D(α)Γ( f − αBx∗ − Ax∗ ), u ˇ(α) = D(α)

n 

(1.16)

ξk (α)ϕk ,

1

ξk (α) = γk , u,

k = 1, . . . , n.

(1.17)

In formulas (1.16) and (1.17), we introduce the following notations: D(α) = (I + αΓB + Γ A)−1 ,

f = f˜ − f,

A = A˜ − A.

The functions ξ1 (α), . . . , ξn (α) are defined from the linear algebraic equations n 

aik (α)ξk = bi (α),

i = 1, . . . , n,

k=1

where aik (α) = ψi , (αB + A)D(α)ϕk , bi (α) = γi , D(α)Γ( f − αBx∗ − Ax∗ ), ψi , Bx∗  = 0, i, k = 1, . . . , n, and we may consider without loss of generality that ψi , Bϕk  = δik . Thus, ([aik (α)] = A(α)): A−1 (α) = α−n [E + α−n (A(α) − αn E)]−1 = [cik (α)|ni,k=1 ,

Auxiliary Information on the Theory of Linear Operators

where 1 cik (α) = α



13

 1 δik − ψi , Aϕk − αBΓBϕk − AΓBϕk + rik , α

|rik | = O[(δ + α0 + β)2 ]. Hence, ξi =

n 

cik (α)bk (α)

k=1

=

1 ψi , f − Ax∗ − αBΓ(−αBx∗ + f − Ax∗ ) α n  ∗ ψi , Aϕk ψi , −BΓBx∗  + α AΓBx  + k=1

+

n 

ψi , BΓBϕk ψk , f − Ax∗  + O[(δ + α0 + β)2 ].

k=1

(1.18) Since u + ˇ u

˜ x − x∗ ≤ ˆ ≤ (1 − α ΓB − δ Γ )−1  ∗

∗

× Γ (δ + α Bx + δ x ) + δ ≤ β, α

n 

 |ξk | ϕk ,

k=1

α ≤ α0 ,

then, finally,

˜ x − x∗ ≤ c1 δ + c2 α0 + c3 β + O[(δ + α0 + β)2 ], where c1 , c2 , c3 are the constant values. The theorem is proved. 1.2.1.



Stabilizing operator and error estimation improvement

The operator B satisfying the conditions of Theorem 1.2 plays the same role as a stabilizing functional in the Tikhonov functional [162]. Therefore, we can call it the stabilizing operator. We provide the

14

Toward General Theory of Differential-Operator and Kinetic Models

precise definition of stabilizing operator suitable in the following more general case. Lemma 1.3 and Theorem 1.2 are generalized to the case of arbitrary stabilizing operator with a total B-Jordan set from null space of A- and B-adjoined operators. However, if a total B-Jordan set contains adjoined vectors, then the error of approximation will be more because the value of regularization parameter α grows with increasing lengths of B-Jordan chains of operator A. Let us consider in detail the estimation (1.15). If c2 > 0 and c3 > 0, then min(c1 δ + c2 α0 + c3 β + r(δ, α0 , β)) = 2(c2 c3 δ)1/2 + O(δ),  δ α0 , β ∈ D = α0 > 0, β > 0, ≤ α0 , β and a minimum is reached at the point   c3 1/2 c2 1/2 δ , β≈ δ . α0 ≈ c2 c3 √ Hence, ˜ x − x∗ = O( δ) in the general case. Nevertheless, the following result holds. Corollary 1.1. Assume that the conditions of Theorem 1.2 are ˜− are solutions of equation (1.14) in satisfied, f ∈ R(A), x˜+ and x which we have the operators +B and −B, respectively. Then, the element x ˜=

1 (˜ x+ + x ˜− ) 2

(1.19)

satisfies the estimation

˜ x − x∗ ≤ c1 δ + r1 (δ, α0 , β),

(1.20)

moreover, r1 (δ, α0 , β) = O[(δ + α0 + β)2 ]. Proof. In order to carry out the proof, we consider the following inequality: 1 1 u(α) + u ˆ(−α) + ˇ u(α) + u ˇ(−α) .

˜ x − x∗ ≤ ˆ 2 2

(1.21)

Auxiliary Information on the Theory of Linear Operators

15

Owing to (1.16) and (1.17), we have

ˆ u(α) + u ˆ(−α) ≤ 2 Γ ( f + Ax∗ ) + r1 (α),

(1.22)

ˇ u(α) + u ˇ(−α) ≤ 2 Γ

B na2 (1 + na2 B )(1 + 2 x∗ )δ + r2 (α), (1.23) where r1 (α) = O[(α0 + δ)2 ],

r2 (α) = O[(δ + α0 + β 2 ].

Combining (1.21), (1.22) and (1.23), one obtains the estimation (1.20). The corollary is proved.  Therefore, the transformation (1.19) improves √ the approximation of the desired solution x∗ . Indeed, if β = α0 = δ, then   √  1 ∗ ∗  (˜ x + x ˜ ) − x − x

= O( δ). (1.24) = O(δ),

˜ x + − ±  2 ˜− , then this indicates that the estimation (1.24) following If x ˜+ = x from (1.15) is found to be crude. The transformation similar to (1.19) has been employed by [49] to the approximation improvement of solutions for the ill-posed systems of linear algebraic equations. With the help of Theorem 1.2, one can obtain a known result of Tikhonov about the computation of normal solution Γf of equation (1.8), when X and Y are Hilbert spaces and A is a Fredholm operator. Corollary 1.2. Let the condition (1) of Theorem 1.2 hold, X and Y be Hilbert spaces and f ∈ R(A). Let α0 (δ) and β(δ) be some continuous positive functions monotonically tending to zero at δ → 0 such that δ2 /β(δ) ≤ α0 (δ), and α > 0 be an arbitrary number satisfying the inequalities δ2 ≤ α ≤ α0 (δ). β(δ)

(1.25)

x = A˜∗ f˜ (A˜∗ A˜ + αI)˜

(1.26)

Then, the equation

16

Toward General Theory of Differential-Operator and Kinetic Models

has a unique solution x ˜ and the following estimation holds:

˜ x − Γf ≤ c(1) δ + c(2) α0 + c(3) β + r2 (δ, α0 , β),

(1.27)

where c(1) , c(2) , c(3) are constants, and r2 (δ, α0 , β) = O[(δ + α0 + β)2 ]. Since equation (1.26) represents a special case of equation (1.14), then the proof follows from Theorem 1.2. But, by virtue of the specific character of equation (1.10), one may define the estimations of number values more exactly. Indeed, now, x∗ = Γf . In formula (1.18), for finding ξi instead of the term 1 ψi , f − Ax∗ , α which can have a singularity at α → 0 (this term requires the coordination of parameter α with a value of error δ), there will be the term 1 ϕi , A˜∗ f˜ − A∗ f − (A˜∗ A˜ − A∗ A)Γf . α Since ˜ ), A˜∗ f˜ − A∗ f − (A˜∗ A˜ − A∗ A)Γf = A˜∗ (f˜ − AΓf    1 ˜ )  ϕi , A˜∗ (f˜ − AΓf  α    1 =  (A˜ − A)ϕi , f + (f˜ − f ) − AΓf − (A˜ − A)Γf  α 2 δ , =O α then in the proof of Theorem 1.2, we can change δ/α into δ2 /α. As a result, we obtain the estimation (1.27). If α = α0 = β = δ, then

˜ x − Γf = O(δ). 1.3.

Principal Theorem of Regularization of Linear Equations by the Perturbation Method

Let A be a closed linear operator with dense domain D in a Banach space X and with range R(A) in a Banach space Y .

Auxiliary Information on the Theory of Linear Operators

17

Let us consider the equation Ax = f,

f ∈ R(A).

(1.28)

We do not assume that N (A) = {0}, i.e., the uniqueness of solution of equation (1.28). During computations, we need to consider the approximate equation ˜ = f˜. Ax

(1.29)

Let A˜ and f˜ be δ-approximations of A and f , respectively, which we understand in the following sense (a > 0): ˜ − Ax ≤ δ( x + a Ax )

Ax

(1.30)

f˜ − f ≤ δ.

(1.31)

for all x ∈ D, and

If A ∈ L(X, Y ), then D = X, and one can put a = 0. In the general case, the construction problem of some solution for equation (1.28) is unstable. In Section 1.1, we considered a special case in which A is a bounded Fredholm operator. In this section, we consider the general case also by means of the perturbation of an equation by the operator function B(α), B : [0, α0 ) → L(X, Y ). More precisely, along with equation (1.28), the equations (A + B(α))x = f,

(1.32)

(A + B(α))x = f˜,

(1.33)

(A˜ + B(α))x = f˜

(1.34)

are considered. The operator function B(α) and the parameter α = α(δ) are selected such that A + B(α) are continuously invertible: (A + B(α))−1 ∈ L(X, Y ). Often, it is possible to accept B(α) = B0 + αB1 with B0 = 0 or B1 = 0. In equation (1.34), we can always ˜ include the calculation errors of B(α) to the operator A. Lemma 1.4. Let x∗ be a certain solution of equation (1.28). In order that a solution of equation (1.32) x(α) → x∗ at α → +0, it

18

Toward General Theory of Differential-Operator and Kinetic Models

is necessary and sufficient that S(α, x∗ ) = (A + B(α))−1 B(α)x∗ → 0

(1.35)

at α → +0. The proof follows from the identity (A + B(α))−1 f − x∗ = −(A + B(α))−1 B(α)x∗ . Definition 1.7. The condition (1.35) is referred as the condition of stabilization, the operator B satisfying (1.35) as the stabilizing operator, and a solution x∗ (obviously unique) as the B-normal solution. Lemma 1.5. Let x(α) and x ˇ(α) be the solutions of equations (1.32) and (1.33), respectively. If α = α(δ) > 0 is selected such that α(δ) → +0 and δ A + B(α))−1 → 0

(1.36)

at δ → 0, then x ˇ(α) − x(α) → 0 at δ → 0. The proof follows from the estimation (1.31) and the inequality

x(α) − x ˇ(α) ≤ (A + B(α))−1

f˜ − f . Definition 1.8. The condition (1.36) is called the coordination condition of the regularization parameter α with the calculation error. Lemma 1.6. Let us assume α(δ) → +0 at δ → 0 and the condition (1.36) holds. Let us fix q ∈ (0, 1) and find δ0 > 0 such that an inequality aδ + (1 + a B(α) )δ (A + B(α))−1 ≤ q will be satisfied at δ ≤ δ0 . Then, the operator A˜ + B(α) is continuously invertible and the estimations

(A˜ + B(α))−1 ≤

1

(A + B(α))−1 , (1 − q)

(1.37)

Auxiliary Information on the Theory of Linear Operators

19

(A˜ + B(α))−1 f ≤ (A + B(α))−1 f + δ(A + B(α))−1 (1 − q)−1 × {a f + (1 + a B(α) ) (A + B(α))−1 f } (1.38) are valid. Proof.

For any f ∈ Y , by inequality (1.30), we obtain

(A˜ − A)(A + B(α))−1 f

≤ δ{ (A + B(α))−1 f + a (A + B − B)(A + B(α))−1 f } ≤ δ{a f + (1 + a B(α) ) (A + B(α))−1 f }.

(1.39)

Therefore,

(A˜ − A)(A + B(α))−1 ≤ q,

(A˜ − A)(A + B(α))−1 ∈ L(Y, Y ).

If we now consider that q < 1, A˜ + B(α) = [I + (A˜ − A)(A + B(α))−1 ](A + B(α)), then the existence and the boundedness of (A˜ + B(α))−1 and also the estimation (1.37) follow from the theorem on inverse operator. Furthermore, having used the identities C −1 = D−1 − D−1 [I + (C − D)D−1 ]−1 (C − D)D−1 , C = A˜ + B(α),

D = A + B(α),

(1.40)

and the inequality (1.39), we arrive at the estimation (1.38). The  lemma is proved. Principal Theorem. Let the conditions of Lemma 1.6 hold. Then equation (1.34) has a unique solution x ˜. If, moreover, the element x∗ satisfies equation (1.28), then the following estimation is valid:

˜ x − x∗ ≤ S(α, x∗ ) + δ (A + B(α))−1 (1 − q)−1 {1 + a f + x∗

+ a B(α)x∗ + (1 + a B(α) )S(α, x∗ )}, where S(α, x∗ ) is defined by formula (1.35).

(1.41)

20

Toward General Theory of Differential-Operator and Kinetic Models

If, furthermore, x∗ is B-normal solution, then x ˜ → x∗ at δ → 0 with a degree of convergence evaluated by the estimation (1.41). If A, B ∈ L(X, Y ), then x ≤ ˜ x − x∗

A −1 f − A˜ ≤ S(α, x∗ ) + (1 + x∗ + S(α, x∗ )) ×

δ

(A + B(α))−1 . (1 − q)

Proof. Existence and uniqueness of solution x ˜ have been proved in Lemma 1.6. Since (A˜ + B(α))(˜ x − x∗ ) = f˜ − f − (A˜ − A)x∗ − B(α)x∗ , then by (1.37) and (1.38),

˜ x − x∗ ≤ (A˜ + B(α))−1 { f˜ − f

+ (A˜ − A)x∗ } + (A˜ + B(α))−1 B(α)x∗

≤ (1 − q)−1 (A + B(α))−1 (δ + δ( x∗ + a Ax∗ )) + S(α, x∗ ) + δ (A + B(α))−1 (1 − q)−1 {a B(α)x∗

+ (1 + a B(α) )S(α, x∗ )}, where Ax∗ = f . The theorem is proved.



Let us consider applications of the principal theorem to the special classes of equations. Let A be a Fredholm operator. According to Schmidt’s lemma, there exists the bounded operator −1  n  γi , ·zi Γ= A+ i=1

given on Y , and the function x = Γf +

n 

ci ϕi

i=1

is a general solution of equation (1.28), where ci represent the arbitrary constants. Let B(α) = B0 + αB1 . At first, let n  γi , ·zi , B1 = 0. B0 = i=1

Auxiliary Information on the Theory of Linear Operators

21

Then the stabilization condition (1.35) has the form (A + B0 )−1 B0 x∗ = 0, i.e., γi , x∗  = 0, i = 1, . . . , n. If f ∈ R(A), then γi , Γf  = 0, i = 1, . . . , n and B0 -normal solution is a x∗ = Γf -normal solution in terms of Tikhonov. Now, let detψi , B1 ϕk |ni,k=1 = 0.

B0 = 0,

Then the element (1.12) will be αB1 -normal solution with condition of stabilization (1.35) of the form

(A + αB1 )−1 αB1 x∗ = O(α) because B1 x∗ ∈ R(A). Here,

(A + αB1 )−1 = O(1/α). Consequently, for the case of Fredholm operators, it follows from the principal theorem. Corollary 1.3. Let A be the Fredholm operator, f ∈ R(A),

B(α) ≡

n 

γi , ·zi ,

γi = zi = 1,

i = 1, . . . , n.

1

Then equation (1.28) has a normal solution x∗ = Γf . If δ ≤ q{a + (1 + na) Γ }−1 ,

0 < q < 1,

then a unique solution of equation (1.34) is defined by the formula

−1 n  f˜ γi , ·zi x ˜ = A˜ + i=1

and satisfies the estimation

˜ x − Γf ≤ (1 − q)−1 Γ (1 + a f + Γf )δ. Since S(α, x∗ ) ≡ 0, then the proof directly follows from the estimation (1.41).

22

Toward General Theory of Differential-Operator and Kinetic Models

Corollary 1.4. Let A be the Fredholm operator, f ∈ R(A), B(α) = αB1 , where B1 ∈ L(X, Y ) and √ detψi , B1 ϕk |ni,k=1 = 0, α = δ. Then equation (1.28) has a αB1 -normal solution, the conclusions of principal theorem are true, moreover, √ (1.42)

˜ x − x∗ = O( δ). In Section 1.2, for bounded Fredholm operators, the results of Corollaries 1.3 and 1.4 are obtained. Let the operator A not be a Fredholm one, but N (A) = {0} and f ∈ R(A), c

(A + αB)−1 ≤ . (1.43) α Let Bx∗ ∈ R(A),

(1.44)

where c is a constant, x∗ = A−1 f . Then S(α, x∗ ) = α(A + αB)−1 (A + αB − αB)A−1 BA−1 f

≤ A−1 BA−1 f (1 + c B )α and the element x∗ is a αB-normal solution. 1.4.

Regularization of Linear Equations Based on the Perturbation Theory in Hilbert Spaces

In Section 1.3, we have offered the regularization of linear equations (1.28) based on the perturbation theory. Regulating algorithms of that method have the form x ˜α = (A˜ + B(α))−1 f˜

(1.45)

if B(α) is the stabilizing operator (SO). In this section, the construction problem of SO in Hilbert space is solved for equation (1.28). 1. Let us consider the question on regularization of linear equation with a self-adjoint operator. Let A be self-adjoint operator from

Auxiliary Information on the Theory of Linear Operators

D ⊂ H to H, generally speaking, unbounded:  ¯ D = H, H = H+ ⊕ H− ⊕ H0 , A =

+∞

−∞

λdpλ ,

23

(1.46)

where H+ and H− represent the invariant pairwise orthogonal subspaces of operator A, N (A) = N (A) = H0 , pλ ≡ p−r for λ ∈ [−r, 0). Let (Ax, x) > 0 for all x ∈ H+ ∩ D,

x = 0,

(1.47)

(Ax, x) ≤ −r(x, x) for all x ∈ H− ∩ D. Then Ax = A+ x+ + A− x− for all x ∈ D. Here,



A− =

−r −∞

 x− = P− x =

−r −∞

 λdpλ , dpλ x,

A+ =

+∞ 0

(1.48)

λdpλ , 

x+ = P+ x =

0

+∞

dpλ x,

(1.49)

i.e., P− and P+ are projectors onto H− and H+ , R(A− ) = H− , 0 R(A+ ) ⊆ H+ . Let P0 = −r dpλ be a projector H onto N (A). Then x = x+ + x− + x0 x− = P− x,

for all x ∈ H,

x+ = P+ x,

x0 = P0 x.

Example 1.2. Let A ∈ L(H, H) be a completely continuous selfadjoint operator with a finite number of negative eigenvalues. We assume that N (A) = {0}. Then, the spectral integral of operator A represents a series  +∞ λdpλ x Ax = −∞

=

−1 

λi x, ei ei +

i=−n

= A− x− + A+ x+

+∞ 

λi x, ei ei + P0 x

i=1

for all x ∈ H,

(1.50)

Toward General Theory of Differential-Operator and Kinetic Models

24

where −1 

A− =

λi ·, ei ei , A+ =

i=−n

ei , ej  = δij ,

+∞ 

λi ·, ei ei ,

i=1

i, j = −n, −n + 1, . . . , −1, +1, +2, . . . ,

λ−n ≤ λ−n+1 ≤ · · · ≤ λ−1 = −r < 0, λ1 ≥ λ2 ≥ · · · ≥ λn ≥ · · · > 0, inf

i=+1,+2,...

P+ =

λi ≥ 0,

{e−n , e−n+1 , . . . , e−1 } = H− , {e1 , e2 , . . .} = H+ ,

+∞ −1   ·, ei ei , P− = ·, ei ei , R(A) ⊆ R(A+ ) ⊕ R(A− ), i=−n

1

R(A+ ) ⊆ H+ ,

R(A− ) = H− , N (A) = (H+ ⊕ H− )⊥ = H0 .

If {ϕ , ϕ , . . .} is an orthonormal basis in H0 , then P0 +∞ 1 2 1 ·, ϕi ϕi .

=

Lemma 1.7. Let A be a self-adjoint operator admitting the spectral resolution of the form (1.48)–(1.49). Then    +∞ λ−2 d(pλ f, f ) < +∞ , (a) R(A) = f : f ∈ N (A)⊥ ,   k (b) R(A+ ) = f : f ∈ H+ ,

0

+∞ 0

−2k

λ

 d(pλ f, f ) < +∞

for all k ∈ (−∞, +∞). If f ∈ R(A), then the function  −r  +∞ 1 1 dpλ f + dpλ f x∗ = λ λ −∞ 0 is a unique solution of equation (1.28) in N (A)⊥ . Proof.

Thanks to (1.50), one has the identity Ax = A+ x+ + A− x−

for all x ∈ D. Since f = f+ + f− + f0 = P+ f + P− f + P0 f, R(A) ⊆ R(A+ ) ⊕ R(A− ) ⊆ H+ ⊕ H− = N (A)⊥ ,

(1.51)

Auxiliary Information on the Theory of Linear Operators

25

then one needs f0 = 0 if equation (1.28) is solvable. Let equation (1.28) be solvable. Then (1.28) is equivalent to the system  −r A− x− ≡ λdpλ x− = f− , (1.52) −∞

 A+ x+ ≡

+∞

0

λdpλ x+ = f+ ,

(1.53)

where A−1 − ∈ L(H− , H− ), f− ∈ H− and hence,  −r 1 ∗ dpλ f− . (1.54) x− = −∞ λ  +∞ Since Ak+ = 0 λk dpλ , then    +∞ k −2k λ d(pλ f, f ) < +∞ R(A+ ) = f : f ∈ H+ , 0

for k ∈ (−∞, +∞). Consequently, x∗+ = 

+∞ 0

 +∞ 0

1/λdpλ f+ if

λ−2 d(pλ f+ , f+ ) < +∞, x∗ = x∗− + x∗+ .

(1.55)

Therefore, the conclusions (a) and (b) of Lemma 1.7 are valid, and identity (1.51) follows from the formulas (1.54) and (1.55). The  lemma is proved. Theorem 1.3. Let A be a self-adjoint operator admitting a spectral resolution of the form (1.48)–(1.49), the conditions (1.30) and (1.31) are satisfied, δ/α(δ) → +0 at δ → +0. If, moreover, f ∈ R(A), then the operator B(α) ≡ αI (α > 0) is a stabilizing operator, and the algorithm (1.45) is a regulating algorithm for the computation of normal solution (1.51) of equation (1.28). Proof. For proof, it is enough to control the fulfillment of conditions (1.35) and (1.36). Since (A + αI)−1 x = (A+ + αP+ )−1 x+ + (A− + αP− )−1 x− +

1 x0 α

Toward General Theory of Differential-Operator and Kinetic Models

26

for all x ∈ H, then

(A + αI)−1 L(H,H) = O(1/α) if 0 < α < r. Therefore, δ (A + αI)−1 → 0 at δ → 0, if δ/α → 0 when δ → 0,  −r  +∞ 1 1 −1 ∗ ∗ (A + αI) αx = α dpλ x− + α dpλ x∗+ , λ+α −∞ λ + α 0 P0 x∗ = 0. Thus, S(α, x∗ ) = (A + αI)−1 αx∗

 +∞    1 α ∗ ∗

x− + α  dpλ x+  ≤  , | − r + α| λ+α 0

where   2 α 

0

+∞

2  +∞  1 α2 ∗ dpλ x+  = d(pλ x∗+ , x∗+ ) ≤ x∗+ 2 2 λ+α (λ + α) 0

for any α > 0 because α/λ + α ≤ 1. Hence, given  > 0, there are a() > 0 and A() < +∞ (A() > a()) such that  a()  +∞  α2 α2 ∗ ∗ d(p x , x ) + d(pλ x∗+ , x∗+ ) ≤ , J1 = λ + + 2 2 (λ + α) (λ + α) 2 0 A() independent of α. Therefore, there exists α0 depending on , a() and A() such that  A()  α2 d(pλ x∗+ , x∗+ ) ≤ J2 = 2 2 a() (λ + α) for 0 < α ≤ α0 = b(, a(), A()). Then  +∞ 2   1 2 ∗ dpλ x+  ≤ J1 + J2 ≤  α  λ+α 0 for 0 < α ≤ α0 . Therefore, S(α, x∗ ) → 0 at α → +0. The theorem is  proved.

Auxiliary Information on the Theory of Linear Operators 1+q)

Remark 1.3. If P+ f ∈ R(A+    

+∞

−1

(λ + α)

0

27

), q > 0, then

2 

dpλ x∗+  

 2   ≤ α−2+2q A−1−q f + , +

and by the estimation (1.41), ˜ xα − x∗ ≤ c(α, δ), where c(α, δ) = c1 α + c2 αq + c3

δ + c4 δ + c5 δα + c6 δαq−1 + c7 δαq , α

c1 , . . . , c7 are constants, moreover, minα c(α, δ) is reached at the point    c3 1/1+q 1/1+q + o(1) . α ¯=δ qc2  +∞ 2. Now, in equation (1.28), let A = −∞ λdpλ be the arbitrary selfadjoint operator, the element  +∞ ∗ λ−1 dpλ f ∈ {H  N (A)}, x = −∞

and it satisfies (1.28) (the point λ = 0 is eliminated by integration for N (A) = {0}). For computation of the solution of (1.28), we construct the RA of the form (1.45). Let us here put  +∞ √ (a(λ, α) − λ + b(λ, α) −1)dpλ , 0 ≤ α < α0 , B(α) = −∞

where a and b are some real functions of λ and α, bounded and measurable by all measures generated by the functions σ(λ) = (pλ x, x), x ∈ X. Let (1) supλ {λ2 (a2 + b2 )−1 , (a2 + b2 )−1 } = k(α) < +∞ for α = 0, (2) there exist constants r and c such that supλ,α A(λ, α) ≤ c < +∞, where A(λ, α) = (a2 +b2 )−1 [(a−λ)2 +b2 ],

α ∈ [0, α0 ),

λ ∈ (−r, +r),

(3) A(λ, α) → 0 at α → +0, λ ∈ (−∞, +∞)/[−ρ, +ρ] uniformly in λ for every ρ ∈ (0, r].

28

Toward General Theory of Differential-Operator and Kinetic Models

Then S(α, x∗ ) =



+∞ −∞

λ−2 A(λ, α)d(pλ f, f ) → 0

at α → +0, B(α) is SO, and x∗ is a B-normal solution. If, furthermore, α(δ) is selected so that δ2 K(α) → 0 at δ, α → 0, then (1.45) is RA owing to the principal theorem on regularization by the perturbation method. Note that formula (1.45) determines RA and when (a2 + b2 )−1 [(a − λ)2 + b2 ] = c(λ, α) + D(λ, α),  +∞ λ−2 D(λ, α)d(pλ f, f ) = 0, −∞

where the function c(λ, α) satisfies conditions (2) and (3) and δ2 k(α) → 0 at δ, α → +0. Theorem 1.4. Let A be a self-adjoint nonnegative operator, and B be a self-adjoint positive definite operator, i.e., Ax, x ≥ 0, Bx, x ≥ γx, x for any x ∈ H, where γ is a constant. Let max( A˜ − A , f˜ − f ) ≤ δ, δ/α → +0 at δ → +0. Then, the equation (A˜ + αB)x = f˜,

α > 0,

(1.56)

has a unique solution x ˜α . If, moreover, the equation (1.28) has a ∗ ∗ solution x with Bx ∈ N (A)⊥ , then x∗ is a B-normal solution, and algorithm (1.45) is the RA of this solution. Proof. Since (A + αB)x, x ≥ αγx, x, then (A + αB)−1 L(H,H) ≤ 1/αγ and (1.56) has a unique solution x ˜α . Denote by pλ the spectral function of operator A + αB. Then S(α, x∗ ) = (A + αB)−1 αBx∗

1/2  A+αB  2 α d(pλ Bx∗ , Bx∗ ) . = λ 0 Since λ ≥ αγ, Bx∗ ∈ H  N (A), then just as in the case of the previous theorem, it is easy to show that S(α, x∗ ) → 0 at α → +0.  The theorem is proved.

Auxiliary Information on the Theory of Linear Operators

29

Remark 1.4. Let A = A∗ , f ∈ R(A), and one of the conditions hold: (1) N (A) = {0}; (2) ϕi |n1 is a basis for N (A) and detBϕi , ϕk |ni,k=1 = 0. Then equation (1.28) has a solution x∗ such that Bx∗ ∈ N (A)⊥ . Theorem 1.4 allows the generalization for closed, densely selfadjoint operators A and B. Corollary 1.5. Let, in the conditions of Theorem 1.4 with A = K ∗ K, K ∈ L(H, H), x∗ be a unique solution of equation Kx = y, x∗ ∈ D(B). Then algorithm (1.45) is the RA of this solution. 1 Example 1.3. Let the equation 0 K(t, s)x(s)ds = y(t) have a ˚ 1 , where unique solution x∗ (t) ∈ W 2  1 1 K 2 (t, s)dsdt < ∞, y(t) ∈ L2 [0, 1]. 0

0

Then, from Corollary 1.5, the known result of Tikhonov [162] follows: as the regularized solution, one can take the solution of equation      1 d dx q1 (t) − q0 (t)x = f˜(t), A(t, s)x(s)ds − α dt dt 0 x(0) = x(1) = 0, where q1 (t) > 0, q0 (t) ≥ 0, q1 (t), q0 (t) ∈ C[0,1] ,  1  1 ˜ ˜ ˜ ˜ t)˜ K(ξ, t)K(ξ, s)dξ, f (t) = K(ξ, y (ξ)dξ. A(t, s) = 0

0

3. For the equation with no self-adjoint operators, we give one result. Theorem 1.5. (1) Let X = H1 , Y = H2 , A be closed linear operator from D ⊂ H1 ¯ = H1 , and N (A) be a null-subset of operator A. into H2 , D (2) Let (A + αB)−1 L(H2 ,H1 ) ≤ c/α, where B ∈ L(H1 , H2 ), c and a are constant values. (3) Let the conditions (1.30) and (1.31) hold, δ/α → +0 at δ → +0. (4) Let equation (1.28) have a solution x∗ , Bx∗ ∈ R(A).

30

Toward General Theory of Differential-Operator and Kinetic Models

Then x∗ is an αB-normal solution of equation (1.28), αB is the stabilizing operator, and formula (1.45) determines the RA of computation of this solution with α > 0. Proof. Since N (A) is a subspace in H1 , then the contraction Aˆ of operator A onto H1  N (A) ∩ D has an inverse operator Aˆ−1 defined on R(A) (may be unbounded). Hence, AˆAˆ−1 f = AAˆ−1 f = f if f ∈ R(A). Since Bx∗ ∈ R(A), then (A + αB)−1 αBx∗ = (A + αB)−1 αAAˆ−1 Bx∗ = α(A + αB)−1 (A + αB)Aˆ−1 Bx∗ − α2 (A + αB)−1 B Aˆ−1 Bx∗ . Thus, S(α, x∗ ) ≤ α(1 + c B ) Aˆ−1 Bx∗

and S(α, x∗ ) → 0 at α → +0. The theorem is proved.



Corollary 1.6. Let X = Y = H, B = I, A a Fredholm operator, ϕi |n1 a basis for N (A), ψi |n1 a basis for N ∗ (A), A with no adjoint vectors, ψi , f  = 0, i = 1, . . . , n, and (ψi , ϕj  = δij , i, j = 1, . . . , n). Then

−1 n  ∗ ψi , ·ϕi f, x = A+ i=1

αI is a normal solution, αI is SO, and equation (1.45) is RA. Proof. Since detψi , ϕk |ni,k=1 = 0, then (A + αI)−1 = O(1/α). Furthermore, Γk f ∈ R(A), k = 0, 1, . . . because Γ∗ ψi = ψi , i = 1, . . . , n, where

−1 n  ψi , ·ϕi . Γ= A+ i=1

x∗

∈ R(A). Consequently, all conditions of Theorem 1.5 Therefore,  are valid and the corollary is proved.

Auxiliary Information on the Theory of Linear Operators

1.5.

31

Regularization with Vector Regularizing Parameter for the First Kind Equations

One of the most common problems of scientific applications involves the computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by the application of perturbation method to linear first kind equations Ax = f with bounded operator A. We assume that we ˜ know the operator A˜ and source function f˜ only such that ||A−A|| ≤ ˜ + B(α)x = f˜ possesses δ, ||f˜ − f || < δ. The regularizing equation Ax the unique solution. In contrast with Section 1.4 where α ∈ R1 , now let α ∈ S, S is assumed to be an open space in Rn , 0 ∈ S, α = α(δ). As result of this generalization, we suggest a novel algorithm, providing accurate results even in the presence of a large amount of noise. Let A be bounded operator in Banach space X with range R(A) in a Banach space Y. Let us consider the following linear operator equation: Ax = f,

f ∈ R(A).

(1.57)

We assume domain R(A) can be nonclosed and Ker A = {0}. In many practical problems, one needs to solve an approximate equation ˜ = f˜, Ax

(1.58)

instead of exact equation. Here, A˜ and f˜ are approximations of exact operator A and right-hand side function f , respectively, such that ||A˜ − A|| ≤ δ1 ,

||f˜ − f || ≤ δ2 ,

δ = max{δ1 , δ2 }.

(1.59)

Let us address the regularized processes construction by the introduction of the following perturbed equation: Axα + B(α)xα = f.

(1.60)

It is to be noted that the regularization method based on perturbed equation was first proposed by Lavrentiev in the case of completely

32

Toward General Theory of Differential-Operator and Kinetic Models

continuous self-adjoint and positive operator A and B(α) ≡ α, α ∈ R1 . We select the stabilizing operator (SO) B(α) to make the solution xα unique and provide computation stability. Let us call α ∈ S ⊂ Rn the vector parameter of regularization. Here, S is an open set, zero belongs to the boundary of this set (briefly, S-sectoral neighborhood of zero in Rn ), limSα→0 B(α) = 0. Parameter α is adjusted to the data error level δ. Previously, only the simple case was addressed with B(α) = B0 + αB1 , α ∈ R+ . Such SO was employed in the development and justification of iterative methods of Fredholm points λ0 calculation, zeros and the elements of the generalized Jordan sets of operator functions, for the construction of approximate methods in the theory of branching of solutions of nonlinear operator equations with parameters, of construction of solutions of differential-operator equations with irreversible operator coefficient in the main part. Here, we propose the novel theory for operator system regularization. First, we obtained sufficient conditions when the perturbed equation (1.60) enables a regularization process. Let us formulate and prove the fundamental theorem of regularization with vector parameter by using the perturbation method. Then, the choice of SO B(α) will be made. An important role is played by a classic Banach–Steinhaus theorem. Apart from equations (1.57), (1.58), and (1.60), let us introduce the equations (A + B(α))x = f˜, (1.61) (A˜ + B(α))x = f˜.

(1.62) ˜ Operator B(α) errors can always be included into operator A. Equation (1.62) is called regularized equation (RE) for the problem (1.58). The following estimates are assumed to be fulfilled: ||(A + B(α))−1 || ≤ c(|α|),

(1.63)

||B(α)|| ≤ d(|α|),

(1.64) Rn ,

0 ∈ S, where c(|α|) is a continuous function, α ∈ S ⊂ ∗ lim|α|→0 c(|α|) = ∞, lim|α|→0 d(|α|) = 0. If x is the solution to

Auxiliary Information on the Theory of Linear Operators

33

equation (1.57), then (A + B(α))−1 f − x∗ = (A + B(α))−1 B(α)x∗ . Therefore, we have the following lemma. Lemma 1.8. Let x∗ be some solution to equation (1.57), and x(α) satisfy and equation (1.60). Then, in order to arrive at xα → x∗ for S  α → 0, it is necessary and sufficient to have the following equality fulfilled: S(α, x∗ ) = ||(A + B(α))−1 B(α)x∗ || → 0

for S  α → 0.

(1.65)

Definition 1.9. We call the condition (1.65) a stabilization condition, operator B(α), a stabilization operator if it satisfies the condition (1.65), and solution x∗ a B-normal solution of equation (1.57). Obviously, the limit of the sequence {xα } is a unique one in normed space, and therefore, equation (1.57) can have only one Bnormal solution. From estimates (1.63) and (1.64), we have the following lemma. ˆα be the solutions of equations (1.60) and Lemma 1.9. Let xα and x (1.61), respectively. If parameter α = α(δ) ∈ S is selected, such that δ→0 |α(δ)| → 0

and

δc(|α(δ)|) → 0,

(1.66)

ˆα || = 0. then limδ→0 ||xα − x Definition 1.10. Condition (1.66) is called the coordination condition of vector parameter α with error level δ. The coordination conditions play a principal role in all regularization methods for the ill-posed problems. The coordination condition is assumed to be fulfilled. In the following, we also assume α depends on δ, but for the sake of brevity, we omit this fact. Lemma 1.10. Let the estimates (1.63) and (1.64) as well as the coordination condition for the regularization parameter (1.66) be satisfied. Then, we select q ∈ (0, 1) and find δ > 0 such that for

Toward General Theory of Differential-Operator and Kinetic Models

34

δ ≤ δ0 the following inequality will be fulfilled: δc(|α|)) ≤ q.

(1.67)

Then A˜ + B(α) is continuously invertible operator and the following estimates are fulfilled: ||(A˜ + B(α))−1 || ≤

||(A + B(α))−1 || , 1−q

||(A˜ + B(α))−1 f || ≤ ||(A + B(α))−1 f || + δ

Proof.

(1.68)

c(|α|) ||(A + B(α))−1 f ||. 1−q (1.69)

Based on estimate (1.59) for all f , we have

||(A˜ − A)(A + B(α))−1 f || ≤ δ||(A + B(α))−1 f ||.

(1.70)

Hence, taking into account the estimates (1.63), (1.64) and (1.67), we have the following inequality: ||(A˜ − A)(A + B(α))−1 f || ≤ δc(|α|) ≤ q||f ||.

(1.71)

−1 )(A+ ˜ ˜ Now, since q ≤ 1, we have A+B(α) = (I +(A−A)(A+B(α)) −1 ˜ B(α)), then the existence of inverse operator (A + B(α)) , as well as estimate (1.68), follows from the known inverse operator theorem. Then, we employ the following operator identity: C −1 = D−1 − D −1 (I + (C − D)D−1 )−1 (C − D)D−1 , where C = (A˜ + B(α)), D = A+B(α) and, based on inequalities (1.70) and (1.71), we get estimate  (1.69).

Theorem 1.6 (Main Theorem). Let the conditions of Lemma 6.3 be fulfilled, i.e., parameter α is coordinated with noise level δ. Then RE (1.62) has a unique solution x ˜α . Moreover, if in addition x∗ is the solution of the exact equation (1.57), then the following estimate is fulfilled: ||˜ xα − x∗ || ≤ S(α, x∗ ) +

δc(|α|) (1 + ||x∗ || + S(α, x∗ )). 1−q

(1.72)

xα } If x∗ is also the B-normal solution of equation (1.57), then {˜ converges to x∗ at a rate determined by bound (1.72) as δ → 0.

Auxiliary Information on the Theory of Linear Operators

35

Proof. The existence and uniqueness of the sequence {˜ xα } as solution of RE (1.62) for α ∈ S are proved in Lemma 6.3. Since (A˜ + B(α))(˜ xα − x∗ ) = f˜ − f − (A˜ − A)x∗ − B(α)x∗ , then we obtain the desired bound (1.72) ||˜ xα − x∗ || ≤ ||(A˜ + B(α))−1 ||(||f˜ − f || + ∗ ||(A˜ − A)x || + ||(A˜ + B(α))−1 B(α)x∗ ||) ≤ S(α, x∗ ) + δc(|α|) 1−q (1 + ∗ ∗ ||x || + S(α, x )) based on the proved estimates (1.68), (1.69) and (1.64). Since x∗ is B-normal solution, then limα→0 S(α, x∗ ) = 0. Thanks to parameter α coordinated with noise level δ, we have xα −x∗ || = 0, limδ→0 δc(|α|) = 0. Hence, due to bound (1.72), limδ→0 ||˜  which completes the proof. It should also be mentioned that for practical applications of this theorem, one needs recommendations on the choice of SO B(α) and B-normal solution existence conditions. It is also useful to know the necessary and sufficient conditions of the existence of B-normal solutions x∗ to the exact equation (1.57). These issues are discussed in the following section. 1.5.1.

Stabilizing operator selection, B-normal solution existence and problem correctness class

If A is a Fredholm operator, {φi }n1 a basis in basis in N ∗ (A), then one may assume B(α) ≡ {γi }, {zi } are selected such that  1 det[φi , γk ]ni,k=1 = 0, zi , ψi  = 0

N (A), and {ψi }n1 a n i=1 ·, γi zi , where if i = k, if i = k.

Herewith, the equation Ax = f −

n  f, ψi zi

(1.73)

i=1

is resolvable for arbitrary source function f. Let us now recall f˜, which is δ-approximation of f. Then, the perturbed equation Ax +

n  i=1

x, γi zi = f˜ −

n  f˜, ψi zi i=1

36

Toward General Theory of Differential-Operator and Kinetic Models

has a unique solution x ˜ such that ||˜ x −x∗ || → 0 for δ → 0, where x∗ is a unique solution of exact solution (1.73) for which x∗ , γi  = 0, i = 1, . . . , n. Thus, in the case of a Fredholm operator A as a stabilizing  operator, one can take finite-dimensional operator B = n1 ·, γi zi , which does not depend on the parameter α. For SO B, it is required to have information about the kernel of the operator A and its defect subspace. Therefore, it is of interest to give the following recommendations on the choice of SO B(α) without the use of such information. It is also important to prove the existence theorem of B-normal solution. Theorem 1.7. Let ||(A + B(α))−1 || ≤ c(|α|), ||B(α)|| ≤ d(|α|) for α ∈ S ⊂ Rn , c(|α|) and d(|α|) be continuous functions, lim|α|→0 c(|α|) = ∞, lim|α|→0 d(|α|) = 0. Let lim|α|→0 c(|α|)d(|α|) < ∞, N (A) = 0, R(A) = Y. Then unique solution x∗ of equation (1.57) is B-normal solution and operator B(α) is its SO. Proof. First, let B(α)x∗ ∈ R(A) for α ∈ S. Then there exists element x1 (α) such that Ax1 (α) = B(α)x∗ . Then (A + B(α))−1 B(α)x∗ = (A + B(α))−1 (Ax1 (α) + B(α)x1 (α) − B(α)x1 (α)) = x1 (α) − (A + B(α))−1 B(α)x1 (α). Since B(0) = 0, N (A) = {0}, then limSα→0 x1 (α) = 0. It is to be noted that, by condition ||(A + B(α))−1 B(α)|| ≤ c(|α|)d(|α|), c(|α|)d(|α|) is a continuous function such that the limit, lim|α|→0 c(|α|)d(|α|), is finite. Then α-sequence {||(A + B(α))−1 B(α)x∗ ||} is infinitesimal when S  α → 0. The sequence of operators {(A + B(α))−1 B(α)} converges pointwise to the zero operator on the linear manifold L0 = {x | B(α)x ∈ R(A)}. Thus, we have proved that the theorem is true when B(α)x∗ ∈ R(A). Since by condition  supα∈S c(|α|)d(|α|) < ∞, then α-sequence  −1 ||(A + B(α)) B(α)|| is bounded. Therefore, the sequence of linear operators {(A+B(α))−1 B(α)} in space X converges pointwise to the zero operator on the linear manifold L0 = {xB(α)x ∈ R(A)}. Then, by the Banach–Steinhaus theorem, we have pointwise convergence of this operator sequence to the zero operator and on the closure L0 , i.e., when B(α)x∗ ∈ R(A). Since R(A) = Y and B(α) ∈ L(X → Y ), then B(α)x∗ ∈ Y and Theorem 1.7 is  proved.

Auxiliary Information on the Theory of Linear Operators

37

The conditions of Theorem 1.7 can be relaxed. Corollary 1.7. If R(A) ⊂ Y, limSα→0 x1 (α) = 0, then the solution x∗ of exact equation (1.57) is B-normal if and only if B(α)x∗ ∈ R(A). In this corollary condition, N (A) = {0} is not used. The set L = {x|B(α)x ∈ R(A)} in conditions of corollary defines maximum correctness class. It is to be noted that in Theorem 1.7, we used the assumption on the finite limit limSα→0 ||(A + B(α))−1 ||||B(α)||. We can also relax this limitation. Theorem 1.8. Let ||(A + αB)−1 || ≤ c(α), where α ∈ R1 , c(α) : (0, α0 ] → R+ is continuous function. Suppose that there is a positive integer n ≥ 1 such that limα→0 c(α)αi = ∞, i = 0, . . . , n − 1, limα→0 c(α)αn < ∞. Let x0 satisfy equation (1.57) and in the case of n ≥ 2, there exist x1 , . . . , xn−1 , which satisfy the sequence of equations Axi = Bxi−1 , i = 1, . . . , n − 1. Then, x0 is B-normal solution to equation (1.57) if and only if Bxn−1 ∈ R(A). Proof. Since Axi = Bxi−1 , we have an equality (A + αB)−1 αBx0 = α(A + αB)−1 (Ax1 + αBx1 − αBx1 ) = αx1 − α2 (A + αB)−1 Bx1 = · · · = αx1 − α2 x2 + · · · − (−1)n αn (A + αB)−1 Bxn−1 , where if α → 0 then first n − 2 terms located on the right-hand side are infinitesimal. Using the Banach–Steinhaus theorem, let us make sure that  n  α ||(A + αB)−1 Bxn−1 || is infinitesimal. Indeed, if Bxn−1 ∈ R(A), then there exist xn such that Axn = Bxn−1 . But, in this case, αn (A + αB)−1 Bxn−1 = αn (A + αB)−1 (A + αB − αB)xn = αn xn − αn+1 (A + αB)1 Bxn , where αn+1 ||(A + αB)1 Bxn || ≤ αn+1 c(α)||Bxn ||, limα→0 αn+1 c(α) = 0. Consequently, {||αn (A + αB)1 Bxn−1 ||} is infinitesimal, and the sequence of linear operators {αn (A + αB)1 B} pointwise converges to zero operator on the linear manifold L = {x|Bx ∈ R(A)} and the sequence {||αn (A+αB)1 B||} is bounded. Since I = {x | Bx ∈ R(A)}, we complete the proof by the reference to the Banach–Steinhaus theorem. 

Chapter 2

Volterra Operator Equations with Piecewise Continuous Kernels: Solvability and Regularized Approximate Methods 2.1.

Theory of the Volterra Operator Equations with Piecewise Continuous Kernels

In the plane s, t, we introduce the triangular domain D = {s, t : 0 < s < t < T } and define continuous functions s = αi (t), i = 0, . . . , n, having continuous derivatives for t ∈ (0, T ). It is assumed that 0 =: α0 (t) < α1 (t) < · · · < αn−1 (t) < αn (t) := t for t ∈ (0, T ), where αi (0) = 0, i = 0, . . . , n. For any h > 0, there exists a  > 0 such that inf (t − αn−1 (t)) ≥ ,

h 0, 1+ or let the conditions in Remark 2.1 hold. Then (1)

αi (t) ≤

αi : Ij →

j−1 

Ik ,

j = 1, . . . , m.

k=0

In view of this property, we can apply the step method, which is well known in the theory of functional differential equations, to construct the solution x(t).

Volterra Operator Equations with Piecewise Continuous Kernels

45

Indeed, at the first step, let us calculate x0 (t) ∈ C(I0 ;E1) , constructing the sequence xn (t) = −A(t)xn−1 − Kxn−1 + f¯(t), 0

x00 (t)

0

0

= f¯(t),

t ∈ I0 .

Since A(0)L(E1 →E1 ) ≤ D(0) < 1, it follows that, by inequality (2.5), we have the estimate A + KL(C(I

0 ;E1 )

→C(I0 ;E1 ) )

0, we have the identity x∗ (t) ≡ 0 in view of the estimate A + KL(C(I

0 ;E1 )

→C(I0 ;E1 ) )

< 1.

Thus, the restriction of the solution x∗ (t) of the homogeneous equation to the interval I0 ∪ I1 will have the form

0, t ∈ I0 , x ¯1 (t) = x1 (t), t ∈ I1 . Obviously, it follows that x1 (t) must satisfy the homogeneous integral Volterra equation of the second kind, and hence, x1 (t) ≡ 0. Therefore, x∗ (t) ≡ 0 for t ∈ I0 ∪ I1 . Extending this process, we can verify that the continuous solution x∗ (t) of the homogeneous equation vanishes  for any t ∈ [0, T ]. The theorem is proved. Using the step method and Lemma 2.1, we obtain the following corollary from the proof of Theorem 2.1. Corollary 2.1. Let all the functions αi (t) satisfy the assumptions

of Lemma 2.1 (or Remark 2.1), and let [0, T ] = m k=0 Ik . If equation (2.3) has a continuous local solution on a small interval [0, h], then this solution can be extended continuously to the whole interval [0, T ], i.e., equation (2.3) will be solvable in the class C([0,T ];E1) . In the following, under conditions of additional smoothness of the kernels Ki (t, s), a method for constructing a local solution of equation (2.3) and the asymptotics of this solution will be proposed for the general case in which inequality (2.5) does not hold.

Volterra Operator Equations with Piecewise Continuous Kernels

47

Example 2.1 (E1 = E2 = Rm). The system of integral equations  t/2  t K1 (t − s)x(s)ds + K2 (t − s)x(s)ds = f (t), 0 ≤ t ≤ T, t/2

0

where K1 (t − s) = K2 (t − s) + E, K1 and K2 are m × m matrices, E is the unit matrix, K2−1 (0)L(Rm →Rm ) < 2, and the matrix K2 (t) and the vector function f (t) = (f1 (t), . . . , fm (t)) , f (0) = 0, have continuous derivatives with respect to t, satisfies the assumptions of Theorem 2.1. Indeed, in this case, 1 K −1 (0)L(Rn →Rn ) < 1. 2 2 Therefore, the system has a unique continuous solution. D(0) =

◦ (2)

Example 2.2 (E1 = C [0,1], E2 = C[0,1]). The boundary value problem ⎧ t/2  2   t 2 ∂ x(t, y) ∂ x(t, y) ⎪ ⎪ ⎪ + x(t, y) dt + dt ⎪ 2 ⎨ ∂y ∂y 2 0 t/2 = f (t, y), 0 ≤ t < T, 0 ≤ y ≤ 1, ⎪ ⎪ ⎪ ⎪ ⎩ x(t, 0) = 0, x(t, 1) = 0, where the function f (t, y), f (0, y) = 0, is continuous in y and has a continuous derivative with respect to t, and K2 =

◦ (2) ∂2 ∈ L(C [0,1] → C[0,1] ), 2 ∂y

satisfies the assumptions of Theorem 2.1. Now, we have  1 1 |G(y, ξ)|dξ < 1, D(0) = max 2 0≤y≤1 0

(ξ − 1)y, y ≤ ξ, where G(y, ξ) = (y − 1)ξ, ξ ≤ y. Differentiating the equation with respect to t and inverting the operator K2 by using the Green function G(y, ξ), we obtain the

Toward General Theory of Differential-Operator and Kinetic Models

48

equivalent equation   1 1 1 G(y, ξ)x(t/2, ξ)dξ + G(y, ξ)fh (t, ξ)dξ x(t, y) = − 2 0 0 with the corresponding contraction operator. Therefore, the boundary value problem has a unique continuous solution. 2.1.2.

Existence theorem for the continuous parametric solution

Consider the formalism for constructing the asymptotic approximations of the solutions in the form of logarithmic polynomials. Let the following condition hold: There exist operator polynomials Pi =

N 

Kiνμ tν sμ ,

i = 1, . . . , n,

ν+μ=0

where Ki,ν,μ ∈ L(E1 → E2 ) are linear continuous operators, a vector function, f N (t) =

N 

f ν tν ,

ν=1

and polynomials, αN i (t) =

N 

αiν tν ,

i = 1, . . . , n − 1,

(2.10)

ν=1

where 0 < α11 < α21 < α31 < · · · < αn−1,1 < 1, such that, as t → +0, s → +0, the following estimates hold: Ki (t, s) − Pi (t, s)L(E1 →E2 ) = O((t + s)N +1 ),

i = 1, . . . , n,

f (t) − f N (t)E2 = O(tN +1 ), N +1 ), |αi (t) − αN i (t)| = O(t

i = 1, . . . , n − 1.

Thus, in order to construct the initial approximation (asymptotics) of the solution, we impose additional smoothness conditions on the

Volterra Operator Equations with Piecewise Continuous Kernels

49

kernel and the given functions. Under these conditions, we can indicate the cases in which the solution can be nonunique. The expansions in powers of t, s given in condition (2.10) will be called the Taylor polynomials of the corresponding elements. Let us consider equation (2.4), which is equivalent to equation (2.3). In contrast to previous paragraph, we shall not assume further that condition (2.5) holds. Now, the case D(0) ≥ 1 is possible. Following [137] (see also [138, 139]), we now search for the initial approximation of the solution of the inhomogeneous equation (2.4) in a neighborhood of zero as a polynomial of the form x ˆ(t) =

N 

xj (ln t)tj .

(2.11)

j=0

Let us show that the coefficients xj with values in E1 depend, in the general case, on ln t and the free parameters. This is consistent with the existence of nontrivial solutions to the homogeneous equation. We introduce the j-parametric family of linear operators B(j) = Kn (0, 0) +

n−1  i=1

(αi (0))1+j (Ki (0, 0) − Ki+1 (0, 0)).

The operator B(j) corresponding to the principal “functional” part of equation (2.4) and considered for j ∈ 0 ∪ N will be called the characteristic operator of equation (2.3). In the calculation of the coefficients xj , an important role is played by the characteristic operator, and regular and nonregular cases are possible. Definition 2.1. The number j ∗ is a regular point of the operator B(j) if the operator B(j ∗ ) has a bounded inverse and j ∗ is a nonregular point otherwise. 2.1.2.1.

Regular case: The characteristic operator B(j) has a bounded inverse for j ∈ (0, 1, . . . , N )

If D(0) < 1 (see condition (2.5)), then the operator B(j) has a bounded inverse for any j ≥ 0. Thus, under the assumptions

50

Toward General Theory of Differential-Operator and Kinetic Models

of Theorem 2.1, we always have only the regular case. In the regular case, the coefficients xj are constant vectors from E1 . Indeed, substituting the expansion (2.11) into equation (2.4), using the indeterminate coefficient method, and taking condition (2.10) into account, we obtain the following recurrence sequence of linear equations for the vectors xj : B(0)x0 = f  (0), B(j)xj = Mj (x0 , . . . , xj−1 ),

(2.12) j = 1, . . . , N.

(2.13)

The vector Mj is expressed in a suitable way in terms of the solutions x0 , . . . , xj−1 of the previous equations and the coefficients of the “Taylor polynomials” from condition (2.10). Since the operators B(j) are invertible in the regular case, it follows that the vectors x0 , . . . , xN are independent of ln t, uniquely defined, and the polynomial (2.11) will be constructed. If the assumption of Theorem 2.1 holds, then this polynomial will be the asymptotics of the corresponding unique continuous solution. 2.1.2.2.

Nonregular case: the operator B(j) has nonregular points in the array (0, 1, . . . , N )

Let us introduce the following definitions. Definition 2.2. The number j ∗ is a singular Fredholm point of the operator B(j) of index 1 if B(j ∗ ) is a Fredholm operator [22, p. 219], det[B (1) (j ∗ )φi , ψl ]ri,l=1 = 0, where {φi }r1 is a basis in N (B(j ∗ )), {ψi }r1 is a basis in N (B  (j ∗ )), B  (j ∗ ) is the adjoint operator to B(j ∗ ), and B (1) (j ∗ ) is the derivative of the operator with respect to j calculated for j ∗ . Definition 2.3. Let B(j ∗ ) be a Fredholm operator. We call j ∗ a singular Fredholm point of index k + 1 if N (B(j ∗ )) ⊂

k 

N (B (i) (j ∗ )),

i=1

det[B (k+1) (j ∗ )φi , ψl ]ri,l=1 = 0,

k ≥ 1.

Volterra Operator Equations with Piecewise Continuous Kernels

51

Note that n−1 

(αi (0))1+j aki (Ki (0, 0) − Ki+1 (0, 0)),

B (k) (j) =

i=1

where ai =

ln αi (0).

Remark 2.2. If E1 = E2 = R1 , then B(j) will be an ordinary function of the argument j. In this case, Definition 2.2 implies that j ∗ is a simple root of the equation B(j) = 0, while Definition 2.3 implies that j ∗ is a (k + 1)-multiple root of this equation. Let us show that, in the nonregular case, the coefficients xj will be polynomials in powers of ln t and depend on arbitrary constants. The degree of the polynomials and the number of arbitrary constants are related to the indices of the singular points of the operators B(j ∗ ) and the dimension N (B(j)). Indeed, since, in the nonregular case, the coefficient x0 can depend on ln t, it follows that by using the indeterminate coefficient method, we must search for x0 as a solution of the difference equation Kn (0, 0)x0 (z) +

n−1  i=1

αi (0)(Ki (0, 0) − Ki+1 (0, 0))x0 (z + ai ) = f  (0), (2.14)

where ai = ln α (0) and z = ln t. Here, the following three cases are possible. Case 1. The operator B(0) has a bounded inverse. Then the coefficient x0 is independent of z and is uniquely determined from equation (2.12). Case 2. Let j = 0 be a simple singular Fredholm point of the operator B(j). We search for the coefficient x0 (z) using the difference equation (2.14) in the form of the linear vector function x0 (z) = x01 z + x02 .

(2.15)

Substituting (2.15) into (2.14), we obtain the following two equations for defining the vectors x01 and x02 : B(0)x01 = 0,

(2.16)

B(0)x02 + B (1) (0)x01 = f  (0).

(2.17)

52

Toward General Theory of Differential-Operator and Kinetic Models

Let {φi }r1 be a basis in N (B(0)). Then, x01 =

r 

ck φk .

k=1

)

The vector c = (c1 , . . . , cr is uniquely determined from the solvability conditions for equation (2.17), i.e., from the system of linear algebraic equations r  B (1) (0)φk , ψi ck = f  (0), ψi ,

i = 1, . . . , r

k=1

with nonsingular matrix. Further, the coefficient x02 is determined by equation (2.17), up to span(φ1 , . . . , φr ), by the formula x02 =

r 

dk φk + Γ(f  (0) − B (1) x01 ).

k=1

Here, d1 , . . . , dr are arbitrary constants, and  −1 r  ·, γk zk , Γ = B(0) + k=1

where φi , γk = δik ,

zi , ψk = δik ,

γk ∈ E1∗ ,

zi ∈ E2 ,

is the Trenogin regularizer. Thus, in Case 2, the coefficient x0 (z) is linear with respect to z and depends on r arbitrary constants. Case 3. Let j = 0 be a singular Fredholm point of the operator B(j) of index k + 1, where k ≥ 1. We search for the solution x0 (z) of the difference equation (2.14) in the form of the polynomial x0 (z) = x01 z k+1 + x02 z k + · · · + x0k+1 z + x0k+2 .

(2.18)

Substituting the polynomial (2.18) into system (2.14), taking into account the identity n−1

 dk B(j) = (αi (0))1+j aki (Ki (0, 0) − Ki+1 (0, 0)), k dj i=1

Volterra Operator Equations with Piecewise Continuous Kernels

53

where ai = ln αi (0), and equating the coefficients of the powers z k+1 , z k , . . . , z, z 0 to zero, we obtain the following recurrence sequence of linear operator equations for the coefficients x01 , x02 , . . . , x0k+2 : B(0)x01 = 0,   k + 1 x01 = 0, B(0)x02 + B (1) (0) k     k + 1 k B(0)x0l+1 + B (l) (0) x01 + B (l−1) (0) x02 k+1−l k+1−l   k + 1 − l + 1 + · · · + B (1) (0) x0l = 0, l = 1, . . . , k, k+1−l B(0)x0k+2 + B (k+1) (0)x01 + B (k) (0)x02 + · · · + B (1) (0)x0k+1 = f  (0).

(2.19)

In the case under consideration, by the conditions of Definition 2.3, we have the inclusion    k  di B(j)  N . N (B(0)) ⊂ dj i j=0 i=1

Therefore, B (i) (0)x0i+1 = 0, i = 0, . . . , k and the coefficients x01 , . . . , x0k+1 are determined from the homogeneous equation B(0)x = 0 by the formulas x0i =

r 

cij φj ,

i = 1, . . . , k + 1.

j=1

Equation (2.19) takes the form B(0)x0k+2 + B (k+1) (0)x01 = f  (0).

(2.20)

Since B(0) is a Fredholm operator and det[B (k+1) (0)φi , ψk ]i,k=1,...,r def

= 0, it follows that the vector c1 = (c11 , . . . , c1r ) is uniquely

54

Toward General Theory of Differential-Operator and Kinetic Models

determined from the solvability conditions of equation (2.20). Thus, x0k+2 =

r 

ck+2j φj + x ˆk+2 ,

j=1

where x ˆk+2 is a particular solution of equation (2.20). Just as the def

vectors ci = (ci1 , . . . , cir ) , i = 2, . . . , k + 1, the vector ck+2 = (ck+2,1 , . . . , ck+2,r ) , remains arbitrary. Thus, in Case 3, the coefficient x0 (z) is a polynomial of degree (k+1) in the argument z and depends on r × (k + 1) arbitrary constants. Applying the indeterminate coefficient method and taking into account the identities  k  k(k − 1) · · · (k − (s − 1)) k−s (−1)s ln t, tj lnk tdt = tj+1 (j + 1)s+1 s=0

we can construct difference equations for determining the coefficient x1 (z) (z = ln t) and the subsequent coefficients of the asymptotic approximation (2.11). Indeed, in view of the definition of the operator F (see equation (2.4)), we have the representation  n−1  (αi (0))2 (Ki (0, 0) F (x)|x=x0 (z)+x1 (z)t = Kn (0, 0)x1 (z) + i=1



− Ki+1 (0, 0))x1 (z + ai ) + P1 (x0 (z)) t + r(t) (2.21) with the estimate r(t) = o(t). Here, P1 (x0 (z)) is a known polynomial in z whose degree is equal to the index of the singular Fredholm point j = 0 of the operator B(j). In view of the estimate r(t) = o(t) as t → 0, it follows from the relation (2.21) that the coefficient x1 (z) must satisfy the difference equation Kn (0, 0)x1 (z) +

n−1  i=1

(αi (0))2 (Ki (0, 0)

− Ki+1 (0, 0))x1 (z + ai ) + P1 (x0 (z)) = 0.

(2.22)

If j = 1 is a regular point of the operator B(j), then equation (2.22) has a solution x1 (z) by the polynomial of the same order as the index

Volterra Operator Equations with Piecewise Continuous Kernels

55

of the singular Fredholm point j = 0 of the operator B(0). If j = 1 also turns out to be a singular Fredholm point of the operator B(j), then the solution x1 (z) is constructed in the form of a polynomial of degree k0 + k1 , where k0 and k1 are the indices of the singular Fredholm points j = 0 and j = 1, respectively, of the operator B(j). The coefficient x1 (z) will depend on r0 k0 + r1 k1 arbitrary constants, where r0 = dim N (B) and r1 = dim N (B(1)). Let us introduce the following condition: (A) The operator B(j) in the array (0, 1, . . . , N ) has regular points and singular Fredholm points j1 , . . . , jν of indices ki and dim N (B(ji )) = ri , i = 1, . . . , ν. Then, similarly, we can calculate the other coefficients ˆ(t) of the solution x2 (z), . . . , xN (z) of the asymptotic approximation x of equation (2.4) from the sequence of difference equations of the form Kn (0, 0)xj (z) +

n−1 

(α (0))1+j (Ki (0, 0) − Ki+1 (0, 0))xj (z + ai )

i=1

+ Pj (x0 (z), . . . , xj−1 (z)) = 0,

j = 2, . . . , N.

The foregoing implies the following statement. Lemma 2.2. Let conditions (2.10) and (A) hold. Then there exists a vector function x ˆ(t) =

N 

xi (ln t)ti

such that F (ˆ x(t))E2 = o(tN ),

i=0

where the operator F is defined by formula (2.4). Here, the coefficients xi (ln t) are polynomials in ln t of increasing powers not ex ceeding the sum of indices j kj of the singular Fredholm points j ∈ {0, 1, 2, . . . , N } of the characteristic operator B(j). The coefficients  xi (ln t) depend on ij=0 dim N (B(j))kj arbitrary constants. Remark 2.3. If B(0) is a Fredholm operator and dim N (B(0)) ≥ 1, then the coefficient x0 (ln t) can, possibly, be a linear function of ln t and the vector function x ˆ(t) will unboundedly increase as t → +0 (briefly, x ˆ ∈ C((0,T ];E1 ) ).

Toward General Theory of Differential-Operator and Kinetic Models

56

Let us construct a local solution of equation (2.3). Since 0 ≤ αi (0) < 1, αi (0) = 0, i = 1, . . . , n − 1, it follows that, for any 0 <  < 1, there exists an h ∈ (0, T ] such that max

i=1,...,n−1,t∈[0,h]

|αi (t)| ≤ 

and

αi (t) ≤ . t i=1,...,n−1,t∈(0,h] sup

We introduce the following condition: The operator Kn (t, t) has a bounded inverse for t ∈ [0, T ] and N ∗ is chosen so large such that the following inequality holds: sup N

t∈(0,h)

∗

n−1  i=1

(1)

|αi (t)|

× Kn−1 (t, t)(Ki (t, αi (t))

− Ki+1 (t, αi (t)))L(E1 →E1) ≤ q < 1. (2.23)

Let us formulate the existence of a local solution as follows. Lemma 2.3. Let the condition (2.23) hold. Suppose that, in the class C([0,h];E1) of vector functions continuous for t ∈ [0, h], there exists an element x ˆ(t) such that, as t → +0, F (ˆ x(t))E2 = o(T N ),

N ≥ N ∗.

Then, for the class C(0,h];E1 ) , equation (2.3) has the solution ∗

x(t) = x ˆ(t) + tN u(t),

(2.24)

where u(t) is uniquely determined by successive approximations. Proof. By substituting (2.24) into equation (2.4), we obtain the following integro-functional equation for the function u(t):  ∗  n−1  αi (t) N  αi (t) Kn (t, t)u(t) + t i=1

× (Ki (t, αi (t)) − Ki+1 (t, αi (t)))u(αi (t)) n  αi (t)  s N ∗  F (ˆ x(t)) (1) Kit (t, s) u(s)ds + = 0. + N∗ t t αi−1 (t) i=1

(2.25)

Volterra Operator Equations with Piecewise Continuous Kernels

57

We introduce the linear operators def

Lu =

Kn−1 (t, t)

n−1 

αi (t)



i=1

αi (t) t

N ∗

× {Ki (t, αi (t)) − Ki+1 (t, αi (t))}u(αi (t)), n  αi (t)  s N ∗  def (1) Kn−1 (t, t)Kit (t, s) u(s)ds. Ku = t αi−1 (t) i=1

Then system (2.25) can be rewritten in the compact form u + (L + K)u = γ(t), ∗

x(t))/tN is a continuous vector function. where γ(t) = Kn−1 (t, t)F (ˆ Let us introduce the Banach space X of continuous (in t) vector functions u(t) with values in the Banach space E1 with the norm ul = max e−lt u(t)E1 ,

l > 0.

0≤t≤h

Then, for all l ≥ 0, in view of the inequalities supt∈(0,h] αi (t)/t ≤  < 1 and condition (2.23), the norm of the linear functional operator L satisfies the estimate LL(X→X) ≤ q < 1. In addition, for a sufficiently large l, the integral operator K satisfies the estimate KL(X→X) ≤ q1 < 1 − q. Therefore, for a sufficiently large l > 0, L + KL(X→X) < 1, i.e., the linear operator L + K is a contraction operator in the space X. Therefore, the sequence {un }, where un = −(L + K)un−1 + γ(t), converges. The lemma is proved.

u0 = γ(t), 

58

Toward General Theory of Differential-Operator and Kinetic Models

The following statement is valid. Theorem 2.2 (Main Theorem). Let conditions (2.10), (A) and (2.23) hold, and let f (0) = 0. Let the operator B(0) have a bounded inverse. Then, in the space C([0,T ];E1) , equation (2.3) has a solution, ν depending on i=1 ri ki arbitrary constants. Further, the following asymptotic estimate holds: ∗

x(t) − x ˆ(t)E1 = O(tN )

for t → +0,

where x ˆ(t) is the logarithmic polynomial (2.11). Proof. In view of the assumptions of the theorem, using Lemma 2.2, we can construct the asymptotic approximation x ˆ(t) of the local solution in the form of the logarithmic polynomial N 

xi (ln t)ti ,

i=0

where the coefficients xi (ln t) depend on the number of arbitrary ∗ constants. In view of Lemma 2.3, the function x(t) = x ˆ(t) + tN u(t) satisfies equation (2.3) on a small interval [0, h], where 0 < h ≤ T ; here, u(t) is a continuous function that can be constructed by the successive approximation method. Thus, equation (2.3) has a family  of local solutions depending on νi=1 ri ki arbitrary constants. In view of Corollary 2.1, for fixed values of arbitrary constants, this local solution can be extended from the interval [0, h] to the whole interval  [0, T ]. The theorem is proved. Remark 2.4. If j = 0 is a singular Fredholm point of the operator B(j), then the operator B(0) has no inverse. In this case, the ˆ(t) can, possibly, be a linear coefficient x0 (ln t) in the asymptotics x function of ln t, i.e., the solution will unboundedly increase as t → +0. Thus, we can dispense with the invertibility condition for the operator B(0) in Theorem 2.2, allowing for the unbounded increase of the solution as t → +0.

Volterra Operator Equations with Piecewise Continuous Kernels

59

Example 2.3 (E1 = E2 = C[0,1]). Consider the equation 

t/2  1 0

0



K(y, y1 )x(s, y1 )dy1 ds t



+ t/2

0

1

 K(y, y1 )x(s, y1 )dy1 − 2x(s, y) ds = f (y, t),

where 0 ≤ t < T, 0 ≤ y ≤ 1, K(y, y) is a continuous symmetric kernel, f (y, t), f (y, 0) = 0, is a function continuous in y and analytic in t. Here, the characteristic operator is of the form  1 j K(y, y1 )[·]dy1 . B(j) := (−2 + (1/2)) [·] + 0

If the terms of sequence 2−(1/2)j , j = 0, 1, 2, . . . , are not eigenvalues of the kernel K(y, y1 ), then we have the regular case and, in view of the proof of Theorem 2.2, the equation has an analytic (in t) solution. Let 1 be an eigenvalue of the kernel K(y, y1 ) of rank r, and let φ1 (y), . . . , φr (y) be the corresponding orthonormal system of eigenfunctions. Let the other numbers of the sequence 2 − (1/2(, 2 − (1/2)2 ), . . . be regular numbers of the kernel K(y, y1 ). Then, by Theorem 2.2, the equation has an r-parametric family of solutions. In particular, if f (y, t) = g(y)t, then the solution is of the form r  ln t  1 φi (y)φi (y1 )g(y1 )dy1 x(t, y) = − ln 2 0 i=1

+ c1 φ1 (y) + · · · + cr φr (y) + x0 (y), where c1 , . . . , cr are arbitrary constants and x0 (y) is a particular solution of the Fredholm integral equation of the second kind  x(y) = 0

1

K(y, y1 )x(y1 )dy1 − g(y) +

r  i=1

 φi (y)

1 0

φi (y1 )g(y1 )dy1 .

60

2.1.2.3.

Toward General Theory of Differential-Operator and Kinetic Models

Strengthening of Theorem 2.2

Let the numbers {j1 , . . . , jν } ∈ N ∪ {0} be the Fredholm points of the characteristic operator B(j). Let us construct generalized Jordan sets in the sense (see [21, Section 30]) of the characteristic operator B(j) at these points. Then Theorem 2.2 can be strengthened. Indeed, let j ∗ be a singular Fredholm point of the operator B(j). Suppose that (l) we have constructed the elements {φi }, i = 1, . . . , r, l = 1, . . . , pi , satisfying the equalities B(j ∗ )φi = 0   p + 1 i (2) (1) B(j ∗ )φi + B (1) (j ∗ ) φi = 0 p .. .   pi + 1 (l+1) (1) φi + B (l) (j ∗ ) B(j ∗ )φi pi + 1 − l   pi + 1 − l + 1 (l) (1) ∗ φi = 0, + · · · + B (j ) pi + 1 − l i = 1, . . . , r, l = 1, . . . , pi − 1. (1)

Further, let

 (1) (2) det B (pi ) (j ∗ )φi + B (pi −1) (j ∗ )φi  (p ) + · · · + B (1) (j ∗ )φi i , ψj

i,j=1,...,r

= 0,

(2.26)

(2.27)

where {φi }r1 is a basis in N (B  (j ∗ )). Then by analogy with the wellknown theory of the Jordan sets of linear operators [21, p. 422], we say that at the point j ∗ operator B(j) has a complete B(j) Jordan (l) set {φi }i=1,...,r,l=1,...,pi , where the numbers pi are called the lengths of the Jordan chains (1)

(pi )

(φi , . . . , φi   pi

), 

i = 1, . . . , r.

Note that there exists a complete Jordan set at the point j ∗ if j ∗ is a singular Fredholm point of index p in the sense of Definition 2.3.

Volterra Operator Equations with Piecewise Continuous Kernels

In this case, B (l) (j ∗ )φi chains

(1)

61

= 0, i = 1, . . . , r, l = 0, . . . , p − 1, the Jordan

(1)

(1)

(ψ , . . . , ψ ),  i  i 

i = 1, . . . , r,

p

are stationary, i.e., have identical length p and condition (2.27) takes the form (1)

det[B (p) φi , ψj ]i,j=1,...,r = 0. Condition (A) used in Theorem 2.2 will also hold, i.e., we obtain the assertion of Theorem 2.2. Instead of condition (A), we introduce a weaker condition, namely, the following one: (A1) The characteristic operator B(j) in the array (0, 1, . . . , N ∗ ) has exactly ν singular Fredholm points (j1 , . . . , jν ) with complete generalized Jordan sets and all the other numbers in this array are regular. Note that in contrast to condition (A), the Jordan chains in condition (A1) can be nonstationary. If condition (A1) holds, then the asymptotics (required in Lemma 2.1) of the solution of equation (2.4) can be constructed. Indeed, let the operator B(0) be Fredholm, (1) (l) let {φi }ri=1 be a basis in N (B(0)), and let {φi }i=1,...,r,l=1,...,pi be the corresponding complete Jordan set satisfying equation (2.26) and condition (2.27), where we put j ∗ = 0. Then, the first coefficient x0 (z) of the unknown approximation x ˆ satisfying the difference equation (2.14) is constructed in the form of the polynomial x0 (z) =

r  i=1

ci

pi  l=1

φi z pi +1−l + x0 , (l)

(2.28)

where the constants c1 , . . . , cr and x0 ∈ E1 must be determined. Substituting (2.28) into (2.14), we obtain the following linear equation for the element x0 : r  (1) (2) (p ) ci (B (pi ) (0)φi + B (pi −1) (0)φi + · · · + B (1) (0)φi i ) B(0)x0 + i=1



= f (0).

(2.29)

Toward General Theory of Differential-Operator and Kinetic Models

62

In view of inequality (2.27) for j ∗ = 0, the vector (c1 , . . . , cr ) is determined from the solvability condition for equation (2.29). Further, using the Trenogin regularizer, we construct the solution x0 of the inhomogeneous equation (2.29) up to the basis (φ1 , . . . , φr ) in N (B(0)). Similarly, in view of condition (A1), we can calculate the other coefficients x2 (z), . . . , xN (z) of the asymptotics x ˆ(z), satisfying Lemma 2.1. Using Lemma 2.2, Theorem 2.2 is strengthened due to the weakening of condition (A). Theorem 2.3. Let conditions (2.10), (2.23), and (A1) hold, and let f (0) = 0. Then, for the class C([0,T ];E1) , equation (2.3) has a parametric family of solutions. 2.1.2.4.

Open problems

In Theorems 2.1 and 2.3, an important role is played by the method for constructing asymptotics which uses the solutions of difference equations. In the case of one equation when E1 = E2 = R1 , the given method for solving difference systems coincides with Gelfond’s well-known method [54] of constructing solutions of inhomogeneous difference equations with polynomial on the right-hand side. Thus, the methods from functional analysis [167, 169] allowed us to apply ideas of this method to linear Volterra equations of the first kind with piecewise continuous operators acting in Banach spaces. If f (0) = 0, then equation (2.1) is not solvable for the class of continuous functions. In this case, using results from [141, Section 10], one can construct the solutions in the class of distributions. The method presented in this section can be of interest to those who wish to solve a number of classes of nonstandard integro-differential systems with degeneracies [145]. 2.2.

Numerical Methods

In this section, the numerical methods for the solution of the Volterra integral equation of the first kind with piecewise continuous kernels are presented. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays)

Volterra Operator Equations with Piecewise Continuous Kernels

63

which starts at the origin. In order to linearize these equations, the modified Newton–Kantorovich iterative process is employed. Two direct quadrature methods based on the piecewise constant and piecewise linear approximation of the exact solution are proposed for linear solutions. The accuracy of proposed numerical methods is O(1/N ) and O(1/N 2 ), respectively. A certain iterative numerical scheme satisfying the regularization properties is suggested. Furthermore, generalized numerical methods for nonlinear equations are adduced. The midpoint quadrature rule in all the cases is employed. In conclusion, several numerical examples are applied in order to demonstrate the efficiency of proposed numerical methods. The numerical solution of linear integral equations of the first kind is of course a classical problem and has been addressed by numerous authors. But only few authors studied these equations in the case of jump discontinuous kernels. In general, VIE of the first kind can be solved by the reduction to equations of the second kind, regularization algorithms developed for Fredholm equations as well as direct discretization methods can also be applied. On the other hand, it is known that solutions of integral equations of the first kind can be unstable and this is a well-known ill-posed problem. This is due to the fact that the Volterra operator maps the considered solution space into its narrow part only. Therefore, the inverse operator is not bounded. It is necessary to assess the proximity of the solutions and the proximity of the right-hand side using the different metrics. In addition, the proximity of the righthand side should be in a stronger metric. Moreover, as shown in Section 2.1, solutions of the VIE can contain arbitrary constants and can be unbounded as t → 0. The theory of integral models of evolving systems was initiated in the works of Kantorovich, Solow and Glushkov in the mid-20th century. Here, readers may refer to the papers of [86, 159]. It is well known that Solow publication [159] on vintage capital model led him to the Nobel prize in 1987 for his analysis of economic growth. Such theory employs the VIEs of the first kind where bounds of the integration interval can be functions of time. Here, readers may refer, e.g., to Refs. [73–75] and references therewith. These models take into

64

Toward General Theory of Differential-Operator and Kinetic Models

account the memory of a dynamical system when its past impacts its future evolution. The memory is implemented in the existing technological and financial structure of physical capital (equipment). The memory duration is determined by the age of the oldest capital unit (e.g., equipment) still employed. Reference [25] is devoted to the construction of iterative numerical algorithm for the systems of nonlinear Volterra-type equations related to the Vintage Capital Models (VCMs): ⎧  t ⎪ ⎪ ⎪ H(t, τ, x(τ ))dτ, x(t) = ⎪ ⎨ y(t) t ∈ [t0 , T ), t0 < T  ∞,  t ⎪ ⎪ ⎪ ⎪ K(t, τ, x(τ ))dτ = f (t), ⎩ y(t)

with unknown functions x(t) and y(t) satisfying the initial conditions: y(t0 ) = Y0 < t0 , x(τ ) ≡ ϕ0 (τ ), τ ∈ (−∞, t0 ]. First results in the studies of the Volterra equations with discontinuous kernels were formulated by Evans [48] in the beginning of 20th century. Results in the spectral theory of integral operators with discontinuous kernels were obtained by Khromov [89]. Some results concerning the general approximation theory for integral equations with discontinuous kernels are presented in Anselone’s seminal paper [7]. There are several approaches available for the numeric solution of Volterra integral equations of the first kind. One of them is to apply classical regularization algorithms developed for Fredholm integral equations of the first kind. However, the problem reduces to solving algebraic systems of equations with a full matrix, an important advantage of the Volterra equation is lost and there is a significant increase in the arithmetic complexity of the algorithms. The second approach is based on a direct discretization of the initial equations. Here, one may face an instability of the approximate solution because of the errors in the initial data. The regularization properties of the direct discretization methods are optimal in this sense, where the discretization step is the regularization parameter associated with the error of the source data. However, only low-order quadrature formulas (midpoint quadrature or trapezoidal formulas) are suitable

Volterra Operator Equations with Piecewise Continuous Kernels

65

for the approximation of the integrals. The Newton–Cotes formulas, Gregory and others (the second order and higher orders) generate divergent algorithms. It should be noted that it is very difficult to apply these algorithms to solve equation (2.30) by (2.32) because of the kernel discontinuities (2.31) described in Section 2.1. The adaptive mesh should depend on the curves of the jump discontinuity for each number N of divisions of the considered interval, and therefore this mesh cannot be linked to the errors in the source data. It is needed to correctly approximate the integrals. In the following, two approaches are proposed for the numeric solution of the Volterra integral equations of the first kind with piecewise continuous kernels. The first approach involves a direct discretization based on piecewise constant and piecewise linear approximations of the exact solution (the first and second orders of accuracy, respectively). The second approach is based on the preliminary determination of the two acceleration values of the unknown function followed by the special regularizing iterative procedure. 2.2.1.

Problem statement

The object of interest is the following integral equation of the first kind we introduced in Section 2.1 in abstract settings:  t K(t, s)x(s) ds = f (t), t ∈ [0, T ], (2.30) 0

where the kernel K(t, s) is discontinuous along continuous curves αi (t), i = 1, 2, . . . , n − 1, and is of the form ⎧ K1 (t, s), α0 (t) < s < α1 (t), ⎪ ⎪ ⎪ ⎨K2 (t, s), α1 (t) < s < α2 (t), (2.31) K(t, s) = .. ⎪ . ⎪ ⎪ ⎩ Kn (t, s), αn−1 (t) < s < αn (t), where α0 (t) ≡ 0, α0 (t) < α1 (t) < · · · < αn (t) ≡ t, f (0) = 0. Let us assume that the kernels Ki (t, s) and the right-hand side f (t) in equation (2.30) are continuous and sufficiently smooth functions.

Toward General Theory of Differential-Operator and Kinetic Models

66

The functions αi (t) ∈ C 1 [0, T ] are nondecreasing. Moreover, α1 (0) ≤ α2 (0) ≤ · · · ≤ αn−1 (0) < 1. Let us rewrite equation (2.30):   α1 (t) K1 (t, s)x(s) ds + 0

 +··· +

t

αn−1 (t)

α2 (t) α1 (t)

K2 (t, s)x(s) ds

Kn (t, s)x(s) ds = f (t), t ∈ [0, T ].

(2.32)

It is to be noted that conventional Glushkov integral model of evolving systems is the special case of this equation where all the functions Ki (t, s) are zeros except Kn (t, s). For more details concerning evolving (developing) system modeling using integral models with discontinuous kernels, readers may refer to [8, 141]. A dynamic analysis of energy storage with renewable generation using such Volterra models is fulfilled in [140]. 2.2.2.

Direct discretization

2.2.2.1.

Piecewise constant approximation

Let us introduce the mesh nodes (not necessarily uniform) to construct the numeric solution of equation (2.32) on the interval [0, T ] (if the unique continuous solution exists) 0 = t0 < t1 < t2 < · · · < tN = T, h = max (ti − ti−1 ) = O(N −1 ).

(2.33)

i=1,...,N

The approximate solution is determined as the following piecewise constant function:

N  1, t ∈ Δi = (ti−1 , ti ], xi δi (t), t ∈ (0, T ], δi (t) = xN (t) = 0, t ∈ / Δi , i=1 (2.34) with the undefined coefficients xi , i = 1, . . . , N . The differentiation of both parts of equation (2.32) with respect to t leads to the following

Volterra Operator Equations with Piecewise Continuous Kernels

equality: 

f (t) =

n 



αi (t)

αi−1 (t)

i=1

67

∂Ki (t, s) x(s) ds + αi (t)Ki (t, αi (t))x(αi (t)) ∂t 

− αi−1 (t)Ki (t, αi−1 (t))x(αi−1 (t)) . From the last expression, x0 can be obtained as follows: f  (0)   .  i=1 Ki (0, 0) αi (0) − αi−1 (0)

x0 = n

(2.35)

Here, it is assumed that the denominator of (2.35) must not be zero. Let us make the notation fk := f (tk ), k = 1, . . . , N and write the initial equation in the point t = t1 to define the coefficient x1 : n  αi (t1 )  Ki (t1 , s)x(s) ds = f1 . (2.36) i=1

αi−1 (t1 )

Since, at this stage, the lengths of all integration intervals αi (t1 ) − αi−1 (t1 ) in (2.36) do not exceed h, then based on the midpoint quadrature rule, the coefficient x1 can be calculated as follows: f1

 . αi (t1 )+αi−1 (t1 ) (α (t ) − α (t ))K , t i 1 i−1 1 i 1 i=1 2

x1 =  n

(2.37)

Suppose the values x2 , x3 , . . . , xk−1 are known. Then equation (2.30) can be written as follows:  tk−1  t K(t, s)x(s) ds = f (t) − K(t, s)xN (s) ds (2.38) tk−1

0

and require that the last equality holds for the point t = tk  tk−1  tk K(tk , s)x(s) ds = fk − K(tk , s)xN (s) ds. tk−1

0

Taking into account (2.34), one can obtain  tk  k−1  K(tk , s) ds = fk − xj xk tk−1

j=1

tj tj−1

K(tk , s) ds.

(2.39)

Toward General Theory of Differential-Operator and Kinetic Models

68

Thus, xk =

fk −

! tj j=1 xj tj−1 K(tk , s) ! tk tk−1 K(tk , s) ds

k−1

ds

.

(2.40)

! tj K(tk , s) ds in (2.40) can Herewith, the integrals of the form tj−1 be calculated using the midpoint quadrature formulas with auxiliary mesh nodes related to the curves αi (t) of the kernels K(t, s) for each value of N . It is easy to note that the error of the method is   1 . (2.41) εN = x(t) − xN (t)C[0,T ] = O N 2.2.2.2.

Piecewise linear approximation

Let us suppose that the approximate solution is a piecewise linear function as follows: xN (t) =

N  

xi−1 +

i=1

 xi − xi−1 (t − ti−1 ) δi (t), ti − ti−1

t ∈ (0, T ], (2.42)

where δi (t) =

1 for t ∈ Δi = (ti−1 , ti ], 0 for t ∈ / Δi .

The objective is to determine the coefficients xi , i = 1, . . . , N, of the approximate solution. Determining by (2.35) the coefficient x0 and taking into account the equality (2.39), the following equality can be obtained:   tk  xk − xk−1 (s − tk−1 ) K(tk , s) ds xk−1 + tk − tk−1 tk−1 = fk −

k−1   j=1

tj tj−1



 xj − xj−1 (s − tj−1 ) K(tk , s) ds. xj−1 + tj − tj−1

Volterra Operator Equations with Piecewise Continuous Kernels

69

Thus, exclude xk as follows:

! tj k−1  K(tk , s) ds − j=1 xj−1 tj−1 K(tk , s) ds  x −xj−1 ! tj (s − tj−1 )K(tk , s) ds + tjj −tj−1 tj−1 ! tk , 1 tk −tk−1 tk−1 (s − tk−1 )K(tk , s) ds

fk − xk−1 xk = xk−1 +

! tk

tk−1

(2.43) where k = 1, 2, . . . , N. The integrals in (2.43) can be approximated using the midpoint quadratures based on auxiliary mesh nodes so that the values of the functions αi (tj ) are a subset of the set of these mesh points at each particular value of N . The error of this approximation method is   1 . (2.44) εN = x(t) − xN (t)C[0,T ] = O N2 2.2.3.

Iterative method

Let the kernels Ki (t, s) be symmetric functions in their domains, i.e., Ki (t, s) = Ki (s, t),

i = 1, 2, . . . , n.

The objective is to find the approximation solution of equation (2.30) at the mesh (2.33) as a piecewise constant function (2.34). To do this, the initial values of x0 and x1 with formulas (2.35) and (2.37) can be defined. Rewrite equation (2.30)  t1  t K(t, s)x(s) ds = f (t) − K(t, s)xN (s) ds, (2.45) t1

! t1

0

and designate g(t) = f (t) − 0 K(t, s)xN (s) ds. To define the values xk of the required approximation solution (2.34), the following iterative process can be used:    t (m+1) (m) (m) (t) = x (t) + γ g(t) − K(t, s)x (s) ds , x t1

m = 0, 1, . . . ,

(2.46)

where γ is a positive regularization parameter and m is a number of the iteration.

70

Toward General Theory of Differential-Operator and Kinetic Models

The initial approximate value x(0) (t) can be given using the aprioristic data (if available) of the exact solution or it can be supposed x(0) (t) ≡ g(t). Obviously, if the functional sequence x(m) (t) converges to a function x ˜γ (t), then this function satisfies (2.46) for all γ = 0. The values xk , k = 2, 3, . . . , N, can be defined successively as    tk (m+1) (m) (m) = xk + γ g(tk ) − K(tk , s)xN (s) ds , xk t1

m = 0, 1, . . . .

(2.47)

Herewith, to calculate the integrals in (2.47), the midpoint quadrature or trapezoidal formulas can be employed using auxiliary mesh nodes related to the curves αi (t) of the kernels K(t, s) for each value of N . In practice, one may choose the optimal value of the regularization parameter γ with the following condition: n   αi (t) (m) (m) Ki (t, s)xN (s) ds − f (t) → min (2.48) εN = αi−1 (t) i=1

C[0,T ]

for large enough m. 2.2.4.

Nonlinear equations

In this section, the following nonlinear equation is addressed:  t K(t, s, x(s)) ds = f (t), t ∈ [0, T ], f (0) = 0, (2.49) 0

where ⎧ ⎪ ⎨K1 (t, s)G1 (s, x(s)), .. K(t, s, x(s)) = . ⎪ ⎩ Kn (t, s)Gn (s, x(s)),

t, s ∈ m1 , (2.50) t, s ∈ mn ,

where mi = {t, s | αi−1 (t) < t, s < αi (t)}, α0 (t) = 0, αn (t) = t, i = 1, . . . , n. The functions f (t) and αi (t) have continuous derivatives with respect to t in the corresponding domains mi , Kn (t, t) = 0,

Volterra Operator Equations with Piecewise Continuous Kernels

71

αi (0) = 0, 0 < α1 (t) < α2 (t) < · · · < αn−1 (t) < t. The functions Ki , i = 1, . . . , n are continuously differentiable with respect to t continuation into compacts mi . The functions α1 (t), . . . , αn−1 (t) should increase in small neighborhood 0 ≤ t ≤ τ at least. The following theorem states the existence and uniqueness conditions of the solution of equation (2.49). The proof is similar with the proof of [141, Theorem 3.2]. Theorem 2.4. Let the following conditions occur for t ∈ [0, T ]: Ki (t, s), Gi (s, x(s)) are continuous, i = 1, . . . , n, αi (t) and f (t) have continuous derivatives for t, Kn (t, t) = 0, 0 = α0 (t) < α1 (t) < · · · < αn−1 (t) < αn (t) = t for t ∈ (0, T ], αi (0) = 0, f (0) = 0. Let the functions Gi (s, x(s)) satisfy Lipschitz condition Gi (s, x1 (s)) − Gi (s, x2 (s)) − (x1 (s) − x2 (s))  qi x1 − x2 ,

∀x1 , x2 ∈ R1 ,

  −1 qn + n−1 i=1 αi (0)|Kn (0, 0) (Ki (0, 0) − Ki+1 (0, 0))|(1 + qi ) < 1. Then, there exists τ > 0 such that equation (2.49) has a unique local solution in C[0,τ ] . Furthermore, if minτ ≤t≤T (t − αn−1 (t)) = h > 0, then the solution can be constructed on [τ, T ] using the step method combined with successive approximations. Thereby, equation (2.49) has the unique global solution in C[0,T ] . 2.2.4.1.

Linearization

In order to approximate solutions for equation (2.49), the nonlinear integral operator is introduced: n  αi (t)  Ki (t, s)Gi (s, x(s)) ds − f (t). (2.51) (F x)(t) ≡ i=1

αi−1 (t)

Equation (2.49) can be written in an operator form as follows: (F x)(t) = 0.

(2.52)

In order to construct an iterative numerical method to equation (2.49), the operator (2.51) can be linearized according to a modified

72

Toward General Theory of Differential-Operator and Kinetic Models

Newton–Kantorovich scheme [85]: xm+1 = xm − [F  (x0 )]−1 (F (xm )),

m = 0, 1, . . . ,

(2.53)

where x0 (t) is the initial approximation. Then, the approximate solution of (2.52) could be determined as the following limit of sequence: x(t) = lim xm (t). m→∞

(2.54)

The derivative F  (x0 ) of the nonlinear operator F at the point x0 is defined as follows: F (x0 + ωx) − F (x0 ) ω→0 ω  n 1  αi (t) = lim Ki (t, s) ω→0 ω αi−1 (t)

F  (x0 ) = lim

i=1

× [Gi (s, x0 (s) + ωx(s)) − Gi (s, x0 (s))] ds. Implementing the limit transition under the integral sign, one may obtain n  αi (t)   Ki (t, s)Gix (s, x0 (s))x(s) ds, F (x0 )(t) = i=1

αi−1 (t)

 ∂Gi (s, x(s))  . (2.55) where Gix (s, x0 (s)) =  ∂x x=x0 Thus, the operator form of Newton–Kantorovich scheme is obtained as follows: F  (x0 (t))Δxm+1 (t) = −F (xm ),

Δxm+1 = xm+1 − xm ,

or in the extended form n  αi (t)  Ki (t, s)Gix (s, x0 (s))Δxm+1 (s) ds i=1

αi−1 (t)

= f (t) −

n   i=1

αi (t)

αi−1 (t)

Ki (t, s)Gi (s, xm (s)) ds.

(2.56)

Volterra Operator Equations with Piecewise Continuous Kernels

73

The latter can be written as follows: n  αi (t)  Ki (t, s)Gix (s, x0 (s))xm+1 (s) ds = Ψm (t), i=1

αi−1 (t)

m = 0, 1, 2, . . . ,

(2.57)

where Ψm (t) = f (t) +

n   i=1

αi (t) αi−1 (t)

Ki (t, s)

× [Gix (s, x0 (s))xm (s) − Gi (s, xm (s))] ds. Equation (2.57) is now linear Volterra equation of the first kind with respect to the unknown function xm+1 (t). Note that the kernels Ki (t, s)Gix (s, x0 (s)), i = 1, . . . , n, remain constant during each iteration m. Since equation (2.57) has the form (2.32), then the methods suggested in the previous sections to solve them numerically can be applied. Thus, solving equation (2.57), a sequence of approximate functions xm+1 (t) is gained. Then, using formula (2.54), the approximate solution of (2.49) is obtained with an accuracy, depending on m. 2.2.5.

The convergence theorem

Let C[0, T ] be a Banach space of continuous functions equipped with the standard norm xC[0,T ] = maxt∈[0,T ] |x(t)|. The following theorem of convergence (based on the general theory proposed in the classical monograph [85]) for iterative process (2.57) occurs Theorem 2.5. Let the operator F have a continuous second derivative in the sphere Ω0 (x − x0   r) and the following conditions hold: (1) Equation (2.57) has a unique solution in [0, T ] for m = 0, i.e., there exists Υ0 = [F  (x0 )]−1 . (2) Δx1   η. (3) Υ0 F  (x)  L, x ∈ Ω0 .

74

Toward General Theory of Differential-Operator and Kinetic Models √

√

If also h = Lη < 12 and 1− h1−2h η  r  1+ h1−2h η, then equation (2.49) has a unique solution x∗ in Ω0 , process (2.57) converges to x∗ , and the velocity of convergence is estimated by the inequality √ η x∗ − xm   (1 − 1 − 2h)m+1 , m = 0, 1, . . . . h In order to prove this theorem, one must show that equation (2.57) is uniquely solvable (including the case m = 0), i.e., condition (1) of the   theorem holds. Then the boundedness of the second derivative F (x0 ) (x)) should be verified for estimating the constant L in condition (3). One can  verify that the necessary condition for the second derivative F  (x0 ) (x) to be bounded is a differentiability of the initial approximation x0 (t) as well as the functions Ki with respect to second variable. 2.2.5.1.

Generalized numerical method for nonlinear equations

In this section, the generic numerical method for nonlinear weakly regular Volterra equations is suggested using midpoint quadrature. To find numeric solution of equation (2.49) on the interval [0, T ] again, the following mesh (the mesh can be nonuniform) can be introduced: 0 = t0 < t1 < t2 < · · · < tN = T, h = max (ti − ti−1 ) = O(N −1 ).

(2.58)

i=1,...,N

The approximate solution of equation (2.49) can be constructed as the following piecewise constant function: xN (t) =

N 

xi δi (t),

i=1

δi (t) =

t ∈ (0, T ],

1 for t ∈ Δi = (ti−1 , ti ],

(2.59)

0 for t ∈ / Δi

with coefficients xi , i = 1, . . . , N , under determination. In order to find x0 = x(0), both sides of equation (2.49) can be differentiated

Volterra Operator Equations with Piecewise Continuous Kernels

with respect to t: 

f (t) =

n 



αi (t)

αi−1 (t)

i=1

75

∂Ki (t, s) Gi (s, x(s)) ds ∂t

+ αi (t)Ki (t, αi (t))Gi (αi (t), x(αi (t)))



− αi−1 (t)Ki (t, αi−1 (t))Gi (αi−1 (t), x(αi−1 (t))) . Thereby, f  (0) =

n  i=1



0

0

∂Ki (0, 0) Gi (0, x(0)) ds + αi (0)Ki (0, 0)Gi (0, x(0)) ∂t 

− αi−1 (0)Ki (0, 0)Gi (0, x(0)) . In the last expression, the coefficient x0 appears in the case of nonlinear dependency. To find the coefficient x0 , the Van Wijngaarden– Dekker–Brent method can be used. Again, let fk := f (tk ), k = 1, . . . , N . The mesh point of the mesh (2.58) which coincides with αi (tj ) is still denoted as vij , i.e., αi (tj ) ∈ Δvij . Obviously, vij < j for i = 0, . . . , n − 1, j = 1, . . . , N . It is to be noted that αi (tj ) does not always coincide with any mesh point. Here, vij is used as index of the segment Δvij such that αi (tj ) ∈ Δvij (or its right-hand side). Let us now assume that the coefficients x0 , x1 , . . . , xk−1 are known. Equation (2.49) defined in t = tk as n  αi (tk )  Ki (tk , s)Gi (s, x(s)) ds = fk i=1

αi−1 (tk )

can be written as follows: I1 (tk ) + I2 (tk ) + · · · + In (tk ) = fk , where I1 (tk ) =

v1,k −1  t j  j=1

tj−1



α1 (tk )

+ tv1,k −1

K1 (tk , s)G1 (s, x(s)) ds K1 (tk , s)G1 (s, x(s)) ds,

76

Toward General Theory of Differential-Operator and Kinetic Models

 In (tk ) =

tvn−1,k αn−1 (tk )

Kn (tk , s)Gn (s, x(s)) ds 

k 

+

tj

j=vn−1,k +1 tj−1

Kn (tk , s)Gn (s, x(s)) ds.

(1) If vp−1,k = vp,k , p = 2, . . . , n − 1, then  tv p−1,k Kp (tk , s)Gp (s, x(s)) ds Ip (tk ) = αp−1 (tk )



vp,k −1



+

tj

j=vp−1,k +1 tj−1



αp (tk )

+ tvp,k −1

Kp (tk , s)Gp (s, x(s)) ds

Kp (tk , s)Gp (s, x(s)) ds.

(2) If vp−1,k = vp,k , p = 2, . . . , n − 1, then  αp (tk ) Kp (tk , s)Gp (s, x(s)) ds. Ip (tk ) = αp−1 (tk )

The number of terms in each line of the last formula depends on an array vij , defined using the input data: functions αi (t), i = 1, . . . , n − 1, and fixed (for specific N ) mesh. Each integral term is approximated using the midpoint quadrature rule, e.g.,  αp (tk ) Kp (tk , s)Gp (s, x(s)) ds tvp,k −1

 αp (tk ) + tvp,k −1 ≈ αp (tk ) − tvp,k −1 Kp tk , 2    αp (tk ) + tvp,k −1 αp (tk ) + tvp,k −1 , xN . × Gp 2 2 "

#



Moreover, on those intervals where the desired function has been already determined, one may select xN (t) (i.e., t  tk−1 ). On the rest of the intervals, an unknown value xk appears in the last terms. Then this value is defined explicitly and proceeds in the loop for k.

Volterra Operator Equations with Piecewise Continuous Kernels

77

The number of these terms is determined from the initial data vij analysis. To find the coefficient x0 also, Van Wijngaarden–Dekker– Brent method is used. The maximum pointwise error of proposed x(ti ) − xh (ti )| has the order of numerical method εN = max0≤i≤N |¯ "1# O N . 2.2.6.

Numerical examples

2.2.6.1.

Linear equations

Let us consider the following three problems using the uniform 1 is meshes only. The two-mesh difference D N for h = N1 and h = 2N N N defined on [0, T ]. The order of convergence p based on the D is calculated as follows: DN =

max

0≤i≤N,0≤j≤2N

|xhN (ti ) − xh2N (tj )|

(2.60)

with ti = tj , i = 2j, pN = log2

DN . D2N

(2.61)

Here D N and pN are used to estimate the order of convergence for problems with unknown exact solutions. Let us first address the equation  t  t/3 (1 + t + s)x(s) ds − x(s) ds t/3

0

√ 5 7t2 4 3(2t + 3) 2 (2t + 1) + − − , t ∈ [0, 2] = 3 45 18 15 √ with known solution x ¯(t) = 2t + 1 − 1. Table 2.1 shows the x(ti )−xh (ti )|, computed maximum pointwise errors εN = max0≤i≤N |¯ the two-mesh difference D N and the order of convergence pN in double precision arithmetic applied to problem for various values of h. Let us now consider the equation  3t/8  t  t/8 (1 − t · s)x(s) ds + (t + s)x(s) ds − x(s) ds 3 2

t/8

0

=−

67t4 121t3 t5 + − , 16384 3072 384

3t/8

t ∈ [0, 2],

78

The errors for various stepsizes h of the first example.

Piecewise constant approximation h 1/32 1/64 ε 0.097245 0.037330 0.078452 0.027076 DN 1.534798 0.535009 pN

1/128 0.020360 0.018686 1.052673

1/256 0.013031 0.009008 1.330913

1/512 0.005846 0.003580 0.716153

1/1024 0.003016 0.002179 1.354821

1/2048 0.001540 0.000852 0.949923

1/4096 0.000781 0.000441 —

Piecewise linear approximation h 1/32 1/64 ε 9.9445E − 4 3.7228E − 4 DN 9.3805E − 4 3.3986E − 4 1.464704 1.718816 pN

1/128 1.1005E − 4 1.0325E − 4 2.231901

1/256 2.4769E − 5 2.1979E − 5 1.704639

1/512 7.6136E − 6 6.7433E − 6 2.020961

1/1024 1.7694E − 6 1.6615E − 6 1.919694

1/2048 4.8759E − 7 4.3915E − 7 1.782557

1/4096 1.4031E − 7 1.2764E − 7 —

1/128 0.022213 0.019514 1.055158

1/256 0.013809 0.009391 1.272992

1/512 0.006252 0.003886 0.805129

1/1024 0.003126 0.002224 1.308369

1/2048 0.001475 0.000898 1.003216

1/4096 0.000766 0.000448 —

Iterative method h 1/32 ε 0.114115 0.088355 DN 1.469484 pN

1/64 0.044888 0.031906 0.709318

Toward General Theory of Differential-Operator and Kinetic Models

Table 2.1.

Volterra Operator Equations with Piecewise Continuous Kernels

79

where x ¯(t) = t2 is exact solution. Table 2.2 shows εN , D N , pN for various values of h. Finally, the results obtained for the following equation are demonstrated:  t/8  t/2 (1 + t + s)x(s) ds + (2 + ts)x(s) ds 0



t/8

3t/4

+ t/8



t

(t + s − 1)x(s) ds − 4

x(s) ds

3t/4

 1 1 − 4 − (16t + 69t2 + 15t3 ) − exp(t/4)(t2 − 13t + 12) = 128 8 + exp(t)(4t2 − 16t + 28) + exp(3t/2)(14t + 20)  − 32 · exp(2t) , t ∈ [0, 2], with known solution x ¯(t) = for various h. 2.2.7.

exp(2t)−1 . 8

Table 2.3 shows εN , DN , pN

Nonlinear equation

Let us consider the following equation with known solution x ¯(t) = t + π:  t/4  t/8 (t − s) sin x(s) ds + t (2 cos x(s)) ds 0



t/8

t

+ t/4

(−1)(sin2 x(s) + 1) ds

17t 7t + cos(t/8) + (1 + 2t) sin(t/8) − 2t sin(t/4) 8 8 1 1 − sin(t/2) + sin(2t), t ∈ [0, 2]. 4 4

=−

Table 2.4 shows the computed maximum pointwise errors εN = x(ti ) − xh (ti )| for various h using the direct method. max0≤i≤N |¯ Finally, the following equation with known solution x∗ (t) = t2 is

80

The errors for various stepsizes h of the second example.

Piecewise constant approximation h 1/32 1/64 ε 0.152263 0.084389 0.073562 0.041074 DN 0.840735 0.915910 pN

1/128 0.043314 0.021769 1.039741

1/256 0.021544 0.010589 0.939704

1/512 0.011043 0.005520 0.976789

1/1024 0.005522 0.002805 0.978239

1/2048 0.002759 0.001423 0.992681

1/4096 0.001385 0.000715 —

Piecewise linear approximation h 1/32 1/64 ε 2.4351E − 3 8.3747E − 4 DN 2.4588E − 3 8.3227E − 4 1.562823 2.328444 pN

1/128 1.7396E − 4 1.6570E − 4 1.371611

1/256 8.0309E − 5 6.4038E − 5 1.116090

1/512 2.7166E − 5 2.9543E − 5 0.910813

1/1024 1.3062E − 5 1.5713E − 5 0.931088

1/2048 6.3391E − 6 8.2413E − 6 0.917034

1/4096 3.1053E − 6 4.3645E − 6 —

1/128 0.043315 0.021770 1.039775

1/256 0.021545 0.010589 0.939564

1/512 0.011043 0.005521 0.976928

1/1024 0.005523 0.002805 0.978051

1/2048 0.002760 0.001424 0.991917

1/4096 0.001385 0.000716 —

Iterative method h 1/32 ε 0.152264 0.073563 DN 0.840754 pN

1/64 0.084389 0.041074 0.915884

Toward General Theory of Differential-Operator and Kinetic Models

Table 2.2.

Table 2.3.

The errors for various stepsizes h of the third example.

1/128 0.127251 0.062537 0.988773

1/256 0.064714 0.031512 0.961195

1/512 0.033201 0.016186 0.945483

1/1024 0.017015 0.008404 0.980078

1/2048 0.008610 0.004260 0.986308

1/4096 0.004350 0.002150 —

Piecewise linear approximation h 1/32 1/64 ε 2.1312E − 2 8.5079E − 3 DN 1.2804E − 2 5.6290E − 3 1.185647 0.828102 pN

1/128 4.6961E − 3 3.1706E − 3 0.880824

1/256 2.0226E − 3 1.7218E − 3 0.367916

1/512 1.8494E − 3 1.3342E − 3 1.404220

1/1024 7.2204E − 4 5.0412E − 4 0.852152

1/2048 3.8049E − 4 2.7926E − 4 1.195077

1/4096 1.5079E − 4 1.2197E − 4 —

1/128 0.119002 0.057691 0.872399

1/256 0.061311 0.031513 0.961200

1/512 0.033202 0.016186 0.945426

1/1024 0.017016 0.008405 0.980055

1/2048 0.008611 0.004261 0.986184

1/4096 0.004350 0.002151 —

Iterative method h 1/32 ε 0.397955 0.169924 DN 0.640177 pN

1/64 0.228031 0.109029 0.918293

Table 2.4.

The errors for various stepsizes h of the fourth example.

Piecewise constant approximation h 1/32 1/64 1/128 ε 0.596074 0.301323 0.130566

1/256 0.065174

1/512 0.029381

1/1024 0.014691

1/2048 0.007345

1/4096 0.003672

Volterra Operator Equations with Piecewise Continuous Kernels

Piecewise constant approximation h 1/32 1/64 ε 0.449935 0.248336 0.201599 0.121084 DN 0.735482 0.953223 pN

81

82

Toward General Theory of Differential-Operator and Kinetic Models

Table 2.5.

The errors for various stepsizes h of the fifth example.

Piecewise constant approximation h 1/32 1/64 1/128 m 5 5 5 ε 0.0286877 0.0152708 0.00730057 Piecewise linear approximation h 1/32 1/64 m 6 9 ε 0.00149694 0.000832073

1/128 9 0.000460489

1/256 5 0.0044031

1/512 5 0.00386043

1/256 9 0.000235685

1/512 10 0.000118614

considered. Here, t ∈ [0, 1].  t  1t 2 (1 + t + s)(x(s) + x2 (s))ds + (1 + 2t)x(s)ds 0

1 t 2

123 4 1 5 17 6 1 t + t + t . = t3 + 3 192 160 1920 Table 2.5 shows the computed maximum pointwise errors for the Newton–Kantorovich scheme. Remark 2.5. Results presented in this section were derived by Muftahov, Tynda and Sidorov [118]. Readers may also refer to [119]. Remark 2.6. Theory and numerical methods described in this section were applied for dynamic analysis of energy storage with renewable and diesel generation by Sidorov et al. [140].

Chapter 3

Nonlinear Differential Equations Near Branching Points In this chapter we describe an algorithm for the construction of parametric families of small branching solutions of nonlinear differential equations of nth order in the neighborhood of branching points. We propose a new technique based on the methods of the analytical theory of branching solutions of nonlinear equations and the theory of differential equations with singular point of the first kind. We illustrate the general existence theorems on the example of nonlinear differential equations appearing in the magnetic insulation problem. This chapter is organized as follows. In Section 3.1, we provide the problem statement and outline the state of the art in the field. In Section 3.1, the method for the construction of the principal theorem of the solution is described. Also, in this section, we reduce the problem to a system of differential equations with a regular singular point with respect to the parametric family of solutions with the special structure, depending on the behavior of the roots of characteristic polynomial. In Section 3.2, we consider the technique for constructing the asymptotics of parametric families of solutions of the equation obtained in Section 3.1 and adduce the existence theorems for the small solution of the main equation. In Section 3.3, we demonstrate the application of this method for the magnetic insulation problem.

83

Toward General Theory of Differential-Operator and Kinetic Models

84

3.1.

Problem Statement

Consider a continuous function F (xn , xn−1 , . . . , x1 , x, t) defined in a neighborhood of origin. Here, F =

N 

Fik (¯ x)tk + R(x, t),

|i|+k=1

x) = Fin ,in−1 ,...,i1 ,i0 xinn ..xi11 xi0 , Fik (¯

|i| = in + in−1 + · · · + i1 + i0 .

The function R(x, t) has the bound |R(x, t)| = o((|x| + |t|)N ). The main equation to be addressed in this chapter is as follows: F (x(n) (t), x(n−1) (t), . . . , x(1) (t), x(t), t) = 0,

0 ≤ t ≤ p.

(3.1)

Our objective here is to find a solution to equation (3.1) such that, for t → 0, ti x(i) (t) → 0 for i = 0, 1, . . . , n. Definition 3.1. If a solution x(t) is representable in the form x = tε (x0 + v(t)),

(3.2)

where ε = rs is a rational positive number, x0 = 0, ε, x0 are defined nonuniquely, the function v(t) tends to zero as t → 0 and possibly depends on free parameters, ti x(i) (t) → 0 as t → 0, i = 0, . . . , n, then we call t = 0 the branching point of the small solution to equation (3.1). Classes of differential equations (ordinary and partial ones) which are not resolved with respect to major derivatives have been studied in recent years by many mathematicians. See, for example, [143,167,169] for various approaches and related references. In many applications (see, e.g., [1]), it is necessary to construct solutions near branching points. Our objective is to find small solutions to (3.1) based on the analytical branching theory (readers may refer here to [167, Chapter 9; 178]) and the theory of differential equations with a regular singular point (see, e.g., [169] or [35]). For calculating the principle term tε x0 of solution (3.2), we use the method of Newton diagram (here readers may refer for example to

Nonlinear Differential Equations Near Branching Points

85

[169, 167, Chapter 9; 28]). The determination of the function v(t) in solution (3.2) is reduced to the study of nonlinear differential equations in the form   d L t v = M (tn−1 v (n−1) , . . . , tv (1) , v, t1/s ), (3.3) dt n

n−1

d n−1 d where L = tn dt n + an−1 (ε)t dtn−1 + · · · + a0 (ε) is the Euler differential operator. We construct c-parametric families of solutions v(t, c) → 0 as t → 0 to equation (3.3). The structure of the family v(t, c) depends on the roots of the characteristic polynomial

L(λ) = λ(λ − 1) · · · (λ − (n − 1)) + an−1 (ε)λ(λ − 1) · · · (λ − (n − 2)) + · · · + a0 (ε)

(3.4)

of the Euler differential operator and on the form of the Jordan normal form of the matrix A expressed in terms of coefficients of the Euler operator. On this basis, we propose a technique for constructing the asymptotics of the function v as logarithmic power sums of the Fuchs–Frobenius type (for more details, readers may refer to [35, Chapter 4]). Then we use the asymptotics as the initial approximation in the method of successive approximations. In the analytical case, we expand the corresponding solution (3.2) in powers of t1/s , t ln t, tReλi cos(Im λi ln t), tRe λi sin(Im λi ln t), where λi are roots of the characteristic polynomial L(λ), Re λi > 0. 3.1.1.

Principal term of the solution: Problem reduction N x, t) = x)tk and two Introduce the denotation F N (¯ |i|+k=1 Fik (¯ conditions: A. There exist rational numbers ε = rs > 0, and θ = rs11 such that the expansion of F N can be reduced to the form  x, t) = Fik (¯ x)tk . F N (¯ ε|i|+k−i1 −2i2 −···−nin ≥θ

B. R(tε x, t) = o(tθ ) with any x in a neighborhood of origin.

86

Toward General Theory of Differential-Operator and Kinetic Models

In concrete cases, one can easily calculate ε, θ by plotting integer points (|i|, k − i1 − 2i2 − · · · − nin ) that correspond to x) on the coordinate plane nonzero |i|-homogeneous forms Fik (¯ and by constructing the Newton diagram for these points. We set the desired value of ε to tan φ, where φ is the angle of the inclination of one of the segments of the diagram to the negative direction of the abscissa. The corresponding value of θ equals the ordinate of the point of intersection of the continuation of this segment with the coordinate axis. θ can be negative. In the latter case, condition B is satisfied automatically due to the bound R(x, t) = [(|x| + |t|)N ]. Evidently, condition B is also satisfied with any positive θ if the Newton diagram is located below the straight line that goes through points (0, N ) and (N, 0). Since the Newton diagram can have several segments, the choice of ε and θ can be nonunique. Let us introduce the Euler differential operators of the kth order lk (u) = tk u(k) + c1k εtk−1 u(k−1) + c2k ε(ε − 1)tk−2 u(k−2) + · · · + ε(ε − 1) · · · (ε − (k − 1))u,

k = 1, . . . , n.

It is to be noted that (tε u)(k) = tε−k lk (u). Then in view of condition A by changing the variable x(t) = tε u(t), we obtain the expansion F (x(n) , . . . , x(1) , x, t) = tθ



Fik (ln (u), . . . , l0 (u))

ε|i|+k−i1 −2i2 −···−nin =θ

+ r(tn u(n) , . . . , tu(1) , u, t). Here, for the sake of symmetry, we denote l0 (u) = u. Due to the choice of numbers ε and θ, we have the following estimate: |r(tn u(n) , . . . , tu(1) , u, t)| = o(tθ ).

(3.5)

Nonlinear Differential Equations Near Branching Points

87

In order to determine the coefficient x0 in the desired solution (3.2), we introduce the polynomial Q(ln (x), . . . , l0 (x))  =

Fik (ln (x), . . . , l0 (x)),

ε|i|+k−i1−2i2 −···−nin =θ

where lk (x) = ε(ε − 1) . . . (ε − (k − 1))x, k = 1, . . . , n. We assume that apart from the conditions A and B, the following condition is also satisfied: C. The polynomial Q(ln (x), . . . , l0 (x)) has a root x0 = 0, and ∂Q ∂ln (x) |x=x0 = 0. Then by changing the variable by formula (3.2) and dividing (3.1) by tθ , we reduce it to the following equation with respect to the function v(t): An ln (v) + · · · + A0 l0 (v) + P (tn v (n) , . . . , tv (1) , v, t1/s ) = 0. (3.6) Here, Ai = ∂Q ∂li |x=x0 , i = 0, . . . , n, |P | = o(1) as t → 0, li (v) are the Euler differential operators. Since An = 0, by substituting differential operators li (v) in (3.6), we obtain the equation tn v (n) (t) + an−1 (ε)tn−1 v (n−1) (t) + · · · + a0 (ε)v(t) n (n) (t), . . . , tv (1) , v(t), t1/s ) = 0, + A−1 n P (t v

(3.7)

where an−1 (ε) = c1n ε + A−1 n An−1 , an−k (ε), k = 2, . . . , n, are certain k-order polynomials of ε. Thus, in order to determine functions v(t), we obtain equation d )v of nth order in the main part. (3.1) with Euler operator L(t dt It is to be noted that the characteristic polynomial L(λ) in (3.4) is the characteristic polynomial of just this Euler operator. Since |P | = o(1) as t → 0 with any v, equation (3.1) allows one to use the method of successive approximations for determining the element tn v (n) in the neighborhood of origin as function of v(t), tv (1) , . . . , tn−1 v (n−1) , t1/s .

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Toward General Theory of Differential-Operator and Kinetic Models

As a result, the problem on determining the function v in our statement is reduced to an equation in the form (3.3). Note that in (3.3), the function M and its first derivatives in v, tv (1) , . . . , tn−1 v (n−1) vanish at zero. The replacement v = v1 , tv (1) = v2 , . . . , tn−1 v (n−1) = vn turns equation (3.3) into the following system of n nonlinear differential equations with a singular point of the first kind at t = 0: t

d¯ v = A¯ v + f (¯ v , t1/s ), dt

where v¯ = (v1 , . . . , vn ) ,   0 1 0   0 1 1   0 0 2  A =  . . .. ..  .. .   0 0 0   −a0 (ε) −a1 (ε) −a2 (ε)

... ... ... .. . ··· ···

(3.8)

    . 0   . 0  , .. ..   . .   n−2 1  −an−2 (ε)) n − 1 − an−1 (ε))  .

0

f (0, . . . , 0, M (vn , vn−1 , ..v1 , t1/s )) , M (0, . . . , 0) = 0,  ∂M  = 0, i = 1, . . . , n. ∂vi v1 =···=vn =t=0 Thus, the calculation of the function v in the representation of the desired small solution (3.2) to (3.1) is reduced to finding a solution v → 0 as t → 0 from system (3.8). Remark 3.1. One can easily verify the identity det(−λE + A) = (−1)n L(λ), where L(λ) = λ(λ − 1) · · · (λ − (n − 1)) + an−1 ()λ(λ − 1) · · · (λ − (n − 2)) + · · · + a0 () is the characteristic polynomial of the Euler differential operator L in the main part of equation (3.3). Due to the structure of the matrix A, we have rank(−λE + A) ≥ n − 1 with any λ.

Nonlinear Differential Equations Near Branching Points

89

If rank(−λE + A) = n − 1, then vector e¯ that satisfies the homogeneous system λ¯ e = A¯ e has a Jordan chain of length p, where p is multiplicity of the root λ of characteristic polynomial L(λ). Theorem 3.1. Let conditions A−C be satisfied. Fix N > sA and assume that among the roots of characteristic polynomial (3.4), there are no numbers si , i = 1, . . . , N. Then equation (3.1) has a solution in the form x(t) =

N 

xi t

r+i s

+ o(t(r+N )/s ).

(3.9)

i=0

Proof. Choose numbers r/s and x0 in accordance with conditions A−C. We seek for a solution to the main equation (3.1) in the form x = tr/s (x0 + v(t)). Here, v(t) is the first component of the vector v¯, which satisfies system (3.8). We seek for the vector v¯ in the form v¯ =

N 

v¯i ti/s + tN/s ω ¯ (t),

(3.10)

i=0

where ω ¯ → 0 as t → 0. Substituting (3.10) into (3.8) and taking into account the fact that, by assumption of Theorem 3.1, det(nE − sA) = 0 with n ∈ N, we recurrently determine coefficients v¯i from the linear systems   i E − A v¯i = mi (v¯1 , . . . , vi−1 ¯ ), i = 1, . . . , N. (3.11) s The right-hand sides in (3.11) are constructed with the method of undetermined coefficients. Since v¯i = (vi1 , . . . , vin ) , in formula (3.9), we set xi = vi1 , i = 1, . . . , N. For determining the vector function w(t), ¯ we obtain the following system   N dw ¯ = − E+A w ¯ + g(w, ¯ t1/s ), (3.12) t dt s ¯ ≤ r, 0 ≤ t ≤ p, where q = O(t1/s ) with w ¯2 . g(w¯1 , t1/s ) − g(w¯2 , t1/s ) ≤ lw¯1 − w It remains to be proved that with sufficiently large N system (3.12) has a unique solution w ¯ → 0 as t → 0. One can find this solution

Toward General Theory of Differential-Operator and Kinetic Models

90

from the equivalent integral equation    t t N −1 g(w(τ ¯ ), τ 1/s ) dτ ≡ Φ(w, τ exp − E + A ln ¯ t), w(t) ¯ = s τ 0 (3.13) using the method of successive approximations with the zero initial approximation w¯0 = 0. Indeed, since N > SA, ln( τt ) ≥ 0 with τ ≤ t, sign τ = sign t, there exists a constant value C such that with τ ≤ t the following bound is valid:       N exp − N E + A ln t  ≤ C(t/τ )− s +A .  s τ L(Rn →Rn ) Therefore, with N > sA, we obtain the following bound:  t      t N C −1  τ exp − E + A ln dτ  ≤ N .  s τ n n − A L(R →R )

0

Fix q ∈ (0, 1) and choose N such that cl ≤ q. N s − A

S

(3.14)

¯ with Introduce the space C[0,p] of continuous vector functions w(t) norm w =

max

0≤t≤p, 1≤i≤n

|wi (t)|.

In this space, we define the set S = {w ¯ ≤ r, |wi (t)| ≤ rt1/s , i = 1, . . . , n}. Let us seek for a small solution w(t) ¯ to equation (3.13) in set S. With sufficiently large N in view of bound (3.14), we have ¯2 , t) ≤ qw ¯1 − w ¯2 . Φ(w ¯1 , t) − Φ(w Therefore, the operator Φ is contractive with w ¯ ≤ r. Furthermore, with w ¯ ≤ r, we have Φ(w, ¯ t) ≤ Φ(w, ¯ t) − Φ(0, t) + Φ(0, t)

  

t t N



g(w(τ ¯ ), τ 1/s )dτ

τ −1 exp − E + A ln ≤ qr + max

0≤t≤p 0

s τ ≤ qr +

N s

c max q(0, τ 1/s )Rn . 0≤τ ≤p − A

Rn

Nonlinear Differential Equations Near Branching Points

91

Since g(0, τ 1/s ) → 0 as τ → 0, with given q and N , we can choose c max0≤τ ≤p g(0, τ 1/s )RN ≤ p > 0 so as to fulfill the inequality N −A s

(1 − q)r. Therefore, the operator Φ maps the ball w ¯ ≤ r to itself. Moreover, Φ(w, ¯ t) = O(t1/s ) because Φ(w, ¯ t) ≤

N s

c q(w, ¯ t1/s ), − A

where g(w, ¯ t1/s ) = O(t1/s ). Therefore, the contractive operator Φ maps the set C[0,p] to itself, while system (3.12) has a unique solution w ¯ → 0 as t → 0, provided  that N is sufficiently large. When solving concrete equations under the assumptions of Theorem 3.1, one can consider for a solution immediately in the form of series (3.9); in the analytical case, it converges in a neighborhood of zero. Let us weaken the conditions of Theorem 3.1, assuming that the characteristic polynomial L(λ) has roots in the form i/s. Then the obtained result becomes more interesting, namely, in this case, equation (3.1) has a c-parametric family of solutions in the form (3.9), where the coefficients, beginning with some i, are functions of ln t depending on p arbitrary constants, p is the multiplicity of the root si of the characteristic polynomial. Let us describe the way for constructing the family of small solutions in this case. Introduce the following auxiliary linear system: dv = Bv + f (z), (3.15) dz where B is a constant matrix and f (z) is the polynomial of the order m. Consider the construction of polynomial solutions to this system. If det B = 0, then system (3.15) in the class of polynomials m i has the only solution v = i=0 ai z . Coefficients am , am−1 , . . . , a0 are calculated in the indicated order by the method of undetermined coefficients. If det B = 0, then for constructing a solution to system (3.15) in the class of polynomials, it is convenient to use the Jordan

92

Toward General Theory of Differential-Operator and Kinetic Models

normal form of the matrix B. Really, let T BT −1 = J, where J = {λ1 E1 + H1 , . . . , λk Ek + Hk } is the normal Jordan set, λi Ei + Hi are Jordan cells of the order pi . Some of λi can coincide. Let rank B = r. Then, without loss of generality, we can assume that λ1 = · · · = λn−r = 0, λi = 0, i = n − r + 1, . . . , k. In this case, the following lemma is valid. Lemma 3.1. Let rank B = r and let a vector f (z) be a polynomial of the order m. Then system (3.15) has a polynomial solution of the order m + max(p1 , . . . , pn−r ), depending on p1 + · · · + pn−r arbitrary constants. Proof. By putting u = T −1 ω in system (3.15) and multiplying the result by T , we reduce it to the following form: dw = Jω + T f (z). (3.16) dz In accordance with the structure of the matrix J, we divide system (3.16), into k independent subsystems: dwi = (λi Ei + Hi )wi + pi (z), i = 1, . . . , k. dt Since λ1 = · · · = λn−r = 0, we calculate the coordinates of vectors wi , i = 1, . . . , n − r, successively pi times integrating (from zero to z) the source part of the equation. Therefore, vectors wi (z), i = 1, . . . , n − r, appear to be polynomials of the order m + pi , depending on pi constants of integration. The rest of the vectors wi (z), i = n − r + 1, . . . , k, are uniquely defined by means of the method of undetermined coefficients as polynomials of the mth order because  det(λi Ei + Hi ) = 0 for i = n − r + 1, . . . , k. Theorem 3.2. Let conditions A−C be satisfied. Assume that some of the numbers 1s , 2s , · · · Ns , where N ≤ sA are roots of the characteristic polynomial L(λ), denote the least of them by l/s. Then equation (3.1) has a solution in the form x(t) =

l−1  i=0

xi t

r+i s

+

N  i=l

xi (ln t)t(r+i)/s + o(t

r+N s

).

(3.17)

Nonlinear Differential Equations Near Branching Points

93

Proof. The proof of Theorem 3.2 is analogous to the proof of Theorem 3.1, but one should take into account the fact that coefficients in solution (3.17) depend on ln t. Therefore, instead of the algebraic system (3.17), we obtain   i E − A vi = mi (v1 , . . . , vi−1 ), i = 1, . . . , l − 1 s and the following systems of differential equations (z = ln t):   i dvi = − E + A vi + mi (v1 , . . . , vi−1 ), i = l, l + 1, . . . , N. dz s Since

 det

 i E − A = 0 s

with i = 1, . . . , l − 1, the values v1 , . . . , vl−1 are determined unambiguously and are independent of z. By conditions of Theorem 3.2, we have det(− sl E + A) = 0, and in view of Remark 3.1   l rank − E + A = n − 1. s The corresponding eigenvector has a Jordan chain of length pl , where pl is the multiplicity of the root sl of the characteristic polynomial L(λ). Therefore, according to the lemma, vl (z) is the polynomial of the order pl depending on pl arbitrary constants of integration. One can also construct coefficients vl+1 (z), . . . , vN (z) as polynomials, whose orders are determined in accordance with Lemma 3.1. These coefficients can contain new arbitrary constants if l/s is not the only root of the characteristic polynomial L(λ) among numbers (l/s, (l +  1)/s, . . . , N/s). 3.2.

Open Problems and Generalizations

If among roots of the characteristic polynomial there are λ, possibly, complex ones with positive real parts, which do not belong to the set (l/s, (l + 1)/s, . . . , N/s), then the class of small solutions to equation (3.1) is wider than the set of those constructed above in

94

Toward General Theory of Differential-Operator and Kinetic Models

the theorems; it can be extended in the class of complex-valued functions. Since coefficients of the characteristic polynomial L(λ) are real, the polynomial L(λ), together with a root λ, has the conjugate root λ. Therefore, partial sums of the corresponding expansions of small solutions can contain functions in the form tReλ cos(Im λ ln t), tRe λ sin(Im λ ln t). Consider the construction of small solutions in this case. Introduce the vector λ = (λ1 , . . . , λl ), where λi are the roots of the characteristic polynomial, Re λi > 0. Assume that roots  λi , are extracted such that λi = lj=1, j=i mj λj + m with natural m, mj . For simplicity, let the function R(x, t) in equation (3.1) be infinitely differentiable near zero and s = 1 in formula (3.2). We will seek for a solution to the reduced system (3.8) in the form v=

∞  j=1

j

v0j (ln t)t +

∞ 

vij (ln t)t(λ,i)+j .

(3.18)

j=0, |i|≥1

Note that the second sum contains no integer powers of the argument t due to the choice of the vector λ. By means of the method of undetermined coefficients, we obtain the following recurrent sequence of linear differential equations with respect to voj (z), vij (z), (z = ln t): dv0j = (−jE + A)voj + moi (v01 , . . . , v0j−1 ), dz dvij = ((−(λ, i) + j)E + A)vij + mij (vrs ), dz

j = 1, 2, . . . ,

|r| + s < |i| + j, i = 1, 2, . . . , j = 0, 1, . . . . Note that mi0 = 0 with |i| = 1 and in view of Remark 3.1, rank(−(λ, i)E + A) = n − 1, |i| = 1. Therefore, coefficients vi0 with |i| = 1 are determined (accurate to an arbitrary constant) as solutions to the corresponding homogeneous systems (−(λ, i)E + A)vi0 = 0,

|i| = 1.

This lemma also allows one to obtain the rest of the coefficients vij (z) in expansion (3.18) as polynomials of z of increasing orders. As a result, we obtain the following family of small solutions to

Nonlinear Differential Equations Near Branching Points

equation (3.1): ⎛ x(t) = tr ⎝x0 +

∞  j=0

v0j (ln t)tj +

∞  ∞ 

95

⎞ vij (ln t)t(λ,i)+j ⎠ .

(3.19)

j=0 |i|=1

Coefficients vij are functions of ln t and depend on l free parameters, where l is the number of roots with positive real parts of the characteristic polynomial L(λ), taking into account their multiplicities. As in Theorem 3.1, using the principle of contracted mappings, one can prove that partial sums of the formal solution (3.19) are asymptotic approximations of the family of small solutions in the form (3.2) to main equation (3.1). In the analytical case, series (3.19) converges in a neighborhood of origin. 3.3.

Magnetic Insulation Model Example

Consider the following differential equation:  d2 φ (1 + φ)2 − 1 2 = j(1 + φ). dx This model equation occurs in the analysis of the magnetic insulation problem of a vacuum diode. Here, φ is the potential of the electric field and j is the current. Here, readers may refer to [1]. Let us seek for a small continuous solution subject to φ(0) = 0, φ (0) = 0. In this example, the Newton diagram has one segment with vertices at points (3/2, −2), (0, 0). Replacement (3.2) takes the form φ(x) = x4/3 (φ0 + 9 v(x)), where φ0 = ( 4√ j)3/2 , while v(x) satisfies the equation 2 8 dv 2 d2 v + x + v = M (v, x1/3 ) 2 dx 3 dx 3 with the Euler differential operator of the second order in the main part, M (0, 0) = 0, Mv (0, 0) = 0. The corresponding characteristic polynomial λ2 + 53 λ + 23 = 0 has two negative roots: λ1 = −2/3 and λ2 = −1. Therefore, based on Theorems 3.1 and 3.2 in a neighborhood of the point x = 0, exactly one small real solution 9 j)2/3 x4/3 + O(x5/3 ) satisfies our equation with arbitrary φ(x) = ( 4√ 2 current j. x2

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Chapter 4

Nonlinear Operator Equations with a Functional Perturbation of the Argument 4.1.

Nonlinear Operator Equations

Let E1 and E2 be Banach spaces. Consider the nonlinear operator equation F (x(t), x(α(t)), t) = 0,

(4.1)

where F : E1 × E1 × R → E2 is an analytical nonlinear mapping in a neighborhood of zero, i.e., F (x(t), x(α(t)), t) =

∞ 

Fijk xi (t) xj (α(t)) tk ,

(4.2)

i+j+k≥1

where the Fijk are power-law operators [169] with respect to x(t), and  i x(α(t)), and α(t) = ∞ 1 αi t , |α1 | < 1 is a functional perturbation of the argument (FPA) t of neutral type. Analytic solutions of such equations for the case E1 = E2 = R were constructed in [10]: readers may also refer to [33]. Definition 4.1. If F100 = 0, F010 = 0 in the expansion (4.2), then equation (4.1) is considered to be quasilinear. Let us construct small branching solutions x(t) → 0 as t → 0 of equation (4.1). The substitution x(t) = tε (u0 + u(t)),

97

u0 = 0,

(4.3)

98

Toward General Theory of Differential-Operator and Kinetic Models

where ε = r/s is a positive rational number found by the Newton diagram method, permits one to reduce the construction of small solutions x(t) → 0 as t → 0 to a quasilinear equation for the function u(t) with FPA of neutral type: Au(t) − Bu(α(t)) + R(u(t)),

u(α(t), t) = 0,

(4.4)

where A and B are linear bounded operators depending on u0 and corresponding to the Fr´echet derivative of operator (4.2), and R(u(t), u(α(t)), t) =

∞ 

Rij0 ui (t) uj (α(t))

i+j=2 ∞  ∞ 

+

Rijk ui (t) uj (α(t)) tk/s . (4.5)

i+j=0 k=1

The element u0 is found from the following truncation equation:   εj Fijk ui+j (4.6) P (u0 ) := 0 (α (0)) = 0. (i+j)ε+k=θ

Therefore, the Newton diagram method permits one to reduce equation (4.1) to the quasilinear equation (4.4) with FPA of neutral type and the truncation equation (4.6). Theorem 4.1. Let the truncation equation (4.6) corresponding to a certain segment of the Newton diagram of the operator (4.2) for the chosen ε > 0 have a solution u0 = 0, let A be a continuously invertible operator, and let B ≡ 0. Then equation (4.1) has a solution x(t) =

∞ 

xi ti/s ,

xr = u0 .

i=r

Proof. Since A is a continuously invertible operator and B ≡ 0, it follows that coefficients xi can be uniquely found by the method of undetermined coefficients. The convergence of the series in a neighborhood of the point t = 0 can readily be proved with the  use of the contraction mapping principle.

Nonlinear Operator Equations with a Functional Perturbation of the Argument

99

It is to be noted that the substitution t = τ s permits one to eliminate fractional powers from equation (4.4). Therefore, to simplify the exposition, we assume that s = 1 in equation (4.4). We consider equation (4.4) in the general case where B = 0. We assume that A is a continuously invertible operator. We prove that, in the general case, equation (4.1) has a branching solution that can be represented by logarithmic power series. Let us introduce the linear bounded operator C := A − B. Let C : E1 → E2 be a Fredholm operator and have a complete B-Jordan (i) set {φi }, i = 1, . . . , n, j = 1, . . . , pi ; i.e., the following conditions are satisfied: (1)

Cφi

= 0,

(j+1)

i = 1, . . . , n, (j)

= Bφi ,

Cφi

(pi )

det Bφi

i = 1, . . . , n,

j = 1, . . . , pi − 1,

, ψj ni,j=0 = 0,

where C ∗ ψj = 0, j = 1, . . . , n. Consider the triangular system of N operator equations of the form Lx = β,

(4.7)

where L = [Lik ]l,k=1,...,N . If k > i, then Lik = 0. If k ≤ i − 1, then Lik = −Baik , where ai,i−1 = 0. If k = i, then Lik = C. Vector β belongs to E2 ×· · ·×E2 , x ∈ E1 ×· · ·×E1 , and N ≥ maxi=1,...,n {pi }. Lemma 4.1. The operator L acts from E1N to E2N and is a Fredholm operator. In addition, dim N (L) = dim N (L∗ ) = k, where k = p1 + · · · + pn is a B-root number of the Fredholm operator C. The proof is based on the generalized Schmidt lemma, and the properties of complete Jordan sets introduced in Chapter 1. Corollary 4.1. If the column vector β has the form (0, . . . , 0, βmax{pi }+1 , . . . , βmax{pi }+m ) , where m = N − max{pi }, then the system Lx = β is solvable.

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Toward General Theory of Differential-Operator and Kinetic Models

To construct the solutions of (4.4), one has to solve several differential-operator equations with polynomial on the right-hand side of the form Ax(z) − Bx(z + a) = P (z), (4.8)  i where z = ln |t|, a ∈ R, P (z) = m i=0 Pi z is a polynomial of degree m in the argument z, and the coefficients Pi belong to E2 , i = 1, . . . , m. The solution x(z) is constructed in the form of a polynomial of z. Lemma 4.2. Let C be a continuously invertible operator. Then equation (4.8) has a unique solution x(z) =

m 

xi z i .

(4.9)

i=0

To prove this assertion, it suffices to substitute the suggested solution into the original equation. For the coefficients of the solution, we obtain a triangular operator system with an invertible operator on the diagonal, which uniquely determines all coefficients of the solution. Lemma 4.3. Let dim N (C) = dim N (C ∗ ) = n, and let {φi }, i = 1, . . . , n, j = 1, . . . , pi , be a complete B-Jordan set of the operator C. Then equation (4.8) has a solution (j)

x(z) =

m+p 

xi z i ,

(4.10)

i=0

where p = maxi=1,...,n{pi }, which depends on k = p1 + · · · + pn arbitrary constants. Proof. The proof of Lemma 4.3 follows from Lemma 4.1. To this end, it suffices to substitute the suggested solution into the original equation. As a result, for the coefficients xi of the solution (4.10), we obtain a triangular operator system of the form (4.7) with a Fredholm operator on the diagonal, which was considered in Lemma 4.1 and Corollary 4.1. Therefore, the solvability of this system and Lemma 4.1

Nonlinear Operator Equations with a Functional Perturbation of the Argument

101

will follow from Corollary 4.1. The coefficients xi of the expansion (4.10) can be computed successively by the method of indeterminate coefficients. Here, the coefficient of the higher power of z is computed first. The arbitrary constants arising in the computations are found from the solvability conditions for the subsequent equations of the  system. Let us return to the problem of constructing a small solution of the quasilinear equation (4.4) with s = 1. Lemma 4.4. Suppose that numbers q ∈ (0, 1) and Q satisfy the inequality q Q ||A−1 B|| ≤ q,

(4.11)

and there exists a function u∗Q (t) satisfying the estimate ||Au∗Q (t) − Bu∗Q (α(t)) − R(u∗Q (t), u∗Q (α(t)), t)|| = o(|t|Q ), as t → 0. (4.12) Then there exist numbers  > 0 and r > 0 such that equation (4.4) has a solution x(t) = u∗Q (t) + tQ v(t)

(4.13)

in the neighborhood |t| < , where v(t) → 0 as t → 0 and ||v|| := max ||v(t)||E1 ≤ r. |t|≤

(4.14)

Proof. The desired assertion follows from the contraction mapping principle such that the substitution (4.13) leads to an equation for the function v(t) with contraction operator for sufficiently  large Q. The function u∗Q (t) used in the substitution (4.13) can be constructed by the polynomial u∗Q (t)

=

Q 

us (ln |t|) ts ,

(4.15)

s=1

where the coefficients us (ln |t|) are polynomials of increasing degrees in ln |t| and can be computed from recursion sequences of difference

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Toward General Theory of Differential-Operator and Kinetic Models

equations by Lemmas 4.1–4.3. We have thereby proved the main theorem of this chapter. Theorem 4.2. Suppose that the operator A in (4.4) is continuously invertible, α(0) = 0, |α (0)| < q < 1, Q satisfies the inequality |α (0)|Q ||A−1 B|| < q < 1, the functions α(t) and R(u(t), u(α(t)), t) are analytic or have smoothness of order Q in some neighborhood of zero, and the number λ = 0 is a regular point or an isolated Fredholm point of the operator family ˜ := A + (λ − α (0)i )B, C(i)

i = 1, 2, . . . , Q.

(4.16)

Then equation (4.4) has a solution of the form u(t) =

Q 

ui (ln |t|) ti + tQ v(t).

(4.17)

i=1

˜ If λ = 0 is a regular point of all operators C(i), then the coefficients ui are independent of ln |t| and are uniquely determined. If λ = 0 is ˜ a regular point of the operators C(i), i = 1, . . . , j − 1, j ≤ Q, and ˜ an isolated Fredholm point of the operator C(j), then the coefficients u1 , . . . , uj−1 are also uniquely determined, and the subsequent coefficients uj (ln |t|), . . . , uQ (ln |t|) are polynomials in the argument ln |t| and depend on free parameters. The function v(t) occurring in the solution (4.17) can be constructed by the successive approximation method and satisfies the estimate ||v|| = o(1) as t → 0. Therefore, the above-represented method permits one to prove existence theorems and construct parametric families of real solutions of equation (4.1) in the form x(t) = t

r/s

 u0 +

Q  i=1

ui (ln(|t|)) t

i/s

Q/s

+ o(|t|

 ) .

Nonlinear Operator Equations with a Functional Perturbation of the Argument

4.2.

103

Conclusion

In this section, we constructed small solutions x(t) → 0 as t → 0 of nonlinear operator equation F (x(t), x(α(t), t)) = 0 with functional perturbation α(t) of the argument. By the Newton diagram methods, we have the problem of quasilinear operator equations with functional perturbation of the argument. It is demonstrated that solutions of such equations can have not only algebraic but also logarithmic branching points and contain free parameters. We also note that the number of free parameters and the solution form depend on the properties of the Jordan structure of the operator coefficients of the equation. The results of this chapter is published in [152, 153] in Russian language.

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Chapter 5

Nonlinear Systems’ Equilibrium Points: Stability, Branching, Blow-Up This chapter focuses on the nonlinear dynamic model formulated as the system of differential and operator equations. This system is assumed to satisfy an equilibrium point. The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process. Sufficient conditions of the global classical solution’s existence and stabilization at infinity to the equilibrium point are formulated. Solution can be constructed by the method of successive approximations. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some solutions can blow up in a finite time, while others stabilize to an equilibrium point. The special case of considered dynamic models are differentialalgebraic equations which model various nonlinear phenomena in circuit analysis, power systems, chemical processes and many other processes. Let us consider the system A

dx = F(x, u), dt 0 = G(x, u).

(5.1) (5.2)

Here, linear operator A : D ⊂ X → E has a bounded inverse, nonlinear operators F : X  U → E, G : X  U → U are continuous in the neighborhoods ||x||X ≤ r1 , ||u||U ≤ r2 of real Banach spaces 105

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X, U , and E is the linear real normed space. The following operator decompositions are assumed: F(x, u) = F(0, 0) + A1 x + A2 u + R(x, u),

(5.3)

||R(x, u)|| = o(||x|| + ||u||),

(5.4)

G(x, u) = G(0, 0) +

n 

  dk G(0, 0); (x, u) + r(x, u),

(5.5)

k=1

  ||r(x, u)|| = o (||x|| + ||u||)n .

(5.6)

Here, the derivatives and Fr´echet differentials calculated in point (0, 0) as follows: A1 :=

∂F(0, 0) ∂F(0, 0) ∂G(0, 0) ∂G(0, 0) , A2 := , A3 := , A4 := , ∂x ∂u ∂x ∂u k

d (G(0, 0); (x, u)) =

 i+j=k

  xi uj , ∂xi ∂uj x=x0 ,u=u0

∂ Cki

k G(0, 0) 

∂ k G(x, u) · ·  U → U. : X · ·  X  U   ·   · ∂xi ∂uj i times j times It is assumed that point (0, 0) satisfies operator equations F(x, u) = 0, G(x, u) = 0. Therefore, (0, 0) is the equilibrium point of system (5.1)–(5.2). The special cases of system (5.1)–(5.2) with initial condition x(0) = Δ

(5.7)

are considered in many publications. Here, Δ is an element from the neighborhood of equilibrium point. Solution x(t), u(t) on semiaxis [0, +∞) is constructed such that limt→+∞ (||x(t)|| + ||u(t)||) = 0. Functions u(t) are selected such that solution x(t), u(t) is stabilized to the equilibrium point (0, 0) as t → +∞. Such a problem is important for the solution of various automatic control problems. It is also important for nonlinear dynamic systems’ mathematical

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modeling using the “input”–“output” approach in power system stability, when u is the “input”, and y = h(x, u) is the “output”. In the case of closed loop, the output must satisfy the given criteria in the form of equation G(y) = 0. In the works on the power system stability, models (5.1) and (5.2) with ordinary differential equations have been intensively studied. Calculations were performed to demonstrate the problem complexity caused by stability analysis and possible blow-up of the solution. Constructed theory enables the unified consideration of “input– output” models involving both differential and integro-differential equations. Let F(0, 0) = 0 and G(0, 0) = 0. Definition 5.1. We call the ball ||x|| ≤ r the basin of attraction of equilibrium point (0, 0) of system (5.1)–(5.2) such that, for arbitrary Δ from this ball, there exist solutions x : [0, +∞) → X and u : [0, +∞) → U with initial condition x(0) = Δ stabilizing to zero on the positive semiaxis. Definition 5.2. If the basin of attraction of an equilibrium point is nonempty (r > 0), then the stationary solution (0, 0) of system (5.1)–(5.2) is called asymptotically stable. The objective of this chapter is the construction of sufficient conditions for the nonempty basin of attraction of the equilibrium point, the proof of the existence and uniqueness theorem for the solution (5.1)–(5.2) with the initial condition x(0) = Δ and the development of the method of successive approximations of the solution of the Cauchy problem on semiaxis [0, +∞). Sufficient conditions are formulated for the Cauchy problem’s solution branching for system (5.1)–(5.2) with the stability analysis of individual branches of this solution. The main part of this section concentrated on the case when the linear operators A and A4 have a bounded inverse. Spectrum of linear bounded operator acting from X to X, M = A−1 (A1 − A2 A−1 4 A3 ),

(5.8)

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will be used. It is assumed that Re λ ≤ −l < 0 for λ ∈ σ(M). The existence and uniqueness theorem on the semiaxis [0, ∞) as well as the asymptotic stability of the Cauchy problem for sufficiently small norm ||x(0)|| are proved. It is shown that the solution branching of the Cauchy problem can occur. Some of the branches extend to the whole semiaxis [0, +∞) and they stabilize to zero as t → +∞, and others can collapse (go to infinity). The illustrative examples are given. 5.1.

Reduction of a Nonlinear System in the Neighborhood of an Equilibrium Point to a Single Differential Equation

Let G(x, u) = A3 x + A4 u + r(x, u), where ||r(x, u)|| = o(||x|| + ||u||). The Implicit Mapping Theorem allows us to prove the following lemmas. Lemma 5.1. Let operator A4 have a bounded inverse such that A4 = ∂G(0,0) is a Fr´echet derivative. Then for arbitrary ball S1 : ||x|| ≤ r1 , ∂u there exists ball S2 : ||u|| ≤ r2 such that for any x ∈ S1 , equation (5.2) has a unique continuous solution u(x) in ball S2 . In this case, the following asymptotic representation of the solution is valid: u = −A−1 4 A3 x + o(||x||), as ||x|| → 0, where A3 =

∂G(0,0) ∂x

(5.9)

is the Fr´echet derivative.

From Lemma 5.1, we obtain the following lemma. Lemma 5.2. There exists neighborhood ||x|| ≤ r, ||u|| ≤ r2 such that system (5.1)–(5.2) can be uniquely reduced to the following differential equation: A

dx = f (x), dt

(5.10)

where mapping f : S(0, r) ⊂ X → E2 := F(x, u(x)) is defined as follows: f (x) = A1 x − A2 A−1 4 A3 x + L(x).

(5.11)

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Nonlinear mapping L : X → E satisfies the estimate ||L(x)|| = o(||x||). After the determination of x(t), one may find the approximate u(t). Indeed, let x(t) be a solution to differential equation (5.10) constructed in Lemma 5.2. Then function u(t) is constructed by following asymptotics u(t) ∼ −A−1 4 A3 x(t) as ||x|| → 0 using Lemma 5.1. 5.1.1.

A priori estimate of the Cauchy problem’s solution

Let us consider system (5.1)–(5.2) with initial Cauchy condition x(0) = Δ. Based on Lemmas 5.1 and 5.2 and the Gronwall–Bellman inequality, the following lemma can be proved. Lemma 5.3. Let x : [0, +∞) → X be a solution to the Cauchy problem for system (5.1)–(5.2) and ||Δ|| in the initial condition be sufficiently small. Let Re λ ≤ −l < 0 for all λ ∈ σ(M). Then there exist C ≥ 1 and ε ∈ (0, l) such that || exp Mt||L(X→X) ≤ Ce−lt and ||x(t)||X ≤ C||Δ||e(ε−l)t for t ∈ [0, +∞). A priori assessment of the solution justifies the continuation of a local continuous solution of the Cauchy problem to the entire interval [0, +∞), see Theorem 5.1. 5.1.2.

Global existence, uniqueness, and asymptotic stability

Theorem 5.1. Let (0, 0) be the equilibrium point of system (5.1)– (5.2). Let Re λ ≤ −l < 0 for all λ ∈ σ(M) and ||Δ|| be sufficiently small. Then, system (5.1)–(5.2) with condition x(0) = Δ has a unique global solution x : [0, +∞) → X, u : [0, +∞) → U. Moreover, limt→+∞ (||x(t)|| + ||u(t)||) = 0. Proof. By virtue of Lemmas 5.1 and 5.2 and the obvious validity of Picard’s theorem for equation (5.10), the Cauchy problem (5.1)– (5.2), x(t0 ) = x0 , has a unique local solution for all t0 ∈ [0, ∞). Therefore, the set of values of the arguments t, for which the local solution continuously extends, is open in any interval [t0 , +∞). Since

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the solution of the Cauchy problem for sufficiently small ||x(0)|| on the basis of Lemma 5.2 satisfies the a priori estimate x(t) < δ for all t ∈ (0, +∞) and does not reach δ, then the set of values of the argument t, on which the solution can be continued, will be closed. Therefore, by the known facts about the method of continuation with respect to a parameter, the solution continuously extends to the entire interval [0, +∞). In view of the above-proved lemmas, the desired functions x(t), u(t) stabilize as t → +∞ to point (0, 0) for sufficiently small initial value Δ. The theorem is proved.  Example 5.1. ⎧ ∂x(t, z) ⎪ ⎪ = −x(t, z) + x3 (t, z) + u2 (t, z), ⎪ ⎪ ∂t ⎪ ⎨ x(0, z) = Δ(z), t ∈ [0, +∞), z ∈ [0, 1], ⎪  1 ⎪ ⎪ ⎪ ⎪ ⎩ u(t, z) + zsu(t, s) ds + x2 (t, z) + u2 (t, z) = 0, 0

|Δ(z)| ≤ ε, ε is sufficiently small. Here, A= I, A1 = −I, A3 = 0, 1 X = E = U = C[0,1] . Operator A4 = I + 0 zs[·] ds has a bounded  3 1 inverse A−1 4 = I − 4 0 zs[·] ds, M = −I. If |x(t, z)| is sufficiently small, then sequence {un (t, z)}, where un (t, z) = −x2 (t, z) −  3 1 2 2 un−1 (t, z) + 4 0 zs{x (t, s) + u2n−1 (t, s)} ds, u0 (t, z) = 0, converges and enables the algorithm for the construction of solution u(t, z) of integral equation as function of x(t, z). Substituting this solution into a differential equation, we reduce the problem to the following = −x(t, z) + O(||x||2 ), x(0, z) = Δ(z). differential equation: ∂x(t,z) ∂t 2 Here, ||u|| = O(||x|| ). Therefore, this model, consisting of differential and integral equations, satisfies the conditions of Theorem 5.1 and on the semiaxis [0, ∞) has a unique continuous solution x(t, z), u(t, z), stabilizing as t → +∞ to the equilibrium point (0, 0) if max0≤z≤1 |Δ(z)| is sufficiently small.

Nonlinear Systems’ Equilibrium Points

5.2.

111

The Construction of a Solution of a Nonlinear System by the Successive Approximations Method

Under the conditions of Theorem 5.1, the desired solution x(t), u(t) of system (5.1)–(5.2) with condition x(0) = Δ can be constructed without prior system’s reduction to one differential equation. Indeed, we introduce two sequences {xn (t)}, {un (t)} with conditions xn (0) = Δ, n = 0, 1, . . . , where ||Δ|| is sufficiently small. Let u0 = 0, and ||Δ|| be sufficiently small, xn (t) the solution to the Cauchy problem n A dx dt = F(xn , un−1 ), xn (0) = Δ, n = 1, 2, . . . . Obviously, solution xn (t) exists and is unique for t ≥ 0 due to Theorem 5.1. Then, let us construct functions un using the iterations un = un−1 + wn , where u0 = 0. Due to the invertibility of the operator A4 , functions wn can be found from the linear equation A4 wn + G(xn , un−1 ) = 0, n = 1, 2, . . . . Then, under Theorem 5.1, assumptions limn→∞ xn (t) = x(t), limn→∞ un (t) = u(t), limt→∞ (||x(t)|| + ||u(t)||) = 0. It is essential to have a small ||Δ||, otherwise the solution to nonlinear differential equation may blow up in the point t∗ (refer to the following examples). Example 5.2. Let us consider the system ⎧ x(t) ⎨ dx(t) =− − u(t) + x2 (t), dt 2 ⎩ 0 = 2u(t) − x(t) + 2u(t) sin u(t) − x(t) sin u(t), with the initial condition x(0) = Δ, 0 ≤ t < +∞. The replacement u(t) = x(t) 2 will reduce this system to Cauchy problem  x(t) ˙ = −x(t) + x2 (t), x(0) = Δ. It is easy to verify that the latter model has an exact solution x(t) = Δ Δ ∗ et (1−Δ)+Δ . Let us demonstrate that point t = ln Δ−1 may appear to be a blow-up of the constructed solution. We consider the following four cases.

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Toward General Theory of Differential-Operator and Kinetic Models

Fig. 5.1.

Δ ∈ (0, 1).

Case 1. If Δ ∈ (0, 1), then the blow-up point t∗ is complex, solution is continuous for t ∈ (0, +∞) and stabilizing to the equilibrium point x = 0 as t → +∞ (see Fig. 5.1). Case 2. If Δ ∈ (−∞, 0), then the blow-up point is negative, and on semiaxis [0, +∞) solution is continuous and stabilizing to the equilibrium point x = 0 as t → +∞ (see Fig. 5.2). Case 3. If 1 < Δ < ∞, then the solution blows up for t∗ = Δ Δ ln Δ−1 , where Δ−1 > 0. If t > t∗ , then the solution is continuous and also stabilizing to the equilibrium point x = 0 as t → +∞ (see Fig. 5.3). Case 4. For Δ = 0 and Δ = 1, we get the stationary solutions. Based on the above results, the following conclusion can be drawn. In Example 2, for Δ ∈ (−∞, 1), there exists the unique solution to the Cauchy problem as t ≥ 0, and stabilizing to the equilibrium

Nonlinear Systems’ Equilibrium Points

Fig. 5.2.

113

Δ < 0.

point as t → +∞. It is to be mentioned that for Δ > 1, the Cauchy Δ problem’s solution will blow up in the finite time ln Δ−1 . Remark 5.1. The absence of real equilibrium points can generate solutions with a countable set of blow-up points. Example 5.3.

⎧ dx ⎪ ⎪ = αx + βu + u2 + x3 , ⎪ ⎪ dt ⎨ x(0) = Δ, ⎪ ⎪ ⎪ ⎪ ⎩ 2 ax + 2bxu + u2 = 0, 0 ≤ t < ∞,

where (0, 0) is the equilibrium point. Under assumption u = cx, 2 where c is a constant, we get the following quadratic equation: √ c + 2 2bc + a = 0. Then, u is double-valued u1,2 = x(t)(−b ± b − a). Let a < b2 . Let us substitute the determined values of u into the differential equation. Then, the problem of determining the function

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Toward General Theory of Differential-Operator and Kinetic Models

Fig. 5.3.

Δ > 1.

x(t) is reduced to the solution of the two Cauchy problems ⎧   ⎨ dx± = (α + β(−b ± b2 − a)x + (−b ± b2 − a)2 x2 + x3 , ± ± ± dt ⎩ x± (0) = Δ. √ Let α + β(−b − b2 − a) < 0. Then there exists branch x− (t) for small |Δ| for t ≥ 0 and stabilizing to zero as t → +∞. 5.3.

Open Problems

In this work, until now, only the autonomous systems were considered. This can be relaxed. For example, if the system ⎧ ⎨ A dx = (A + A˜ (t))x(t) + (A + A˜ (t))u(t) + R(x, u, t), 1 1 2 2 dt ⎩ 0 = A3 x + A4 u + r(x, u, t),

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where A˜1 (t) → 0, A˜2 (t) → 0 as t → +∞, ||R(x, u, t)|| = o(||x||+||u||) and ||r(x, u, t)|| = o(||x|| + ||u||) for ||x|| + ||u|| → 0, converges uniformly to t ≥ 0, then the results of the main theorem remain correct. In the theory of system (5.1)–(5.2), the most difficult case ∂ G(x, u) at the is the case of the irreversible Fr´echet derivative ∂u equilibrium point, and therefore, the Implicit Operator Theorem is not fulfilled for the map G(x, u) = 0. This case needs the results of the state-of-the-art analytic theory of branching solutions of nonlinear operator equations and weakly regular equations considered by many authors. It is also interesting to consider system (5.1)–(5.2) with a discontinuity in the equilibrium points’ neighborhood, when the stability condition in the first approximation is not satisfied, and more advanced methods must be used. For example, methods related to Lyapunov function construction can be employed to evaluate the location of potential blow-up points using the method of convex majorants of Kantorovich used in [154]. In this case, when one constructs the algorithms for analyzing stability and constructs the estimates of the regions of attraction of the equilibrium points of the power systems, it is necessary to use the methods based on the Lyapunov vector function theory. Finally, it is interesting to consider system (5.1)–(5.2) with equilibrium points for an irreversible operator A. In this case, the standard Cauchy problem has no classic solutions and it is necessary to introduce other initial conditions. If the irreversible operator A allows a finite-length skeleton decomposition, then new correct initial conditions for the problem (5.1)–(5.2) can be formulated.

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Chapter 6

Nonclassic Boundary Value Problems in the Theory of Irregular Systems of Equations with Partial Derivatives Consider the linear equation     ∂ ∂ u = L1 u + f (x), BL ∂x ∂x

(6.1)

where linear bounded operator B acting from linear space E to E has no inverse operator. Differential operators    ∂n ∂ k0 +···+km ∂ = n+ ak0 ...km (x) , L ∂x ∂t ∂tk0 ∂xk11 . . . xkmm 

L1

∂ ∂x

k0 +k1 +···+km ≤n−1



=



bk0 ...km (x)

k0 +k1 +···+km ≤n1

∂ k0 +···+km ∂tk0 ∂xk11 . . . xkmm

, n1 < n

are defined. Here, coefficients ak0 ...km : Ω ⊂ Rm+1 → R1 , bk0 ...km : Ω ⊂ Rm+1 → R1 are sufficiently smooth and defined in Ω, 0 ∈ Ω. The domains of definition of operators L, L1 consist of linear manifolds, E∂ , sufficiently smooth functions in Ω with their ranges in E, which satisfy a certain system of homogeneous boundary conditions. Abstract function f : Ω ⊂ Rm+1 → E of argument x = (t, x1 , . . . , xm ) is assumed to be given and our problem is to find the solution u : Ω ⊂ Rm+1 → E∂ . Operator B is assumed independent of x. If the operator B has bounded inverse operator, then equation (6.1) is called regular and 117

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otherwise it is called irregular equation. If E = RN and det B = 0, then equation (6.1) is the system of linear partial differential equations (PDE) of Kovalevskaya type, and we have a well-known regular problem of the PDE theory. The foundation of many branches of modern general theory of PDE systems was constructed by academician Petrovskii. In the regular case, the initial conditions for equation (6.1) can be defined as follows:  ∂ i u  = ϕi (x1 , . . . , xm ), i = 0, 1, . . . , n − 1. (6.2) ∂ti t = 0 Here, functions ϕi are analytical functions in Ω. If f is an analytic function in t, x1 , . . . , xm in Ω, then the Cauchy problem (6.1)– (6.2) is not only solvable but also well-posed in the class of analytic functions. The well-posedness of Cauchy problem is a challenging issue even for linear PDE systems in spaces of nonanalytic functions. They are usually solved in the class of functions satisfying certain estimates. Irregular models enable the study of system behavior in critical situations. At present, the basis of relevant theory is constructed for certain classes of equations. For example, the theoretical and numerical methods for differential-algebraic equations have been constructed. Intensive studies of more complex theory of irregular PDE and abstract irregular differential operator equations have been conducted but there are still a lot of unexplored problems. If B is a normally solvable operator, x ∈ R1 , then the approach in the theory of equation (6.1) can be based on the expansion of the Banach space into direct sum in accordance with the Jordan structure of operator B [146, 154, 169] and results from the theory of semigroup [160]. In this field, the analytical methods were proposed for constructing classical and generalized solutions of Cauchy problem for ordinary differential-operator equations for x ∈ R1 in Banach spaces with irreversible operator in the main part. The theory of irregular differential-operator PDEs in Banach spaces in the multi-dimensional case for x ∈ Rn , n ≥ 2, is yet to

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119

be constructed. There are only initial results in this field published in preprints. Therefore, the construction of the general theory of equation (6.1) with irreversible operator B is of theoretical interest. It is also important for the modern mathematical models of complex systems. It is to be outlined that classical initial Cauchy conditions (6.2) for equation (6.1) play a very limited role. Indeed, because of the irreversibility the operator B, the time direction is characteristic and functions ϕi cannot be arbitrally selected in the initial conditions (6.2)! Then the question of reasonable formulation and methods of solution of nonclassic boundary problems appear for system (6.1), taking into account the structure of the operator B. The objective of this work is to solve this problem. It is solved for the irreversible operator B which satisfies skeleton decomposition B = A1 A2 , A1 ∈ L(E → E1 ), A2 ∈ L(E1 → E), where E1 is the normed space. The remainder of the section is organized as follows. First, we present the introduction concerning the skeleton chains of linear operators. The concept of a regular and singular skeleton chains is introduced. It is proved that operator B must be nilpotent in case of singular skeleton chain. It is assumed that noninvertible operator B generates a skeleton chain of linear operators of finite length p. It is demonstrated that the irregular equation (6.1) can be reduced to the recurrent sequence of p+1 equations. It is to be noted that each equation of this sequence is regular under the natural restrictions on differential operators L, L1 and certain initial boundary conditions. Therefore, if operator B has a skeleton chain of length p, then the solution of irregular equation (6.1) can be reduced to a regular system from (p + 1)th equation. The proposed approach can be employed for a wide range of concrete problems (6.1) due to the finite length of skeleton chain of finite-dimensional operator B. The formulas relating the solution of equation (6.1) with the solution of reduced regular system are derived. This result allows us to set new well-posed nonclassic boundary conditions for equation (6.1) for which the equation has a unique solution. For applications,

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it is important that this solution can be found by solving the sequence of regular problem proposed in this chapter. Corresponding results and examples conclude this section. Finally, open problems concerning the generalization to nonlinear equations are given.

6.1.

Skeleton Chains of Linear Operators

Let B ∈ L(E → E), and B = A1 A2 , where A2 ∈ L(E → E1 ), A1 ∈ L(E1 → E), E1 , E are linear normed spaces. The following definitions can be introduced. Decomposition B = A1 A2 is called skeleton decomposition of operator B. Let us introduce linear operator B1 = A2 A1 . Obviously, B1 ∈ L(E1 → E1 ). If operator B1 has bounded inverse or it is null operator acting from E1 to E1 , then B generates the skeleton chain {B1 } of length 1. Then operator B1 can be called as skeleton-attached operator to operator B. This chain is called singular if B1 = 0 and regular if B1 = 0. If B1 is an irreversible nonnull operator, then it is assumed to have skeleton decomposition B1 = A3 A4 , whereA4 ∈ L(E1 → E2 ), A3 ∈ L(E2 → E1 ), where E2 is the new linear normed space. Obviously, in this case, A2 A1 = A3 A4 and operator B2 = A4 A3 ∈ L(E2 → E2 ) can be introduced. If it turns out that B2 has bounded inverse or B2 ≡ 0, then B has the skeleton chain {B1 , B2 } of length 2. Chain {B1 , B2 } is singular if B2 = 0 and regular otherwise. Therefore, this process can be continued for a number of linear operator classes by introduction of the normed linear spaces Ei , i = 1, . . . , p and by bounded operator construction A2i ∈ L(Ei−1 → Ei ), A2i−1 ∈ L(Ei → Ei−1 ), which satisfy the following equalities: A2i A2i−1 = A2i+1 A2i+2 ,

i = 1, 2, . . . , p − 1.

(6.3)

Equation (6.3) defines the sequence of linear operators {B1 , . . . , Bp } as follows: Bi = A2i A2i−1 ,

i = 1, 2, . . . , p.

(6.4)

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121

Obviously, Bi ∈ L(Ei → Ei ). Here, operator Bp either has bounded inverse or Bp is null operator acting from Ep to Ep . This process can be formalized as per the following definition. Definition 6.1. Let B = A1 A2 and operators {Ai }2p i=1 satisfy equality (6.3). Let operators {B1 , . . . , Bp } defined by formula (6.4), and operators {B1 , . . . , Bp−1 } be noninvertible, and let operator Bp have bounded inverse or null operators acting from Ep to Ep . Then operator B generates the skeleton chain of linear operators {B1 , . . . , Bp } on length p. If Bp = 0, then the chain is regular and if Bp = 0, then the chain is called singular. Operators {B1 , . . . , Bp } are called skeleton-attached to operator B. The most important linear operators generating skeleton chains of the finite lengths are considered as follows: (1) Let E = Rm . Then the square matrix B : Rm → Rm with det B = 0 obviously has skeleton chain {B1 , . . . , Bp } of decreasing dimensions. The final matrix Bp will be regular or null matrix, det Bi = 0, i = 1, . . . , p − 1. (2) Let E be an infinite-dimensional normed space. Then the finite  operator B = ni=1 ·, γi zi , where {zi } ∈ E, γi ∈ E ∗ , has skeleton chain consisting of finite number of matrices {B1 , . . . , Bp } of decreasing dimensions. Here, B1 = ||zi , γj ||ni,j=1 is the first element of this chain, det Bi = 0, i = 1, . . . , p − 1. Bp is null matrix or det Bp = 0. Here, according to Definition 6.1, we have the length of the chain p = 1 if det[zi , γj ]ni,j=1 = 0 or zi , γj = 0, i, j = 1, 2, . . . , n. In the general case, the chain always consists of finite number of matrices. Using (6.3), (6.4) and Definition 6.1, the following result can be formulated. Lemma 6.1. If the operator B has a skeleton chain of length p, then B n = A1 A3 . . . A2n−1 Bn−1 A2n−2 A2n−4 . . . A2 ,

n = 1, . . . , p + 1, (6.5)

where B1 , B2 , . . . Bp are elements of the skeleton chain of operator B. From Lemma 6.1, we have the following corollary.

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Corollary 6.1. If the operator B has a singular skeleton chain of length p, then B is a nilpotent operator of index p + 1. To proof the corollary, it is sufficient to put n = p + 1 and demonstrate that degree B p+1 is null operator because Bp is null operator due to above, introduced definition of singular skeleton chain. 6.2.

Abstract Irregular Equation Reduction to the Sequence of Regular Equations

The operator B and linear operators {Ai }2p i=1 from the skeleton chain of operator B are assumed independent of x and commutative with linear operators L and L1 . For the sake of clarity, it is assumed that operators L and L1 can be different from the above, ∂ ∂ introduced differential operators L( ∂x ), L1 ( ∂x ) and the equation can be considered in abstract form BLu = L1 u + f.

(6.6)

Equation (6.1) can be considered as a special case of equation (6.6). Obviously, the introduced commutativity condition is fuilfilled for equation (6.1) with linear operator B independent of x and intro∂ ∂ ), L1 ( ∂x ). duced differential operators L( ∂x Let us reduce equation (6.6) to system of p + 1 equations which are regular in certain conditions imposed on operators L, L1 . Let us start with simple case p = 1. We introduce the system of two equations B1 Lu1 = L1 u1 + A2 f, L1 u = −f + A1 Lu1 ,

(6.7) (6.8)

where u ∈ E and u1 ∈ E1 . The decomposed system (6.7)–(6.8) can be obtained by formal multiplication of equation (6.6) on operator A2 from the skeleton decomposition of operator B and making notation u1 = A2 u. It is to be noted that system (6.7)–(6.8) is split and B1 is an invertible operator. Therefore, if operators B1 L − L1 and L1 have bounded

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123

inverse operators, then the unique solution can be constructed. Of course without additional conditions, there remains an open question: Does the constructed solution u(x) satisfy equation (6.6)? Let us introduce two lemmas establishing the link between equation (6.6) and system (6.7)–(6.8) to answer that question. Lemma 6.2. Let u∗ satisfy equation (6.6) and let operator L1 have left inverse. Then the pair u∗1 = A2 u∗ and element u∗ satisfy system (6.7)–(6.8). Proof. Based on the conditions of the lemma, the following equality is satisfied: A1 A2 Lu∗ = L1 u∗ + f.

(6.9)

From (6.9), because of the commutativity condition, the following equality is valid: A1 LA2 u∗ = L1 u∗ + f,

(6.10)

A2 A1 LA2 u∗ = L1 A2 u∗ + A2 f.

(6.11)

and The latter equality demonstrates that u∗1 = A2 u∗ is a solution of equation (6.7). Substitution of u∗1 into the right-hand side of equation (6.8) yields the following equation with known right-hand side: L1 u = −f + A1 A2 Lu∗ with respect to u. Here, the operator commutativity property is employed. The solution exists for such an equation. Indeed, due to equation (6.9) the right-hand side of the equation is equal to L1 u∗ . Hence, for given u∗ and u∗1 = A2 u∗ , the right-hand side belongs to the range of operator L1 . Therefore, L1 u = L1 u∗ . Since operator L1 has left inverse, it follows that u∗ is a unique solution of equation (6.8)  for u1 = A2 u∗ . the Lemma 6.2 is proved. Lemma 6.3. Suppose that there exists a pair (u∗1 , u∗ ) solution to equations (6.7) and (6.8). Let the operator L1 have right inverse ∗ L−1 1 . Then the element u satisfies equation (6.6).

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Proof. Element −f + A1 Lu∗1 belongs to the range of operator L1 , because pair u∗1 , u∗ satisfies the system of equations (6.7)–(6.8). ∗ ∗ Hence, u∗ = L−1 1 (−f + A1 Lu1 ) because −f + A1 Lu1 ∈ R(L1 ), where R(L1 ) is the range of operator L1 . It is to be demonstrated that the constructed element u∗ satisfies equation (6.6) since by hypothesis u∗1 satisfies equation (6.7). Indeed, substitution of the constructed u∗ into equation (6.6), where B = A1 A2 , yields ∗ ∗ A1 A2 LL−1 1 (−f + A1 Lu1 ) = A1 Lu1 .

Taking into account operator commutativity, the following equality is valid: ∗ ∗ A1 L{A2 L−1 1 (−f + A1 Lu1 ) − u1 } = 0.

(6.12)

Since A1 and L are linear operators, it remains to be verified in equation (6.12) that the element in braces is zero. Since u∗1 ∈ E1 satisfies equation (6.7), where B1 = A2 A1 , then the following equality is valid: A2 (A1 Lu∗1 − f ) = L1 u∗1 .

(6.13)

∗ Hence, A2 (A1 Lu∗1 − f ) ∈ R(L1 ) and u∗1 = L−1 1 A2 (A1 Lu1 − f ), where L−1 1 is right inverse to operator L1 . Since A2 L1 = L1 A2 , then −1 −1 −1 A2 = L−1 1 A2 L1 . This yields A2 L1 = L1 A2 L1 L1 . From equality −1 L1 L−1 1 = I, it follows that the right inverse L1 is also commutative

with operator A2 . Then equation (6.13) can be represented as ∗ ∗ A2 L−1 1 (A1 Lu1 − f ) − u1 = 0.

Thus, we have shown that the expression in braces in equation (6.12)  is zero. The Lemma 6.3 is proved. Let us concentrate on the general case of skeleton chain of arbitrary finite length p. We assume that operator B has skeleton chain {B1 , . . . , Bp }, p ≥ 1, and linear operators {Ai }2p i=1 represent decomposition of Bi in which the skeleton is attached to B. Introduce i  A2j u, i = 1, . . . , p, (6.14) ui = j=1

where ui ∈ Ei , ui = A2i ui−1 , u0 := u.

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125

If u0 satisfies equation (6.6), then for p ∈ N by Definition 6.1, we get equalities Bp Lup = L1 up +

p 

A2j f,

(6.15)

j=1

L 1 ui = −

i 

A2j f + A2i+1 Lui+1 ,

(6.16)

j=1

L1 u = −f + A1 Lu1 .

(6.17)

For p ≥ 2, there is a connection between solution of equation (6.6) and system (6.15)–(6.17). In particular, the following two lemmas can be formulated. Lemma 6.4. Let u∗ satisfy equation (6.6) and operator L1 have left  inverse. Then elements u∗i = ij=1 A2j u∗ , i = p, p − 1, . . . , 1 satisfy equations (6.15)–(6.15), and u∗ satisfies equation (6.17). Lemma 6.5. Let elements u∗p , u∗p−1 , . . . , u∗1 , u∗ satisfy equations (6.15)–(6.17) and operator L1 have right inverse. Then the element u∗ determined from equation (6.17) of split system (6.15)–(6.17) is the solution to equation (6.6). Proof of Lemmas 6.4 and 6.5 for any natural number p can be reduced to employment of the skeleton chain via operators {Aj }2p j=1 and repeats stages of the proofs of lemmas for the case of p = 1. Based on Lemmas 6.1–6.5, the following main result can be formulated. Main Theorem. Let the irreversible bounded operator B have skeleton chain {B1 , . . . , Bp }. Operator Bp L − L1 has bounded inverse and its definition domain is in Ep . Let the operator L1 be defined on domains Ei , i = 1, . . . , p, and E. Let the operator L1 have bounded inverse operator on Ei , i = 1, . . . , p, or E. Then system (6.15)–(6.17) has a unique solution {u∗p , u∗p−1 , . . . , u∗1 , u∗ } defined as follows: u∗p = (Bp L − L1 )−1

p  j=1

A2j f,

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126

⎧ ⎫ i ⎨  ⎬ −1 ∗ ∗ ui = L1 A2j f + A2j+1 Lui+1 , − ⎩ ⎭

i = p − 1, . . . , 1,

j=1

∗ u∗ = L−1 1 {−f + A1 Lu1 }.

Moreover, the element u∗ satisfies equation (6.6) and u∗i = A2j u∗ , i = 1, . . . , p.

i

j=1

By setting the initial boundary conditions to ensure the reversibility of the operators L1 and Bp L − L1 with specific differential operators L and L1 and using the Main Theorem, the existence and uniqueness theorems can be derived. Moreover, the formula obtained in the theorem can effectively build the desired classical solution of equation (5.1) with sufficient smoothness of f : Ω ⊂ Rm+1 → E and the coefficients of the differential operators L and L1 . Such applications of the theory are discussed further. 6.2.1.

The existence and methods of constructing solutions of nonclassic BVP with partial derivatives

6.2.1.1.

Problem I

Consider the system B



k1 +k2 ≤n

ak1 k2

∂ k1 +k2 u(x, t) = ∂tk1 ∂xk2

 k1 +k2 ≤m

ck1 k2

∂ k1 +k2 u(x, t) + f (x, t). ∂tk1 ∂xk2 (6.18)

Here, m < n, B is a constant (N × N )-matrix, det B = 0, ak1 k2 , ck1 k2 are numbers, and an0 = 0, a0n = 0, c0m = 0, cm0 = 0. Vector functions u(x, t) = (u1 (x, t), . . . , uN (x, t))T , f (x, t) = (f1 (x, t), . . . , fN (x, t))T are supposed to be defined and are analytical for −∞ < x, t < ∞. Let rank B = r < N. Then based on [52] B = A1 A2 , where A1 is N × r matrix and, A2 is (r × N )-matrix. Let us introduce the r × r matrix B1 = A2 A1 and assume det B1 = 0. Then the solution to the system (6.18) can be reduced to the successive solution of

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127

equations (6.7)–(6.8), which are as follows in this case: 

B1

ak1 k2

k1 +k2 ≤n



=

k1 +k2 ≤m



ck1 k2

k1 +k2 ≤m

∂ k1 +k2 u1 (x, t) ∂tk1 ∂xk2

ck1 k2

∂ k1 +k2 u1 (x, t) + A2 f (x, t), ∂tk1 ∂xk2

(6.19)

∂ k1 +k2 u(x, t) ∂tk1 ∂xk2

= −f (x, t) + A1

 k1 +k2 ≤n

ak1 k2

∂ k1 +k2 u1 (x, t) , ∂tk1 ∂xk2

(6.20)

where det B1 = 0, u1 (x, t) = (u11 (x, t), . . . , u1r (x, t))T , r < N, u1 = A2 u. Since by hypothesis an0 = 0, c0m = 0, then for system (6.18), one may introduce the initial conditions  ∂ i u(x, t)  = 0, i = 0, 1, . . . , m − 1, (6.21) ∂xi x=0  ∂ i u(x, t)  = 0, i = 0, 1, . . . , n − 1. (6.22) A2 ∂ti t=0 Vector function u1 (x, t), based on Kovalevskaya theorem, can be defined as a unique solution to system (6.19) with initial conditions:  ∂ i u1 (x, t)   = 0, i = 0, 1, . . . , n − 1. ∂ti t=0 By substituting the vector u1 (x, t) on the right-hand side of system (6.20), the desired vector u(x, t) can be found as a unique solution to the Cauchy problem (6.20)–(6.21). 6.2.1.2.

Problem II

Consider the system   2 ∂ ∂ n u(x, t) 2 ∂ − a = u(x, t) + f (x, t), n ≥ 3. B ∂xn ∂t ∂x2

(6.23)

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As in system (6.18), B is singular (N × N )-matrix with rank B = r < N, B = A1 A2 , B1 = A2 A1 , det B1 = 0. Let f (x, t) = (f1 (x, t), . . . , fN (x, t))T be vector function defined for 0 ≤ x ≤ 1, 0 < t < ∞, continuous with respect to x and analytical by t, u = (u1 , . . . , uN )T . The objective is to construct a solution of system of equations (6.23) in Ω = {0 ≤ x ≤ 1, 0 < t < ∞}. Based on Lemma 6.4 and Main Theorem, we introduce the system of two equations (u1 = A2 u)  2  ∂ ∂ n u1 (x, t) 2 ∂ −a B1 = u1 (x, t) + A2 f (x, t), (6.24) ∂xn ∂t ∂x2   2 ∂ n u1 (x, t) ∂ 2 ∂ −a (6.25) u(x, t) = −f (x, t) + A 1 ∂t ∂x2 ∂xn with initial boundary conditions  ∂ i u1 (x, t)  = 0, ∂xi x=0

i = 0, 1, . . . , n − 1,

u(x, t)|t=0 = 0, u(x, t)|x=0 = 0,

u(x, t)|x=1 = 0.

(6.26) (6.27) (6.28)

Since det B1 = 0, then the vector function u1 (x, t), based on Kovalevskaya theorem, can be defined as a unique solution of the Cauchy problem (6.24)–(6.26). By substituting u1 (x, t) on the righthand side of (6.25), the unique solution of the first boundary value problem (BVP) (6.25), (6.27), (6.28) is constructed for the heat equation using known formula with the source function. Constructed solution u(x, t) will be a classic unique solution of system (6.23) in the domain Ω = {0 ≤ x ≤ 1, 0 < t < ∞}. This solution satisfies the initial conditions  ∂ i u(x, t)  = 0, i = 1, . . . , n − 1 A2 ∂xi x=0 and conditions (6.27)–(6.28).

Nonclassic BVPs

6.3.

129

Skeleton Decomposition in the Theory of Irregular Ordinary Differential Equation in Banach Space

Consider the simplest irregular ordinary differential equation du(t) = u(t) + f (t), (6.29) dt f (t) : [0, ∞) → E, B ∈ L(E → E). Let {B1 , . . . , Bp } be skeleton chain of operator B. Then from Main Theorem, the following results can be formulated. B

Theorem 6.1. Let {B1 , . . . , Bp } be a regular skeleton chain, function f (t) (p − 1)-times differentiable. Then equation (6.29) with initial condition p  A2j u(t)|t=0 = c0 , c0 ∈ Ep (6.30) j=1

has a unique classic solution u0 (t, c0 ). Here, du1 , dt where u1 is defined uniquely (see Main Theorem). u0 (t, c0 ) = −f (t) + A1

(6.31)

Let us outline the scheme for construction function u1 (t, c0 ) in solution (6.31) to problem (6.29)–(6.30) as follows: (1) If p = 1, then u1 (t, c0 ) satisfies the regular Cauchy problem: ⎧ ⎨ B du1 = u + A f (t), 1 1 2 dt ⎩ u1 (0) = c0 . (2) If p ≥ 2, then the function u1 (t, c0 ) can be constructed by the following recursion: ⎧ p  ⎪ ⎨ B dup = u + A2j f (t), p p dt j=1 ⎪ ⎩ up (0) = c0 ,

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Toward General Theory of Differential-Operator and Kinetic Models i

ui (t, c0 ) = A2i+1

dui+1 (t, c0 )  A2j f (t), i = p − 1, p − 2, . . . , 1. − dt j=1

Theorem 6.2. Let {B1 , . . . , Bp−1 , 0} be a singular chain of length p, and 0 as null operator acting from Ep to Ep be singular skeleton chain of length p ≥ 1. Then B is a nilpotent operator and the homogeneous equation B du dt = u has only a trivial solution. In this case, if the function f (t) is p-times differentiable, then the unique classic solution of equation (6.29) can be constructed as follows: d un−1 (t), u0 (t) = −f (t), n = 1, 2, . . . , p. dt Here, the function up (t) is a unique classic solution of equation (6.29). un (t) = −f (t) + B

Chapter 7

Epilogue for Part I The generalized Jordan sets and skeleton chains of linear operators enable efficient regularization of equations and study the wide classes of the problems with noninvertible operator in the main part. Based on these results, the new types of initial and boundary value problems were tackled and solved. The results of this part have demonstrated the wide possibilities of using the proposed methods for linear and nonlinear problems in nonstandard cases with bifurcation. We assume that special attention could be paid to the studies of bifurcation and chaos appearing in discontinuous and continuous systems [50] using the presented methods.

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Part II

Lyapunov Methods in Theory of Nonlinear Equations with Parameters

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Chapter 8

Lyapunov Convex Majorants in the Existence Theorems The majorants in semi-ordered K-spaces have been considered by Kantorovich, Vulich and Pinsker [87, Chapter 12] for the investigation of functional equations in Bk -spaces. At first, the method of “Kantorovich majorants” might seem to be mainly of theoretical interest since it takes for granted the existence of such functions (see [124]). These majorants can be used to investigate the implicit transformations. In this connection, for the transformations in BK -spaces, there exists the construction problem of majorants depending on the parameters and satisfying the Kantorovich conditions. A solution of this problem really provides the ability to obtain nonlocal existence theorems of implicit functions and estimate the range of parameters (under which the method of successive approximations is convergent) contained in the equation. This problem wipes out the important problem of the solution of nonlinear equations with parameters, when it is required to find the solution on a possibly wide domain or at least estimate its measure from below. One has to point out the results of Grebennikov, Ryabov, and Trenogin. Related majorants depending on parameters and satisfying the Kantorovich conditions are called the Lyapunov majorants. In this section, in the general case, the methodology of majorants is studied. The goal is the investigation of implicit transformations in BK -spaces in which K = Rn . The convex majorants depending on the parameters in the equation only are considered. We introduce

135

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Toward General Theory of Differential-Operator and Kinetic Models

the “condition of potentiality” which is adapted to simply construct the majorants. Majorants constructed provide the ability and skill necessary to prove the nonlocal existence theorems of implicit transformations having estimated a parametric domain in that the method of successive approximations is convergent. Let U be linear space normed by means of n-dimensional Rn space, i.e., for every element u ∈ U , the vector t = |u|U ∈ Rn+ ⊂ Rn is related; moreover, the standard axioms (see [87]) of normalization of linear set by the elements of semi-ordered space are observed and the completeness condition of space is given. Thus, in this section, the space U is the space of BK -(Banach–Kantorovich) type, where K = Rn . This type of result is also worked out for the case in which K is the abstract linear semi-ordered space. Among others, one notes the cases arising from applications: (1) U = U1 × · · · × Un , where Ui are Banach spaces. Then the norms in Ui induce the “vector norm” |u|U = (u1 U1 , . . . , un Un ) for the space U , and U is related to the spaces of type BRn . (2) U = U1 × · · · × Ul , where Ui are the complete spaces normed by means of semi-ordered Rni spaces. Then U is a complete space of type BK , where K = Rn1 +···+nl . Then, we consider the following equation: u = Φ(u, ), where u ∈ U,  ∈ Λ, U is a BK -space and Λ is a normed space. Required solutions u() → 0 at  → 0 are computed by the method of successive approximations. The range of the parameter , for which there is a solution and the method of successive approximations is convergent, is estimated by the convex majorants. In the case of differentiable operators, we offer the method of finding of such majorants which requires the potentiality of matrix rows. The final section presents the results for more general equations of the form F (u, ) = 0.

Lyapunov Convex Majorants in the Existence Theorems

8.1.

137

Parameter-Independent Majorants

Consider the equation u = Φ(u, ),

(8.1)

where u ∈ U,  ∈ Λ, and Λ is a normed space. The nonlinear operator Φ is defined and continuous with respect to  in the neighborhood V = {(u, ) : |u|U ≤ r,  ≤ ρ, r = (r1 , . . . , rn ), ri > 0, ρ > 0} with Φ(0, 0) = 0. It is required to find a solution u() → 0 at  → 0. Furthermore, we will omit the subscript U in the notation of the norm |u|U in space U . 

Let us introduce the “interval” [0, R] = {t ∈ Rn+ : 0 ≤ ti ≤ Ri , i = 1, . . . , n}, Ri ≥ ri and the vector function f : [0, R] → Rn+ which is monotone increasing and continuous on [0, R]. Therefore, if 0 ≤ t ≤ t ≤ R, then f (t) ≤ f (t ). If 0 ≤ tn ≤ R and tn  t, then f (tn )  f (t). Assume that in the domain  ≤ ρ, (1) sup≤ρ |Φ(0, )| ≤ f (0). Introduce the family of sets D(t1 , t2 ) = {(v, w) : |v| ≤ t2 , |w| ≤ t1 , |v − w| ≤ t2 − t1 , 0 ≤ t1 ≤ t2 ≤ r} in the domain |u| ≤ r. It is easy to check that D(t1 , t2 ) are convex sets. Let us assume, furthermore, that (2) sup≤ρ |Φ(v, ) − Φ(u, )| ≤ f (t2 ) − f (t1 ) for any (u, v) ∈ D(t1 , t2 ); (3) the monotone increasing sequence {tn } is convergent, limn→∞ tn = t∗ , where tn = f (tn−1 ), t0 = 0 with t∗ ≤ r.

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Definition 8.1. The vector function f which satisfies the conditions (1)–(3) is considered to be a Kantorovich majorant of the operator Φ in domain |u| ≤ r,  ≤ ρ. A solution t∗ from (3) is considered to be the principal solution of the majorant equation (m.e.) t = f (t). Lemma 8.1. If there exists t+ ∈ (0, r) such that f (t+ ) ≤ t+ , then condition (3) holds. The proof follows from the upper boundedness of the monotone increasing sequence in (3) using vector t+ . Without additional conditions on the operator Φ, the open question on the construction of the convex majorant f satisfying (2) remains open. For the solution of this problem, we introduce some restrictions to operator Φ. Assume that (4) sup≤ρ |Φ(0, )| ≤ a, where a ∈ (0, r) ⊂ Rn+ . Consider the question of the differentiation of the operators in BRn -spaces. Take a fixed element u0 from BRn with |u0 | < r and assume that there exists additive and regular operator A(u0 , ) in the sense of Kantorovich [87] such that, for all u with |u| = (1, . . . , 1), lim

k→0

Φ(u0 + ku, ) − Φ(u0 , ) = A(u0 , )u, k

(8.2)

uniformly in u. Then owing to the known definition of regular operator in space BK , |A(u0 , )u| ≤ U |u|,

(8.3)

where U is a positive [n × n]-matrix. In this case, we call the operator A(u0 , ) to be a Fr´echet derivative of operator Φ(u, ) at the point u0 . Since the regular operator is homogeneous, A(u0 , ) is a linear operator. We give a more exact definition for the differentiability of operator Φ(u, ).

Lyapunov Convex Majorants in the Existence Theorems

139



Definition 8.2. Let λ = (λ1 , . . . , λn ) ∈ Rn+ , δ ∈ R1+ . If, for every λ > 0, there exists δ = δ() > 0 such that, for all u with |u| = (1, . . . , 1),     Φ(u0 + ku, ) − Φ(u0 , )   − A(u0 , )u (8.4)  ≤ λ,  k with |k| < δ, where additive operator A(u0 , ) satisfies (3), then operator A(u0 , ) is considered to be a Fr´echet derivative of operator Φ(u, ) and we denote it by Φu (u0 , ). Remark 8.1. From the inequality (8.4), it follows that the Fr´echet differentiability of the operator in BRn -space implies the existence of additive and regular operator A(u0 , ) such that for any λ > 0 there exists δ = (δ1 , . . . , δn ) ∈ Rn+ , by which from the inequality |h| < δ, we have the inequality |Φ(u0 + h, ) − Φ(u0 , ) − A(u0 , )h| ≤ λ|h|. 8.1.1.

Derivative of the composition transformations in BRn -spaces

Lemma 8.2. Let X, Y, Z be the spaces of BRni type, i = 1, 2, 3, U (x0 ) the neighborhood of the point x0 ∈ X, F : U (x0 ) → Y, y0 = F (x0 ), and V (y0 ) the neighborhood of the point y0 ∈ Y ; G : V (y0 ) → Z. Let F be differentiable at the point x0 and G differentiable at the point y0 . Then the transformation H = GF is differentiable at the point x0 and H  (x0 ) = G (y0 )F  (x0 ). Proof. Let us introduce the notation y = F (x0 + tx) − F (x0 ). Then F (x0 + tx) = y + y0 , H(x0 + tx) − H(x0 ) t F (x0 + tx) − F (x0 ) GF (x0 + tx) − GF (x0 ) = G (y0 ) = t t  G(y0 + y) − G(y0 ) − G (y0 ) y . (8.5) + t

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The first term has the limit G (y0 )F  (x0 ) in (8.5) uniformly in x at t → 0, |x| = (1, . . . , 1) because      G (y0 )F  (x0 )x − G (y0 ) F (x0 + tx) − F (x0 )    t       F (x0 + tx) − F (x0 )      = G (y0 ) F (x0 )x −   t      F (x0 + tx) − F (x0 )   , ≤ U    F (x0 )x − t where U is a positive matrix, and F (x0 + tx) − F (x0 ) = F  (x0 )x, lim t→0 t uniformly in x. The second term in (8.5) tends to zero at t → 0 because    G(y0 + ty) − G(y0 ) − G (y0 )ty   = 0,   lim  t→0  t uniformly in y for y = ty, |y| = (1, . . . , 1). The lemma is proved.



By Lemma 8.2, the Lagrange formula (8.6) holds. Lemma 8.3. Let the transformation Φ(u, ) be differentiable in a domain D(t1 , t2 ) with respect to u with u ∈ BRn ,  ∈ Λ. Then, for any u, v ∈ D(t1 , t2 ),  1 Φ (u + θ(v − u), )dθ(v − u). (8.6) Φ(v, ) − Φ(u, ) = 0

The result follows from the proof of the Lagrange formula taking account of the convexity of domain D(t1 , t2 ). We return to equation (8.1). (5) Let the transformation Φ(u, ) be Fr´echet differentiable in u for |u| ≤ r,  ≤ ρ, and moreover, sup |Φu (y, )h| ≤ U(|u|)|h|,

(8.7)

≤ρ

where U(|u|) is a positive [n × n]-matrix whose elements are monotone increasing functions.

Lyapunov Convex Majorants in the Existence Theorems

141

(6) Assume that the rows in matrix U(t), t ∈ Rn+ are potential vectors, i.e., (ai1 (t), . . . , ain (t)) = ∇fi (t). Then aij = ∂fi (t)/∂tj , i, j = 1, . . . , n. Hence, by the Lagrange formula of mean values for functions fi (t), we can consider  1 f (t) = a + U (θt)tdθ, (8.8) 0

up to the constant a ≥ 0. The vector function (8.8) is a required majorant of transformation Φ. Lemma 8.4. Let the conditions (4)–(6) be satisfied and the vector function f be defined by (8.8). Then f satisfies conditions (1) and (2). Proof.

By the Lagrange formula and inequality (8.7), we have   1     Φ (u + θ(v − u), )dθ(v − u) |Φ(v, ) − Φ(u, )| =     ≤

0

1

U (|u + θ(v − u)|) dθ|(v − u)|

0

 ≤

1

U (|u| + θ|(v − u)|) dθ|(v − u)|

0

 ≤

1

U (t1 + θ(t2 − t1 ))dθ(t2 − t1 )

(8.9)

0

for any pair (v, u) ∈ D(t1 , t2 ). On the other hand,  1 ∇fi (t1 + θ(t2 − t1 )), t2 − t1 dθ fi (t2 ) − fi (t1 ) = 0

 =

1 1

0 j=1

 =

0

1

aij (t1 + θ(t2 − t1 ))(t2j − t1j )dθ

Ui (t1 + θ(t2 − t1 ))dθ(t2 − t1 ),

i = 1, . . . , n. (8.10)

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According to (8.9) and (8.10), we obtain |Φ(v, ) − Φ(u, )| ≤ f (t2 ) − f (t1 ), and (2) holds. Since f (0) = a, owing to condition (4), sup |Φ(0, )| ≤ f (0),

≤ρ

and (1) is also valid. The lemma is proved.



This lemma allows the following generalization for the case, when the operator Φ contains the nondifferentiable component. Lemma 8.5. Let the operator Φ satisfy (4); moreover, Φ(u, ) = Φ1 (u, ) + Φ2 (u, ). Let the operator Φ1 satisfy conditions (5) and (6), and |Φ2 (v, ) − Φ2 (u, )| ≤ B|v − u| for |v| ≤ r, |u| ≤ r,  ≤ ρ, where B is a positive [n×n]-matrix. Then the vector  1 U (θt)tdθ + Bt f (t) = a + 0

satisfies conditions (1) and (2). If here the eigenvalues of a positive matrix B + U (0) belong to the unit disk, then condition (3) is valid for 0 ≤ a < λ, where λ = (λ1 , . . . , λn ), λ > 0 is small enough. Theorem 8.1. Let there be a majorant f (t) for which the conditions (1)–(3) are satisfied. Then (8.1) has at least one solution u∗ () in the domain |u| ≤ t∗ at  ≤ ρ, the sequence {um } with um = Φ(um−1 , ), u0 = 0 converges to that solution for  ≤ ρ. If the operator Φ is continuous in u and  or t∗ ≤ r/3 in (3) and a majorant f (t) is continuous, then a solution u∗ () is continuous in  and u∗ () → 0 at  → 0. Proof. (a) Existence of the solution. We show that the sequence {um } is fundamental in Theorem 8.1 at  ≤ ρ; moreover,

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143

|um | ≤ t∗ . Apart from {um } we consider sequence {tm }, where tm = f (tm−1 ), t0 = 0. Owing to conditions (1) and (3) |u1 ()| ≤ t1 ≤ t∗ ≤ r

and |u1 () − u0 ()| ≤ t1 − t0 .

We apply the method of mathematical induction. Let 0 ≤ k ≤ l ≤ N |ul − uk | ≤ tl − tk

(8.11)

for 0 ≤ k ≤ l ≤ N . Then |ul ()| ≤ tl , l = 0, . . . , N . We verify that inequality (8.11) still also holds for 0 ≤ k ≤ l ≤ N + 1. Indeed, considering (2), we have, for 0 ≤ k ≤ N + 1, |uN +1 () − uk ()| = |Φ(uN (), ) − Φ(uk−1 (), )| ≤ f (tN ) − f (tk−1 ) = tN +1 − tk , |uN +1 ()| ≤ |uN +1 () − u1 ()| + |u1 ()| = |Φ(uN , ) − Φ(u0 , )| + |Φ(u0 , )| ≤ f (tN ) − f (t0 ) + f (t0 ) = f (tN ) = tN +1 ≤ t∗ ≤ r. Hence, for  ≤ ρ, 0 ≤ k ≤ l < ∞, |ul () − uk ()| ≤ tl − tk ,

|uk ()| ≤ tk .

Since the monotone sequence {tm } is convergent, owing to (8.11), {um } is a fundamental sequence in the complete space U with |um | ≤ tm ≤ t∗ ≤ r. Hence, for  ≤ ρ, there exists lim um () = u∗ ().

m→∞

Next, |u∗ − Φ(u∗ , )| ≤ |u∗ − um | + |Φ(um−1 , ) − Φ(u∗ , )| ≤ |u∗ − um | + f (t∗ ) − f (tm−1 ) = |u∗ − uM | + t∗ − tm . Passing to the limit at m → ∞, we obtain u∗ − Φ(u∗ , ) = 0 for  ≤ ρ.

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(b) Uniqueness of the solution in a ball |u| ≤ t∗ . Introduce the initial approximations u ¯0 , t¯0 such that 0 < |¯ u0 | ≤ t∗ , t¯0 = |¯ u0 |. m m m m−1 Consider the sequences {¯ u }, {t¯ }, where u ¯ = Φ(¯ u , ), t¯m = f (t¯m−1 ). Then, |¯ u0 − u0 | = |¯ u0 | = t¯0 = t¯0 − t0 . Let |¯ uk − uk | ≤ t¯k − tk for k = 1, . . . , N . Then owing to condition (2), uN , ) − Φ(uN , )| |¯ uN +1 − uN +1 | = |Φ(¯ = |Φ(uN + (¯ uN − uN ), ) − Φ(uN , )| ≤ f (tN + (t¯N − tN )) − f (tN ) = f (t¯N ) − f (tN ) = t¯N +1 − tN +1 . By condition (3) lim (t¯N +1 − tN +1 ) = 0,

N →∞

therefore, ¯N = lim uN = u∗ (), lim u

N →∞

N →∞

u0 | ≤ t∗ , we i.e., starting with any initial approximation u ¯0 with |¯ ∗ obtain the solution u () for any  ≤ ρ. (c) Continuity of the solution u∗ (). Since u0 = 0, a zero approximation of sequence {um ()} is continuous. Let u1 (), . . . , uN () be continuous at every point 0 such that  < ρ. If the operator Φ is continuous in u and , obviously uN +1 () will also be continuous. We prove continuity of uN +1 () at the point 0 assuming that Φ is continuous in , t∗ ≤ r/3 in condition (3) and f (t) is a continuous majorant. In this case, |uN +1 () − uN +1 (0 )| = |Φ(uN (), ) − Φ(uN (0 ), 0 )| ≤ |Φ(uN (), ) − Φ(uN (0 ), )| + |Φ(uN (0 ), ) − Φ(uN (0 ), 0 )|. (8.12)

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145

Since |uN ()| ≤ |uN (0 )| + |uN () − uN (0 )| ≤ 3t∗ ≤ r and |uN () − uN (0 )| ≥ 0, then |Φ(uN (), ) − Φ(uN (0 ), )| ≤ f (|uN ()| + |uN () − uN (0 )|) − f (|uN (0 )|).

(8.13)

Since a majorant f (t) is continuous (it suffices that there exists the limit limt→t0 +0 f (t) = f (t0 )), from (8.13), we have lim |Φ(uN (), ) − Φ(uN (0 ), )| = 0,

→0

i.e., the first term on the right-hand side of (8.12) tends to zero at  → 0 . By continuity of the operator Φ in , the second term on the right-hand side of (8.12) also tends to zero at  → 0 . Hence, uN +1 () is continuous at the point 0 , where  ≤ ρ. The corresponding sequence {un ()} is uniformly convergent in  owing to (8.11) which is true for  ≤ ρ. Therefore, the solution u∗ () is continuous. The  theorem is proved. Note that it is possible to prove the continuity of a solution u∗ () directly: |u∗ () − u∗ (0 )| ≤ |u∗ () − un ()| +|u∗ (0 ) − un (0 )| + |un () − un (0 )| ≤ 2(t∗ − tn ) + |un () − un (0 )| ≤ 2(t∗ − tn ) + λ for  − 0  ≤ δ, δ = δ(λ) > 0. Considering that n is arbitrary and t∗ − tn → 0 at n → ∞, then |u∗ () − u∗ (0 )| ≤ λ for  − 0  ≤ δ. Remark 8.2. If the operator Φ in Theorem 8.1 is analytic with respect to u and , then a solution u∗ () will also be analytic. Theorem 8.2. Let conditions (4)–(6) be valid, the vector function (8.8) satisfy condition (3) and Φ(0, 0) = 0. Then equation (8.1) has

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at least one solution u∗ () in the domain |u| ≤ t∗ for  ≤ ρ, and the method of successive approximations is convergent to that starting with u0 = 0 for  ≤ ρ. The proof follows from Theorem 8.1. Example 8.1.



ui (x) =

b

a

Ki (x, s, u1 (s), . . . , un (s)) ds + i bi (x),

|i | ≤ ρ, i = 1, . . . , n. where Ki (x, s, u) =  max

a≤x≤b a



Kij (x, s)uj ,

(8.14) uj = uj11 , . . . , ujnn ,

|j|≥1 b

|Kij (x, s)|ds ≤ kij ,

max |bi (x)| ≤ βi .

a≤x≤b

Conditions (4)–(6) hold; moreover, a = (ρβ1 , . . . , ρβn ) , fi (t, ρ) = ρβi +

N 

kij tj ,

i = 1, . . . , n.

|j|≥1

If the eigenvalues of the matrix (kij )i=1,...,n,|j|=1 belong in the unit disk, for small enough ρ > 0, there exists t+ > 0 such that f (t+ , ρ) ≤ t+ . Hence, the majorant equation t = f (t, ρ) has a principal solution. The system (8.14) for |i | ≤ ρ+ , ρ+ > 0 has a solution u∗ (x, ) continuous in , |u∗ (x, )| ≤ t+ , u∗ (0, ) → 0 at  → 0. The desired ρ+ locates the existence domain of the principal solution of the majorant system and it is an upper bound of positive ρ, of which the sequence {tn } with tn = f (tn−1 , ρ), t0 = 0 is convergent. Note that the desired ρ+ can be found by searching for positive solutions of the system ⎧ N  ⎪ ⎪ ⎪ = kij tj + ρβi , i = 1, . . . , n, t ⎪ i ⎪ ⎪ ⎨ |j|≥1 ⎞⎤ ⎡ ⎛ ⎪ N ⎪  ⎪ ⎪ det ⎣I − ∂/∂t ⎝ ⎪ kij tj ⎠⎦ = 0. ⎪ ⎩ |j|≥1

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147

If {ρi , ti }, i = 1, . . . , n, are positive solutions of that system, ρ+ = maxi ρi . The above-discussed example shows that condition (3) depends on the value of the parameter . If (8.1) has no solutions for all  ≤ ρ, then there is no majorant f (t) satisfying condition (3), while majorants with conditions (1) and (2) are readily calculated. Thus, in Section 8.2 we consider the sharper majorants f (t, ) depending on parameters. 8.2.

Majorants Depending on a Parameter

Instead of conditions (1) and (2), we introduce the following conditions: (7) |Φ(0, )| ≤ f (0, ), (8) |Φ(v, ) − Φ(u, )| ≤ f (t2 , ) − f (t1 , ) for all v, u ∈ D(t1 , t2 ),  ≤ ρ. Lemma 8.6. Let Φ(u, ) = L(u, ) + F (u, ), |Φ(0, )| ≤ a(), and the operator F be differentiable in u; moreover, |Fu (u, )h| ≤ U(|u|, )|h|, |L(u, ) − L(v, )| ≤ B()|u − v| for |u| ≤ r, |v| ≤ r,  ≤ ρ. Here, U (t, μ) is a positive matrix of (n × n)-dimension, where its rows are potential in t and monotone increasing in t and μ; and B(μ) is a positive matrix of (n × n)dimension, where its elements are monotone increasing functions. Set  1 U (θt, μ)t dθ + B(μ)t. f (t, μ) = a(μ) + 0

Then conditions (7) and (8) hold. The proof follows from Lemmas 8.2–8.5. (9) Assume that in conditions (7) and (8), f (t, μ) is continuous, monotone increasing in all arguments, f (0, 0) = 0, derivatives

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148

∂fi /∂tj , i, j = 1, . . . , n, are monotone increasing in t, μ in domain |t| ≤ R, |μ| ≤ ρ , R > r, ρ ≥ ρ . (10) The majorant equation t = f (t, μ)

(8.15)

has a principal solution t(μ) for μ ∈ [0, μ+ ] and t(μ+ ) = t+ ≤ r. If there exist μ+ > 0, t+ > 0 such that f (t+ , μ+ ) ≤ t+ ≤ r, then (8.10) is satisfied. Such t+ and μ+ always exist if a positive matrix U (0, 0) + B(0) is trivial or even if its eigenvalues belong in the unit circle. Theorem 8.3. Let conditions (7)–(10) hold and operator Φ be continuous with respect to u, . Then (8.1) for  ≤ μ+ has a continuous solution u∗ () → 0 at  → 0. This solution is unique in the domain |u| ≤ t∗ and the method of successive approximations is convergent for u0 = 0. Proof. By condition (9) for t2 ≥ t1 ,  ≤ μ+ , the following inequality holds:  1 2 1 ∇f (t1 + θ(t2 − t1 ), ), t2 − t1 dθ f (t , ) − f (t , ) = 0

 ≤

1

f (t1 + θ(t2 − t1 ), μ+ ), t2 − t1 dθ

0

= f (t2 , μ+ ) − f (t1 , μ+ ),

f (0, ) ≤ f (0, μ+ ).

Hence, the majorant f (t) = f (t, μ+ ) satisfies conditions (1) and (2) for  ≤ μ+ . By condition (10), this majorant also satisfies condition (3). Hence, by Theorem 8.1, equation (8.1) has at least one continuous solution u∗ () for  ≤ μ+ , to which the method of successive approximations is convergent. The theorem is proved.  8.2.1.

Estimation of the existence domain of a principal solution of majorant equation

The existence domain of the desired solution u∗ () incorporates the existence domain of a principal solution of majorant equation. Therefore, we then estimate the existence domain of a principal

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149

solution t∗ (μ), i.e., an upper bound μ+ such that, for μ ∈ [0, μ+ ], sequence {tn (μ)}, t0 = 0, is convergent. Proposition 8.1. Let positive t , μ be selected so that f (t , μ ) ≤ t . Then we have μ+ ; moreover. for 0 ≤ μ ≤ μ , equation (8.15) has a principal solution t∗ (μ) and t∗ (μ) ≤ t for 0 ≤ μ ≤ μ . Proposition 8.2. Let positive t , μ selected so that f (t , μ ) ≤ t . Then we have μ+ ≥ μ ; moreover, for 0 ≤ μ ≤ μ , equation (8.15) has a principal solution t∗ (μ) and t∗ (μ) ≤ t for 0 ≤ μ ≤ μ . Proposition 8.3. Let (ti , μi ), i = 1, . . . , n, be a set of nonnegative solutions of the systems ⎧ ⎪ ⎨ ti = fi (0, . . . , 0, ti , 0, . . . , 0, μ), ∂f (0, . . . , 0, ti , 0, . . . , 0, μ) ⎪ ⎩1 − i = 0, ∂ti i = 1, . . . , n. Let μ = min(μ1 , . . . , μn ). Then μ+ ≤ μ . Proof. We introduce the plane (y, ti ). On this plane, a curve y(ti ) = fi (0, . . . , 0, ti , 0, . . . , 0, μ) lies above the line y = ti for μ > μ . Since the function fi (t, μ) is monotone increasing in all arguments, a curve yˆ(ti ) = fi (t∗1 , . . . , ti , . . . , t∗n , μ) on the plane (y, ti ) lies above a curve y(ti ) for μ > μ and any t∗1 , . . . , t∗i−1 , t∗i+1 , . . . , t∗n , and hence above the line y = ti . Therefore, the equation ti = fi (t∗1 , . . . , ti , . . . , t∗n , μ), for μ > μ , has no positive solution ti for arbitrary t∗i ≥ 0. Hence, equation (8.15) has no positive solutions for μ > μ . Since by definition of μ+ , equation (8.15) must have a  positive solution, then μ = μ+ . The proposition is proved. Let us introduce the following condition.

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(11) Let {ti , μi }, i = 1, . . . , n be the set of all nonnegative solutions of the system  t = f (t, μ), (8.16) det[E − ∂f /∂t] = 0. + = max μi . Then by Let (t+ , μ+ ) ∈ {ti , μi }N i i=1 ; moreover μ Proposition 8.1, equation (8.15) has a principal solution t∗ (μ) for 0 ≤ μ ≤ μ+ and limn→∞ tn (μ) = t∗ (μ) with tn = f (tn−1 , μ), t0 = 0.

Proposition 8.4. Let condition (11) hold, the eigenvalues of matrix [∂fi (0, 0)/∂tj ] belong to the unit circle, or that matrix be trivial. Then the following two cases are possible. (1) If t+ = t∗ (μ+ ), then μ∗ = μ+ and the existence domain of a principal solution t∗ (μ) of (8.15) is a closed set, namely, the interval [0, μ+ ]. (2) If t+ = t∗ (μ+ ), then the interval [0, μ+ ] is included in the existence domain of a principal solution t∗ (μ) of (8.15); moreover, the domain is an open set. In both cases, there is implicit transformation u() for  ≤ μ+ ; moreover, in the first case, we cannot improve this estimation using a majorant f (t, μ). Proof. (1) Let M be the existence domain of t∗ (μ). Since (t+ , μ+ ) satisfies (8.15), then [0, μ+ ] ⊆ M . Let M = [0, μ+ ]. Then the principal solution is extended in the domain μ > μ+ . Since the principal solution is monotone increasing, in this case, there are ¯ > μ+ satisfying (8.15), i.e., the following equality points t¯ > t+ , μ holds: t¯ = f (t¯, μ ¯). We rewrite last one in the form ¯ − μ), (I − ft (t+ , μ+ )(t¯ − t+ ) = r(t¯ − t+ , μ

(8.17)

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151

where ¯ − μ) = fμ (t+ , μ+ )(¯ μ − μ) r(t¯ − t+ , μ 2   1 ∂ 1 + ¯ (¯ μ − μ) (1 − θ) ∂/∂t(t − t ) + + 2 0 ∂μ × f (t+ + θ(t¯ − t+ ), μ + θ(¯ μ − μ))dθ, and r > 0 for t¯ > t+ , μ ¯ > μ+ . The matrix ft (t, μ) is positive for t > 0, μ > 0, and for t = 0, μ = 0, its eigenvalues belong in the unit circle. Set in this matrix t = t∗ (μ), i.e., to the principal solution. Since the principal solution is monotone increasing, the maximum eigenvalue of the positive matrix does not decrease with increase of μ; here, readers may refer to [52, Chapter 13]. By condition μ = μ+ , it becomes equal to 1. Therefore, by the Perron–Frobenius theorem [52], 1 is the maximum eigenvalue of matrix ft (t+ , μ+ ) and the positive eigenvector corresponds to it. Hence, a conjugate matrix [ft (t+ , μ+ )] also possesses the positive ¯ eigenvector d. If the principal solution is extended in the domain μ > μ+ , there exist t¯ > t+ , μ ¯ > μ+ satisfying (8.15). But if we scalar multiply both ¯ we are led to equality sides of (8.15) by vector d, ¯ − μ+ ), d = 0, r(t¯ − t+ , μ ¯ > μ+ and d¯ > 0. The last one is clearly where r > 0 for t¯ > t+ , μ not satisfied. Hence, in the first case, the existence domain of the principal solution of (8.15) is the interval [0, μ+ ]. (2) Since t+ = t∗ (μ∗ ), then   ∂f (t+ , μ+ ) = 0, det E − ∂t and hence, the principal solution exists on the interval [0, μ+ + ) with  > 0, i.e., it is extended through μ+ . Let a principal solution ¯ > μ+ . Since t∗ (μ) exist for a certain μ   ∂f (t(¯ μ), μ ¯) = 0 det E − ∂t

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for μ ¯ > μ+ , by implicit function theorem, the solution also exists in the neighborhood (¯ μ −1 , μ ¯ +1 ). So, in the second case, the existence domain of the principal solution of (8.15) is an open set.  The proposition is proved. Example 8.2. Let U = BR1 be a standard Banach space and the operator Φ(u, ) be analytic in neighborhood u ≤ r,  ≤ ρ, sup u 0; (III) f (0, 0) = 0; (IV) there are some t+ , μ+ such that M f (t+ , μ+ ) ≤ t+ .

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Lemma 8.7. For equation (8.19), let the conditions (I)–(IV) hold. Then the system of successive approximations tk = M f (tk−1 , μ),

t0 = 0,

k = 1, 2, . . .

(8.20)

converges to the solution of (8.19) t∗ = t∗ (μ) in the domain 0 ≤ t ≤ t+ , 0 ≤ μ ≤ μ+ . Proof. Note that the sequence {tk }∞ 0 is monotone increasing. Indeed, by condition (II), T n+1 = M f (t0 , μ) = f (0, μ) > 0 = t0 for k = 1. For k = 2, . . . , n let tn > tn−1 hold. Then, by condition (I), tn+1 = M f (tn , μ) > M f (tn−1 , μ) = tn for k = n + 1. Thus, tk+1 > tk for all k, and the sequence {tk } is monotone increasing. Then, the sequence {tk } is bounded. Indeed, for k = 1 by conditions (I) and (IV), we have t1 = M f (0, μ) < M f (t+ , μ) < M f (t+ , μ+ ) < t+ . For k = 2, . . . , n, let tn < t+ hold. Then for k = n + 1, by conditions (I) and (IV) and the last inequality, we obtain tn+1 = M f (tn , μ) < M f (t+ , μ) < M f (t+ , μ+ ) ≤ t+ . Hence, tk < t+ for all k and the sequence {tk } is upper bounded. In this case, there exists the limit lim tk (μ) = t∗ (μ) ≤ t+ .

k→∞

Passing to the limit in (8.20), by condition (I), we obtain   k+1 k k = lim M f (t , μ) = M f lim t , μ , lim t k→∞

k→∞

k→∞

i.e., t∗ = M f (t∗ , μ) and t∗ is a solution of (8.19). The lemma is  proved.

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155

Theorem 8.4. Let equation (8.19) satisfy conditions (I)–(IV) and majorize (8.18) according to (A) and (B). Then the successive approximations uk = uk−1 − A−1 F (uk−1 , ),

u0 = 0,

k = 1, 2, . . .

(8.21)

are convergent to a unique continuous solution of (8.18) u∗ = u∗ () : u∗ (0) = 0 in domain |u| ≤ t+ ,  ≤ μ+ . Proof.

We replace (8.18) by the equation u = Φ(u, ),

(8.22)

where Φ(u, ) = u − A−1 F (u, ). From the estimations (A) and (B) for (8.22), we have the following inequalities: (A ) |Φ(0, )| = |A−1 |F (0, )| ≤ |A−1 ||F (0, )| ≤ M f (0, μ), (B ) |Φ(u2 , ) − Φ(u1 , )| = |u2 − u1 − A−1 (F (u2 , ) − F (u1 , ))| ≤ |A−1 ||F (u2 , ) − F (u1 , ) − A(u2 − u1 | ≤ M (f (t2 , μ) − f (t1 , μ)) = M f (t2 , μ) − M f (t1 , μ), where |ui | ≤ ti , i = 1, 2, |u2 − u1 | ≤ t2 − t1 . In this case, by (A ) |u1 | = |Φ(0, )| ≤ M f (0, μ) = t1 , |u1 − u0 | = |u1 | ≤ t1 = t1 − t0 . Let the following inequalities hold for k = 2, . . . , n: |un | ≤ tn ,

|un − un−1 | ≤ tn − tn−1 .

Then, owing to (B ), |Φ(un , ) − Φ(un−1 , )| ≤ M f (tn , μ) − M f (tn−1 , μ).

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Hence, |un+1 − un | ≤ tn+1 − tn . Then, |un+1 | ≤ |un+1 − un | + |un | ≤ tn+1 − tn + tn = tn+1 . Thus, the successive approximations (8.21) are majorized by (8.20) for all k and remain in the domain |uk | ≤ t+ . Since the monotone sequence {tk } is convergent for all μ ≤ μ+ , then for n, m → ∞, n > m,  ≤ μ+ , the following estimation holds: |un − um | ≤ |un − un−1 | + · · · + |um+1 − um | ≤ tn − tn−1 + . . . + tm+1 − tm = tn − tm . The sequence {uk } is fundamental, and owing to the completeness of the space, there exists the limit lim uk () = u∗ ()

k→∞

for  ≤ u+ . By continuity of A−1 , F , passing to the limit, we have   k k−1 k−1 lim u = lim Φ(u , ) = Φ lim u ,  k→∞

k→∞

= lim u k→∞

k−1

−1

−A

 F

k→∞

k−1

lim u

k→∞

 , ,

i.e., u∗ = u∗ − A−1 F (u∗ , ). Hence, u∗ = Φ(u∗ , ) and u∗ is a solution of (3.1), u∗ (0) = 0, u∗ () is continuous in . In domain |u| ≤ t+ ,  ≤ μ+ , let there be a solution u ˜() = ∗ u () for which the conditions (A) and (B) are also valid. Since |˜ u − u0 | = |˜ u| ≤ t+ = t+ − t0 , then |Φ(˜ u, ) − Φ(u0 , )| ≤ M f (t+ , μ+ ) − M f (t0 , μ), and hence, |˜ u − u1 | ≤ t+ − t1 . Continuing such arguments, we obtain the estimation |˜ u − uk | ≤ t+ − tk on the kth step. In this case, for all  ≤ μ+ and every δ > 0, there exists N such that, for all n > N the following estimation holds: |˜ u() − un ()| ≤ t+ − tn < δ.

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157

Hence, limn→∞ |˜ u() − un ()| = 0, and the solution of (8.18) is unique. The theorem is proved.  The existence domain of the desired solution u∗ () involves the existence domain of the principal solution of majorant equation (8.19). Therefore, we need to estimate an upper bound μ+ such that, for all μ ∈ [0, μ+ ], the successive approximations (8.20) are convergent. Also, let us outline that computer algebra systems make it possible to do this in a relatively general situation.

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Chapter 9

Investigation of Bifurcation Points of Nonlinear Equations Consider the equation F (x, λ) = 0,

(9.1)

where F is the nonlinear operator, F : X × I → Y , X, Y are the real Banach spaces, λ is a real parameter, λ ∈ I, I is a finite or infinite interval of the real axis F (x0 (λ), λ) = 0 for λ ∈ I.

(9.2)

Condition (9.2) denotes that, for any λ ∈ I, equation (9.1) has a known (trivial) solution x0 (λ). Definition 9.1. The point λ0 is called a bifurcation point (or branch point) of equation (9.1) (the operator F ) if, for any  > 0, δ > 0, there exist x and λ satisfying (9.1) such that 0 < x − x0 (λ) < , |λ − λ0 | < δ. In this chapter, the conditions are given, which when fulfilled make the point λ0 ∈ I a branch point of equation (9.1) and one may construct the asymptotics of nontrivial branches of solutions for equation (9.1). By a branch, we mean a continuous real solution x(λ) defined in a one-sided neighborhood of the point λ0 , x(λ) → x0 (λ0 ) at λ → λ0 + 0 (λ → λ0 − 0). We obtain a solution of the given problem of investigating the branching system of equations

159

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Toward General Theory of Differential-Operator and Kinetic Models

(BSEq) corresponding to equation (9.1) by means of combination of elementary methods: (1) topological: rotation of finite-dimensional vector fields; (2) algebraic: study of Jordan structure for linearized problem; (3) variational: searching the points of conditional extremum of continuous functions. Sufficient existence conditions of bifurcation points have been first studied by Krasnosel’skii [92] and Vainberg [169] with the help of generalizations of topological and variational methods onto infinite-dimensional spaces. Thus, an essential role was played by requirements such as potentiality, a complete continuity of operator F , and the equations of more special form than equation (9.1) were considered. In these works, the asymptotics of solutions were not constructed. Constructive methods in branching theory can be modeled using the Lyapunov–Schmidt reduction [51, 146, 169]. Since by equation (9.1) by change of variables x = x0 (λ) + u, one can always transform to the form Φ(u, λ) = 0, where Φ(0, λ)|λ∈I = 0, then we can later note that this transformation already was down and F (0, λ) = 0 for all λ ∈ I. Let us introduce the following assumptions: I. The operator F is continuous in x and λ, and continuously differentiable with respect to x in the sense of Fr´echet in the neighborhood Ω of the point (0, λ0 ), i.e., F (x, λ) = Fx (0, λ)x + W (x, λ),

(9.3)

where W (x, λ) = o(x) at x → 0 for any λ ∈ Ω. II. The point λ0 is a singular Fredholm point of operator A(λ), def

A(λ) = Fx (0, λ) = A0 −

q 

Ak (λ − λ0 )k + R(λ),

k=1

where R(λ) = o(|λ − λ0 |q ), and ϕi |n1 is a basis for N (A0 ), and ψi |n1 is a basis for N ∗ (A0 ).

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161

III. For the operator W , the following representation holds: W (x, λ) =

m 

Fs (x, λ) + R(x, λ),

l ≥ 2,

(9.4)

s=1

where R(x, λ) = o(xm ) for any λ ∈ Ω and Fs is uniform operator in x of order s, Fs (tx, λ) = ts Fs (x, λ). If assumption I is valid and the point λ0 is a regular point of operator function Fx (0, λ), then equation (9.1) has a trivial solution only in the sphere x <  for |λ − λ0 | < δ, where  and δ are arbitrarily small. The elementary theorems of bifurcation points allow us to build the asymptotics of nontrivial branches of solutions one may obtain applying the Lyapunov–Schmidt method [169, 177] if it is assumed that a special constructed system of nonlinear algebraic equations has a simple real solution. The method considered in this book allows one to get rid of such restrictive hypothesis. In Section 9.1, the branching equation (BEq) is constructed. In Chapter 10, the general existence theorems on the bifurcation points are obtained on the basis of the application of the singular points theory of finite-dimensional vector fields to BEq. We study BEq with the help of the analytic Lyapunov–Schmidt method that complemented a certain information from the singular points theory and by the variational arguments. Under certain conditions, the asymptotics of continuous branches in a neighborhood of a branch point is constructed in Chapter 11. In the computation of asymptotics by our method, it is necessary to solve a simple problem: find the points of conditional extremum for certain functions on the sphere. 9.1.

Lyapunov–Schmidt Method in the Problem of a Bifurcation Point

Let assumptions I and II hold; moreover, let λ0 be an isolated singular Fredholm point of the operator Fx (0, λ), maxi pi ≤ q,

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where pi is the length of ith GJC of operator Fx (0, λ) and k is the corresponding root number. Putting λ = λ0 + μ, one rewrites equation (9.1) as a system n  F (x, λ0 + μ) + (γi , x − ξi )zi = 0,

(9.5)

1

ξi = γi , x.

(9.6)

Thanks to the implicit operator theorem, equation (9.5) has a unique small continuous solution x = x(μ, ξ) → 0, μ, ξ → 0 (ξ = (ξ1 , . . . , ξn )). One can find this solution by the method of successive approximations from the equation   n  −1 ξk ϕk + Γw(x, μ) , (9.7) x = (I − Γc(μ)) 1

where Γ = Aˆ−1 0 ,

c(μ) =

q 

Ak μk + o(|μ|q ),

(9.8)

1

Aˆ0 = A0 +

n  γi , ·zi , 1

w(x, μ) = −W (x, λ0 + μ) = −F (x, λ0 + μ) + Fx (0, λ0 + μ)x. (9.9) As a result, one obtains the function x=

n 

(I − Γc(μ))−1 ξk ϕk + y(ξ, μ),

1

(9.10)

 where y(ξ, μ) = o(ξ) (e.g., ξ =  n1 ξk ϕk ). By substituting (9.10) into the right part of (9.7), one obtains  n  n   ξk ϕk + Γw (I − Γc(μ))−1 x = (I − Γc(μ))−1 1

⎞⎫ ⎬ × ξk ϕk + y(ξ, μ), μ⎠ . ⎭

1

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163

The condition (9.6), now, gives BSEq  n n   −1 aik (μ)ξk + γi , (I − Γc(μ)) Γw (I − Γc(μ))−1 ξk ϕk k=1

1



+ y(ξ, μ), μ

i = 1, . . . , n,

= 0,

(9.11)

where aik (μ) = γi , (I − Γc(μ))−1 Γc(μ)ϕk ,

det[aik (μ)]ni,k=1 ≈ cμk , c = 0.

Transform the system (9.11). To this end, assuming that the formulas (9.7) of the previous section are satisfied, one obtains the equalities ψi , c(μ)(I − Γc(μ))−1 ϕk    p −1  q k   (n+1) Aj μj + o(|μ|q μn ϕk + o(|μ|pk ) ϕk = ψi , =

ψi ,

1 pk  j=1

n=0

(p +1−j)

Aj ϕk k

= δik μpk + o(|μ|pk ),

μpk + o(|μ|pk )

i, k = 1, . . . , n.

Thus, solving the transformed BSEq with respect to μpi ξi , i = 1, . . . , n, for small |μ|, one has μpi ξi = fi (ξ1 , . . . , ξn , μ), ((f1 , . . . , fn ) = f ),

p1 + · · · + pn = k, i = 1, . . . , n.

(9.12)

There exists BSEq in the problem of a branch point, where λ0 is an isolated singular Fredholm point of the operator Fx (0, λ). Note fi (ξ1 , . . . , ξn , μ), i = 1, . . . , n, are continuous functions in a neighborhood of the point ξ1 = · · · = ξn = μ = 0,  n      fi ϕi  = o(ξ) f (ξ, μ) =    i=1

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164

at ξ → 0 and any μ from its neighborhood,  n   ξk ϕk , λ0 fi (ξ1 , . . . , ξn , 0) = ψi , W ,

i = 1, . . . n.

1

Let, in addition to I and II, assumption III be also valid. Then the structure of functions fi , i = 1, . . . , n may be defined more exactly and BSEq (9.12) has the form pi

μ ξi =

m 

fis (ξ1 , . . . , ξn , μ) + ri (ξ1 , . . . , ξn , μ),

i = 1, . . . , n,

s=l

(9.13)

where fis are uniform forms of the s order in ξ, i.e., fis (tξ1 , . . . , tξn , μ) = ts fis (ξ1 , . . . , ξn , μ), with

ψi , Fs

fis =

 n 

 ξk ϕk , λ0

1

for μ = 0, s = l, l + 1, . . . , 2l − 2, i = 1, . . . , n, r(ξ, μ) = o(ξm ) at ξ → 0. If all pi = +∞, then the system (9.11) is reduced to m 

fi,s (ξ, μ) + ri (ξ, μ) = 0.

(9.14)

s=l

9.2. 9.2.1.

Open Problems BEq in the problem at a branch point of unbounded operator

Let the operator F = A(λ)x + W (x, λ) be not bounded in a neighborhood of Ω, but satisfy the following conditions: (1) A(λ) is a closed operator with dense domain D in X independent of λ; (2) W is defined for x, λ ∈ Sxλ = {x ∈ Sr (0, X) ∩ D, |λ − λ0 | < ρ};

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165

(3) A(λ0 ) is a Fredholm operator, F (Γu, λ) is continuous with respect to u and λ for u < r  < Γ−1 r, |λ − λ0 | < ρ and for any u, v ∈ Sr (0, Y ) W (Γu, λ) − W (Γv, λ) ≤ c(r  , ρ)u − v, where continuous function c(r  , ρ) → 0 at r  → 0. It is to be noted that Γ ∈ L(Y, X) and A(λ)Γ is bounded operator. Then the problem of a branch point for equation (9.1) can be reduced to BEq of the form (9.11). For the proof, it is sufficient to put x = Γy in (9.5) and repeat calculations using the contraction mapping principle. 9.2.2.

BEq in the problem of a branch point for equation (9.1) of the first kind

Let F ≡ A(λ)x + W (x, λ), where A0 is a closed operator with dense domain in X, but its range be not closed. The problem is to construct BEq under the condition where Γ is a closed unbounded operator. It is to be noted that the following remarks, lemma and example can be used to tackle these problems. Assume the operator F (x, λ) − A0 x is defined and continuous for x, λ ∈ Ω and there exists the operator (I + Γ(A(λ) − A0 ))−1 , which is continuous in λ in a neighborhood (one-sided neighborhood) of the  point λ0 . If the operator A(λ) + n1 γi , ·zi is continuously invertible at 0 < λ − λ0 <  (− < λ − λ0 ) < 0, then operator I + Γ(A(λ) − A0 ) will be continuously invertible at 0 ≤ λ−λ0 <  (− < λ−λ0 ≤ 0). We also note the operator (I + Γ(A(λ) − A0 ))−1 exists and is continuous in a neighborhood of the point λ0 if R(A(λ) − A0 ) ⊆ D(Γ) = R(Aˆ0 ). For the construction of BEq, one introduces the condition ΓW (x, λ) − ΓW (y, λ) ≤ c(r, ρ)x − y,

(9.15)

where continuous function c(r, ρ) → 0 at r → 0. Under these conditions, the following assertion holds.

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Lemma 9.1. (1) Let there exist the operator (I + Γ(A(λ) − A0 ))−1 and be continuous in λ in a neighborhood (half-neighborhood) of the point λ0 , (2) Let the condition (9.15) be valid and N (A0 ) ⊂ D{Γ(A(λ) − A0 }. Then equation (9.1) has as many small solutions as small solutions have BSEq (9.11) for λ ∈ I. exists, Proof. Since N (Aˆ0 ) = {0}, then the operator Γ = Aˆ−1 0 although it is not bounded, if operator A0 is not normal solvable. Hence, by conditions (1) and (2), equation (9.7) is transformed to  BSEq (9.11) for λ ∈ I. Remark 9.1. If there exists a total GJS of the operator A(λ) corresponding to the point λ0 , ϕi ∈ D{[Γ(A(λ) − A(λ0 ))]ji },

i = 1, . . . , n,

ji = 1, . . . , pi + 1,

k = p1 + · · · + pn , pi are the lengths of GJC, then one may transform BSEq to the form (9.12) for λ ∈ I. Remark 9.2. If A0 is a Fredholm operator ((9.1) is equation of the second kind), then Γ is a bounded operator, and the formulation of lemma is simplified in a natural way, and the problem (9.1) is reduced to BSEq (9.11) or (9.12) in the conditions of Remark 9.1 in the neighborhood of the point λ0 . (i)

(i)

Remark 9.3. Let (ϕ1 , . . . , ϕki ) be a basis in the subspace of zeros of Fredholm operator A(λi ), i = 1, 2, . . ., |λ1 | ≥ |λ2 | ≥ · · · ≥ |λi | ≥ · · · > 0,

inf |λ| ≥ 0. i

Let there exist a positive integer N such that R{W (x, λ)} ⊆ R{A(λi )},

i = N, N + 1, . . .

for all x ∈ Γ. Then equation (9.1) has a solution x = x(λ, c1 , . . . , cki ) in each point λi , i = N, N +1, . . . depending on ki arbitrary constants c1 , . . . , cki , x(λi , c1 , . . . , cki ) → 0 at c1 , . . . , cki → 0.

Investigation of Bifurcation Points of Nonlinear Equations

167

Note that the conditions of Remark 9.3 will be satisfied for N ≥ k + 1 if X = Y = H, A(λ) = A0 − λI, A0 is a completely continuous is an orthonormalized sequence of its self-adjoint operator, {ϕi }+∞ 1 eigenvectors corresponding to the eigenvalues λi , |λ1 | ≥ |λ2 | ≥ · · · ≥ |λi | ≥ · · · > 0 (k is a constant), R(W (x, λ)) ⊆ N (A0 ) ∪ {ϕ1 , . . . , ϕk }. Example 9.1.  1  1 ∞ cos kπt cos kπs x(s)ds + a(s, x(s))ds. λx(t) = k2 0 0

(9.16)

k=1

Here, a(s, x) is a continuous function of its arguments, a(s, x) = al (s)xl + o(|x|l ),

l ≥ 2,

X = Y = C[0,1] ,

1 , k = 1, 2, . . . . 2k2 The conditions of Lemma 9.1 hold in a negative half-neighborhood (condition (1)) of the point λ0 , and the conditions of Remark 9.3 are valid for the sequence {λk }+∞ 1 . Indeed, λ0 = 0,

ϕ = ψ = 1,

λk =

(I − λΓ)−1 f  1  1 ∞ 2 1 f (s)ds + = 1−λ 0 1 − 2λk2 0 k=1

× cos kπt cos kπsf (s)ds for all f (t) ∈ C[0,1] , λ = λk , k = 1, 2, . . .. Therefore, the operator (I − λΓ)−1 is continuous for λ ≤ 0,     1 0 for λ = 1/2k2 , 1 ξ(λ) − a(s, x(s))ds + x= 1−λ ck cos kπt for λ = 1/2k2 , 0 where ξ(λ) = at BEq

1 0

x(s)ds, ck is a constant. Setting ck = 0, we arrive

λξ =

1 (1 − λ)l



1 0

al (s)dsξ l + o(|ξ|l ),

(9.17)

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which has l small solutions. Consequently, equation (9.16) has l small complex continuous solutions:    λ + o(|λ|1/(l−1) ), k = 1, . . . , l − 1, xk (λ) = l−1  1 0 al (s) ds k

xl (λ) ≡ 0.

(9.18)

For λ < 0, equation (9.16) has no other small solutions (Lemma 9.1 holds). For λ > 0, there are other small solutions. Precisely, there exists a solution x(λk ) = 2k2 b + c cos kπt

(9.19)

at the points λk = 1/2k2 , k = 1, 2, . . ., where b and c are constants connected by the condition  1 a(s, 2k2 b + c cos kπs)ds. (9.20) b= 0

Since equation (9.20) has a small solution b = b(c) → 0 at c → 0, then equation (9.16) has a solution x(λk , c) → 0 at c → 0 at the points λk (Remark 9.3 is valid). The example analyzed illustrates the contents of Lemma 9.1 and Remark 9.3 in the case of equations of the first kind. All results in the following section are based on the study of BEqs (9.12)–(9.14) and are formulated for equation (9.1) of the second kind. Evidently, they remain true for certain classes of unbounded operators.

Chapter 10

General Existence Theorems for the Bifurcation Points In this chapter, BEqs (9.12)–(9.14) are studied. With the help of known results from the singular points theory for the finitedimensional vector fields, sufficient existence conditions for the bifurcation points and continuous branches of the real solutions of equation (9.1) of the second kind are obtained. It is shown that if BEq has the form (9.14) (all pi = +∞), then in the general case, the point λ0 will not be a branch point. At the end of the chapter, it is indicated that the method considered is applicable and when λ ∈ M, where M is a connected bounded or unbounded domain of the space Rm , λ0 = (λ01 , . . . , λ0m ) is an interior point of M. In this case, sufficient existence conditions of bifurcation surfaces passing through the point λ0 are obtained and the equations defining these surfaces are constructed. Theorem 10.1. Let the assumptions I and II hold, where q ≥ max{p1 , . . . , pn } and the root number k of the operator A(λ) corresponding to the point λ0 is odd. Then λ0 is a bifurcation point of equation (9.1). Proof.

Consider the vector field

Φμ (ξ) = {μpi ξi − fi (ξ1 , . . . , ξn , μ)}ni=1 ,  corresponding to BSEq (9.12) on a sphere S = (ξ| ni=1 ξi2 = ρ) (a general case of untransformed system (9.11) is considered similar). Here, 0 < ρ < , |μ| < δ,  > 0, δ > 0 are arbitrary small. Let us

169

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introduce the continuous mapping Φ : S × [0, 1] → Rn , Φ = Φi (ξ, t)|ni=1 = [(2t − 1)δ1 ]pi ξi − fi (ξ1 , . . . , ξn , (2t − 1)δ1 )|ni=1 (0 < δ1 < δ). If Φ(ξ, t) = 0 for t ∈ [0, 1], ξ ∈ S, then a rotation J(Φ(ξ, t), S) is given. Since a rotation is an integer number, then J(Φ(ξ, 0), S) = J(Φ(ξ, 1), S) and sign(−1)k = sign(+1)k !. This is impossible. Hence, there are points t∗ ∈ [0, 1] and ξ ∗ ∈ S such that Φ(ξ ∗ , t∗ ) = 0 and the point λ0 is a bifurcation point. The theorem is  proved. In the conditions of Theorem 10.1, the continuous branch of solution of equation (9.1) corresponds to the point λ0 in the following sense. Denote by G a set of solutions x corresponding to the values λ from (−δ, +δ). Let L be the boundary of arbitrary open set in X containing Θ and situated in a sphere of arbitrary small radius. Definition 10.1. The set G forms the continuous branch of solutions if G ∩ L = ∅. Corollary 10.1. Let the conditions of Theorem 10.1 be valid. Then the continuous nontrivial branch of solutions of equation (9.1), in the sense of Definition 10.1, corresponds to the point λ0 . Corollary 10.1 is evident because in the proof of Theorem 10.1, instead of a sphere S, one may take an arbitrary sufficiently smooth closed surface containing the point ξ = 0. Theorem 10.1 and its corollary show further strengthening of Krasnosel’skii’s theorem. Example 10.1. Consider the boundary value problem   2 ∂u ∂ u , , u, x, λ = 0, x ∈ Ω, F ∂xi ∂xk ∂xi where F (ξ, η, ζ, x, λ) (ξ ∈ R2n , η ∈ Rn , ζ ∈ R1 , x ∈ Rn ) is the continuously differentiable function of ξ, η, ζ for |ξ| + |η| + |ζ| < r,

|ξ| =

n  i,k=1

|ξik |,

|η| =

n  i,k=1

|ηik |,

|η| =

n  i=1

|ηi |

General Existence Theorems for the Bifurcation Points

171

and analytic in λ for λ ∈ I ⊂ R1 , = 0, F (0, 0, 0, x, λ)|x∈Ω,λ∈I ¯ Ω is a bounded domain in Rn with a boundary of class C (1+α) (α ∈ (0, 1))  ∂F  ¯ = aik (x, λ) ∈ Cx(1+α) (Ω), ∂ξik ξ=0,η=0,ζ=0  ∂F  ¯ = ai (x, λ) ∈ Cxα (Ω), ∂ηi ξ=0,η=0,ζ=0  ∂F  ¯ = a(x, λ) ∈ Cxα (Ω), ∂ζ ξ=0,η=0,ζ=0 F (ξ, η, ζ, x, λ) =

n 

aik (x, λ)ξik

i,k=1

+

n 

ai (x, λ)ηi + a(x, λ)η + w(ξ, η, ζ, x, λ),

i=1

w∈

¯ Cxα (Ω),

|w(ξ, η, ζ, x, λ)| = o(|ξ| + |η| + |ζ|).

Let the problem have a trivial elliptic solution u = 0 for any λ ∈ I,  i.e., the quadratic form ni,k=1 aik (x, λ)ξi ξk is uniformly positive for all x ∈ Ω. It is required to explain when λ0 will be a branch point, if u ∈ C (2+α) (Ω). Setting def

A(λ)u = aik (x, λ)uxi xk + ai (x, λ)uxi + a(x, λ)u and introducing Banach spaces X = C (2+α) (Ω, Γ), Y = C α (Ω), one may interpret our problem as the problem on bifurcation points of the operator equation A(λ)u + w(u, λ) = 0, where A(λ) ∈ L(X × I → Y ) is a continuous operator from X to Y . Under the formulated conditions, all singular points of the operator A(λ) will be Fredholm (see [114]). If N (A(λ0 )) = {0}, then there is a neighborhood S of the point λ0 such that in a ball u <  ( is

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arbitrary small), there exists only a trivial solution u = 0 for all λ ∈ S. If a root number k of the operator A(λ) is odd at the point λ0 , then owing to Theorem 10.1, λ0 is a branch point. For example, λ0 is a branch point if A(λ)u =

n 

aik uxi xk + λa(x)u,

i,k=1 n 

aik ξi ξk ≥ c|ξ|2 ,

c = const > 0,

a(x) > 0,

i,k=1

and λ0 is the least eigenvalue of the problem A(λ)ϕ = 0, ϕ|Γ = 0. Example 10.2. A(λ)u ≡

n  ∂ (aik (x, λ)uxk + ai (x, λ)u) ∂xi

i,k=1

+

n 

bi (x, λ)uxi + a(x, λ)u

i=1

= w(uxi , u, x, λ),

x ∈ Ω ⊂ Rn ,

aik = aki ,

u|Γ = 0.

Let the operator A(λ) be elliptic in the bounded domain Ω for all λ ∈ R1 , and its coefficients be measurable bounded functions in x and analytic in λ, and w(ξi , η, x, λ) be continuously differentiable functions in ξi , η, x for |ξ| + |η| =

∞ 

|ξi | + |η| < r,

|λ| < ρ,

¯ x ∈ Ω,

1

¯ |λ| < ρ, and analytic in λ for x ∈ Ω, |w(ξi , η, x, λ)| = O[(|ξ| + |η|)2 ]. We shall call the problem at a branch point λ0 the construction of ◦

the generalized solution u(x, λ) ∈W 12 (Ω) satisfying the identity. Here,

General Existence Theorems for the Bifurcation Points

173

readers may refer to [96].  (aik uxi ηxk + ai uηxi − bi uxi η − auη)dx Ω



= Ω

−w(uxi , u, x, λ)η(x)dx

(10.1)

◦

for all η(x) ∈W 12 (Ω) such that 0 < u

◦

W12 (Ω)

< ,

|λ − λ0 | < δ,

 > 0,

δ > 0.

Transform this problem to the operator equation of the form (9.1), ◦

where X = Y =W 12 (Ω). To this end, following [96], we introduce a new scalar product  aik uxi vxk dx [u, v] = Ω

 ◦ [u, v] in W 12 . Therefore, the equivalent and the norm u 1 = operator equation has the form (see [96]) A(λ)u + R(u, λ) = 0, where A(λ) = I + C(λ), and the bounded operators C(λ), R(u, λ) ◦

acting in W 12 (Ω) are defined by the formulas  [Au, η] =

Ω

(ai uηxi − bi uxi η − auη)dx,

 [R(u, λ), η] =

Ω

w(uxi , u, x, λ)ηdx,

where operator C(λ) is completely continuous. Hence, all singular points of operator A(λ) will be Fredholm and Theorem 10.1 is applicable.

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Example 10.3. 

(−1)α Dα Aα (Dm u, . . . , u, x)

|α|=m



=

(−1)α D α Bα (D m u, . . . , u, x, λ),

|α|≤m−1

(x ∈ Ω ⊂ Rn ),

u|Γ = 0, where 

α

D =

∂ ∂x1

α1

 ···

∂ ∂xn

D m u = {Dm u : |α| = m},

αn

,

|α| = α1 + · · · + αn ,

Aα (0, . . . , 0, x) = Bα (0, . . . , 0, x, λ) = 0.

We shall call the problem at a branch point λ0 the construction of ◦

generalized solutions u(x, λ) ∈W m p (Ω) satisfying the identity ⎡



⎣

Ω



Aα (D m u, . . . , u, x) −

|α|=m

⎤



Bα (Dm u, . . . , u, x, λ)⎦

|α|≤m−1

α

×D ηdx = 0

(10.2)

◦

for all η ∈W m p (Ω) such that 0 < u < , |λ − λ0 | < δ. Let us introduce the following assumptions: (a) Ω is a bounded domain in Rn ; ¯ (ξ = (b) the functions Aα (ξ, x), Bα (ξ, x, λ) are defined for x ∈ Ω M (ξβ |β| ≤ m) ∈ R , M is a number of various multi-indices β of the length not more than m), continuously differentiable by all variables, and analytic with respect to λ; (c) the elliptic condition holds in the form  |α|=|β|=m

⎛ ∂Aα (x, ξ) ηα ηβ ≥ c1 ⎝1 + ∂ξβ

 |β|=m

⎞p−2 |ξβ |⎠

 |α|=m

ηα2 ;

General Existence Theorems for the Bifurcation Points

175

(d) the smoothness conditions and the estimations to increase at |ξ| → +∞ are valid, for example, in the form |Aα (x, ξ)| ≤ c2 (1 + |ξ|)p−1 and similar estimations to Bα and derivatives of the functions Aα , Bα . ◦

◦

m ∗ Let us introduce the operator F :W m p (Ω) → (W p (Ω)) determined by the integral identity (10.2). Based on the works of Skrypnik [157, 158], it is continuously differentiable in the sense of Fr´echet at p ≥ 3; moreover, Fu (u, λ) is defined by the formula   Aαβ (Dm u, . . . , u, x)Dβ U Dα ηdx [Fu (u, λ)U, η] = |α|,|β|=m Ω

−





|α|,|β|≤m−1 Ω

Bαβ (Dm u, . . . , u, x, λ)D β U Dα ηdx,

def

Fu (0, λ) = A (0) + B  (0, λ). In [157], Skrypnik has demonstrated that [A (0)]−1 B  (0, λ) is a completely continuous operator. Hence, the operator Fu (0, λ) is Fredholm, and its singular point λ0 will be a branch point if the corresponding root number k is odd. Theorem 10.2. Let the assumptions I and II hold with q ≥ max1≤i≤n pi and a root number of the operator A(λ) corresponding to the point λ0 be equal to k. Let x = 0 be an isolated solution of the equation F (x, λ0 ) = 0,

sign(λ − λ0 )k = J(f (ξ, 0), S(0, ρ))

(10.3)

(0 < ρ < ,  is arbitrary small) at least in one of the halfneighborhoods of the point λ0 . Then the point λ0 is a bifurcation point, moreover, there also exists a continuous branch of solutions in a neighborhood (half-neighborhood), in which (10.3) is valid. Proof. It is sufficient to show that except zero, the field Φμ (ξ) has another singular point ξ ∗ = ξ ∗ (μ) in a ball S(0, ρ) at |μ| < δ, where

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δ > 0 is arbitrarily small. Since x = 0 is an isolated solution of the equation F (x, λ0 ) = 0, then the point ξ = 0 is an isolated singular point of the field f (ξ, 0). Therefore, the field Φμ (ξ) is homotopy to  the non-degenerate field f (ξ, 0) on a sphere S = (ξ| n1 ξi2 = ρ), (0 < ρ ≤ ) for arbitrarily small |μ| < δ, where δ = δ() > 0. But the field f (ξ, 0) does not depend on μ. Hence, a rotation J(Φμ (ξ), S) is the same number independent of μ for small |μ| < δ, where 0 < δ ≤ δ(), δ() > 0. The index γ0 of the point ξ = 0, as a singular point of the field Φμ (ξ), is equal to sign μk = J(Φμ (ξ), S) in a half-neighborhood I+ (I− ) of the point λ0 in which we have (10.3). It means that in a sphere S, the field Φμ (ξ) has another singular point ξ ∗ (μ) except  zero. In this case, 0 < n1 (ξi∗ )2 (μ) ≤ ρ <  if 0 < |μ| < δ ≤ δ(), μ ∈ I+ (I− ) and λ0 is a bifurcation point. The continuous branch x(λ) is defined in a half-neighborhood of the point λ0 , where the  inequality (10.3) holds. The theorem is proved. Since operator F is differentiable with respect to x in the sense of Fr´echet, then the functions Φi (ξ, t), i = 1, . . . , n, also, seem to be differentiable in ξ1 , . . . , ξn and for the calculation of a rotation J(f (ξ, 0), S), one may use the Kronecker integral   ∂f1  ...  ∂ξ1   n   .. 1  . J(f (ξ, 0), S) =  S1 S i=1   ∂fn . . .  ∂ξ 1

∂f1 ∂f1 f1 ... ∂ξi−1 ∂ξi+1 .. . ∂fn ∂fn fn ... ∂ξi−1 ∂ξi+1

cos(nˆxi ) × n 2 n/2 dS, ( 1 fk ) π n/2 ξi , cos(nˆxi ) = √ , Γ(n/2) + ρ  n n     ξk ϕk , λ0 f (ξ, 0) = ψi , W  .  S1 = 2

i=1

i=1

 ∂f1   ∂ξn  ..  .   ∂fn  ∂ξ  n

General Existence Theorems for the Bifurcation Points

177

Since a rotation is always integer, then we need the calculations to make up to integers. Here, readers may refer to O’Neil and Thomas [123] and others. Under the estimation of rotation, the known properties are also useful: property of being even, homotopy principle of vector field by the own dominant part and so on. Theorem 10.3. Let the assumptions I−III hold with q ≥ max1≤i≤n pi and the root number k of the operator A(λ) be even at the point λ0 . Let the vector field {fil (ξ1 , . . . , ξn , 0)}ni=1 be nondegenerate and l be the even number. Then the point λ0 is a bifurcation point of equation (9.1); moreover, there exists a continuous branch of solution in a neighborhood of the point λ0 . Proof. It is sufficient to show that the conditions of theorem are valid now in a neighborhood of the point λ0 . Indeed, since k is even number, then γ0 = +1. Since the field fl (ξ, 0) = {fil (ξ1 , . . . , ξn , 0)}ni=1 is regular, then J(f (ξ, 0), S(0, ρ)) = J(fl (ξ, 0), S(0, ρ)), where 0 < ρ < ,  > 0 is arbitrarily small. But l is an even number. Therefore, either J(fl (ξ, 0), S(0, ρ)) is an even number or it is equal to zero. Hence, J(f (ξ, 0), S(0, ρ)) = γ0 = sign(λ − λ0 )k in a neighborhood of the point λ0 and the conditions of the theorem now hold in a neighborhood of this point. The theorem is  proved. Remark 10.1. Note that the field {fil (ξ, 0)}ni=1 will be regular if and only if the system fil (ξ, 0) = 0, i = 1, . . . , n only has a trivial solution in Rn . To this end, it is sufficient that at least one of the forms fil (ξ, 0), i ∈ (1, . . . , n) have the definite sign. Simple transformations of BSEq make it possible to obtain the other useful conditions of the existence of bifurcation points. Theorem 10.4. Let the assumptions I−III be valid with q ≥ max1≤i≤n pi , and λ0 be an isolated singular Fredholm point of

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the operator A(λ), whose GJC numbering is ordered in the following way: p1 ≤ · · · ≤ pk < pk+1 = · · · = pm = p < pm+1 ≤ · · · ≤ pn . Let zero be an isolated singular point of the field

Θ1 =

 1, 0,

Φl (ξ) = {Θ1 ξi − Θ2 fil (ξ, 0)}ni=1 ,  i = 1, . . . , m, 0, i = 1, . . . , k, Θ2 = i = m + 1, . . . , n, 1, i = k + 1, . . . , n, Pn

sign(λ − λ0 )

i=m+1

pi −(n−m)p

= J(Φl (ξ), S(0, ρ)),

(10.4)

and at least one of the numbers l or p is even. Then λ0 is a bifurcation point of equation (9.1); moreover, there exists the continuous branch of solution in a neighborhood (one-sided neighborhood) of the point λ0 in which inequality (10.4) is valid. Proof. By substituting ξ = μp/(l−1) η into BSEq, one can obtain the equations μ(pi (l−1)+p)/(l−1) ηi = μ(pl)/(l−1) fil (η1 , . . . , ηn , 0) + o(|μ|(pl)/(l−1) ), i = 1, . . . , k, m + 1, . . . , n, μ(pl)/(l−1) ηi = μ(pl)/(l−1) fil (η1 , . . . , ηn , 0) + o(|μ|(pl)/(l−1) ),

i = k + 1, . . . , m.

Then ηi = μp−pi fil (η1 , . . . , ηn , 0) + o(|μ|p−pi ), ηi = fil (η1 , . . . , ηn , 0) + o(1),

i = 1, . . . , k, p > pi ,

i = k + 1, . . . , m,

μpi −p ηi = fil (η1 , . . . , ηn , 0) + o(1), i = m + 1, . . . , n, p < pi ,

(10.5)

as μ → 0. Index of a singular Pn point η = 0 of the vector field for system (10.5) equals sign μ i=m+1 pi −(n−m)p , and a rotation of this field on the sphere S(0, ρ) is equal to J(Φl (ξ), S(0, ρ)), where 0 < ρ ≤ ,  is arbitrarily small. Therefore, owing to (10.4), the system (10.5) has a nontrivial real solution. Since at least one from the numbers l or p is even, then the corresponding vector ξ will be real. The theorem is  proved.

General Existence Theorems for the Bifurcation Points

179

Theorem 10.5. Let the operator F (x, λ) be analytic in a neighborhood of the point (0, λ0 ) and all GJC of the operator Fx (0, λ) be equal to +∞ at the point λ0 . Let the field {fi2 (ξ, 0)}ni=1 be regular. Then λ0 will not be a bifurcation point. Proof. Let us consider the corresponding BSEq. Thanks to the analyticity, it has the form ∞  i=2 (0)

(0) fi (ξ)

+

∞ 

μ

i=1

i

∞  k=2

(k)

fi (ξ) = 0,

(10.6)

(0)

where f2 (ξ) = {fi2 (ξ, 0)}ni=1 . Since by the condition ξ = 0 we have an isolated solution of the system ∞  i=2

(0)

fi (ξi ) = 0,

then based on the known properties of analytic sets [169], all small solutions of the system (10.6) are represented in the form of convergent series in integer or fractional powers of parameter μ ξ = μ (c + o(1)),

 > 0,

c = 0,

(0)

where c satisfies the system f2 (ξ) = 0. But the last one only has a trivial solution in Rn . Therefore, the point λ0 may not be a branch  point. The theorem is proved.

10.1.

Open Problem

Let us consider the bifurcation problem of solutions for equation (9.1), when λ ∈ M where M is connected bounded or unbounded domain of space Rm and λ0 = (λ01 , . . . , λ0m ) is an internal point of M. In this case, the branch surfaces may pass through the point λ0 . Indeed, let the assumptions I and II hold, where λ0 is a singular

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Fredholm point of the operator A(λ), A(λ) = A0 − c(λ − λ0 ), 0

c(λ − λ ) =

q 

Ak (λ − λ0 )k + R(λ − λ0 ),

k = (k1 , . . . , km ),

|k|=1



(λ − λ0 )k =

(λ1 − λ01 )k1 . . . (λm − λ0m )km ,

k1 +···+km =k 0

R(λ − λ ) = o(|λ − λ0 |q ). Then the problem at a branch point of equation (9.1) can be reduced to BSEq (9.11) with μ = (μ1 , . . . , μm ) = (λ1 − λ01 , . . . , λm − λ0m ). Let

 def Bk () = A(λ)λ =λ0 , i

i

i = 1, . . . , m/k,

λk = , k = 1, . . . , m.

It is evident that if at least one of the operators Bk (), k = 1, . . . , m, has an odd root number at the point ξ = λ0k , then the point λ0 will be a branch point of equation (9.1) according to the definition where  0 now |λ − λ0 | = m 1 |λi − λi | < δ. However, in the multi-parametric problem, the case of branch surface passing through the point λ0 is possible. Definition 10.2 ([154]). Let Λ be a surface in domain M ⊂ Rm and the point λ∗ be an arbitrary point of this surface. We shall call the surface Λ a branch surface of equation (9.1) if for any  > 0 and δ > 0 there exist x and λ satisfying (9.1) such that 0 < x < , |λ − λ∗ | < δ. The proof of existence of branch surface needs detailed studies. The elementary results in this direction are given in the following. Under certain conditions, the branch surface can be described by the equation n L(λ) ≡ detψi , c(λ − λ0 )(I − Γc(λ − λ0 ))−1 ϕk i,k=1 = 0. (10.7)

General Existence Theorems for the Bifurcation Points

181

Lemma 10.1. Let λ0 be a singular Fredholm point of the operator A(λ), λ0 ∈ Λ, and Λ be the surface in Rm . In order that Λ can be a branch surface, it is necessary that L(λ)|λ∈Λ = 0. Proof. Let Λ be a branch surface, λ∗ ∈ Λ, L(λ∗ ) = 0, 0 < |λ0 − λ∗ | < δ. Then the operator A(λ) has a bounded inverse at the point λ∗ if δ is arbitrary small, and equation (9.1) only has a trivial solution in the ball x <  for any λ such that |λ−λ∗ | < δ, , δ are arbitrarily  small. The lemma is proved. Theorem 10.6 ([154]). (1) Let the assumptions I and II hold with λ = (λ1 , . . . , λm ); (2) Let x = 0 be an isolated solution of the equation F (x, λ0 ) = 0; (3) Let equation (10.7) give the smooth oriented surface Λ passing through the point λ0 and dividing their neighborhood onto the two parts S+ , S− such that L(λ)|λ∈S+ > 0, Then Λ will be a branch surface.

L(λ)|λ∈S− < 0.

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Chapter 11

Construction of Asymptotics in a Neighborhood of a Bifurcation Point Let the assumptions I−III hold and let p1 = · · · = pn = 1, detψi , A1 ϕk |ni,k=1 = 0. This is the most standard case of the problem (9.1) when linearized problem has no adjoined vectors and BEq (9.13) has the form μξ =

m 

fs (ξ, μ) + r(ξ, μ).

(11.1)

s=l

Here, r(ξ, μ) = o(ξm ) as ξ → 0 r(ξ, μ) = (r 1 (ξ, μ), . . . , r n (ξ, μ)). In this chapter, the solutions of equation (11.1) are constructed. A number of results are obtained for the general equation (9.13) when not all pi = 1. The existence theorems are not connected with evenness (oddness) of the root number k. The investigation method consists of a combination of analytic Lyapunov–Schmidt method and finite-dimensional variational and topological methods. 11.1.

Analytic Lyapunov–Schmidt Method in the Study of Branching Equations

We consider a solution of equation (11.1) in the following form: ξ = (ξ0 + o(1))|μ|1/(l−1) ,

183

ξ0 = (ξ10 , . . . , ξn0 ) = 0,

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Toward General Theory of Differential-Operator and Kinetic Models

where (sign μ)ξ0 = fl (ξ0 , 0).

(11.2)

There are two systems: one system at μ > 0 and the other system at μ < 0, i.e., the following systems exist: ξ0 = fl (ξ0 , 0),

(11.3)

−ξ0 = fl (ξ0 , 0).

(11.4)

Lemma 11.1. Let ||fl (ξ, 0)|| = 0 as ||ξ|| = 0, i.e., fl (ξ, 0) is a nondegenerate field. (1) Then both systems (11.3)–(11.4) have the nontrivial real solution for even l. (2) Then at least one from the systems (11.3)–(11.4) has a nontrivial real solution ξ0 for odd nl. Proof. Since the index γ0 of the point ξ = 0 for the field sign μξ − fl (ξ, 0) equals sign μn , and the rotation J(sign μξ − fl (ξ, 0), S(0, R)), where R is arbitrarily large, in the first case is even, and the second one is odd, then γ0 = J in the first case in a neighborhood of the point μ = 0, and the second one at least in  one of its half-neighborhoods. The lemma is proved. n n Lemma 11.2. Let i=1 fil (ξ, 0) > 0 for ξ ∈ M = {ξ | ξi ≥ 0, 1 ξi = 1}. Then the system ξ = fkl (ξ, 0) has a nontrivial solution ξ0 . Proof. Let us introduce the continuous function Φ : M → M, where Φ = (Φ1 , . . . , Φn ), Φi =

ξi + fil (ξ, 0)  , 1 + ni=1 fil (ξ, 0)

i = 1, . . . , n.

By Brouwer’s fixed-point theorem, there exists the point ξ ∗ in M such that ξi∗ =

ξi∗ + fil (ξ ∗ , 0)  , 1 + nk=1 fkl (ξ ∗ , 0)

i = 1, . . . , n,

n  i=1

ξi∗ = 1.

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

185

But then, n 

fkl (ξ ∗ , 0)ξi∗ = fil (ξ ∗ , 0),

k=1

 with ni=1 fil (ξ ∗ , 0) = c > 0. Putting ξ ∗ = c1/(l−1) ξ0 , one obtains the  identity ξ0 = fl (ξ0 , 0). The lemma is proved. Case 1: l is even. Changing ξ0 ⇒ −ξ0 leads (11.3) to (11.4). From here, we obtain the following condition: If l is even and there exists a simple solution of equation (11.3), i.e., matrix I − fl (ξ0 , 0) is invertible, then equation (9.1) has a small solution in a neighborhood of the point λ0 of the form   n  0 ξi ϕi + o(1) (λ − λ0 )1/(l−1) . (11.5) x= i=1

Case 2: l is odd. Changing ξ0 ⇒ −ξ0 does not change equations (11.3) and (11.4), and they are independent. If l is odd and equation (11.3) has a simple solution ξ0 , then equation (9.1) has two small solutions of the form   n  ξi0 ϕi + o(1) |λ − λ0 |1/(l−1) (11.6) x= ± i=1

in a neighborhood I+ (λ > λ0 ). If just equation (11.4) has a simple solution ξ0 , then equation (9.1) has two small solutions of the same form (11.6) in a neighborhood I− (λ < λ0 ). It is to be noted that one may change the condition of simplicity for solution ξ0 onto lesser severe constraint: existence of isolated solution ξ0 with index γ0 = 0 for the systems (11.3) and (11.4). It has used the property of the stability of such a solution with respect to small perturbations of equation. The Lyapunov–Schmidt method and methods from Ref. [154] concerning the finite-dimensional topology enables readers to prove the existence theorems and build the main terms of the asymptotic expansions.

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Example 11.1. In [105], the following biochemical model was formulated:  ax − (1 + λ)x + x2 y + c = 0, θay  + λx − x2 y = 0 with conditions x(0) = x(1) = c, y(0) = y(1) = λ/c. Here, a, θ are fixed concentrations of certain chemical substance, x, y are desired concentrations which are functions of r, and λ, c are diffusion coefficients. This system has a trivial solution x = c, y = λ/c. We search for nontrivial solutions as x → c, y − λ/c → 0 as λ → λ∗ , where λ∗ is the desired bifurcation point. Let x = c + u, y = λ/c + v. Then we obtain the following system: A(λ)u + F2 (u, λ) + F3 (u) = 0, where

⎤ d2 +λ−1 c2 ⎥ ⎢a A(λ) = ⎣ dr 2 ⎦, d2 2 −λ θa dr − c 2 ⎡ ⎤ λ   2cuv + u2 u2 v ⎢ ⎥ c , F2 (u, λ) = ⎣ ⎦ , F3 (u) = λ 2 −u2 v −2cuv − u c ⎡

(2)

X = {C[0,1] , u(0) = u(1) = 0},

u=

  u v

,

Y = C[0,1] .

Problem A(λ)φ = 0 has eigenvalues λn = 1 + cθ + π 2 a(n2 + nl 2 )

 and eigenfunctions φ = sin πnr ρ1 , where l2 = c2 π −4 a−2 , θ −1 , ρ = c−2 (1−λ2 +pi2 n2 a) < 0. If l2 = n2 m2 , then ρ = c−2 (1−λn +pi2 n2 a) < 0. If l2 = n2 m2 , then λn is a double eigenvalue. Ifl2 = n2 m2 , then, 1 n is a single eigenvalue. The function ψ = sin πnr −ρ c2 , λn   1 0 , A (λ) = −1 0 2

2

satisfies the related problem. Let l2 = n2 m2 . Then the root number is 1 and λn is the bifurcation point. Then branching equation is as follows: L11 ξμ + L2 (μ)ξ 2 + L3 (μ)ξ 3 + · · · = 0, where μ = λ − λn ,

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

187

L11 = π 2 n2 aθ(2π 2 n2 aθ + 2c2 )−1 , L2 (0) = (1 + (−1)n+1 ) 23 (3ρc2 λn + 2ρ λ2n + 2c4 ρ2 )(cλn πn)−1 , L3 (0) = 38 ρ λnλ+c . If L2 (0) = 0, then there n L11 exists the solution u = − L2 (0) μφ + o(|μ|). If L2 (0) = 0, then for μ > 0, there exist two solutions    μ √ 1 sin nπr + o( μ). u± = ±2 ρ −3ρ In the case of l2 = n2 m2 in system (11.1), ξ = (ξ1 , ξ2 ), n = 2 and the previous results can be applied.

11.2.

Variational Methods in the Study of Branching Equations

In this section, BSEq (11.1) is considered in the assumption of potential lowest vector field fl (ξ, 0).

 The field f (ξ) is potential if and only if matrix ∂f∂ξi (ξ) , i, j = j 1, . . . , n, is symmetric. The corresponding potential U (ξ) is defined as follows: U (ξ) =

n   i=1

0

1

fi (tξ1 , . . . , tξn )ξi dt.

The field fl (ξ, 0) is potential if Fl (x, λ0 ) is a potential operator, X = Y = H, A(λ0 ) = A∗ (λ0 ). The converse statement is not always true. Let us demonstrate this with the following example. Example 11.2.  F (x) =

1 0

K(t, s)(x(s) + xl (s))ds,

K(t, s) = K(s, t), X = Y = L2 [0, 1],  1 K(t, s)ϕi (s)ds, i = 1, . . . , n, ϕi (t) = 0

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Toward General Theory of Differential-Operator and Kinetic Models

fil (ξ)|ni=1 =

n   1  K(t, s) ⎝ ξj ϕj (s)⎠ dsϕi (t)dt  0 j=1  ⎛

1

 0

1 = grad l+1

 0

1

⎛ ⎝

⎞l

n 

n 

i=1

⎞l+1 ξj ϕj (s)⎠

ds.

j=1

The field fl (ξ) is potential. However, the operator Fl (x) = xl (s)ds is not potential because  1 1 K(t, s)xl−1 (s)h1 (s)h2 (t)dsdt 0

0

1 1

 =

0

0

1 0

K(t, s)

K(t, s)xl−1 (s)h2 (s)h1 (t)dsdt

for any h1 , h2 ∈ L2 [0, 1]. Lemma 11.3. Let the potential vector field n fl (x) = fil (x1 , . . . , xn )i=1 , be given in Rn , fil (x1 , . . . , xn ) be uniform continuous functions of the order l. Then, for any ρ > 0, there exists g = 0 such that the system 2gx = fl (x)  has a real solution x on the sphere n1 x2i = ρ, which is an extreme point of the corresponding potential. Proof. Owing to the potential nature of the field fl (x), there exists a continuous function U : Rn → Rn such that ∂U (x) = fil (x1 , . . . , xn ), ∂xi where U (x) is uniform function of the order l + 1. Consider the  function U (x) on the sphere ni=1 x2i = ρ and introduce the Lagrange function   n  x2i − ρ . U (x, g) = U (x) − g i=1

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

189

The function U (x) reaches their max and min on the sphere. Therefore, there exists a Lagrange multiplier g0 such that a real vector x0 be the point mins U (x) and satisfies the system ∂U = 2gxi , ∂xi

i = 1, . . . , n,

n  i=1

x2i = ρ

(11.7)

g0

for g = g0 and there exists such that the point maxs U (x) is a real 0 vector x and also satisfies the system (11.7) for g = g0 . Show that at least one of the numbers g0 , g0 is not equal to zero. Indeed, from  (11.7), it follows that ni=1 xi Uxi = 2gρ if (x, g) is a solution of the system (11.7), and since n 

xi Uxi ≡ (l + 1)U (x)

i=1

for any x, then g=

l+1 U (x) 2ρ

in a stationary point x. Therefore, g0 = 0 if U (x0 ) = 0, g0 = 0 if U (x0 ) = 0. But then, maxs(0,ρ) U (x) = mins(0,ρ) U (x) = 0 for any ρ > 0. Hence, U (x) ≡ 0, which is impossible. The lemma is  proved. From Lemma 11.3, we have the following theorem. Theorem 11.1. (1) Let the assumptions I–III hold. (2) Let a root number of the operator A(λ) be equal to dim N (A(λ0 )) = n at the point λ0 . (3) Let fl (η, 0) = grad U (η). Then each isolated extremum of the potential U (η) on the sphere n 2 i=1 ηi = 1 with U (η0 ) = 0 corresponds to the solution of equation (9.1) of the form   n  ci ϕi + o(1) |λ − λ0 |1/(l−1) . (11.8) x= i=1

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Toward General Theory of Differential-Operator and Kinetic Models

Here,

⎧! " ⎪ sign(λ − λ0 ) 1/(l−1) 0 ⎪ ⎪ ηi , ⎨ (l + 1)U (η0 ) ci = 1/(l−1)  ⎪   1 ⎪ ⎪   ⎩±  ηi0 , (λ − λ0 )U (η0 ) > 0, (l + 1)U (η0 ) 

l is even, l is odd. (11.9)

Proof. form:

We consider a solution of the system (11.1) in the following ! ξ=

μ 2g

"1/(l−1)

η,

 where ni=1 ηi2 = 1, and g = g(μ) is a scalar parameter satisfying the condition μg > 0 for odd l. For finding g and η, one obtains the system ⎧ ⎪ ⎪2gη = fl (η, 0) + Θ(η, g, μ), ⎨ n  (11.10) ⎪ ηi2 = 1, ⎪ ⎩ i=1

(Θ(η, g, μ) = o(1), μ → 0), where ⎧ ⎪ 2g(0)η(0) = fl (η(0), 0), ⎪ ⎨ n  ⎪ ηi (0)2 = 1. ⎪ ⎩

(11.11)

i=1

By Lemma 11.3, the system (11.11) has a real solution (g0 , η0 ) with g0 = 0, and η0 is the extremum of potential U (η) on the sphere n 2 i=1 ηi = 1. By the condition of the theorem, η0 is an isolated extremum. Consider the perturbed vector field  # n  ∂U − Θi (η, g, μ), i = 1, . . . , n, ηi2 − 1 . Φμ (η) = 2gηi − ∂ηi i=1

Let S((g0 , η0 ), ρ) be a sphere in Rn+1 with center at the point (g0 , η0 ), 0 < ρ < and |μ| < δ, , δ be arbitrarily small. Then J{Φμ (η), S((g0 , η0 ), ρ)} = J{Φ0 (η), S((g0 , η0 ), ρ)} = γ0 ,

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

191

where γ0 is the index of the point (g0 , η0 ) which is an isolated extremum. Thus, owing to the remark, γ0 = 0. Hence, J{Φμ (η), S((g0 , η0 ), ρ)} = 0, and the field Φμ (η) has a singular point in the neighborhood (g0 , x0 ) (the critical point (g0 , η0 ) is stable). Since 2g0 = (l + 1)U (η0 ), then the system (11.1) has a real solution ξ = c|μ|1/(l−1) + o(|μ|1/(l−1) ),

c = (c1 , . . . , cn ),

where ⎧! "1/(l−1) ⎪ sign μ ⎪ ⎪ ηi0 , l is even, i = 1, . . . , n, ⎨ (l + 1)U (η0 ) ci = 1/(l−1)  ⎪   1 ⎪ ⎪±   ⎩ ηi0 , μU (η0 ) > 0, l is odd, i = 1, . . . , n.  (l + 1)U (η0 )  By substituting obtained ξ into (9.10), one has the solution (11.8).  The theorem is proved. One may weaken the second condition in Theorem 11.1, having required p1 = · · · = pn = p. In this case, we need to change exponent 1(l − 1) in Theorem 11.1 onto p/(l − 1), ⎧! " ⎪ sign(λ − λ0 )p 1/(l−1) 0 ⎪ ⎪ ηi , l is even, ⎨ (l + 1)U (η0 ) ci = 1/(l−1)  ⎪   1 ⎪ ⎪   ⎩±  ηi0 , (λ − λ0 )p U (η0 ) > 0, l is odd. (l + 1)U (η0 )  Note also that all extrem a U (η, g) are necessarily isolated if the eigenspace grad U (η) is finite dimensional. If the forms fil (η, 0), i = 1, . . . , n are uniform (the operator Fl (x, 0) is l exponential), then one way write the corresponding conditions of finite dimensionality in terms of resultants. 1 Example 11.3. λx(t) = 0 K(t, s)f (s, x(s))ds, where K(t, s) and f (s, x) are continuous functions of their arguments, K(t, s) = K(s, t),

f (s, 0) = 0,

fx (s, 0) = 1,

1/l!fx(l) (s, 0) = a(s) = 0,

fx(i) (s, 0) = 0,

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Toward General Theory of Differential-Operator and Kinetic Models

i = 2, . . . , l−1. Let λ0 be an eigenvalue of the kernel K(t, s) of rank n. Then for the integral equation, the conclusions of Theorem 11.1 are valid with  n l+1  1  λ0 U (ξ) = − a(s) ξi ϕi (s) ds. l+1 0 i=1

Example 11.4. Consider the Karman boundary value problem  2 w = λ[F, w] + [w, f ] in Ω, (11.12) 2 f = −[w, w]  w = wx = wy = 0 on Γ = ∂Ω, (11.13) f = fx = fy = 0 where def

[F, w] = (Fyy wx − Fxy wy )x + (Fxx wy − Fxy wx )y = Fxx wyy + Fyy wxx − 2Fxy wxy , f ∈ W24 (Ω, Γ), w ∈ W24 (Ω, Γ), F is a known smooth function, w and f are unknown functions, W24 (Ω, Γ) are subspaces of W24 (Ω) with elements satisfying the conditions (11.13). Let us represent the problem (11.12)–(11.13) in the form of one abstract equation (9.1). To this end, we introduce a Banach space X → 4 of columns x= [ w f ], where w, f ∈ W2 (Ω, Γ), and a Banach space of such two-dimensional columns with components from L2 (Ω) which we denote by Y . Then one can write the problem (11.12)–(11.13) in the form of the following equation: →

→

→2

F ( x , λ) ≡ [A0 − (λ − λ0 )A1 ] x +F2 x = 0, where F : (X × R → Y ) is a continuous operator,  2    − λ0 [F, ·] 0 [F, ·] 0 , , A1 = A0 = 0 0 0 2       −[w, f ] −[w1 , f2 ] − [w2 , f1 ] → wi →2 → → , 2F2 x 1 x 2 = , x i= , F2 x = +[w, w] fi 2[w1 , w2 ]

193

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

A0 , A1 ∈ L(X, Y ), i = 1, 2. Let the basis of formally self-adjoint boundary value problem 2 ϕ − λ0 [F, ϕ] = 0

in Ω

ϕ = ϕx = ϕy = 0

on Γ = ∂Ω

(11.14)

consist of n elements ϕ1 , . . . , ϕn selected such that    2 ϕi , ϕk  = 2 ϕi (x, y)ϕk (x, y)dxdy = λ0 δik , Ω

i, k = 1, . . . , n, →

operator. Then the elements ϕ i , i = 1, . . . , n and A0 is a Fredholm  → ϕi with ϕ i = 0 generate a basis in N (A0 ); moreover, →

→

A1 ϕ i , ϕ k  = δik ,

i, k = 1, . . . , n.

Therefore, BEq has the form (11.1) with ⎛ ⎞2 %n $  n   → → ξj ϕ j ⎠  f2 (ξ, 0) = ϕ i , F2 ⎝  j=1

≡0

i=1

because ⎛ ⎞2   n  0 → ξj ϕ j ⎠ = F2 ⎝ [U, U ] j=1

U=

Pn 1

, ξj ϕj

⎛ ⎞2 %n n n    → → → ϕ i , −F2 ΓF2 ⎝ ξj ϕ j ⎠ ξj ϕ j  ,  j=1 j=1

$ f3 (ξ, 0) = 2

→  → ˆ where ΓF2 ( nj=1 ξj ϕ j )2 = u is a solution of the problem ⎛ ⎞2 n  → ˆ = F2 ⎝ ξj ϕ j ⎠ A0 u → ˆ in X ∞−n , i.e., u =

2 u



0 Γ1 [U,U ]



i=1

j=1

, Γ1 [U, U ] is a solution of the problem

= [U, U ] in Ω, u = ux = uy = 0 on ∂Ω(Γ1 = Γ∗1 ). Therefore, ⎛ ⎞2   n n   −[Γ1 [U, U ], U ] → → . ξj ϕ j ⎠ ξj ϕ j = 2F2 ΓF2 ⎝ 0 j=1 j=1

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Finally, n n f3 (ξ, 0) = −ϕi , [−Γ1 [U, U ], U ]i=1 = +Γ1 [U, U ], [ϕi , U ]i=1  ∂U (ξ) n = , ∂ξi i=1 where 1 1 U (ξ) = [U, U ], Γ1 [U, U ] = 4 4

  Ω

[U, U ]Γ1 [U, U ]dxdy.

Hence, the problem (11.12)–(11.13) has two small solutions →

x≈

n 

→

ci ϕ i |λ − λ0 |1/2 ,

i=1

1/2  with ci = ± 4U 1(η0 )  ηi0 , η0 = (η10 , . . . , ηn0 ) is the conditional isolated  extremum of the potential U (η) on the sphere n1 ηi2 = 1 defined in a neighborhood of the point λ0 , in which (λ − λ0 )U (η0 ) > 0. Example 11.5. Let us consider the boundary problem A(λ)x ≡ x(4) + λx(2) + 4x = x2 , x(0) = x(π) = x (π) = 0. Here, λ0 = 5 is a Fredholm point of the √ operator A(λ), & 2/π sin t, ϕ2 = 1/ 2π sin 2t. The dim N (A(λ0 )) = 2, ϕ1 = 32 3 conclusions of Theorem 11.1 hold with l = n = 2, U = − 9(2π) 3/2 (η1 +

3/5η1 η22 ).

We obtain through the construction of the real solutions of equation (9.1) corresponding to nonisolated extremum of the potential n  1  fil (tξ1 , . . . , tξn )ξi dt. U (ξ) = i=1

0

We require more smoothness for the operator F (x, λ) because it will be necessary to achieve the terms of equation (9.1) with greater orders of slowness. Therefore, in this section, we furthermore assume in assumption III that the operators Fs (x, λ) are s-power in x of the order s and are differentiable in λ as many as we needed.

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

195

Lemma 11.4. Let U (η, g) = U (η) − g(η12 + η22 − 1) be a Lagrange function with  amn η1m η2n , U (η) = const(η12 + η22 )(l+1)/2 . U (η) = m+n=l+1

Then all stationary points of the function U (η, g) are isolated. Proof. Consider the system  mamn η1m−1 η2n = 2gη1 , m+n=l+1

 m+n=l+1

namn η1m η2n−1 = 2gη2 ,

η12 + η22 = 1. The stationary point (g∗ , η1∗ , η2∗ ) satisfies the equations   mamn mη1m−1 η2n+1 = namn η1m+1 η2n−1 = 2gη1 η2 , m+n=l+1

m+n=l+1

η12 + η22 = 1. Introduce the notation  amn [mη1m−1 η2n+1 − nη1m+1 η2n−1 ] = Φ(η1 , η2 ). m+n=l+1

Let (η1∗ , η2∗ , g∗ ) be a nonisolated stationary point. Then there exists an infinite sequence {η1i , η2i , gi } → (η1∗ , η2∗ , g∗ )

at i → ∞;

moreover, a set {η1i , η2i }, i = 1, 2, . . . generates the infinite zero set M of the function Φ(η1 , η2 ). The point (η1∗ , η2∗ ) is the condensation point of these zeros. If the set M is finite, then all sequences {η1i , η2i , gi } would also be finite because 2gi = (l + 1)U (η1i , η2i ). Let |η1∗ | = 1 = 1, then |η2∗ | = 0 = 1). Then |η1∗ | < 1, the function (if |η1∗ | & Φ(η1 , ± 1 − η12 ) is analytic in a circle |η1 | < 1 and it&has the infinite sequence of zeros {η1i } → η1∗ , i → ∞. Hence, Φ(η1 , ± 1 − η12 ) ≡ 0 in the set |η1 | < 1. Here, the following cases are possible: (1) If l = 2N is even, then a1,2N = a3,2N −2 = · · · = a2N +1,0 = 0, a2N,1 = a2N −2,3 = · · · = a0,2N +1 = 0, and U (η) ≡ 0.

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Toward General Theory of Differential-Operator and Kinetic Models

(2) If l = 2N + 1 is odd, then ai,l+1−i = 0,

i = 1, 3, 5, . . . , 2N + 1,

2a2,2N − (2N + 2)a0,2N +2 = 0, 4a4,2N −2 − 2N a2,2N = 0, .. . (2N + 2)a2N +2,0 − 2a2N −2 = 0, whence a0,2N +2 =

1 1·2 a2,2N = a4,2N −2 N +1 (N + 1)N

= ··· =

(N + 1)! a2N +2,0 , (N +!)!

a0,2N +2 = a2N +2,0 = c,

c = const,

a2,2N = a2N,2 = (N + 1)c, a4,2N −2 = a2N −2,4 =

(N + 1)N c, 2!

a6,2N −4 = a2N −4,6 =

(N + 1)N (N − 1) c, 3!

.. . Therefore, in the second case, U (η1 , η2 ) = c(η12 + η22 )N +1 , where N + 1 = (l + 1)/2. The lemma is proved.  Corollary 11.1. Let U (η1 , η2 ) be the uniform form of the odd order. Then all conditionally stationary points of the function U (η1 , η2 ) will be isolated on a circle. Theorem 11.2. Let the conditions of Theorem 11.1 hold, n = 2, and l even. Then every extremum η0 of the potential U (η) on a circle η12 + η22 = 1 such that U (η0 ) = 0 corresponds to the solution of

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

197

equation (9.1) of the form x=

(η10 ϕ1

+

η20 ϕ2

! + o(1))

λ − λ0 (l + 1)U (η0 )

"1/(l+1)

,

(11.15)

defined in a neighborhood of the point λ0 . Theorem 11.3. Let the conditions of Theorem 11.1 hold, n = 2, and l odd, {fil (η, 0)}2i=1 = const grad(η12 + η22 )(l+1)/2 . Then every extremum η0 of the potential U (η) on a circle η12 + η22 = 1 such that U (η0 ) = 0 corresponds to two solutions of equation (9.1) of the form    λ − λ0 1/(l−1) 0 0   , (11.16) x± = (±η1 ϕ1 ± η2 ϕ2 + o(1))  (l + 1)U (η0 )  defined in that neighborhood of the point λ0 , in which (λ − λ0 ) U (η0 ) > 0. Proof of Theorems 11.2 and 11.3 follows from Theorem 11.1 and Lemma 11.4 directly. The results similar to Theorems 11.2 and 11.3 have been obtained by Sather [135], but for more special classes of equations in Hilbert space and by means of a more complicated method. Theorem 11.4. Let the conditions of Theorem 11.1 be valid, n = 2, and l odd, fil (η, 0) = cηi (η12 + η22 )(l−1)/2 , {fi,l+1 (η, 0)}2i=1 = grad U1 (η), U1 (η) = 0, i = 1, 2, c = const. Then every extremum η0 of the potential U1 (η) corresponds to two solutions of equation (9.1) of the form    λ − λ0 1/(l−1) 0 0  , x± = (±η1 ϕ1 ± η2 ϕ2 + o(1))  c  defined in such a neighborhood of the point λ0 in which (λ−λ0 )c > 0.

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Toward General Theory of Differential-Operator and Kinetic Models

Proof.

In the given case, system (11.10) takes the form ! "1/(l−1) μ 2 2 (l−1)/2 2gηi = cηi (η1 + η2 ) + fi,l+1 (η, 0) + Θi1 (η, g, μ), 2g ' ( |Θi1 (η, g, μ)| = o(|μ|1/(l−1) ) , i = 1, 2. (11.17) η12 + η22 = 1, We set g = c/2 + g1 (μ), where g1 (μ) is not yet defined and consider (11.16) in a neighborhood of the point μ = 0 in which μc > 0. For finding η1 , η2 , g, one has the system 2g1 η = fl+1 (η, 0)a(μ) + Θ1 (η, g, μ),  ! "1/(l−1)  μ a(μ) = η12 + η22 = 1 . c + 2g1

(11.18)

If we put g1 (μ) = (μ/c)1/(l−1) g2 (μ) in (11.18), then we obtain the system 2g2 η = fl+1 (η, 0) + Θ2 (η, g2 , μ),

η12 + η22 = 1.

(11.19)

where Θ2 (η, g2 (μ)) = o(1),

at μ → 0,

the point g2 (0), η(0) satisfies the system 2g2 η = fl+1 (η, 0),

η12 + η22 = 1,

(11.20)

all solutions of which are isolated because l + 1 is an even number. Let (g20 , η0 ) be a solution of this system which is the extremum of the corresponding Lagrange function (now, it is not needed to have g20 = 0). Since l + 1 is an even number, then the extremum is isolated and system (11.19) has a real solution in a neighborhood of the point (g20 , η0 ), if |μ| < δ, δ being arbitrarily small. The theorem is  proved. Corollary 11.2. Let the conditions of Theorem 11.4 hold. Then equation (9.1) has four small continuous solutions x(λ) → 0 at λ → λ0 + 0(λ → λ0 − 0).

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Construction of Asymptotics in a Neighborhood of a Bifurcation Point

The proof is evident because the potential U1 (η) reaches its max and min. Now, let the conditions of Theorem 11.4 hold, but U1 (η) ≡ 0. Then having required the complementary smoothness of the operator F , we may continue our arguments. For example, let F (x, λ) be an analytic operator in a neighborhood of the point (0, λ0 ) and the righthand side of BEq (11.1) be a potential. Then the system (11.10) is transformed: ! " ∞ ∞   sign μ (s−l)/(l−1) s−l+σ(l−1) fsσ (η) ν (sign μ)σ , 2gη = 2g σ=0 s=l

η12 + η22 = 1, where the positive parameter η is introduced by the formula μ = (sign μ)ν l−1 , ) 1 * 1 2 fsσ (η)η1 + fsσ (η)η2 , fsσ (η) = grad Usσ (η), Usσ (η) = (s + 1) s = l, l + 1, . . . , σ = 0, 1, . . .. Putting g = g0 + g1 ν + · · · + gn ν n ,

g0 = 0,

(11.21)

(g0 μ > 0 for odd l) one can obtain the system 2g0 η − grad Ul+1 (η) + (2g1 η − grad Ul+2 (η))ν + (2g2 η − grad Ul+3 (η))ν 2 + · · · + (2gn η − grad Ul+n+1 (η) + o(1))ν n = 0, η12

(11.22)

η22

+ = 1. Here, Ul+i (η), i = 1, . . . , n, are certain algebraic polynomials of the order l + i Ul+1 = Ul,0 (η), Ul+2 = Ul+1,0 (η) +

 0, U2,1 (η),

l > 2, l = 2,

.. . Ul+i (tη) = tl+i Ul+i (η). Let us consider the following generalization of Lemma 11.4.

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Toward General Theory of Differential-Operator and Kinetic Models

Lemma 11.5. Let the system 2gηi =

∂U (η1 , η2 ) , ∂ηi

η12 + η22 = 1,

i = 1, 2

(11.23)

have a solution g∗ , η1∗ , η2∗ which is a nonisolated extremum of holomorphic function U (η1 , η2 ) on the circle η12 + η22 = 1. Then U = ϕ(η12 + η22 ) for η12 + η22 = 1. The proof of the lemma follows from the uniqueness theorem of analytic functions. Let the functions Ul+1 , Ul+2 , . . . , Ul+n−1 have nonisolated extremas on a circle. Then, by Lemma 11.5, Ul+j (η1 , η2 ) = ϕl+j (η12 + η22 ),

j = 1, . . . , n − 1.

Let gj =

l+j Ul+j (1), 2

j = 1, . . . , n − 1.

Then 2gj η − grad Ul+j ≡ 0 for η12 + η22 = 1 and system (11.22) reads 2gn η − grad Ul+n + o(1) = 0,

η12 + η22 = 1.

If η ∗ is a conditional isolated extremum of the function Ul+n on a circle, then " ! ' ( μ 1/(l−1) ∗ η +o |μ|1/(l−1) . η = η ∗ +o(1), gn = gn∗ +o(1), ξ = 2g0 Thus, one may always continue the computations until we do not obtain an isolated solution (gn∗ , η ∗ ). This is possible and is a singular case, when the process is continued with no limit. But then by Lemma 11.5, i  (η) ≡ 2ηi Usσ (η12 + η22 ), fsσ

and using (11.20), we arrive at the following equation for finding g: " ! ∞ ∞   sign μ (s−l)/(l−1)  Usσ (1) (sign μ)σ ν s−l+σ(l−1) . 2g = 2 2g s=l σ=0 (11.24)

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

201

 (1) = 0 (otherwise, f (η) ≡ 0), then (11.24) has a solution Since Ul0 l0  (1) + o(ν) with g = Ul0  (1) = Ul0

l+1 Ul0 (1). 2

For that g, equations (11.20) are satisfied identically in a degenerate case at any point of a circle η12 + η22 = 1. Hence, if BEq (11.1) is potential and n = 2, then equation (9.1) always has a continuous branch of solutions of the form x = (c1 ϕ1 + c2 ϕ2 + o(1))|λ − λ0 |1/(l−1) , where ⎧! " ⎪ sign(λ − λ0 ) l/(l−1) ∗ ⎪ ⎪ ηi , ⎨ (l + 1)Ul+1 (η ∗ ) ci = 1/(l−1)  ⎪   1 ⎪ ⎪   ⎩±  ηi∗ , (λ − λ0 )U (η ∗ ) > 0, ∗ (l + 1)Ul+1 (η ) 

(11.25)

l is even, l is odd,

where η ∗ is the point of conditional isolated extremum of the function Ul+n on a circle η12 + η22 = 1 if previous functions Ul+1 , . . . , Ul+n−1 have the conditional nonisolated extremum on this circle; η ∗ is arbitrary point of a circle η12 + η22 = 1 if i (η)}2i=1 = grad Usσ (η), {fsσ

Usσ (η) = ϕsσ (η12 + η22 )

for all s, σ. It follows from the following statement. Theorem 11.5. Let F (x, λ) be the analytic operator in a neighborhood Ω of the point (0, λ0 ), the root number of the operator Fx (0, λ) be equal to two at the point λ0 , and BSEq (11.1) be potential. Then equation (9.1) has a small continuous solution of the form (11.25). Example 11.6. 1 A(λ)x ≡ x(t) − π

 0

2π

(sin t sin s + cos t cos s)x(s) ds − λx(t) = x2 (t).

Here, λ0 = 0 is a two-multiple eigenvalue, k = 2, ϕ1 = sin t, ϕ2 = cos t, the conditions of Theorem 11.5 hold; moreover, a singular

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Toward General Theory of Differential-Operator and Kinetic Models

case occurs: there exists a continuous solution (λ < 0) " ! 2 1/2 x(t) = (c1 sin t + c2 cos t + o(1)) − λ , 3 where c1 and, c2 are arbitrary constants, c21 + c22 = 1. Theorem 11.6. Let the assumptions I–III hold, the root number of the operator Fx (0, λ) be equal to two at the point λ0 , and BSEq (11.1) be potential. Then λ0 is a bifurcation point of equation (9.1). Proof. Owing to Theorem 11.1 and Lemma 11.4, it is enough to consider a case, when BSEq (11.1) has the form μξi =

∂(Ul+1 (ξ, μ) + R(ξ, μ)) , ∂ξi

i = 1, 2,

ξ = (ξ1 , ξ2 ),

l is odd, where Ul+1 (ξ, μ)|μ=0 = c(ξ12 + ξ22 )(l+1)/2 ,

|R(ξ, μ)| = o[(ξ12 + ξ22 )(l+1)/2 ],

where c is a constant. For definiteness, let c > 0, μ > 0 (when c < 0 is considered similarly, only μ need to be considered negative). Introduce the function μ Φμ (ξ) = (ξ12 + ξ22 ) − cUl+1 (ξ, μ) − R(ξ, μ). 2 It is easy to show, for any > 0, δ > 0, that there are ξ1∗ , ξ2∗ , μ such that 0 < (ξ1∗ )2 + (ξ2∗ )2 < 2 , 0 < μ∗ < δ, dΦμ∗ (ξ1∗ , ξ2∗ ) = 0. Indeed, there exists a sufficiently small δ1 ∈ (0, δ) such that for μ∗ = δ1 there exist the positive numbers 1 , 2 , 3 (0 < 1 < 2 < 3 ≤ ) for which the function Φμ∗ will be positive, monotone increasing in ξ1 , ξ2 in a domain S1 = (ξ1 , ξ2 |0 < ξ12 + ξ22 ≤ 21 ) and negative, monotone decreasing in ξ1 , ξ2 in a domain S2 = (ξ1 , ξ2 | 22 < ξ12 + ξ22 < 23 ). Introduce another domain S3 = (ξ1 , ξ2 | 21 < ξ12 + ξ22 < 22 ). Denote by h the minimal value of the function Φμ∗ (ξ) on a circle ξ12 + ξ22 = 21 . Introduce a number ρ = qh, where q is any fixed number from (0,1). Then there is a closed curve l1 in S1 in which there is the point (0, 0) such that Φμ∗ (ξ)|ξ∈l1 = qh. Otherwise, there would be a path connecting the point (0, 0) with the boundary of the domain

Construction of Asymptotics in a Neighborhood of a Bifurcation Point

203

S1 in one point of which the continuous function does not accept an intermediate value. It is impossible by the connection principle. For the same reason, in a domain S3 , there exists a closed curve l2 in which the domain S1 is situated such that Φμ∗ (ξ)|ξ∈l2 = qh. But then there exists a closed domain D limited by curves l1 and l2 on the boundary of which Φμ∗ (ξ)|Γ=l1 ∪l2 = const; ¯ D. Hence, by Rolle’s theorem, there is the point moreover, (0, 0) ∈ ∗  ξ ∈ D, in which dΦμ∗ (ξ ∗ ) = 0. The theorem is proved. Remark 11.1. If the conditions of Theorem 11.1 hold, where U (η) is a positive definite (negative definite) function, then λ0 will be a bifurcation point of equation (9.1). The proof is similar. Example 11.7. In the Karman problem (11.12)–(11.13), the potential U (η) is a uniform form of definite sign of the fourth order. Therefore, the eigenvalues of the linearized problem (11.14) are the bifurcation points of a bifurcation problem (11.12)–(11.13).

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Chapter 12

Regularization of Computation of Solutions in a Branch Point Neighborhood In order to tackle the real-world problems using the branching theory, one should combine the analytical and numerical methods. Here, we focus on the nonlinear equation F (x, λ) = 0

(12.1)

in a neighborhood of the branch point λ0 . Convergence of the corresponding methods have been proved on the assumption that the equation is precisely given and all computations are precisely developed. However, the computation method is characterized by various errors: (1) input data error; (2) round-off errors; (3) errors caused by the numerical methods. Thus, the problem of nonlinear equation solutions in a neighborhood of the branch point is a typical ill-posed problem: a small error in the initial data can result in much larger errors in the answers. Therefore, we are interested in those methods of construction of approximate solutions which give uniform convergence in λ. In order to construct the efficient analytical–numerical methods, one should take into account all the information concerning the given problem: solution’s nonuniqueness and branching, presence of

205

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Toward General Theory of Differential-Operator and Kinetic Models

small parameters, and requirement of uniform approximation with respect to a parameter. These problems appear in mathematical modeling when it is necessary to construct the real-valued branching solutions. 12.1.

Construction of the Regularization Equation in the Problem at a Branch Point

Let X, Y be real Banach spaces, λ a small real parameter, and F a nonlinear mapping X × R into Y , which is strong continuously differentiable with respect to x in a neighborhood Ω = (x, λ|x < r, |λ| < ρ) of the point x = 0, λ = 0 such that F (0, λ) = 0 for all x ∈ (−ρ, +ρ). It is assumed that the operator Fx (0, 0) is Fredholm. Let us consider equation (12.1). In this section, we give the construction method for the regularization equation (RE) in the problem at a branch point. Let us introduce the following assumptions: A. Let the assumptions I−III hold in a neighborhood Ω = {||x|| < r, ||λ|| < ρ}, q = 1, m = l, λ0 = 0, F ≡ (A0 − A1 λ − R(λ))x + Fl (λ)xl + R(x, λ), R(λ) = O(λ2 ),

R(x, λ) = O(xl+1 ),

R(x, λ) − R(y, λ) ≤ c(r)x − y, c(r) = O(r l ). B. Let δ be a parameter characterizing the absolute error of calculations and the approximate equation read ˜ λ) + w = 0. ˜ F˜ (x, λ) ≡ A(λ)x + F˜l (λ)xl + R(x,

(12.2)

Here, w < cδ. We consider the exact equation (12.1) to be unknown and we need to give the construction method of the function x ˜(λ) such that ˜ x(λ) − x(λ) → 0 at δ → 0 uniformly with respect to λ = 0 in a small neighborhood of the point λ = 0, x(λ) is desired small continuous solution of equation (12.1)).

Regularization of Computation of Solutions in a Branch Point Neighborhood

207

Let x : I → X with I = [0, ρ) (I = (−ρ, 0]) is a branch of solution of equation (12.1). Definition 12.1. We call the equation ˜ α (x, λ) = 0 R

(12.3)

the RE of this branch, where α = α(δ) is auxiliary parameter depending on error δ, if for any  > 0, there exist positive numbers δ0 > 0 and ρ0 > 0 such that equation (12.3) has a real solution x(λ) − x(λ) ≤  for any δ ≤ δ0 , λ ∈ [0, ρ0 ] ∩ [0, ρ) x ˜(λ), supλ ˜ (λ ∈ [−ρ0 , 0] ∩ (−ρ, 0]). If x = x0 (λ) + o(x0 ),

x ˜=x ˜0 (λ) + u ˜(λ),

and ˜ x0 (λ) − x0 (λ) ≤ ,

˜ u(λ) ≤ c(x0 ) + ,

c(x0 ) = o(x0 ), then (12.3) is the RE of the principal term x0 (λ) of asymptotics of this branch. It is clear that, if equation (12.3) is the RE of the branch on the whole, then it is also the RE of the principal term of this branch also. For the detailed description of the analytical methods for the construction of RE (12.3) based on the perturbation method and the bifurcation parameter shift method, readers may refer to [154]. Let us consider the simple solution construction for nonlinear equations in Section 12.2. 12.2.

Definition and Properties of Simple Solutions

Let X, Y be Banach spaces. Consider equation (12.1) where now the operator F : X × R → Y is defined for x, λ ∈ Ω = {x, λ| x ≤ r, |λ| ≤ ρ} and strongly continuously differentiable in x, F (0, 0) = 0. The operator Fx (0, 0) in general is not necessary Fredholm, it has no

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bounded inverse Fx (x, λ) − Fx (y, λ) ≤ L(λ)x − yγ ,

∀x, y, λ ∈ Ω, 0 < γ ≤ 1. (12.4)

Definition 12.2. Let I be a neighborhood (one-sided neighborhood) of the point λ = 0, I ∈ (−ρ, +ρ). We shall call a continuous function x : I → X, x(0) = 0 a simple solution of equation (12.1) if: (a) F (x(λ), λ) = 0 for λ ∈ I; (b) there exist numbers c > 0, l ≤ N such that Fx−1 (xN , λ) ≤ c|λ|−l/s ,

λ ∈ I/0,

(12.5)

 where s ≥ 1 is an integer number, xN = n1 ai λi/s , ai ∈ X is the asymptotic of function x of the N order |λ|−N/s x(λ) − xN (λ) → 0,

λ → 0.

If λ = 0 is a regular point, then in the definition, l = 0 and there exists the bounded inverse operator Fx−1 (0, 0). If λ = 0 is a singular point, then l ≥ 1. Note that if F is an analytic operator in a neighborhood Ω, then (12.5) is satisfied if and only if  −1    l    i/s  = O(|λ|−l/s ),  Ai λ     i=0 where Ai , i = decomposition

0, . . . , l, is the bounded operators from the

Fx

 l  i=1

 ai λ

i/s

,λ

=

∞ 

Ai λi/s .

i=0

If F (x, λ) is an analytic operator and the operator Fx (0, 0) is Fredholm, then condition (b) is equivalent to the other, in practice,

Regularization of Computation of Solutions in a Branch Point Neighborhood

209

 i/s be a small solution more suitable criteria. Indeed, let x = ∞ i=1 ai λ of equation (12.1) and ξ = ξ(λ) a corresponding solution of BSEq f (ξ, λ) = 0,

(12.6)

where f = (f1 , . . . , fn ), fi = ψi , R(x, λ), R(x, λ) = Fx (0, 0)x − F (x, λ), Fx (x(λ), λ)|λ=μs = A(μ),

n = dim N (Fx (0, 0)).

Then, the following theorem is valid. Theorem 12.1 ([154]). For the operator A(μ), the following four conditions are equivalent: (1) operator A(μ) has a continuous inverse at 0 < |μ| < , where  is sufficiently small; (2) there exists a complete GJS of operator A(μ) corresponding to the point μ = 0; ∂fi = a(μ) = 0 at 0 < |μ| <  ; moreover, the (3) det ∂ξ j i,j=1,...,n, λ=μs point μ = 0 is the zero of the k multiplicity of function a(μ), k is a root number of operator A(μ); (4) solution x(λ) is simple and there exists its asymptotics xN = N i/s i=1 ai λ , satisfying the estimation (12.5) for l = N = k. Corollary 12.1. Let the operator Fx (0, 0) be Fredholm and BEq (12.6) of problem (12.1) have a solution ξ=

∞ 

ξi λi/s ;

i=1

moreover,



∂fi det ∂ξj

n i,j=1

= λk/s (a + o(1)),

a = 0.

(12.7)

Then equation (12.1) has a simple solution satisfying the estimation (12.5) for N = l = k. Lemma 12.1. (1) Let there exist a polynomial xN (λ) = estimation (12.5) holds.

N

i/s i=1 ai λ

for which

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210

(2) Let Fx−1 (xN , λ)F (xN , λ) = o(|λ|l/s ). (3) Let in the estimation (12.4), L(λ) = o(|λ|(1−γ)/sl ). Then equation (12.1) has a simple solution x(λ) =

N 

ai λi/s + o(|λ|N/s ).

i=1

The proof follows from the Newton–Kantorovich theorem. Theorem 12.2 ([154]). Let equation (12.1) have a simple solution x∗ (λ), in the estimation (12.4), L(λ) = o(|λ|(1−γ)/sl ). Then the successive approximations xn+1 = xn − Fx−1 (xn , λ)F (xn , λ),

(12.8)

n = 0, 1, 2, . . . , x0 = xN (λ), 0 < |λ| < , converge to the solution x∗ (λ) for sufficiently small . Similar result occurs for the sequence of modified Newton method xn+1 = xn − Fx−1 (x0 , λ)F (xn , λ). 12.3.

Construction of Regularization Equation of Simple Solutions

Let equation (12.1) have a simple solution x : I → X, x(0) = 0. Assume that the operator F is defined with a certain error, i.e., F˜ (x, λ) − F (x, λ) ≤ δ,

F˜x (x, λ) − Fx (x, λ) ≤ δ

(12.9)

for all x, λ ∈ Ω, δ is a maximum absolute calculation error. It is needed to calculate a solution x : I → X, (x(0) = 0) of equation (12.1) by the approximate equation F˜ (x, λ) = 0.

(12.10)

It is easy to choose the prime construction method of RE for the simple solutions of equation (12.1) by the approximate solution (12.10).

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211

Lemma 12.2. Let the estimates (12.4) and (12.9) be valid, where L(λ) = o(|λ|(1−γ)l/s ), and equation (12.1) have a simple solution x : I → X, satisfying the estimate (12.5). Let α0 (δ) and β(δ) be some positive continuous functions which monotonically tend to zero at δ → 0 such that  s/(2l) δ ≤ a ≤ α0 (δ), μ = λ + α ∈ I, α = (sign λ)a. β(δ) (12.11) Then F˜x−1 (x(μ), μ) ≤ c2 |μ|−l/s ,

c2 = const,

F˜x (x, μ) − F˜x (y, μ) ≤ 2δ + c1 |μ|

(1−γ)l/s

(12.12) γ

x − y ,

c1 = const (12.13)

for all x, y, μ ∈ Ω. The proof of inequality (12.13) uses the theorem on inverse operator. Inequality (12.12) directly follows from estimates (12.9) and (12.4). It is to be noted that if conditions of Lemma 12.1 are fulfilled, then equation F˜ (˜ x, λ + α) = 0

(12.14)

is the RE of a simple solution x : I → X of equation (12.1) defined in Lemma 12.1.

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Chapter 13

Iteration Methods, Analytical Initial Approximations, Interlaced Equations 13.1.

Iterations and Uniformization of Branching Solutions

The Newton diagrams and polyhedrons are among the key tools in the branching theory. The polynomials of various types can be employed to construct the principal part of BEq at the first iteration. In this section, a construction method of initial approximation and a selection of a parameter of uniformization are suggested. The method of the solution branch uniformization uses the parametric families of solutions and takes into account the computational errors in the neighborhood of the branch points. As readers can note, in the previous chapters, we employed the auxiliary (regularized) equation. The solution of such a regularized equation is uniformly approximate to the exact solution with respect to a parameter. In addition to these ideas, the modification of a uniformization parameter is permitted. For example, one may use any of the coefficients of projection P x of the unknown solution. This allows the possibility to “go around” some specific types of branch points and to extend the convergence domain of the method. We discuss this here more in detail. Let E1 and E2 be Banach spaces. Consider the equation def

F (x, λ) = Bx − R(x, λ) = 0, 213

(13.1)

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Toward General Theory of Differential-Operator and Kinetic Models

where B : D(B) ⊂ E1 → E2 is a closed Fredholm operator with a dense domain in E1 , dimN (B) = n ≥ 1, and the operator  Rik (x)λk R(x, λ) = R01 λ + i+k≥2

is analytic in a neighborhood of the point x = 0, λ = 0. It is required to build a solution x → 0 at λ → 0. 13.2.

Branching Equation and the Selection of Initial Approximation

Let {ϕ}n1 be a basis in N (B), {ψi }n1 is a basis in N (B ∗ ), and {γi }n1 , and {zi }n1 are, respectively, related biorthogonal systems from E1∗ and E2 . −1  n  ·, γi zi , Γ= B+ 1

P =

n 

·, γi ϕi ,

n  Q= ·, ψi zi .

1

1

The bounded operator Γ is called the Schmidt pseudo-resolvent of operator B. Setting in (13.1) x = ξϕ + Γy, with y ∈ E2∞−n , ξϕ = and y:

n 1

(13.2)

ξi ϕi , we obtain the system for finding ξ

y = R(ξϕ + Γy, λ), y, ψi  = 0,

i = 1, . . . , n.

(13.3) (13.4)

Owing to the theorem on implicit operator, equation (13.3) has a unique small solution   ym0 (ξϕ) + ymn (ξϕ)λn . (13.5) y= m≥2

m≥0 n≥1

Iteration Methods, Analytical Initial Approximations, Interlaced Equations

215

We can compute coefficients ymn by the following recurrence formulas: y20 = R20 (ξϕ), yn0

 n−1    1 dn  i  = R y μ , n = 3, 4, . . . , Γ i0 i0  n! dμn μ=0 i≥2

i=1

y01 = R01 , y0n

 n−1     1 dn  s =7 Rik Γ y0s λ λk  , n = 2, 3, 4, . . . , n n! dλ λ=0 s=1 i+k≥2

yn−j,j =

∂n 1 (n − j)!j! ∂λj ∂μn−j ⎛ n−1   k ⎝ × λ Rik Γ yik μi λk i+k≥2

+

j−1  s=0

i+k≥1

  Γyn−s,sμn−s λs 

λ=0,μ=0

,

n = 2, 3, 4, . . . , j = 1, . . . , n − 1, def

y10 = ξi zi . A sequence yn = R(ξϕ + Γyn−1 , λ),

n = 1, 2, . . .

with y0 = 0 converges to the solution (13.5). Owing to the analyticity of the operator F (x, λ),  n yik (ξ)λk , yn = i,k

it is quite interesting to explain how many “correct” coefficients are n =y . there in iteration yn such that yik ik Let sup R(x, λ) = {i, k| Ri,k (x) = 0} be a support for the related series.

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Toward General Theory of Differential-Operator and Kinetic Models

Lemma 13.1. Let si + qk = Θ be an equation of the supporting line n = y to sup R(x, λ). Then yik ik for i, k ∈ wn , where wn = {i, k| si + qk ≤ θ + (n − 1)(θ − s)}. Proof. Let y(ξ, λ) be a small solution of (13.3). Then the following identity holds: y = R(ξϕ, λ) + Rx (ξϕ, λ)Γy + w(y, ξ, λ),

(13.6)

where w = R(ξϕ + Γy, λ) − R(ξϕ, λ) − Rx (ξϕ, λ)Γy. Introduce a sequence yn = R(ξϕ, λ) + Rx (ξϕ, λ)Γyn−1 + w(yn−1 , ξ, λ),

(13.7)

where n = 1, 2, . . . , y0 = 0, yn → y(ξ, λ) at n → ∞. We recall that  R(x, λ) = si+qk≥Θ Rik (x)λk . Therefore, subtracting from (13.6) the corresponding parts of (13.7), we obtain the following inequality: y − yn  ≤ a(ξ, λ)Γy − yn−1  + b(ξ, λ)y − yn−1 , where

⎛ a(ξ, λ) = O ⎝



⎞ |ξ|i |λ|k ⎠ ,

si+qk=Θ−s

⎛ b(ξ, λ) = O ⎝



(13.8)

⎞ |ξ|i |λ|k ⎠ .

si+qk=Θ−s

 It is evident that y −y1  = o( si+qk=Θ |ξ|i |λ|k ). Now, by induction, method, we establish the correctness of lemma. Indeed, let ⎛ ⎞  |ξ|i |λ|k ⎠ y − yn  = o ⎝ si+kq=Θ+(n−1)(Θ−s)

in a neighborhood of the point ξ = 0, λ = 0. Then by (13.8), a similar  estimation holds for y − yn+1 .

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Iteration Methods, Analytical Initial Approximations, Interlaced Equations

13.2.1.

Geometric interpretation of Lemma 13.1

n = y } of the Let us introduce the domain Dn = {i, k| yik ik “correct” coefficients in a decomposition of n’s iteration yn (ξ, λ). As the supporting lines sup R(x, λ) in Lemma 13.1, we take the sides si+qk = θ of the Newton diagram H1 of operator R(x, λ). Construct the lines si + qk = θ + (n − 1)(θ − s). The diagram Hn corresponds to these lines. Thus, by Lemma 13.1, all integer indices (i, k), which are situated below Hn , enter the domain Dn . By construction of a solution (13.5) and by its substitution into (13.4), we obtain the following BEq:  j  def Lj (ξ, λ) = Lm0 (ξ) + Ljmν (ξ)λν = 0, j = 1, . . . , n, m≥2

m≥0 ν≥1

(13.9) where Ljmν = ymν (ξϕ), ψj  =

 m1 +···+mn =m

Ljm1 ,...,mn ,ν ξ1m1 . . . ξnmn .

Let L(ξ1 , . . . , ξn+1 ) =

 i

i

n+1 Li ξ i , ξ1i1 . . . ξn+1

one of the left-hand sides of (13.9). For symmetry in notations, we put λ = ξn+1 . Without loss of generality, we consider L(ξ1 , . . . , 0i , . . . , ξn+1 ) = 0 for i = 1, . . . , n + 1. Otherwise, we can reduce the corresponding equation in (13.9) by a certain degree of ξi . Let sup L = {i | i ∈ N+n+1 , Li = 0}, where N+ is a set of integer positive numbers. n+1 of the points from sup L is considered to A convex hull in R+ be Newton polyhedron of the function L and we denote this by HL . Definition 13.1. The hyperplane n+1 | (ξ, α) = Θ, α ∈ N+n+1 , Θ ∈ N+ l : ξ ∈ R+

218

Toward General Theory of Differential-Operator and Kinetic Models

is considered to be supporting for sup L if (1) (ξ, α) ≥ Θ for all ξ ∈ sup L, (2) l ∩ sup L = 0. A nonempty set G = HL ∩ l is considered to be a face of polyhedron HL , where l is supporting hyperplane of sup L. Condition (a). In N+ , the numbers α1 , . . . , αn+1 , Θ1 , . . . , Θn are n+1 | (ξ, α) = Θj }, known such that the parallel hyperplanes lj : {ξ ∈ R+ j = 1, . . . , n, are considered to be supporting for sup Lj , j = 1, . . . , n. If a polyhedron HL1 possesses a face (ξ, α) = Θ1 , and other ones HLi , i = 2, . . . , n have parallel faces, possibly smaller dimensions, then Condition (a) will be satisfied. By analyticity, Condition (a) means for ξi = αi ηi , i = 1, . . . , n + 1, Lj = Θj lj (η1 , . . . , ηn+1 ) + rj (η, ), where lj =



Lji η i = 0,

(13.10)

rj = o( Θj ), j = 1, . . . , n.

(i,α)=Θj

A vector function l(η) is considered to be the principal part of BEq for every pair α, Θ. For the construction of α, we need to find the Newton polyhedrons HLi , i = 1, . . . , n, faces, and the corresponding normal cones. The required α lies in the intersection of these normal cones. Here, readers may refer to the monographs by Bruno [28]. If α1 = · · · = αm , αm+1 = · · · = αn+1 , the hyperplanes lj are symmetric with respect to the axes ξ1 , . . . , ξm and ξm+1 , . . . , ξn+1 . In this symmetric situation, vectors α and Θ are easily found by a method of Newton diagrams. In fact, we rewrite (13.9) in the form  j Lik (u, v) = 0. (13.11) i,k

Here, u = (ξ1 , . . . , ξm ), v = (ξm+1 , . . . , ξn+1 ), Ljik are i-uniform forms in u and k-uniform ones in v. Consider the sets of points Qj = {|i|, |k| ∈ N+2 , Ljik = 0},

j = 1, . . . , n.

Iteration Methods, Analytical Initial Approximations, Interlaced Equations

219

For every set Qj , we construct its own Newton diagram Dj . As analytic meaning of Condition (a), see (13.10), the following claim holds. Lemma 13.2. Let the diagrams D1 , . . . , Dl have parallel faces belonging to the lines lj : p |i| + q|k| = Θj , j = 1, . . . , l,

l ≤ n,

(13.12)

with {p, q, Θj } ∈ N+ . If l < n, we take the numbers Θl+1 , . . . , Θn in N+ such that the lines of (13.12) for j = l + 1, . . . , n considered to be supporting for the sets Ql+1 , . . . , Qn . Then Condition (a) will be fulfilled for α1 = · · · = αm = p, αm+1 = · · · = αn+1 = q. Proof of the lemma directly follows from the way of construction of diagrams Dj . Remark 13.1. In Lemma 13.2, a case l < n is assumed for n ≥ 3. If l < n, the diagrams Dl+1 , . . . , Dn have no faces parallel to the lines (13.12). Nevertheless, one may draw the lines parallel to (13.12) through the vertices of any of these diagrams by its convexity without intersection with the diagram. Since there is a finite number of vertices, one has a finite number of unknown values Θl+1 , . . . , Θn in Lemma 13.2. Condition (b). The system lj (η1 , . . . , ηn+1 ) = 0,

j = 1, . . . , n,

(13.13)

0 ): (see (13.10)) has a solution η 0 = (η10 , . . . , ηn+1   def  ∂lj   A(η) =   ∂ηi  , j = 1, . . . , n, i = {1, . . . , n + 1}.

Moreover, det A(η0 ) = 0. A solution η 0 = 0 is considered to be a solution of a total rank of (13.13). Let Conditions (a) and (b) hold. Then by the theorem on implicit function, BEq (13.9) has a solution ξi = αi (ηi0 + o(1)),

i = 1, . . . , n + 1,

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Toward General Theory of Differential-Operator and Kinetic Models

where ξn+1 = λ( ) and ξ∗ = α∗ η∗0 . By substitution of this solution into (13.2) taking account of (13.5), we obtain a desired solution x( ) → 0, λ( ) → 0 at → 0 of (13.1). 0 ) will be used for the A vector ξ 0 = ( α1 η10 , . . . , αn+1 ηn+1 construction of initial approximation in an N -step iteration method.

Chapter 14

Iterative Methods Using Newton Diagrams The sets Mj = {i, k | Rik (x), ψj  = 0}, j = 1, . . . , n, are to be called the support of coefficients of projection QF. For a selection of the integer vectors α, θ in Conditions (a) and (b), we may use the parallel supporting lines of the sets Mj , j = 1, . . . , n. In particular, for simplicity of construction, it is convenient to use the parallel intervals of Newton diagrams of these sets. Let us introduce the following condition: Condition (A). Let the support of coefficients of projection QF have the parallel supporting lines si + qk = Θj , j = 1, . . . , n, Θj > s and let R0i = 0, i = 1, . . . , [s/q] − 1, QR0s/q = 0 for s/q ∈ N+ . Note that we may give analytic form to Condition (A): Condition (A). For all i, k ∈ N+ , such that si + qk < θj , the identities Rik (x), ψj  = 0 hold; among the integer pairs (i, k) such that si + qk = θj for every j ∈ (1, . . . , n), there exists a point (i1 , k1 ) for which Ri1 k1 (x), ψj  = 0. Remark 14.1. If in Condition (A ) for every j, there exist two points (i1 , k1 ), (i2 , k2 ) for which Ri1 k1 (x), ψj  = 0, Ri2 k2 (x), ψj  = 0, then the supporting lines pass through the parallel intervals of Newton diagrams of coefficients of projection QF. Lemma 14.1. Assume Condition (A). Then BEq corresponds to equation (13.1)  def λk Rik (ξϕ + x ˆ0 (λ)), ψj  + ρj (ξ, λ) = 0, (14.1) Lj (ξ, λ) = si+qk=θj 221

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Toward General Theory of Differential-Operator and Kinetic Models

where



0,

s/q = N+ ,

ΓR0s/q λs/q ,

s/q ∈ N+ ,

0

x ˆ (λ) =

(14.2)

lim→0 −θj ρj ( s η, q ηn+1 ) = 0, j = 1, . . . , n. Here, the hyperplanes s(ξ1 + · · · + ξn ) + qξn+1 = Θj , j = 1, . . . , n satisfy Condition (a), and the corresponding principal part of BEq has the form  def k Rik (ηϕ + x ˆ0 (ηn+1 )), ψj ηn+1 = 0, j = 1, . . . , n. lj = si+qk=θj

(14.3) Remark 14.2. If all terms of (14.1) have common multipliers, we must cancel BEq by these ones. Example 14.1. Let R(x, λ) =

∞ 

R0k λk +

Rik (x)λk ,

i=1 k=m

k=1

QR0k = 0,

∞  ∞ 

k = 1, . . . , m,

∗ ψj = 0, R1m

m > 1,

j = 1, . . . , n.

Then a line i + k = m + 1 satisfies Condition (A), x ˆ0 = ΓR01 λ. BEq (14.1) reads as follows: def

Lj = λm (R1m (ξϕ + ΓR01 λ) + R0,m+1 λ, ψj  + o(|ξ| + |λ|)) = 0, where j = 1, . . . , n. Its principal part is quasi-linear, i.e., it has the def m Ξη = 0, where Ξ = aik i=1,...,n;k=1,...,n+1 , form l(η) = ηn+1  R1m ϕk , ψi , k, i = 1, . . . , n, aik = R1m ΓR01 + R0,m+1 , ψi , i = 1, . . . , n, k = n + 1. Let rank Ξ = n, A=



1, . . . , n



1, . . . , ∗ − 1, ∗ + 1, . . . , n + 1

be a rank minor of matrix Ξ. If ∗ = n + 1, Condition (b) will be satisfied for the corresponding principal part of BEq. Following

Iterative Methods Using Newton Diagrams

223

Remark 14.2, we cancel BEq by λm . Now, Condition (b) will be fulfilled for the principal part of BSEq λ−m Lj (ξ, λ) = 0, j = 1, . . . , n for all ∗ ∈ {1, . . . , n + 1}. Thus, BEq has a small solution ξ∗ = , ξi = ηi0 + o( ), i = {1, . . . , n + 1}\∗ where η 0 is a unique solution of the linear system Ξη = 0, η∗ = 1. Hence, (13.1) has a solution: (a) in the case ∗ = n + 1, η∗0 = 1, x=

n  i=1

0 ηi0 ϕi + ΓR01 ηn+1 + o( ),

0 + o( ); λ = ηn+1

(b) in the case ∗ = n + 1, x=λ

n  1

ηi0 ϕi + λΓR01 + o(λ).

Remark 14.3. In the more general case in which  ∞ ∞  ∞   k k R0k λ + Rik (x)λ f (x, λ), R(x, λ) = k=1

i=1 k=m

where f : E1 × R1 → R1 is an analytic functional, QR0k = 0, ∗ ψ = 0, j = 1, . . . , n, BEq also has a general k = 1, . . . , m, R1m j multiplier, the principal part of BEq appears to be quasi-linear again and the results become similar. The relation between Conditions (a) and (A): By Lemma 14.1, Condition (A) suffices for the fulfillment of Condition (a). It is clear that Condition (A) is not necessary for the fulfillment of Condition (a) and it is harder to demand. But Condition (A) gives a chance to write out the principal part of BEq easily and apply a one-step iteration method considered in Section 14.1.

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Toward General Theory of Differential-Operator and Kinetic Models

14.1.

One-Step Iteration Method

Let Conditions (A) and (b) hold, when lj are determined in (14.3). We consider a solution of (13.1) in the form x=

n 

ξi ϕi + x ˆ0 (λ( )) + Γy s ,

(14.4)

i=1 def

λ = ηn+1 q = ξn+1 ( ), where

 ξi = ηi ( )

s ,

(14.5)

i = 1, . . . , n,

q , i = n + 1,

where η0 is a solution of a total rank of (14.3), η∗ ( ) = η∗0 , ∗ ∈ (1, . . . , n + 1), η(0) = η0 . Unknowns y( ) and ηi ( ), i = 1, . . . , n + 1, are continuous at zero and satisfy the system ˆ0 + Γy s , ξn+1 ) − B x ˆ0 ), y = −s (R(ξϕ + x ˆ0 + Γy s , ξn+1 ), ψj  = 0, j = 1, . . . , n −θj R(ξϕ + x for η∗ = η∗0 . We consider this system as the one-operator equation K(u, ) = 0,

(14.6)

where K : Y × R1 → Y, Y = E2 ⊕ Rn , u = (y, η1 , . . . , η∗+1 , . . . , ηn+1 ),   − → η

0 , η0 , . . . , η0 K(u0 , 0) = 0 for u0 = (0, η10 , . . . , η∗−1 ∗+1 n+1 ). In a ball s(u0 , r), operator K is diffrentiable in u   I 0 , Ku (u0 , 0) = ˆ A ·, Ψ

where ˆ n ) , ˆ = (·, Ψ ˆ 1 , . . . , ·, Ψ ·, Ψ  ∗ 0 0 0k Rik (η ϕ + x ˆ0 (ηn+1 ))ηn+1 ψj , ψˆj = Γ∗ si+qk=Θj

225

Iterative Methods Using Newton Diagrams



∂lj (η 0 )



A=

∂ηi j=1,...,n, i={1,...,n+1}\∗ det A = 0. Ku (u0 , 0) has the bounded inverse ⎛ Ku−1 (u0 , 0) = ⎝

I,

0

ˆˆ ·, Ψ, A−1

⎞ ⎠,

where ˆ ˆ ·, Ψ ˆ ∈ L(Y → Rn ), I : Y → Y, 0 : Rn → Y. ˆ = −A−1 Ψ, Ψ Thus, equation (14.6) satisfies the theorem on implicit operator. One can find a desired solution u = (y, η) by the method of successive approximations um = um−1 − Ku−1 (u0 , 0)K(um−1 , ).

(14.7)

From (14.7), we obtain the iteration formulas m−1 m−1 m−1 ˆ0 (ξn+1 ) + s Γym−1 , ξn+1 ) − Bx ˆ0 (ξn+1 )), ym = −s (R(ξ m−1 ϕ + x

(14.8) ˆˆ → − → η m−1 − ym−1 − ym , Ψ − A−1  s ym , E( )Ψ, ηm=−

(14.9)

where E( ) =  θj δij i,j=1,...,n , m = 1, 2, . . .. For a sequence um , there is one  ˆ0 (λm ), xm = ξ m ϕ + Γym s + x (14.10) m λm = ξn+1 for ξ∗m = ξ∗0 , m = 1, 2, . . .. Introduce the elements wm = R(xm−1 , λm−1 ). Then (14.10) reads ⎧ n ⎨ x = s  η m ϕ + Γw + x ˆ0 (λm ) − x ˆ0 (λm−1 ), m i m i (14.11) i=1 ⎩ m , λm = q ηn+1

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Toward General Theory of Differential-Operator and Kinetic Models

m = 1, 2, . . .. Here, wm = R(xm−1 , λm−1 ), ˆˆ → − → η m−1 − −s wm−1 − wm , Ψ − wm , A−1 E( )Ψ, ηm=−

(14.12) (14.13)

η∗m = η∗0 for all m. As an initial approximation, we take  s i = 1, . . . , n, ξi0 = ηi0 i = n + 1, q , w0 = 0, x0 = ξ 0 ϕ + x ˆ0 (λ0 ). From what has been said, the following theorem holds. Theorem 14.1. Assume Conditions (A) and (b). Then (13.1) has a solution (14.4)–(14.5). A sequence {xm , λm } defined by (14.11)– (14.13) converges to this solution in the neighborhood 0 < | | < ρ. Weakening of Condition (b): Let Condition (A) hold; moreover,  ∂k  F (x, λ), ψ  =0 j  ∂λk λ=0 for k = 0, 1, . . . , kj , j = 1, . . . , n and for all x. Then F (x, λ), ψj  = Rj (x, λ), ψj λkj and (14.1) has general multipliers λkj . Therefore, (14.3) has the form def

k

j = 0, lj (η) = Mj (η)ηn+1

j = 1, . . . , n.

Instead we introduce a more weak Condition (b). Condition (b ). The system Mj = 0, j = 1, . . . , n has a solution η 0 = 0 of a total rank,



∂Mj (η 0 )

, A=

∂ηi

j=1,...,n,i=1,...,∗−1,∗+1,...,n+1 det A = 0. 0 = 0, Condition (b) Note that if in Condition (b ) we claim ηn+1 is also fulfilled.

Iterative Methods Using Newton Diagrams

227

Introduce the notations bj (x, λ) = R(x, λ), ψj ,   0,k−k Rik∗ (η 0 ϕ + x ˆ0 (ηn+1 ))ηn+1 j ψj , ; j = 1, . . . , n. ψ˜j = Γ∗ si+qk=θj

In (14.13), we make two changes ˆ ˆ → −A−1 Ψ, ˜ Ψ wm , A−1 E( )Ψ → A−1 E( )b(xm−1 , λm−1 ). Then the results of Theorem 14.1 hold for Conditions (A), and (b ). In the conditions of Theorem 14.1, equation (13.1) has a solution (14.4)–(14.5), to which sequence (14.11)–(14.13) is convergent in some neighborhood 0 < < ρ. We can find a boundary ρ of this neighborhood by means of the construction of a majorant system for the system of successive approximations (14.7). We rewrite (14.7) taking account of the formulas of Theorem 8.4 in the form u = Φ(u, ) ≡ (Φ1 (u, ), Φ2 (u, )), where ⎧ ⎨ Φ1 (u, ) = 1/ s [R( s ϕη + x ˆ0 q ηn+1 ) − B x ˆ0 ( q ηn+1 )], ˆˆ ⎩ Φ (u, ) = η − y − Φ (u, ), Ψ ˆ − A−1  s Φ1 (u, ), E( )Ψ. 2 1 In view of R(x, λ) and assuming that η = ζ + η0 in a domain ⎧ y ≤ t+ ⎪ 1, ⎪ ⎨ |ζ|Rn ≤ t+ 2, ⎪ ⎪ ⎩   ≤ μ+ for Φ(u, ), one can obtain the following majorant estimations: (A) |Φ(Θ, )| ≤ f (Θ, μ),   (B ) |Φu (u, )u| ≤ ft (t, μ)|u|. In the more general case, (B ) has the form

,

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Toward General Theory of Differential-Operator and Kinetic Models

(B) |Φ(u2 , ) − Φ(u1 , )| ≤ f (t2 , μ) − f (t1 , μ), where |ui | ≤ ti , i = 1, 2, |u2 − u1 | ≤ t2 − t1 . Here, | · | is an abstract norm and Θ is n-dimensional trivial vector. A function f (t, μ) satisfies the following conditions: (1) f is given, continuous and monotone increasing by all own arguments. (2) f (Θ, μ) > θ for μ > 0, f (Θ, 0) = 0. (3) There exist t+ , μ+ such that f (t+ , μ+ ) ≤ t+ . (4) There is ft -matrix of (n × n)-dimension whose elements ∂fi /∂tj , i, j = 1, . . . , n are monotone increasing functions by all own arguments. We note that t+ , μ+ is a principal solution to which the system of successive approximations tn = f (tn−1 , μ) is convergent starting from zero initial approximation. 14.2.

N -step Iteration Method

Let Condition (a) hold in which min(α1 , . . . , αn ) = s < min(θ1 , . . . , θn ).

(14.14)

Example 14.2. In the conditions of Lemma 13.2, let l = m = n. Then α1 = · · · = αn = p, αn+1 = q. Since there is not a point (1, 0) on the Newton diagram of BEq and the points (m1 , n1 ), (m2 , n2 ) are different on the ends of any face of the diagram with m1 + m2 = 0, then p < min(θ1 , . . . , θn ). Therefore, equation (14.14) is valid. Lemma 14.2. Let Condition (b) hold, λ = αn+1 ηn+1 , and assume ⎧ f or αn+1 ≥ s + 1, ⎪ ⎨ 0,  0 (14.15) x ˆ = Γ y0k (ηn+1 αn+1 )k f or αn+1 ≤ s, ⎪ ⎩ k:kαn+1 ≤s

Iterative Methods Using Newton Diagrams

229

where y0k are coefficients of (13.5). Then x ˆ0 ∈ E ∞−n ,

F (ˆ x0 + s cϕ, λ( )) = O(| |s+1 )

for any c and ηn+1 . Proof. If αn+1 ≥ s + 1, the lemma is trivial. Therefore, it suffices to consider the case αn+1 ≤ s. By Condition (a), for ξi = αi ηi , i = 1, . . . , n + 1, where ξn+1 = λ, BEq (13.9) takes the form  def Lji η i (i,α) = 0. (14.16) Lj (ξ) = (i,α)≥θj

Here, by (13.10),  = Lji η i = lj (η1 , . . . , ηn+1 ),

j = 1, . . . , n.

(14.17)

(i,α)=θj

Setting ξi = αi ηi in a solution (13.5) of (13.3) and using its analyticity, we write (13.5) in the form y = yˆ0 + r( , η),

(14.18)

where yˆ0 =



y0k (ηn+1 n+1 )k ,

k:kαn+1 ≤s



r( , η) = i≥s+1 yi i . By substituting (14.18) into (13.4), we obtain BEq (14.16). Therefore, it is necessary that y0k ∈ E2∞−n for kαn+1 ≤ s and ys+i , ψj |∀η = 0 for i = 1, . . . , pj − 1, where pj = θj − s. Hence, x ˆ0 ∈ E1∞−n . By (14.18), yθj , ψj  = lj (η1 , . . . , ηn+1 ), j = 1, . . . , n.

(14.19)

Taking into account the fact that (14.18) satisfies (13.3), we write the identity  n   0 αi αn +1 ηi ϕi + y, ηn+1 − r( , η). yˆ = R 1

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Toward General Theory of Differential-Operator and Kinetic Models

Since x ˆ0 ∈ E1∞−n , the following identities hold: F (˜ x0 + s cϕ, αn+1 ηn+1 ) x0 + s cϕ, αn+1 ηn+1 ) = yˆ0 − R(ˆ   n  αi ηi ϕi + Γy, αn+1 ηn+1 = −r( , η) + R 1

− R(ˆ x0 + s cϕ, αn+1 ηn+1 )  n   1  αi 0 s Rx (. . .)dΘ ηi ϕi + Γy − x ˆ − cϕ , = −r( , η) + 0

where

i=1



Rx (. . .) = Rx

x ˆ0 + s cϕ + Θ 

 n 

 αj ηi ϕ + Γy − x ˆ0 − s cϕ ,

i=1

αn+1 ηn+1 . By (14.18), Γy − x ˆ0 = Γr( , η). Since r( , η) = O(| |s+1 ),

αi ≥ s, Rx (. . .) = O(| |)

for all c, ηi , then F (ˆ x0 + s cϕ, αn+1 ηn+1 ) = O(| |s+1 ). The lemma is proved.



Let Conditions (a) and (b) hold. We consider a solution of (13.1) in the form ⎧ n  ⎪ ⎨x = ξi ϕi + x ˆ0 (λ( )) + Γy s , (14.20) i=1 ⎪ ⎩ def α n+1 = ξn+1 , λ = ηn+1 where ξi = ηi αi , ηi (0) = ηi0 , i = 1, . . . , n+1, η∗ = η∗0 , ∗ ∈ {1, . . . , n+1}, see x ˆ0 in (14.15), y(0) = 0. A vector η 0 satisfies Condition (b) Unknowns y( ) and ηi ( ), i = 1, . . . , n + 1, are continuous at zero

231

Iterative Methods Using Newton Diagrams

and satisfy the system def

ˆ0 + Γy s , ξn+1 ) − B x ˆ0 = Φ(y, ξ), s y = R(ξϕ + x y, ψi  = 0,

i = 1, . . . , n.

(14.21) (14.22)

By Lemma 14.2, lim→0 −s Φ(y, ξ( )) = 0. We transform (14.21) and (14.22) in such a way that the conditions of the theorem on implicit operator in the neighborhood of a point 0 0 0 , η∗+1 , . . . , ηn+1 , = 0 y 0 = 0, η10 , . . . , η∗−1

are satisfied for η∗ = η∗0 . To this end, introducing the iterations into (14.21) and (14.22), we pass on to the system s y = Φ(Φ(. . . Φ(y, ξ), . . . , ξ),  

(14.23)

N

Φ(Φ(. . . Φ(y, ξ), . . . , ξ), ξ)ψi  = 0, i = 1, . . . , n,  

(14.24)

N

where ˆ0 Φ(Φ(. . . Φ(y, ξ), . . . , ξ), ξ) = R(ξϕ + ΓR(ξϕ + · · · + ΓR(ξϕ + x     N

N

+ Γy s , ξn+1 ), . . . , ξn ), ξn+1 ) − B x ˆ0 , (14.25) ξi = αi ηi . Unknowns y, η satisfy this system. Expanding the righthand side of (14.25) in a Taylor series in the neighborhood of a point = 0, we obtain  yiN (y, η) i . (14.26) Φ(Φ(. . . Φ(y, ξ), . . . , ξ), ξ) = i≥s+1

 Introduce a sequence yN , where yN = i≥s+1 yiN (0, η) i , N = 1, 2, . . . , y = 0. Since (14.18) satisfies (13.3), the function −s r( , η) =  0 i−s satisfies (14.21). Thus, lim N →∞ yN = r( η) in the i≥s+1 yi neighborhood of a point = 0 and by induction easily follows yiN = yi , i = s + 1, . . . , s + N − 1.

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Toward General Theory of Differential-Operator and Kinetic Models

Let p = max(p1 , . . . , pn ), where pj = θj − s, θj are defined in Condition (a). Then, for N ≥ p + 1, yiN = yi , i = s + 1, . . . , max(θ1 , . . . , θn ). In the course of proof of Lemma 14.2, we obtained the identities ys+i , ψj |∀η = 0 for i = 1, . . . , pj − 1 and the identities yθj , ψj  = lj (η), j = 1, . . . , n. Therefore, we can define unknowns y( ), η( ), i = 1, . . . , n + 1 from the system  y = −s Φ(Φ(. . . Φ(y, ξ), . . . , ξ), ξ), (14.27) −θj Φ(Φ(. . . Φ(y, ξ), . . . , ξ), ξ), ψj  = 0, j = 1, . . . , n for η∗ = η∗0 . We consider (14.27) as the one-operator equation K(u, ) = 0,

(14.28)

where K : Y × R1 → Y, Y = E2 ⊕ Rn , → η ), u = (y, η1 , . . . , η∗−1 , η∗+1 , . . . , ηn+1 ) = (y, − def

K(u0 , 0) = 0

0 , η 0 , . . . , η 0 ). The operator K in a ball for u0 = (0, η 0 , . . . , η∗−1 ∗+1 n+1 s(u0 , r) is differentiable in u,   I, 0 , Ku (u0 , 0) = 0, D

where



∂lj (η 0 )



, D=

∂ηi j=1,...,n, i={1,...,n+1}\∗  Ku−1 (u0 , 0)

=

I,

0

0,

D−1

det D = 0,

 .

Thus, equation (14.28) satisfies the conditions of the theorem on implicit operator. An unknown solution u = (y, η) can be found by the method of successive approximations ˆ0 (λm−1 ) ym = −s (R(ξ m−1 ϕ + . . . + ΓR(ξ m−1 ϕ + x ˆ0 (λm−1 ), (14.29) + Γym−1 s , λm−1 ), . . . , λm−1 )) − B x → → − η m−1 − D−1  s ym , E( )Ψ, (14.30) ηm=−

Iterative Methods Using Newton Diagrams

233

η∗m = η∗0 for all m, m = 1, 2, . . .. Here, E( ) =  −θi δij i,j=1,...,n . For → a sequence ym , − η m , there is one ⎧ n  ⎪ ⎪ ⎨ xm = αi ηim ϕi + x ˆ0 (λm ) + Γym s , (14.31) i=1 ⎪ ⎪ ⎩ m , λm = ηn+1 convergent to an unknown solution (14.20). We transform (14.29)–(14.31) to a more convenient form. Introduce the elements ⎧ def ⎪ = xm−1 , x0 ⎪ ⎨ m−1 m−1 ϕ, j = 1, . . . , N − 1, (14.32) xjm−1 = ΓR(xj−1 m−1 , λm−1 ) + ξ ⎪ ⎪ ⎩ −1 N ≥ p + 1. wm = R(xN m−1 , λm−1 ), Then (14.29) and (14.30) read ˆ0 (λm−1 ), ym s = wm − B x → − → η m−1 − D−1 wm , E( )Ψ, ηm=− where η∗m = η∗0 . Finally, for the construction of the unknown {xm ( )λm ( )}, we have the formula ⎧ n  ⎪ ⎪ ⎨ xm = αi ηim ϕi + Γwm + x ˆ0 (λm ) − x ˆ0 (λm−1 ), ⎪ ⎪ ⎩

i=1

λm =

(14.33) sequence

(14.34)

m αn+1 , ηn+1

where m = 1, 2, . . . , wm , ηim , i = 1, . . . , n + 1 are computed by (14.32) and (14.33). In an initial approximation, we take x0 =

n 

αi ηi0 ϕi + x ˆ0 (λ0 ),

i=1 0 αn+1 , λ0 = ηn+1

η∗m = η∗0 , where η 0 is a solution of a total rank of (13.13).

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Toward General Theory of Differential-Operator and Kinetic Models

Theorem 14.2 follows from what has been stated above. Theorem 14.2. Let Conditions (a) and (b) hold. Then (13.1) has a solution (14.20), to which a sequence xm , λm defined by the formulas (14.32)–(14.34) is convergent for N ≥ p + 1 in the neighborhood 0 < | | < r. Remark 14.4. In the conditions of Theorem 14.2, an operator function Fx (x( ), λ( )), where (x( ), λ( )) is an unknown solution, has a TGJS. A number p1 + · · · + pn , where pj = θi − s ≥ 1, is considered to be the root number of an operator function Fx (x( ), λ( )). Here, Fx−1 (x( ), λ( )) = O(| |− max pi ), N ≥ 2 because p ≥ 1. To yield more sharp estimations of a number of steps N in Theorem 14.2, one may use Lemma 13.1 or the Newton diagram of coefficients of projection OF. 14.2.1.

Basic stages of an N -step method

Stage 1 (Preparation stage). Construction of an initial approximation x0 =

n  1

ξi0 ϕi + x ˆ0 (λ0 ),

0 λ0 = ηn+1 αn+1 ;

Stage 2. Computation of the element wm by (14.32): In this case, it is needed to solve N − 1 linear equations; Stage 3. Computation of the element Γwm ; → Stage 4. Computation of the vector − η m by (14.33); Stage 5. Computation of xm and λm by (14.34). 14.3.

Iteration Method for Nonlinear Equation Invariant Under Transformation Groups

In view of the structure of the matrix E( ), the formulas (14.13) and (14.33) have multipliers −θj . In a concrete situation, we can avoid this singularity by making some transformations and dividing by the corresponding degree of . If we do not succeed in doing this, to provide for computation stability in the neighborhood of

235

Iterative Methods Using Newton Diagrams

a branch point, one can use the concept of regularization in the sense of Tikhonov by making in (14.13) and (14.33) the change ⇒ + sign λδν , 0 < ν < 1/(2p), where δ is a maximum absolute computation error. Then the proposed iteration methods will be regularized algorithms in the sense of Tikhonov. The methods discussed above allow the modification if (13.1) possesses a symmetry-induced BEq which is invariant with respect to a certain group G. In fact, we have the following condition. Condition (B). Let 0 = n0 < n1 < . . . < nl = n, 1 ≤ l ≤ n and BEq (13.9) is potential and possesses a symmetry SO(ni −ni−1 )(O(2) for ni − ni−1 = 2) by ith set of variables ξni−1 +1 , . . . , ξni , i = 1, . . . , l, F (0, λ) = 0. Introduce the projections Pl(a) =

l l   → → − − → → → → − − → ·, ( Γ i , − a i )( Φ i , − a i ), Ql(a) = ·, ( Ψ i , − a i )(→ z i, − a i ), i=1

i=1

where − → → a i) = ( Φ i, − → → − a i) = ( Ψ i, −

ni  s=ni−1 +1 ni 

→ − ϕi ais , ( Γ i , ai ) = → − ψs ais , ( Z i , ai ) =

s=ni−1 +1



γs ais ,

s=ni−1 +1



zs ais ,

s=ni−1 +1

→ → − → a i | = · · · = |− a l | = 1, a i = (aini−1 +1 , . . . , aini ) ∈ Rni −ni−1 , |− Rn = Rn1 −n0 ⊕ Rn2 −n1 ⊕ · · · ⊕ Rnl −nl−1 . We expand the vector (ξ1 , . . . , ξn ) in a sum of l terms (ξ1 , . . . , ξn ) =

l 

(0, . . . , ξni−1 +1 , . . . , ξni , . . . , 0)   i=1 − → ξi

with support in subspaces Rni −ni−1 , i = 1, . . . , l. In every subspace, → − → a i , i = 1, . . . , l. we introduce its own spherical coordinates ξ i = μi −

236

Toward General Theory of Differential-Operator and Kinetic Models

→ − Lemma 14.3. Let Condition (B) hold. Then for (13.9) at ξ i = → a i , i = 1, . . . , l, the following identities hold: μi − ni 

Lj aij = J i (μ1 , . . . , μl , λ)

(14.35)

j=ni−1 +1

− for any → a i ∈ Si (0, 1) ⊂ Rni −ni−1 , i = 1, . . . , l. Proof.

Based on Corollary 2 from [101], Lj (ξ1 , . . . , ξn , λ) =

∂V (I1 (ξ), . . . , Il (ξ)) ∂Is , ∂Is ∂ξj

where j = ns−1 + 1, . . . , ns , s = 1, . . . , l, Is (ξ) = ξn2 s−1 +1 + · · · + ξn2 s = → a i , i = 1, . . . , l. μ2s . We scalar multiply (Lni−1 +1 , . . . , Lni ) by vector − As a result, we obtain (14.35). Introduce BEq J i (μ1 , . . . , μl , λ) = 0,

i = 1, . . . , l.

(14.36)

Condition (C). Equation (14.36) satisfies Conditions (a) and (b); moreover, the corresponding element x ˆ0 (see (14.15)) lies in the ∞−n subspace E1 . Here, in Conditions (a) and (b), n ⇒ l, the element x ˆ0 being constructed by (14.15) in which s = min(α1 , . . . , αl ), λ = αl+1 ηl+1 , α1 , . . . , αl+1 correspond to (14.36).  Theorem 14.3. Assume Conditions (B) and (C). Then (13.1) has → a parametric solution of − a, x=

l  i=1

→ → − αi ηi0 ( Φ i , − a i) + x ˆ0 (λ( )) + o( s ),

0 + o( αl+1 ). λ = αl+1 ηl+1

A sequence xm , λm defined by (14.32)–(14.34), in which the changes n ⇒ l, ξϕ ⇒

l 

→ → − αi ηi ( Φ i , − a i ),

→ → − → →  − Ψ ⇒ (( ψ 1 , − a 1 ), . . . , ( ψ l , − a l ))

1

(14.37) → have been made, is convergent to that solution for any − a ∈ Πl1 si (0, 1).

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Iterative Methods Using Newton Diagrams

Our proof follows along the lines of Theorem 14.2 with the change (14.37) and using conditions → → − a i ) = 0, i = 1, . . . , l y, ( Ψ i , −

(14.38)

instead of (14.22). The conditions (14.38) correspond to (14.36). Thus, if Conditions (B) and (C) hold, an N -step iteration method modified as a result of (14.36) permits us to construct the → − → a -parametric families of solutions for any − a. There is a different approach to the problem when (13.1) is invariant with respect to a certain group G. In this approach, by a method of N -step iterations, we preliminarily construct a solution x( ), λ( ) of (13.1) in the subspace E1∞−n ⊕ span(ϕ1 , . . . , ϕr ),

1 ≤ r < n.

Then a solution extends to E1 by means of the transformation Lg x, where Lg is a group representation G in E1 . In fact, let BEq (13.9) satisfy the following conditions: Condition (D). Lj (ξ1 , . . . , ξn , λ)|ξj =0 = 0, j = r + 1, . . . , n. Condition (E). A system Lj (ξ1 , . . . , ξr , 0, . . . , 0, λ) = 0,

j = 1, . . . , r

(14.39)

satisfies Conditions (a) and (b) (or Conditions (a) and (A)) in which we assumed n ⇒ r,

j = 1, . . . , r,

λ = ξr+1 .

If Conditions (D) and (E) are valid, there is a vector 0 ), η 0 = (η10 , . . . , ηr0 , 0, . . . , 0, ηn+1

satisfying (13.13); moreover,



∂lj (η 0 )

≤ n. r ≤ rank

∂ηi

j=1,...,n, i=1,...,n+1

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Toward General Theory of Differential-Operator and Kinetic Models

Lemma 14.4. Let F : E ∞−n ⊕ span(ϕ1 , . . . , ϕr ) → E2∞−n ⊕ span(z1 , . . . , zr ). Then Condition (D) will be satisfied. If here Condition (E) holds, then



∂lj (η 0 )

= r. rank

∂ηi

j=1,...,n,i=1,...,n+1 The proof follows from the construction of (13.9) and the identities     r  ξi ϕi + Γy, λ , ψj = 0, j = r + 1, . . . , n, R 1

which are valid for any ξ1 , . . . , ξr and any y from E2∞−n ⊕ span(z1 , . . . , zr ). Lemma 14.5. Let (13.9) generate a potential vector field whose potential is an invariant of l-parametric continuous group G having a total system (I1 (ξ), . . . , Ir (ξ)), n − r ≤ l of functionally independent k invariant where all ∂I ∂ξj |ξj =0 = 0 for j = r + 1, . . . , n. Then Condition (D) will be satisfied. The proof follows from the identities Lj =

r  ∂V ∂Ik , j = 1, . . . , n. ∂Ik ∂ξj k=1

Let Conditions (D) and (E) hold and the corresponding vector x ˆ0 ∞−n (see (14.33), where n ⇒ r) is in E2 . Then the following result is true. Theorem 14.4. Let (13.9) satisfy Conditions (D) and (E), P x ˆ0 = 0. Let (14.39) satisfy Conditions (a) and (b), where n ⇒ r. Then (13.1) has a small solution x( ) → 0, λ( ) → 0 at → 0, to which an N -step iteration method converges. If (14.39) satisfies Conditions (a) and (A) where n ⇒ r, then a one-step iteration method is also convergent. Here, x( ) ∈ E1∞−n+r . If in addition, the equation is invariant with respect to l-parametric group for λ = λ( ), to which (13.1) satisfies a

Iterative Methods Using Newton Diagrams

239

parametric solution Lg x( ), where Lg is a group representation of G in E1 , Lg x( ) ∈ E1 . Proof. By Conditions (D) and (E) instead of (14.22), we use the conditions y, ψi  = 0, i = 1, . . . , r. By setting n ⇒ r, we repeat the proof of Theorem 14.2 in the case of Conditions (a) and (b). 

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Chapter 15

Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods Let us consider the following nonlinear operator equation: B(λ)x + R(x, λ) = 0,

(15.1)

where B(λ) is a closed linear operator with a dense (in X) domain independent of the parameter λ ∈ Λ. The nonlinear operator R : X × Λ → Y is continuous in x and λ in a neighborhood of zero and R(0, 0) = 0. Here, X and Y are Banach spaces and Λ is a linear normed space. It is required to construct the solution x(λ), x(λ) → 0 as S  λ → 0, S ∈ Λ (further, briefly, as λ → 0) of the maximal order of smallness (or the “minimal branch” of the small solution) by the method of successive approximations in the nonregular case where the range of the operator B(0) can be nonclosed and the dimension safisfies the inequality dim N (B(0)) ≥ 1. In this chapter, we review the results from [142] and consider the problem of constructing the “minimal branch” of the solution of equation (15.1), assuming that the operator B(λ) has an inverse for λ ∈ S and B −1 (λ) = O(1/a(λ)),

S  λ → 0,

(15.2)

where S is an open set in the space Λ whose boundary contains the point λ = 0 and a(λ) : S ⊂ Λ → (0, +∞)

241

242

Toward General Theory of Differential-Operator and Kinetic Models

is a positive continuous functional, a(0) = 0. The set S is called a sectorial neighborhood of zero. Here, readers may also refer to Chapter 16. Estimates of the form (15.2) are valid in a number of applications. For example, suppose that X = Y = H. Consider the operator B(λ) = B0 + a(λ)B1 , where B0 is a self-adjoint nonnegative operator and B1 is a self-adjoint positive operator, i.e., (B0 x, x) ≥ 0 and (B0 x, x) ≥ γ(x, x)

for all x ∈ H.

Then we have (B0 + a(λ)B1 )−1  ≤

1 , γa(λ)

and estimate (15.2) holds. If B(0) is a Fredholm operator, then in order to construct asymptotically small solutions of equation (15.1), we can use classical results from the theory of branching solution of nonlinear equations [169]. However, it was assumed that Λ = R1 , the range of operator B(0) is closed, and the method for initial approximation selection was complicated. Let us here introduce the set Ω = {(x, λ) ∈ X × Λ, x ≤ a(λ)r, λ ∈ S}.

(15.3)

In this chapter, we prove constructive existence and uniqueness theorems for the minimal branch of the solution of equation (15.1) in domains of the form (15.3) with vector parameter λ in a sectorial neighborhood S in the general case where the operator B(0) can fail to be normally solvable and dim N (B(0)) > 1. The results of the chapter further develop the theory of approximate methods in a neighborhood of branch points of solution of nonlinear equations. On the basis of theorems of this chapter, readers may construct the desired branch of the solution by the method of successive approximations converging in the domain S for the zero initial approximations. Thus, the problem of choosing the initial approximation does not arise in our method. The chapter is organized as follows. In Section 15.1, we study in detail the case in which B(0) is a Fredholm operator and the regularization of the successive approximations method is based on

Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods

243

the application of the generalized Schmidt lemma [169, Section 21]. It is shown that, in the Fredholm case, we can construct the method of successive approximations for the solution of equation (15.1), applying “asymptotic regularizers” [143]. In Section 15.2, we present the existence theorems and the method of successive approximations for the minimal branches of the solutions for equation (15.1), provided that estimate (15.2) holds. The range of operator B(0) is not necessarily closed. The corresponding iterative formulae involve operator B −1 (λ). Therefore, in view of inevitable calculation errors, this method of successive approximations requires a generalization in the sense of the ill-posed problems theory. Such a regularization can be carried out in a neighborhood of the branch point of the solution by shifting parameter λ. In the specific examples of an explicitly constructed operator B −1 (λ) for which the point λ = 0 turns out to be a removable singular point, the regularization is achieved by canceling the admissible power of the parameter λ in the reduced equation. The general existence theorems and the method of successive approximations are illustrated by examples of the solutions of nonlinear integral equations of the first and second kinds and of boundary value problems often encountered in applications. 15.1.

Construction of the Minimal Branch of Solutions of Equation with Fredholm Operator

Suppose equation (15.1) is of the form B(λ)x = R(x, λ) + b(λ).

(15.4)

The linear operator B(λ) satisfies the conditions given in the introduction of this section. B(0) is a Fredholm operator, {φi }n1 is a basis in N (B(0)), and {ψi }n1 is a basis in N ∗ (B(0)). The nonlinear operator R : X × Λ → Y is continuous in x and λ in a neighborhood of zero, R(0, 0) = 0, R(x1 , λ) − R(x2 , λ) ≤ L(r)x1 − x2 , for |λ| ≤ ρ and x1 , x2 from the ball S(0, r), and L(r) = O(r). The function b(λ) : Λ → Y is defined and continuous in a neighborhood

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Toward General Theory of Differential-Operator and Kinetic Models

of the point λ = 0 and b(0) = 0. Since B(0) is a Fredholm operator, it is easy to verify condition (15.2) by constructing the inverse operator B 1 (λ). Indeed, using the generalized Schmidt lemma, we introduce the bounded Trenogin operator −1  n 

·, γi zi , Γ = B(0) + i=1

where φi , γi  = δik and zi , ψk  = δik . We introduce the notation A(λ) := B(λ) − B(0). Suppose that for x ∈ D(B(0)): I. A(λ)x ≤ c(λ)(x) + bB(0)x, where c(λ) : S ⊂ Λ → (0, +∞) is a positive continuous functional defined in a sectorial neighborhood S of zero, c(0) = 0, and b ≥ 0. Note that lim A(λ)Γ = 0,

Sλ→0

and operator I − A(λ)Γ has a bounded inverse, λ ∈ S0 ⊂ S, where A(λ)Γ ≤ q < 1. II. | det (I − A(λ)Γ)−1 A(λ)φi , ψk i,k=1,...,n | = Δ(λ), where Δ(λ) ∼ a(λ) as S  λ → 0 and a(λ) is a positive continuous functional, a(0) = 0. Remark 15.1. If   p−1    det (A(λ)Γ)j A(λ)φi , ψk   j=0

i,k=1,...,n

    ∼ cλα ,  

λ→0

and α ≤ p, then condition II holds for a(λ) = cλα . Lemma 15.1. Suppose that B(0) is a Fredholm operator and conditions I and II hold. Then operator B(λ) has a bounded inverse λ ∈ S and estimate (15.2) holds. Proof. Suppose that we search for the solution of the linear equation B(λ)x = f for λ ∈ S in the form x = Γu +

n  i=1

ci φi ,

Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods

where u, ψk  = 0, k = 1, . . . , n. Then  u = (I − A(λ)Γ)−1

f + A(λ)

n 

245

 ci φi

.

i=1

Here, for λ ∈ S, the vector c = (c1 , . . . , vn ) is determined from the system of linear algebraic equations n 

(I − A(λ)Γ)−1 A(λ)φi , ψk ci i=1

= (I − A(λ)Γ)−1 f, ψk ,

k = 1, . . . , n,

with nondegenerate (for λ ∈ S) matrix by condition II. In view of condition II, the estimate c = O(1/a(λ)) holds, and so lemma is  proved. Further, suppose that B(λ) = B0 − c(λ)B1 , and for x ∈ D(B(0)), B1 x ≤ a(x + bB0 x),

a > 0, b ≥ 0.

Suppose the following condition: (k)

III. The elements {φi }, i = 1, . . . , n, k = 1, . . . , pi , constitute a complete B1 -Jordan block of the Fredholm operator B0 , and (k) functionals {ψi }, i = 1, . . . , n, k = 1, . . . , pi , constitute the complete B1∗ -Jordan block of operator B0∗ . Let us recall (see [169, Chapter 9]) that, in condition III, we can assume without loss of generality that (k)

φi

= (ΓB1 )k−1 φi , (1)

(k)

ψi

= (Γ∗ B1∗ )k−1 ψi , (1)

k = 2, . . . , pi , i = 1, . . . , n, = δik , B1∗ ψi i , φk  = δik , 

−1 n (p ) (p ) , γi = B1∗ ψi i , zi = B1 φi i . Γ = B0 + ·, γi zi (p ) (1)

B1 φi i , ψk 

(p )

(1)

i=1

Definition 15.1. A linear operator Γ(λ) is called a left asymptotic regularizer of the operator function B(λ) if lim Γ(λ)B(λ)x = x

Sλ→0

for all x ∈ D(B).

Toward General Theory of Differential-Operator and Kinetic Models

246

In a similar way, we introduce the right asymptotic regularizer of the operator B(λ). If λ = 0 is an isolated singular Fredholm point of the operator function B(λ), then the asymptotic regularizers can be constructed in explicit form. Lemma 15.2. Suppose that B(λ) = B0 − c(λ)B1 and condition III holds. Then, in a neighborhood of 0 < |c(λ)| < ε, there exists a bounded inverse operator B −1 (λ), as well as left and right regularizers Γl (λ) and Γr (λ) of the operator B(λ) defined by the formulas Γl (λ) = Γ −

pi n   i=1 s=1

Γr (λ) = Γ −

pi n   i=1 s=1

c−s (λ) ·, ψi

(pi +1−s)

c−s (λ) ·, ψi φi (1)

(1)

φi ,

(pi +1−s)

.

Here, B −1 (λ) = Γr (λ)(I − c(λ)B1 Γ)−1 , where B

−1

 (λ) = O

1 p c (λ)

,

p = max (p1 , . . . , pn ).

If B1 is a bounded operator and 0 < |c(λ)|
0 in the neighborhoods of the critical points α0 = n2 . Let for definiteness α0 = 1. Then the cubic equation ξ 3 − 2π 2 λξ + 4π 3 e = 0

(16.13)

is the corresponding approximate branching equation (see [169, p. 508]). If D = 27e2 −2λ3 < 0, then the Cardano formula guarantees that this equation has three real solutions. Using the theorem, one may prove the existence of the corresponding real solutions to the boundary value and the possibility of their construction by successive approximations. Here, the Newton diagram of the operator F (u, λ, 0) consists of the segment passing through the points (3, 0) and (1, 1). Therefore, in our case, r = 1, s = 2, m = 2. Thus, based on the theorem, we must employ the following uniformization:    2 sin x λ1/2 (16.14) u = Γv + c π with boundary conditions (16.12) under the asymptotic compatibility condition e = o(λ3/2 ) on the parameters of the problem. Obviously, here, D < 0, i.e., the approximate branching equation has three real roots. The operator

2 π d2 ˇ sin x sin s[·] ds (16.15) B = 2 +1+ dx π 0 with boundary conditions (16.12) has the bounded inverse Γ ∈ (2) L(C[0,π] → C˚[0,π] ). By the Lagrange method of variation of arbitrary constants, taking into account the degeneracy of the kernel integral ˇ we find the inverse operator Γv = sin x ∗ part of the operator B, (I − P )v + pv − d sin x, where P is the projection and

2 π sin x sin s[·] ds, P = π 0 2 d = (sin x, sin x ∗ (I − P )v). π By Condition C, construct the algebraic equation L(c) ≡ −c3 /(4π)+ c = 0 for defining an initial approximation of the coefficient √ c in (16.14). It has three simple roots ±2 π and 0. Therefore,

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Toward General Theory of Differential-Operator and Kinetic Models

by the theorem, under the fulfillment of the asymptotic estimate e = o(λ3/2 ), the boundary value problem has three real solutions: √ u1,2 = ±2 2λ sin x + r1,2 (x, λ, ε), u3 = r3 (x, λ, ε), where the functions ri have the estimates ri (λ) = o(λ1/2 ) for e = o(λ3/2 ). By the corollary, we can refine the asymptotic approximations of the solutions u1,2,3 by solving linear equations. If e ∼ cλn where n < 3/2, then the approximate solution to the branching equation (16.13) has a unique real solution since then obviously D > 0. In this case, the corresponding unique small real solution to the boundary value problem is as follows: u = −2(4e)1/3 sin x + r(x, λ, e),

||r(x, λ, e)|| = o(e1/3 ),

where, by the corollary, r(x, λ, e) is constructed uniquely by successive approximations. Also, let us remark that in Example 16.2, the branch u3 of maximal order of smallness may be sought with the zero initial approximations in another way, namely, if e = o(λ2 ), then the successive approximations un converge to the minimal branch u3 of the solution to the boundary value problem (16.11)–(16.12), where d2 un + (1 + λ + α)un dx2 d2 un−1 dun−1 + (1 + λ)(un−1 − sin un−1 ) + 4e sin x, = −e + 2e dx2 dx un (0) = un (π) = 0, u0 = 0, n = 1, 2, . . . , and α(δ) is the regularization parameter coordinated with the computation error δ.

Chapter 17

Interlaced and Potential Branching Equation The solutions of nonlinear equations can depend upon one or several free numerical parameters. These parameters have various meanings in different problems and they can belong to both a bounded and an unbounded domain of Euclidean space. In this chapter, we assume that a kernel of linearized operator is nontrivial. Then, the occurrence of free parameters may be related to the range properties of the nonlinear term of the transformation if the problem is invariant relative to a certain group of transformations. Here, readers may refer to [125], where all or some of the free parameters have the group meaning. For example, in the case of spherical symmetry, the free parameters can be regarded as points of a sphere in finite-dimensional Euclidean spaces. In the branching theory of solutions of nonlinear equations, we need to know the location domain of free parameters both in qualitative and asymptotic analysis. The introduction of the interlaced equation enables us to simplify the calculations and consider various classes of branching solutions with a single point of view. In this set of notes, we analyze the occurrence of free parameters in the branching solutions of general nonlinear equations in Banach spaces. We present the new simplifying methods for equations, extending the possibility of efficient algorithmization of branching theory methods.

265

266

Toward General Theory of Differential-Operator and Kinetic Models

In Section 17.1, we obtain sufficient conditions for the inheritance of interlaced property of nonlinear operator by the BSEq. The case in which an interlaced pair (intertwining operators) consists of projectors is considered in Section 17.2. It is shown how in this case one must reduce the number of equations in the branching system of equation (BSEq) and construct parametric solutions by the method of successive approximations. Section 17.3 deals with the case in which interlaced pair consists of parametric families of linear operators. Sufficient conditions for the reduction of the number of equations in BSEq are given. Sufficient conditions of existence of the branching solutions with free numbers of parameters belonging to certain hypersurfaces in Euclidean spaces are obtained. As a corollary, we consider the case of bi-interlacing. Finally, we focus on the construction problem of branching solutions by the method of successive approximations. 17.1.

Property of (S, K)-interlacing of an Equation and Its Inheritance by Branching Equation

Let E1 and E2 Banach spaces and Λ be a normalized space. Consider the equation def

F (x, λ) = Bx − R(x, λ) = 0.

(17.1)

Here, B : D ⊂ E1 → E2 is a closed Fredholm operator with dense domain in E1 , dim N (B) = n ≥ 1. The nonlinear operator R : Ω ⊂ E1 × Λ → E2 is assumed to be defined, continuous and continuously differentiable in the sense of Fr´echet with respect to x in the neighborhood of Ω zero; moreover, R(0, 0) = 0, Rx (0, 0) = 0. Let us consider the construction problem of small solutions x(λ) → 0 at λ → 0 of equation (17.1). Let {ϕi }ni=1 be a basis in N (B), {ψi }ni=1 a basis in N (B ∗ ), {γi }ni=1 , {zi }ni=1 the corresponding biorthogonal systems from E1∗ and E2 : −1  n  ·, γi zi Γ= B+ i=1

Interlaced and Potential Branching Eqaution

267

is the bounded operator. Then all the small solutions of (17.1) can be represented in the form x = (ξ, ϕ) + Γy,

(17.2)

where y = y((ξ, ϕ), λ) is a unique small solution of equation y = R((ξ, ϕ) + Γy, λ),

(17.3)

and parameter ξ ∈ Rn satisfies BSEq def

Li (ξ, λ) = y((ξ, ϕ), λ), ψi  = 0, i = 1, . . . , n.

(17.4)

Let us introduce the linear operators S ∈ L(E1 ), K ∈ L(E2 ) interlaced by operators B and R, i.e., satisfying the condition BS = KB, R(Sx, λ) = KR(x, λ) for

(x, λ) ∈ Ω.

(17.5)

This can be the parametric continuous representations of a G-group in the spaces E1 and E2 if equation (13.1) is invariant with respect to G-group. On the other hand, these can be irreversible operators, for example, projectors onto spaces E1∞−n = E1  span{ϕ1 , . . . , ϕn }, def

def

E2,∞−n = E2  span{z1 , . . . , zn }, ∞−(n−q)

× Λ → E2,∞−(n−q1 ) . or its parts if R : E1 We find the form in which the interlacing property (17.5) is inherited by BEq (17.4). Introduce the notations E1n = span{ϕ1 , . . . , ϕn }, E2,n = span{z1 , . . . , zn }, def

def

E1∗n = span{γ1 , . . . , γn }, def

∗ E2,n = span{ψ1 , . . . , ψn }. def

∗ be invariant subspaces to the Let subspaces E1n , E2,n , E1∗n , E2,n ∗ ∗ operators S, K, S and K , respectively:

Sϕ = T  ϕ,

Kz = D z,

K ∗ ψ = M ψ,

S ∗ γ = Cγ.

(17.6)

Toward General Theory of Differential-Operator and Kinetic Models

268

def

where T, D, M, C are matrices of (n × n) dimension, ϕ = (ϕ1 , . . . , ϕn ), def

z = (z1 , . . . , zn ), etc. We consider solutions of (17.1) in the form x = (ξ, Sϕ) + Γys ,

(17.7)

where ys = y((ξ, Sϕ), λ) is a unique small solution of equation y = R((ξ, Sϕ) + Γy, λ),

(17.8)

and parameter ξ ∈ Rn owing to condition (17.6) satisfies BSEq Li (T ξ, λ) = y((ξ, Sϕ), λ), ψi  = 0, i = 1, . . . , n.

(17.9)

The following theorem establishes a link between the right-hand sides of (17.4) and (17.9). Theorem 17.1. Assume (17.5) and (17.6), where C = D. Then equality L(T ξ, λ) = M L(ξ, λ)

(17.10)

holds in the neighborhood of zero. Proof.

By condition (17.6), we have equalities K

n 

·, γi zi = (·, γ, D z),

i=1 n  S·, γi zi = (·, S ∗ γ, z) = (·, Cγ, z) = (·, γ, C  z), i=1

in which the right-hand sides are equal because C = D. Therefore, the right-hand sides are also equal. Hence,     n n   ·, γi zi = B + ·, γi zi S and SΓ = ΓK. K B+ i=1

i=1

Interlaced and Potential Branching Eqaution

269

Equations (17.3) and (17.8) at ξ, λ → 0 owing to the theorem on implicit operator, have the unique small solutions y = y((ξ, ϕ), λ)

(17.11)

ys = y((T ξ, ϕ), λ),

(17.12)

and

respectively. By substituting (17.11) into (17.3), we obtain the identity. Having operated, by K on this identity taking account of (17.5) and (17.6) and equality SΓ = ΓK, we obtain Ky((ξ, ϕ), λ) = R((T ξ, ϕ) + ΓKy((ξ, ϕ), λ), λ). Hence, equation (17.8) has the solution Ky((ξ, ϕ), λ) together with the solution y((T ξ, ϕ), λ). Because of the uniqueness of small solution of (17.8), these solutions are coincided. Thus, the coefficients of its projections are equal on E2,n , i.e., y((T ξ, ϕ), λ), ψ = K(y(ξ, ϕ), λ), ψ. The left-hand side of this equality is the vector L(T ξ, λ), and the right-hand side is the vector M L(ξ, λ) because K ∗ ψ = M ψ.  The theorem is proved. Corollary 17.1. If det M = 0, then BEqs (17.4) and (17.9) are equivalent. If in this case, det T = 0, then all small solutions of (17.1) can be written in the form of (17.7) with condition (17.9). In the condition of Theorem 17.1, let equation (17.1) be invariant with respect to G-group, and operators S and K be, respectively, its representations in E1 and E2 . Then Theorem 17.1 establishes that equation (17.4) inherits the group symmetry of (17.1), where det M (α) = 0, det T (α) = 0. Let in condition (17.5) operator S be idempotent, i.e., S 2 = S. Then from (17.5), it follows that for (x, λ) ∈ Ω, vector F (Sx, λ) is a fixed point of operator K. Corollary 17.2. Let conditions (17.5) and (17.6) be satisfied and C = D,S 2 = S. Then for any ξ, λ from the neighborhood of zero, the vector L(T ξ, λ) is a fixed point of matrix M .

Toward General Theory of Differential-Operator and Kinetic Models

270

Proof. By equalities S 2 = S, Sϕ = T  ϕ, T 2 = T  holds. But then T 2 = T, and setting ξ = T ξ in (17.10), we obtain M L(T ξ, λ) = L(T ξ, λ).

(17.13) 

The corollary is proved. Replacing (17.5) by the condition BS = KB, KR(Sx, λ) = R(Sx, λ) for (x, λ) ∈ Ω,

(17.14)

we may sharpen Corollary 17.2, giving up the direct requirement of idempotence. Theorem 17.2. Assume (17.6), (17.14) and C = D. Then vector L(T ξ, λ) is a fixed one of matrix M for any ξ, λ from the neighborhood of zero. Proof.

Consider the sequence ym = R((ξ, Sϕ) + Γym−1 , λ),

m = 1, 2, . . . ,

(17.15)

where y0 = 0. The sequence (17.15) converges to the unique small solution y((ξ, Sϕ), λ) of (17.8). As shown in the proof of Theorem 17.1, SΓ = ΓK. By the method of mathematical induction, we prove equalities ym = Kym , m = 1, 2, . . . . Indeed, by (17.14), y1 = R((ξ, Sϕ), λ) = Ky1 . Let ym = Kym. Then by (17.14) and the equality SΓ = ΓK, ym+1 = R((ξ, Sϕ) + ΓKym , λ) = R(S((ξ, ϕ) + Γym ), λ) = Kym+1 . Therefore, by condition (17.6), the following equalities hold: ym , ψ = M ym , ψ,

m = 1, 2, . . . .

(17.16)

Passing to the limit in (17.16) at m → ∞, we obtain the desired identity L(T ξ, λ) = M L(T ξ, λ).  The theorem is proved. Definition 17.1. If condition (17.5) holds for the pair of operators (S, K) then we say (17.1) is (S, K)-interlaced (α-parametric interlaced for S = S(α), K = K(α), where parameter α ∈ G). If (17.5) is

Interlaced and Potential Branching Eqaution

271

valid for α-parametric pair (S(α), K(α)), and for a pair (S, K), which does not appear in this family, we consider (17.1) to be bi-interlaced. Definition 17.2. If condition (17.14) holds for a pair of operators (S, K), we consider the operator R(x, λ) to be (S 2 , K)-interlaced. Definition 17.3. If (17.10)–(17.13) are valid, we consider the branching equation (17.4) to be (T, M )-interlaced (or (T 2 , M )interlaced). If, doing so, T (α), M (α) are α-parametric matrices, we consider BEq (17.4) to be α-parametric interlaced. 17.2.

(T , M )-interlaced and (T 2 , M )-interlaced Branching Equation

Lemma 17.1. Let equation (17.4) be (T, M )-interlaced. Let rank M = q, {e∗i }ri=1 be a basis in N (M ∗ ), r = n − q,   1, . . . , r e∗ k1 , . . . , kr be a rank minor of the matrix e∗ij j=1,...,n i=1,...,r . Then, we can decrease on r units the number of equations in BSEq (17.9). So, if (ξ, λ) satisfies q equations Li (T ξ, λ) = 0, i ∈ {1, . . . , n}\{k1 , . . . , kr },

(17.17)

then (ξ, λ) satisfies other equations of (17.9). Proof. hold:

By identity (17.10) for any ξ, λ, the following identities (L(T ξ, λ), e∗i ) = 0, i = 1, . . . , r.

We rewrite last ones in the coordinate form   e∗is Ls (T ξ, λ) = − s∈{k1 ,...,kr }

s∈{1,...,n}\{k1 ,...,kr }

i = 1, . . . , r.

(17.18)

e∗is Ls (T ξ, λ),

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Toward General Theory of Differential-Operator and Kinetic Models

From here, by (17.17), 

e∗is Ls (T ξ, λ) = 0,

i = 1, . . . , r,

s∈{k1 ,...,kr }

where det e∗is  = 0. Hence, if (ξ, λ) satisfies (17.17), Ls (T ξ, λ) = 0 also for s ∈ {k1 , . . . , kr }. The lemma is proved.  Theorem 17.3. Let the conditions of Lemma 17.1 be satisfied. Then equation (17.1) has a solution x = (ξ, Sϕ) + Γy,

(17.19)

where ξ satisfies q BEqs (17.17) and y ∈ E2,∞−n . Proof. We consider the solutions of (17.1) in the form of (17.19), assuming that y ∈ E2,∞−n . Then y is defined as a unique small solution of equation (17.8) by the method of successive approximations for y0 = 0. In this case, the parameter ξ must satisfy the conditions y, ψi  = 0, i = 1, . . . , n, for (17.9). By Lemma 17.1, we can decrease the number of equations in (17.9) on r units.  Lemma 17.2. Let equation (17.4) be (T 2 , M )-interlaced. Let rank M=q and rank (M − E) = l. Then, we can decrease on m = max{n − q, l} units the number of equations in BSEq (17.9). Proof.

Owing to (T 2 , M )-interlacing, the following identity holds: (M − E)L(T ξ, λ) ≡ 0.

(17.20)

Let max{n − q, l} = n − q. Then (17.18) in the proof of Lemma 17.1 are valid. Therefore, system (17.9) is reduced to (17.17), consisting of q equations. Now, let max{n − q, l} = l. Since rank(M − E) = l, then for the matrix M − E, there corresponds n − l elementary divisors λp1 , . . . , λpn−1 , and its normal Jordan form has the type J={H p1 , . . . , H pn−l , λpn−l+1 E pn−l+1 + H pn−l+1 , . . . , λpm E pm + H pm }.

Interlaced and Potential Branching Eqaution

273

Here, J = P (M − E)P −1 , P is transformation matrix ⎛ ⎞ 0 1 ⎜ ⎟ .. ⎟, H pi = ⎜ . 1 ⎝ ⎠ 0 λi = 0 are eigenvalues of the matrix M − E, i = pn−l+1 , . . . , pm . Introduce the vector def

l(T ξ, λ) = P L(T ξ, λ). Then identities (17.20) with the aid of the nonsingular matrix P = Pik ni,l=1 are reduced to the equality Jl(T ξ, λ) = 0.

(17.21)

The vector l ∈ Rn takes the form l = (lp1 , . . . , lpm ) , where lpi ∈ Rpi . In this connection, by (17.21), li = 0 for any ξ, λ, i = pn−l+1 , . . . , pm , because the corresponding detλpi E pi + H pi  = 0. By (17.21), all coordinates in vectors li , i = p1 , . . . , pn−l vanish for all ξ, λ except the last ones. Thus, for a pair (ξ, λ) to be satisfied in system (17.9), it is enough that the last coordinates of vectors lpi ∈ Rpi , i = 1, . . . , n − l vanish at the point (ξ, λ). Hence, equation (17.9) is equivalent to the system n 

Pik Lk (T ξ, λ) = 0, i = p1 , p1 + p2 , . . . , p1 + . . . + pn−l , (17.22)

k=1

consisting of n − l equations. The lemma is proved.



Theorem 17.4. Let the conditions of Lemma 17.2 hold. Then equation (17.1) has a solution (17.19), where the vector ξ satisfies q BEqs (17.17) if m = n−q or n−l BEqs (17.22) if m = l, respectively. To prove Theorem 17.4, it is enough to use Lemma 17.2 in a scheme of proof of Theorem 17.3. Corollary 17.3. Let the conditions of Lemma 17.2 hold, det(M − E) = 0. Then (17.1) has a ξ-parametric solution (17.19), where the

274

Toward General Theory of Differential-Operator and Kinetic Models

sequence {yn } converges to y with yn = R((ξ, Sϕ) + Γyn−1 , λ),

y0 = 0.

The solution (17.19) of equation (17.1) depends on m arbitrary parameters with m = rank T . Proof. Since (M − E)L(T ξ, λ) = 0 for any (ξ, λ) and det(M − E) = 0, then L(T ξ, λ) = 0 for any (ξ, λ). Hence, the vector ξ in (17.19) remains arbitrary. Therefore, taking account of the condition rank T = m, there are m free parameters in (17.19). The corollary  is proved. Example 17.1. Introduce the projectors S=I−

q 

n  ·, γi ϕi , K = I − ·, ψi zi ,

i=1

i=1 ∞−(n−q)

where 1 ≤ q ≤ n. Let in equation (17.1) R : E1 i.e., for any x, λ,

×Λ → E2,∞−n ,

n  R(Sx, λ), ψi zi = 0. i=1

Then in conditions (17.14)–(17.6), S 2 = S, M = 0, T = diag{0, . . . , 0, 1, . . . , 1}, rank T = n − q, L(T ξ, λ) = 0 for any ξ, λ, and the result of Corollary 17.3 holds. 17.3.

α-Parametric Interlaced Branching Equation

Let G be a domain of Euclidean space with 0 ∈ G. In this section, we consider that for all α ∈ G and ξ, λ from the neighborhood of zero, the following identity is valid: L(T (α)ξ, λ) = M (α)L(ξ, λ),

(17.23)

where T (α) and M (α) are parametric families of matrices such that T (0) = E, det M (α) = 0 for α ∈ G. Assume Rqn = {c ∈ Rn : cni = 0, i = q + 1, . . . , n}, n = {c ∈ Rn : cni = 0, i = 1, . . . , q}, Rn−q

Interlaced and Potential Branching Eqaution

275

where 1 ≤ q ≤ n − 1, n1 , . . . , nn is a rearrangement of the first n natural numbers. The set o(c) = {M (α)c : α ∈ G} is considered to be a trajectory of the vector c from Rn corresponding to matrix M (α). Definition 17.4. We consider that the trajectory o(c) of vector c from Rn passes through subspace Rnq if there is αc ∈ G such that M (αc )c ∈ Rnq . Definition 17.5. We consider that the trajectory o(c) of vector c from Rnn−q lies in Rnn−q if M (α)c ∈ Rnn−q for all α ∈ G. But if there exists αc ∈ G such that M (αc )c ∈ / Rnn−q , then we say the trajectory o(c) does not remain in Rnn−q . In E1 , we introduce the projectors P1 =

q 

·, γni ϕni ,

i=1

P2 =

n 

·, γni ϕni ,

i=q+1

generating the direct decomposition E1n = E1q ⊕ E1n−q , where E1q = span{ϕn1 , . . . , ϕnq },

E1n−q = span{ϕnq+1 , . . . , ϕnn }.

The set o(ϕ) = {S(α)ϕ : α ∈ G} is considered to be a trajectory of element ϕ from E1n corresponding to operator S(α). Definition 17.6. We consider that the trajectory o(ϕ) of element ϕ from E1n passes through the subspace E1q if there is αϕ ∈ G such that S(α)ϕ ∈ E1q . Remark 17.1. If S(α) is a group of linear operators acting by q-stationary in E1n , then the trajectory of any element ϕ from E1n passes through E1q . Definition 17.7. We consider that the trajectory o(ϕ) of element ϕ from E1n−q lies in E1n−q if S(α)ϕ ∈ E1n−q for all α ∈ G. But if there is αϕ ∈ G such that S(αϕ )ϕ ∈ / E1n−q , then the trajectory o(ϕ) does n−q not remain in E1 .

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Toward General Theory of Differential-Operator and Kinetic Models

Property 17.1. Let S(α)ϕi =

n 

Mji (α)ϕj , i = 1, . . . , n, α ∈ G.

(17.24)

j=1

Then the following propositions are valid: The trajectory o(c) of any vector c from Rn passes through subspace Rnq if and only if the trajectory o(ϕ) of any element ϕ from E1n passes through subspace E1q . The trajectory of any nontrivial vector c from Rnn−q does not remain in Rnn−q if and only if trajectory of any nontrivial element ϕ from E1n−q does not remain in E1n−q . From the matrix M (α), we extract the minors     nq+1 , . . . , nn n1 , . . . , nq , M2 . M1 1, . . . , n nq+1 , . . . , nn We denote by M1 (α) and M2 (α) the corresponding matrices of dimensions (n−q)×n and q×(n−q). From definitions and the form of the matrices M1 (α), M2 (α), we obtain the following two properties. Property 17.2. The trajectory of any vector c from Rn passes through subspace Rnq if and only if there is αc ∈ G for any vector c from Rn such that M1 (αc )c = 0. Property 17.3. Assume one of the following conditions: (1) q ≥ n/2 and there exists α0 ∈ G such that rank M2 (α0 ) = n − q; (2) q ≤ n − 2 and if only one row of matrix M2 (α) consists of the linear independent functions; (3) the trajectory o(c) of any nontrivial vector c from Rn passes through Rnq . Then the trajectory of any nontrivial vector c from Rnn−q does not remain in Rnn−q .

Interlaced and Potential Branching Eqaution

277

Example 17.2.   cos α  M (α) =   − sin α

 sin α   . cos α 

We set M1 (α) = (− sin α, cos α), M2 (α) = sin α. Then for any c ∈ R2 there exists αc such that M1 (αc )c = 0. Besides for all α different from πm, M2 (α) = 0. 17.4.

Interlaced Branching Equation of Potential Type

The branching equation (17.4) satisfying the condition L(ξ, λ) = d gradξ

U (ξ, λ)

in the neighborhood of zero with det d = 0 is called BEq of potential type, and the function U is its potential. Property 17.4. Let (17.4) be α-parametric interlaced. Let there exist α0 ∈ G, such that det M (α0 ) = 0 and L(T (α0 )ξ, λ) = gradξ U (ξ, λ). Then (17.4) is BEq of potential type with d = M (α0 )−1 . Theorem 17.5. Let conditions (17.4)–(17.5) hold, where C(α0 ) = D(α0 ), det M (α0 ) = 0. Let the matrix Rx (I − ΓRx )−1 S(α0 )ϕk , ψi ni,k=1 be symmetric in the neighborhood of zero. Then (17.4) is BEq of potential type with d = M (α0 )−1 . Proof.

By definition,

Li (T (α0 )ξ, λ) = y((ξ, S(α0 )ϕ), λ), ψi , i = 1, . . . , n,

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Toward General Theory of Differential-Operator and Kinetic Models

where y is a unique small solution of equation y = R((ξ, S(α0 )ϕ) + Γy, λ).

(17.25)

From (17.25), by differentiation of the superposition, we obtain

∂y ∂y , (17.26) = Rx S(α0 )ϕk + Γ ∂ξk ∂ξk hence in the neighborhood of zero, S(α0 )ϕk + Γ

∂y = (I − ΓRx )−1 S(α0 )ϕk . ∂ξk

(17.27)

By substituting (17.27) into (17.26), we obtain ∂y = Rx (I − ΓRx )−1 S(α0 )ϕk . ∂ξk According to the condition of the theorem, the matrix    n  ∂y   ∂Li (T (α0 )ξ, λ) n    = , ψi     ∂ξk ∂ξk i,k=1 i,k=1 is symmetric in the neighborhood of zero, and hence, L(T (α0 )ξ, λ) = gradξ U (ξ, λ). Therefore, from Property 17.4, L(ξ, λ) = M (α0 )−1 gradξ U (ξ, λ). 

The theorem is proved.

Corollary 17.4. Let the conditions (17.5) and (17.6) be valid, where C(α0 ) = D(α0 ) and det M (α0 ) = 0. Let all matrices  n n      Mis (α0 )ψs  , m = 0, 1, . . . (17.28)  Rx (ΓRx )m ϕk ,   s=1

i,k=1

be symmetric in the neighborhood of zero. Then (17.4) is BEq of potential type with d = M (α0 )−1 .

Interlaced and Potential Branching Eqaution

279

In view of the equalities, ∞ 

Rx (I − ΓRx )−1 S(α0 ) = K(α0 )Rx

(ΓRx )m ,

m=0

K ∗ (α0 )ψi =

n 

Mis (α0 )ψs , i = 1, . . . , n,

s=1

the proof is simply evident. Symmetrized operators enable us to provide sufficient conditions for the symmetry of the matrices (17.28) in the general case when E2 = E1∗ . Let A : D(A) ⊂ E1 → E2 , J : E2 → E1∗ be linear operators, and P : E1 → E1 be the projector such that P (D(A)) ⊂ D(A). Lemma 17.3. Let J = P ∗ J on R(A). Then the operator JA is symmetric on D(A) if and only if (1) JA = JAP on D(A); (2) P ∗ JAP is symmetric on D(A). Proof. If (1) and (2) hold, then on D(A), the equality JA = JAP = P ∗ JAP is valid, where P ∗ JAP is a symmetric operator on D(A). Thus, the sufficiency of conditions (1) and (2) is proved. We prove the necessity. Let the operator JA be symmetric on D(A). Then, for all x1 , x2 ∈ D(A), we have x1 , JA(I − P )x2  = (I − P )x2 , JAx1  = x2 , (I − P ∗ )JAx1 , where (I − P ∗ )J = 0. Hence, for any x1 , x2 ∈ D(A), x1 , JA(I − P )x2  = 0, i.e., JA = JAP on D(A). Thus, the necessity of condition (1) is proved. Next, by what has been proved, one has JA = JAP = P ∗ JAP , where JA is a symmetric operator on D(A). Hence, the  necessity of (2) is proved.

280

Toward General Theory of Differential-Operator and Kinetic Models

Lemma 17.4. Let C : D(C) ⊂ E1 → E2 be continuously invertible linear operator and operator JC be symmetric on D(C). Then C −1 y1 , Jy2  = C −1 y2 , Jy1 ,

y1 , y2 ∈ E2 .

(17.29)

If here R(C −1 A) ⊂ D(A), and the operator JA is symmetric on D(A), all operators JA(C −1 A)m , m = 0, 1, . . . , are symmetric on D(A). Proof. Since x1 , JCx2  = x2 , JCx1  for x1 , x2 ∈ D(C) and operator C is continuously invertible, then setting x1 = C −1 y1 , x2 = C −1 y2 , we obtain (17.29). We establish the second claim of the lemma by the method of mathematical induction. Indeed, the operator JA is symmetric by the condition. Let it be proved now that the operator JA(C −1 A)m is symmetric. Then x1 , JA(C −1 A)m+1 x2  = x1 , JA(C −1 A)m C −1 Ax2  = C −1 Ax2 , JA(C −1 A)m x1  = C −1 A(C −1 A)m x1 , JAx2  = x2 , JAC −1 A(C −1 A)m x1  = x2 , JA(C −1 A)m+1 x1 . 

The lemma is proved.

In the case of Fredholm operators, the most complete results follow from Lemmas 17.3 and 17.4 with direct evidence of symmetrized operator J. We set ˆ=B+ B

n  ·, γi zi , i=1

P =

n n   ·, γi ϕi , Q = ·, ψi zi . i=1

i=1

Interlaced and Potential Branching Eqaution

281

Corollary 17.5. Let J=

n 

Mik ·, ψi γk ,

i,k=1

ˆ is symmetric on D(B), and where Mik = Mki . Then the operator J B ∗ the operator J Γ is symmetric on E2 . Proof.

Since  x1 , J

n 

 x2 , γi zi

=

i=1

n 

Mik x1 , γk x2 , γi ,

i,k=1

ˆ is symmetric on D(B). Since the where Mik = Mki , the operator J B ˆ is continuously invertible and there is an operator operator B ∗

J =

n 

Mik ·, γi ψk ,

i,k=1

then, by Lemma 17.4, the operator J ∗ Γ is symmetric on E2 , and the corollary is proved.  Corollary 17.6. Let B be a Fredholm operator and there exist an operator J ∈ L(E2 , E1∗ ) such that ˆ is symmetric on D(B); (1) the operator J B (2) the operator JRx (x, λ) is symmetric on D(B) for all (x, λ) from the neighborhood of zero (or QRx (I − P ) = 0 and the operator JRx is symmetric on N (B)); (3) J ∗ ϕ = M ψ. Then all matrices  n n      Mis ψs   Rx (ΓRx )m ϕk ,   s=1

, m = 0, 1, . . .

i,k=1

are symmetric for all (x, λ) from the neighborhood of zero.

(17.30)

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Toward General Theory of Differential-Operator and Kinetic Models

Note that owing to Lemma 17.4, the operator J ∗ Γ is symmetric on E2 . Consider two cases. Let, first, the operator JRx be symmetric on D(B). Then, by Lemma 17.4, x1 , JRx (ΓRx )m x2  = x2 , JRx )m x1 , m = 0, 1, . . .

(17.31)

for all x1 , x2 from D(B). While taking into account condition (3) and setting x1 = ϕi , x2 = ϕk , we obtain the desired symmetry of matrices (17.30). Let now QRx (I − P ) = 0 and the operator JRx be symmetric on N (B). Then ΓRx (N (B)) ⊂ N (B). Hence, the identities (17.31) hold true for x1 , x2 from N (B), and the proof is carried out as in the first case. Example 17.3. Let J=

n 

Mik ·, ψi γk ,

i,k=1

where Mik = Mki . Then conditions (1) and (3) of corollary are valid; moreover, the matrix M is symmetric. Here, condition (2) denotes the symmetry of matrix  n n      Mjk ψk  ,  Rx ϕi ,   k=1

i,j=1

and one of the two additional conditions (I − Q)Rx P = 0 or QRx (I − P ) = 0. Theorem 17.6. Let the conditions (17.5) and (17.6) be satisfied, where C(α0 ) = D(α0 ) and the matrix M (α0 ) is invertible. Then if all the conditions of corollary hold, BEq (17.4) is of potential type. Let us consider the properties of potential to BEq of potential type.

Interlaced and Potential Branching Eqaution

283

Property 17.5 (T -invariance of potential). Let BEq (17.4) be (T, M )-interlaced of potential type, where matrices T, M are nonsingular. Then the potential U is T -invariant if and only if M dT  = d.

(17.32)

Proof. Without loss of generality, we consider U (0, λ) = 0. By potentiality of equation (17.4) and the Lagrange mean value theorem,  1 U (ξ, λ) = (d−1 L(θξ, λ), ξ) dθ. (17.33) 0

Furthermore, by potentiality and (T, M )-interlacing, d gradξ U (ξ, λ) = M −1 L(T ξ, λ). While applying the Lagrange theorem again, we obtain  1 (d−1 M −1 L(θT ξ, λ), ξ) dθ. U (ξ, λ) =

(17.34)

0

In (17.33) let us set ξ = T ξ. Then  1 (d−1 L(θT ξ, λ), T ξ) dθ. U (T ξ, λ) =

(17.35)

0

The potential U is T -invariant if U (ξ, λ) = U (T ξ, λ) for all ξ, λ. The last one, by (17.34) and (17.35), will be satisfied if and only if  1 ((T  d−1 − d−1 M −1 )L(θT ξ, λ), ξ) dθ = 0 0

for all ξ, λ, i.e., if T  d−1 − d−1 M −1 = 0. This is equivalent to (17.32).  The property is proved.

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Chapter 18

Epilogue for Part-II The theory of interlaced branching equations is also presented in detail in original papers [102, 147, 148] in the Russian language. This theory is connected with the representation theory and group analysis methods. Methods of these papers and methods presented in this part enable the construction of the parametric asymptotics of branching solutions in various problems in mathematical physics and mechanics. Here, readers may refer to [143, Chapter 5; 107]. Using these methods, the new problems appearing in mathematical modeling (e.g., considered in [44]) can be formulated and studied.

285

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Part III

Kinetic Models

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Chapter 19

The Family of Steady-State Solutions of Vlasov–Maxwell System 19.1.

Ansatz of the Distribution Function and Reduction of Stationary Vlasov–Maxwell Equations to Elliptic System

Let us consider stationary VM system   ∂ qi 1 ∂ fi = 0, E+ v×B v fi + ∂r mi c ∂v

i = 1, 2,

(19.1)

rot E = 0,

(19.2)

div B = 0,

(19.3)

div E = 4π

2  j=1

 qj

R3

fj (v, x) dv,

 2 4π  qj vfj (v, x) dv, rot B = c R3 j=1   fi (v, x) dx dv = 1. R3

(19.4)

(19.5)

(19.6)

Ω1

We assume that q1 > 0 and q2 < 0, which means that f1 (v, x) and f2 (v, x), respectively, are the ion and electron distribution functions. Taking into account condition (19.6), we also find the stationary distribution functions of the form fi (v, x) = fi (−αi v 2 + ϕi , vdi + ψi ) = fˆ(R, G), 289

(19.7)

290

Toward General Theory of Differential-Operator and Kinetic Models

where R = −αi v 2 + ϕi , G = vdi + ψi and the corresponding electromagnetic fields E(x), B(x) satisfy the system of equations (19.2)–(19.5). Here, we assume ϕi : R3 → R,

ψi : R3 → R,

x ∈ Ω1 ⊆ R3 ,

αi ∈ R+ = [0, ∞),

di ∈ R3 .

v ∈ R3 ,

Let us build the system of equations determining the functions ϕi , ψi . After substituting (19.7) into (19.1) we arrive at the relationships mi ∇ϕi , 2αi qi mi c ∇ψi , B(x) × di = − qi E(x) =

(E(x), di ) = 0.

(19.8) (19.9) (19.10)

Moreover, it follows from (19.8) and (19.10) that (∇ϕi , di ) = 0.

(19.11)

On the other hand, from (19.9), we obtain (∇ψi , di ) = 0.

(19.12)

Let us note that (B(x), di ) = λi , where λi (x) is an arbitrary function at this moment. Thus, for definition of B(x), we need the joint solution of equations (19.9) and (19.12) taking into account a fact that (B(x), di ) = λi (x). Vector B(x) takes the form B(x) =

mi c λi (x) di − di × ∇ψi , 2 di qi d2i

(19.13)

where d2i = d21i + d22i + d23i . The most outstanding fact is that fields E(x) and B(x), determined by means of formulas (19.8) and (19.9), do not depend on index i. Hence, the functions ϕi , ψi can be searched in the form ϕi = ϕˆi + ϕi (x),

ψi = ψˆi + ψi (x),

(19.14)

where ψˆi and ϕˆi are constants; functions ϕi (x) and ψi (x), respectively, satisfy the relations ϕ2 (x) = l1 ϕ1 (x) and ψ2 (x) = l2 ψ1 (x).

The Family of Steady-State Solutions of Vlasov–Maxwell System

291

Parameters l1 and l2 , with respect to (19.8) and (19.9), are connected with the relations m1 m2 = l1 , (19.15) α1 q1 α2 q2 q2 q1 l2 d1 = d2 . (19.16) m1 m2 In this case, l1 =

m1 α2 q2 < 0, m2 α1 q1

mi > 0,

αi > 0,

q1 q2 < 0.

As it follows from (19.16), vectors d1 and d2 are linearly dependent. Since ϕˆi and ψˆi defined in (19.14) are constants, then ∇ϕi = ∇ϕi (x), ∇ψi = ∇ψi (x). By substituting (19.7) and (19.8) into (19.4), we obtain the equations  2  mi ϕi (x) = 4π qj fj (v, x)dv. (19.17) 2αi qi R3 j=1

Since div[di × ∇ψi ] = 0 and taking into account (19.12), substituting (19.13) into (19.3) gives (∇λi (x), di ) = 0.

(19.18)

By substituting (19.13) into (19.5), we obtain the system of equations  2 mi c 4π 2  d di ψi (x) + qj vfj dv. ∇λi × di = qi c i R3 j=1

The above system is solved if and only if functions ψi (x) satisfy the equations  2 4π  mi c ψi (x) = qj (v, di )fj dv. (19.19) − qi c R3 j=1

In this case,    2 γi 4π  qj di × vfj dv ∇λi (x) = 2 di + c di R3 j=1

(19.20)

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Toward General Theory of Differential-Operator and Kinetic Models

because [di × dj ] = 0, i, j = 1, 2. Finally, combining (19.20) and (19.18) and di = 0, we obtain    2 4π  qj di × vfj dv . ∇λi (x) = c R3

(19.21)

j=1

Relations (19.17) and (19.19) are the desired system of elliptic equations related to the functions ϕi (x) and ψi (x), respectively. Thus, the problem of finding the steady-state solutions of the VM system (19.1)–(19.5) led to a joint study of equations (19.17) and (19.19) with conditions of orthogonality (19.11)–(19.12) and normalization condition (19.6). Systems (19.17) and (19.19) will be studied, assuming fi (x, v) = e−αi v

2 +ϕ

i (x)+di v+ψi (x)

.

(19.22)

In this case, 

 R3

fj (v, x)dv =

π αj

3/2

ϕi +ψi

e

e

d2 j 4αj

,

(19.23)

 (di , dj ) (v, di )fi (v, x)dv = fj (v, x)dv, 2αj R3 R3   di vfj dv = di × fj (v, x)dv, di × 2αj R3 R3



(19.24) (19.25)

where i, j = 1, 2. Since vectors d1 and d2 are linearly dependent, then the right-hand sides in (19.25) are equal to zero. Therefore, if fi (v, x) is defined by (19.22), then due to (19.21), ∇λi (x) = 0. Hence, λi (x) = βi is constant in (19.13). The right-hand sides of (19.17) and (19.19) are completely defined due to (19.23) and (19.24) and the last one takes the form  3/2 d2 2 j  π mi c ϕj +ψj 4αj ϕi = 4π qj e e , (19.26) 2αi qi αj j=1

  2 d 4π  (di , dj ) π 3/2 ϕj +ψj 4αjj mi c ψi = qj e e , − qi c 2αj αj 2

j=1

(19.27)

293

The Family of Steady-State Solutions of Vlasov–Maxwell System

where i = 1, 2, ϕi = ϕˆi + ϕi (x), ψi = ψˆi + ψi (x), (d1 , d2 ) = kd21 ,

d22 = k2 d21 ,

α2 l2 . α1 l1

k=

Rewriting (19.26)–(19.27) in the vector form ϕ = Af (ϕ + ψ),

(19.28)

ψ = Bf (ϕ + ψ),

(19.29)

where

     a a b     2α1 cy1  x1 x1  , B =  A =   b  a  a   x x2 2α1 cy2 k 2     ψ  ϕ   1  1 ϕ =  , ψ =  , ψ2  ϕ2    eϕ1 +ψ1    f (ϕ + ψ) =  ϕ +ψ , 2 2  e  a = 4πq1 xi =

mi , 2αi qi

π α1

3/2

e

yi = −

d2 1 4α1

 bk   2α2 cy1  , b  2α cy  2

 ,

mi c , qi d2i

b = 4πq2

2

π α2

3/2

e

d2 2 4α2

,

i = 1, 2,

and introducing notations,

  Φ   1 G = A + B, Φ =  , Φ2 

Φi = ϕi + ψi ,

after summation, one obtains equation Φ = Gf (Φ),

(19.30)

instead of (19.28)–(19.29). For studying (19.26)–(19.27), we use the following results. Lemma 19.1. Let Φ satisfy the system of equations (19.30). Then the system (19.28)–(19.29) has a solution ϕ = ϕ0 +u1 , ψ = −ϕ0 +u2 ,

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Toward General Theory of Differential-Operator and Kinetic Models

where ϕ0 is an arbitrary harmonic vector function, u1 and u2 are vector functions satisfying the linear Poisson equations u1 = Af (Φ),

u2 = Bf (Φ).

Moreover, if det G = 0, then u1 = AG −1 Φ,

u2 = BG −1 Φ.

With the study of (19.30), let us consider two cases: (1) det G =  0, (2) det G = 0. Lemma 19.2. If det G = 0, then solution of the system (19.30) of the form   Φ   ˆ 1 + u Φ=  ˆ 2 + u Φ corresponds to the solution of algebraic system   1 ˆ = I, I =  , Gf (Φ) 1 ˆ = −I), (Gf (Φ)

(19.31) (19.32)

where function u satisfies the Liouville equation u = eu

(u = −eu ).

Remark 19.1. Analogous result occurs in the case, when G is the nonsingular matrix of the n-th order. Lemma 19.3. If det G = 0, then a general solution λ | cc12 |, where ˆ = 0, has a λ, an arbitrary constant of the uniform system Gf (Φ) correspondence to the solution family of the system (19.30) of the form    Φ  ˆ1 + u  (19.33) Φ= , ˆ 2 + lu Φ where u satisfies the equation of the type u = λc3 (eu − elu ), l, c1 , ˆ i = ln λci . c2 , c3 are constants and Φ

The Family of Steady-State Solutions of Vlasov–Maxwell System

Proof.

295

Since det G = 0, then there exists a constant l such that    a  11 a12  G= . la11 la12 

Therefore substituting (19.33) into (19.30), one obtains ˆ

ˆ

u = a11 eΦ1 +u + a12 eΦ2 +lu , where

   ˆ   1  eΦ1       ˆ  = λ  a11  eΦ2  − a  12

if a12 = 0. Since a11

ˆ eΦ1

+ a12

ˆ

ˆ eΦ2

= 0, then

ˆ

a11 eΦ1 +u + a12 eΦ2 +lu = λa11 [eu − elu ]. 19.1.1.



Boundary value problem

In this section, we aim at constructing the distribution functions fi (v, x), electromagnetic fields E(x), B(x) and setting the adequate boundary value problems. We will consider the distribution functions fi (v, x) and the fields E(x) and B(x) corresponding to equations (19.22), (19.8), (19.13), where functions ϕi and ψi satisfy (19.26), (19.27), (19.14)–(19.16) and conditions (19.11) and (19.12). Let us consider two cases: (1) l2 = (2) l2 =

m1 α2 q2 m2 α1 q1 , m1 α2 q2 m2 α1 q1 ,

i.e., l2 = l1 ; i.e., l2 = l1 .

Case (1) refers to Lemma 19.3. In fact, since det G = −

ab (l2 − l1 )2 , 2cα1 l1 x1 y1

then det G < 0 for l2 = l1 and det G = 0 for l2 = l1 . Thus, in the first case, det G = 0. Therefore, it is possible to use Lemma 19.3. 1 α2 q2 We note, due to (19.15) and (19.16), l2 = l1 = l = m m2 α1 q1 < 0, α2 d1 = α1 d2 .

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Toward General Theory of Differential-Operator and Kinetic Models

In this case, matrix G can be transformed as      1 1 a b − G=  . x1 2α1 cy1 la lb    Φˆ 1 +u  Since Φ =  ˆ , then (19.30) degenerates into one equation: Φ2 +lu



u =

1 1 − x1 2α1 cy1



ˆ

ˆ

(aeΦ1 +u + beΦ2 +lu ).

(19.34)

ˆ 2 are defined from system ˆ 1 and Φ Moreover, Φ  ˆ  eΦ1    G= ˆ =0 eΦ2  due to Lemma 19.3. Since we are interested in real solutions, sign ˆ 1 = ln λ, Φ ˆ 2 = ln | − λ a |, where λ ∈ R+ is ab = sign q1 q2 < 0, then Φ b an arbitrary parameter. Equation (19.34) takes the form u = λa11 (eu − elu ),

(19.35)

where d1 2πq12 4α 2 2 2 1 (4α c − d ) = e 1 1 α1 m1 c2 2

a11



π α1

3/2

,

l ∈ R− .

Let x ∈ Ω2 ⊂ R3 , Γ = ∂Ω. We search for the nontrivial solution of (19.35) satisfying boundary condition u|Γ = 0

(19.36)

(∇u, d1 ) = 0.

(19.37)

and relation

With respect to (19.37), one needs   y y z x − , − u=u d11 d12 d12 d13

The Family of Steady-State Solutions of Vlasov–Maxwell System

297

if d1k = 0, k = 1, 2, 3. Besides, the three-dimensional problem (19.35)–(19.37) is transformed into a two-dimensional one. Dirichlet problem (19.35)–(19.37) has a trivial solution u = 0. Nonlinear Dirichlet problem (19.35)–(19.37) can have a small nontrivial real solution u → 0 for λ → λ0 in the neighborhood of eigenvalue λ0 of linearized problem u = λa11 (1 − l)u, x ∈ Ω2 ⊂ R3 ,  uΓ = 0, (∇u, d1 ) = 0. Assume without loss of generality that λ0 is the smallest eigenvalue. Then if it is unique, in the semi-neighborhood of the point λ0 , there exists a small nontrivial solution u → 0 at λ → λ0 (λ → −λ0 ). We note that Lyapunov–Schmidt branching equation of nonlinear problem has the form L(μξ) = ∇ξ U (ξ, μ), μ = λ − λ0 . This equation is potential for any eigenvalue of linearized problem. Therefore, any eigenvalue λ0 will be a bifurcation point. If λ0 is odd multiple, then the real solution exists at least in the semi-neighborhood of the point λ0 . The detailed description of the domain Ω2 allows one to build the asymptotic behavior of the corresponding branches of nontrivial solutions u of (19.35). For determining the functions ϕi and ψi , we use Lemma 19.1. ˆ 1 + u, Knowing vector function Φ, we construct functions ϕ1 + ψ1 = Φ ˆ 2 + lu, substituting them in the right-hand side of the ϕ2 + ψ2 = Φ first equation of (19.26). As a result, we obtain the linear equation m1 ϕ1 (x) = λa(eu − elu ), 2α1 q1 ϕ1 (x)|Γ = 0,

(∇u, d1 ) = 0.

We find ϕ1 (x) in the form ϕ1 (x) = Θu, where Θ is a constant. Then we arrive at the equation Θ

m1 u = λa(eu − elu ), 2α1 q1

u|Γ = 0,

(∇u, d1 ) = 0.

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Toward General Theory of Differential-Operator and Kinetic Models

1 q1 a Assuming Θ = 2α m1 a11 , we obtain identity because u is a solution of (19.35) under conditions (19.36) and (19.37). Hence,

ˆ 1 + Θu + ϕ0 (x), ϕ1 (x) = Φ where ϕ0 (x) is an arbitrary harmonic function. Since E(x) = m1 m1 2α1 q1 ∇ϕ1 (x), then 2α1 q1 ϕ1 (x) is a potential of the required electric field. Assume that the value of this potential P0 is given on the boundary Γ, then function ϕ0 (x) is defined concretely. For its determination, we obtain linear Dirichlet boundary value problem ϕ0 (x) = 0, ϕ0 (x)|Γ = P0 −

m1 ˆ Φ. 2α1 q1

ˆ 1 + u, then having obtained ϕ1 , we construct ψ1 Since ϕ1 + ψ1 = Φ ˆ u, ϕ1 , ψ1 and using by formula ψ1 = (1 − Θ)u − ϕ0 (x). Knowing Φ, (19.8), (19.13) and ( 19.22), we find the required E(x), B(x), fi : fi (v, x) = e−αv +vdi +Φi (x) , m1 ∇(Θu + ϕ0 (x)), E(x) = 2α1 q1 2

B(x) =

m1 c β1 d1 − [d1 × ∇[(1 − Θ)u − ϕ0 (x)]], 2 d1 q1 d21

where Φ1 (x) = ln λ + u, Φ2 (x) = ln| − λ ab | + lu, d1 ∈ R3 ,

d2 =

α2 d1 , α1

αi ∈ (0, ∞),

l=

m1 α2 q2 < 0. m2 α1 q1

Further, we consider system of Vlasov–Maxwell equations (19.1)– (19.5) with condition  (19.38) (E, n0 )Γ = e0 . Finding stationary distributions in the form (19.7), as above, we derived (19.17), (19.19) and (19.20).

The Family of Steady-State Solutions of Vlasov–Maxwell System

299

Let us assume that the following condition holds: A fi (v, x) = fi (−αi v 2 + vdi + ϕi + ψi ),   fi (v, x)dv < +∞, vfi (v, x)dv < +∞. R3

R3

Systems (19.17) and (19.19) possess the specific symmetry expressed by the following property. Property 19.1. The second equations in systems (19.17) and (19.19) coincide with the first ones. For the proof of Property 19.1, it suffices to take into consideration that ϕ2 (x) = l1 ϕ1 (x),

ψ2 (x) = l2 ψ1 (x),

taking into account (19.15) and (19.16). Lemma 19.4. If for some function Φ(x), α ∈ R+ , d ∈ R3 , the conditions  f (−αv 2 + vd + Φ(x))dv < +∞, R3

 R3

vf (−αv 2 + vd + Φ(x))dv < +∞

are satisfied, then   d 2 vf (−αv + vd + Φ(x))dv = f (−αv 2 + vd + Φ(x))dv. 2α R3 R3 (19.39) d , we have identity Proof. Since under change of variables v = ξ + 2α     d2 2 2 + Φ dξ vf (−αv + vd + Φ(x))dv = ξf −αξ + 4α R3 R3    d2 d 2 + Φ(x) dξ, f −αξ + + 2α R3 4α

300

Toward General Theory of Differential-Operator and Kinetic Models

then   2 vi f (−αv + vd + Φ(x))dv =

 d2 + Φ dξ ξi f −αξ + 4α R3    di d2 2 + Φ(x) dξ, + f −αξ + 2α R3 4α

R3



2

ξ ∈ R3 , i = 1, 2, 3. Introducing the spherical coordinates  ≥ 0, 0 ≤ ϕ ≤ π, 0 ≤ Θ ≤ 2π, ξ1 =  sin ϕ cos Θ, ξ2 =  sin ϕ sin Θ, ξ3 =  cos ϕ, it is easy to see that    d2 2 + Φ(x) dξ = 0, i = 1, 2, 3. ξi −αξ + 4α R3 Thus,



 R3

Since ξ = v −

vi f d 2α ,

  di d2 + Φ(x) dv = −αv + f dξ. 4α 2α R3 2



then (19.39) is satisfied.

Lemma 19.5. Let the distribution functions f1 and f2 satisfy condition (A). Then system (19.17), (19.19) and (19.20) are transformed to the form pϕ(x) = q1 A1 (λ1 + ϕ(x) + ψ(x)) + q2 A2 (λ2 + l1 ϕ(x) + l2 ψ(x)), (19.40) ℘ψ(x) = q1 A1 (λ1 + ϕ(x) + ψ(x)) ×(λ + l1 ϕ(x) + l2 ψ(x))

d21 + q2 A2 2α1

(d1 , d2 ) , 2α2

(19.41)

where ϕ(x) = ϕ1 (x),

ψ(x) = ψ1 (x),

λ1 = c11 + c21 ,

p=

λ2 = c12 + c22 ,

Functions A1 and A2 are defined in (19.42).

m1 , 8πα1 q1 ℘=−

m1 c2 . 4πq1

301

The Family of Steady-State Solutions of Vlasov–Maxwell System

Proof. Due to Property 19.1, the second equations in (19.17) and (19.19) can be omitted. Since we have identities  fi (−αi v 2 + vdi + ϕi + ψi )dv = Ai , i = 1, 2 (19.42) R3

from Lemma 19.4, where 

A1 = A1 (λ1 + ϕ + ψ),



A2 = A2 (λ2 + l1 ϕ + l2 ψ),

then the first equations in (19.17) and (19.19) can be written in the form of (19.40) and (19.41). Since   dk vfk dv = − fk dv, di × dk = 0, i, k = 1, 2, 2αk R3 R3 

then the lemma is proved.

As already mentioned, during the study of systems (19.17) and (19.19), two cases were detected. Case 1. l2 = α2 q2 m1 /α1 q1 m2 . Then l2 = l1 , α2 d1 = α1 d2 . System (19.40)–(19.41) takes a form

ϕ(x) = q1 A1 (λ1 + ϕ + ψ) + q2 A2 (λ2 + l(ϕ + ψ)),

(19.43)

μψ(x) = q1 A1 (λ1 + ϕ + ψ) + q2 A(λ2 + l(ϕ + ψ)), 

l1 = l2 = l,

=

m1 , 8πα1 q1

μ=−

α1 c2 m1 . 2πq1 d21

(19.44)

Let us consider the nonlinear equation u = a(q1 A1 (λ1 + u) + q2 A2 (λ2 + lu)), a=

2πq1 (4α21 c2 − d21 ), α1 c2 m1

u = ϕ(x) + ψ(x).

(19.45)

If u∗ satisfies (19.45), then the system of equations (19.43)–(19.44) has a solution ϕ(x) = Θu∗ (x) + ϕ0 (x),

ψ(x) = (1 − Θ)u∗ (x) − ϕ0 (x),

302

Toward General Theory of Differential-Operator and Kinetic Models

where Θ = 4α21 c2 /(4α21 c2 − d21 ), ϕ0 is an arbitrary harmonic function. Since solution of (19.43)–(19.44) is expressed via the solution of equation (19.45) and harmonic function, then in Case 1, the VM system is reduced to a study of “resolving” equation (19.45). Hence, we have the following theorem. Theorem 19.1. Let distribution functions f1 and f2 satisfy condition (A). Then the corresponding solution of system (19.1)–(19.5) can be written in the form f1 (v, x) = f1 (−αv 2 + vd1 + λ1 + u∗ (x)),

d1 ∈ R3 ,

f2 (v, x) = f2 (−αv 2 + vd2 + λ2 + lu∗ (x)), d2 ∈ R3 , m1 ∇(Θu∗ (x) + ϕ0 )), E(x) = 2α1 q1 m1 c γ1 [d1 × ∇((1 − Θ)u∗ (x) − ϕ0 )], B(x) = 2 d1 − d1 q1 d21

(19.46)

where u∗ (x) is defined from (19.45), γ1 , λ1 , and λ2 are arbitrary constants, ϕ0 (x) is an arbitrary harmonic function, (∇u∗ , d1 ) = 0, and (∇ϕ0 (x), d1 ) = 0. Let us consider solutions of “resolving” equation (19.45). If arbitrary constants λ1 and λ2 are connected by means of equation q1 A1 (λ1 ) + q2 A2 (λ2 ) = 0,

(19.47)

then (19.45) has a trivial solution u∗ (x) = 0. In this case, the construction of nontrivial solutions of (19.45) is a well-known problem about the bifurcation point for nonlinear equations. For the solution to this problem, we need to use the boundary condition (19.38). Due to (19.38), one has   2α1 q1 ∂ ∗ (Θu (x) + ϕ0 (x)) = e0 (x). ∂n m1 Γ Assuming

 ∂ ∗  u (x) = 0, ∂n Γ

  ∂ 2α1 q1 ϕ0 (x) = e0 (x), ∂n m1 Γ

The Family of Steady-State Solutions of Vlasov–Maxwell System

303

we obtain linear boundary value problem ϕ0 (x) = 0,   2α1 q1 ∂ ϕ0 (x) = e0 (x), ∂n m1 Γ

(∇ϕ0 (x), d1 ) = 0,

(19.48)

and nonlinear boundary value problem u(x) = a(q1 A1 (λ1 + u) + q2 A2 (λ2 + lu)),   ∂ u(x) = 0, (∇u(x), d1 ) = 0, ∂n Γ

(19.49)

a = 2πq1 (4α21 c2 − d21 )/α21 c2 m1 , where arbitrary constants λ1 and λ2 satisfy (19.47). Since (19.49) would have the nontrivial solution u → 0, λ → λ∗ , where λ = (λ1 , λ2 ) satisfies (19.47), it is necessary that linearized problem ϕ(x) = a[q1 A1 (λ∗1 ) + q2 A2 (λ2 )]ϕ(x),   ∂ ϕ(x) = 0, (∇ϕ(x), d1 ) = 0 ∂n Γ

would have the nontrivial solution. For the construction of nontrivial branches of solution u → 0, λ → λ∗ can be used in the detailed bifurcation methods of the solutions of nonlinear equations. Case 2. l2 = α2 q2 m1 /α1 q1 m2 . In this case, we restrict to the construction of solutions ϕ(x) and ψ(x) of (19.40) and (19.41) satisfying the condition ϕ(x) + ψ(x) = l1 ϕ(x) + l2 ψ(x).

(19.50)

Let us assume that the condition is satisfied. (B) There are constants λ1 and λ2 , which satisfy the identity   d21 (1 − l2 ) q1 A1 (λ1 + u) 4α1 (1 − l1 ) − α1 c2   (d1 , d2 ) (l2 − 1) = q2 A2 (λ2 + u) 4α1 (l1 − 1) − α2 c2 for functions f1 and f2 together with condition (A) and some u ∈ R1 .

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Toward General Theory of Differential-Operator and Kinetic Models

Lemma 19.6. Let condition (B) is satisfied. Then the solution of (19.40)–(19.41) with (19.50) has the form ϕ(x) =

l2 − 1 ∗ u (x), l2 − l1

ψ(x) =

1 − l1 ∗ u (x), l2 − l1

where u∗ (x) is a solution of equation u = bA1 (λ1 + u), b=

8πq12 (l2 − l1 )(d21 α2 − (d1 , d2 )α1 ) . m1 (4α1 α2 c2 (l2 − 1) − (d1 , d2 )(l2 − 1))

(19.51)

Proof. By means of nondegenerate linear change of variables u1 = ϕ + ψ, u2 = l1 ϕ + l2 ψ, the system (19.40)–(19.41) is transformed into u1 = (a1 − a2 )q1 A1 + (a1 − a3 )q2 A2 , u2 = (l1 a1 − l2 a2 )q1 A1 + (l1 a1 − l2 a3 )q2 A2 , where a1 =

8πα1 q1 , m1

a2 =

2πq1 d21 , α1 c2 m1

a3 =

2πq1 (d1 , d2 ) . α2 c2 m1

The right-hand sides of the last system are equal to each other in view of condition (B). Due to the same condition, equations over  u1 and u2 are reduced to (19.51). Theorem 19.2. Let conditions (A), (B), l2 = system (19.1)–(19.5), (19.38) has a solution fi (v, x) = fi (−αi v 2 + vdi + λ1 + u∗ (x)),

α2 q2 m1 α1 q1 m2

hold. Then

i = 1, 2,

E(x) =

m1 (l2 − 1) ∇u∗ (x), 2α1 q1 (l2 − l1 )

B(x) =

m1 c(1 − l1 ) γ1 d1 × ∇u∗ (x), d1 − 2 d1 2α1 d21 (l2 − l1 )

where γ1 is an arbitrary constant. Function u∗ satisfies (19.51) with the conditions  ∂ ∗  2α1 q1 (l2 − l1 ) ∗ u e0 (x). = (∇u , d1 ) = 0, ∂n Γ m1 (l2 − 1)

305

The Family of Steady-State Solutions of Vlasov–Maxwell System

Let us show the application of general Theorems 19.1 and 19.2. Assume that it is necessary to determine distributions f1 (v, x) = e−α1 v

2 +vd +Φ (x) 1 1

,

d1 ∈ R3 ,

f2 (v, x) = e−α2 v

2 +vd +Φ (x) 2 2

,

d2 ∈ R3 ,

(19.52)

v ∈ R3 , x ∈ Ω2 ∈ R3 and the corresponding fields E and B satisfy VM system (19.1)–(19.5) with boundary condition (19.38). Due to Lemma 19.5, unknown functions Φ1 (x) and Φ2 (x) can be represented as Φ1 (x) = λ1 + ϕ(x) + ψ(x),

Φ2 (x) = λ2 + l1 ϕ(x) + l2 ψ(x).

We suppose that parameters α1 , α2 , d1 , d2 , l1 , l2 satisfy conditions (19.15) and (19.16). Consider two cases. Case 1. l2 = α2 q2 m1 /α1 q1 m2 . Then for finding parameters λ1 and λ2 and functions ϕ(x) and ψ(x), equation (19.45) should be resolved and therefore, boundary value problems (19.48) and (19.49) are solved regarding Theorem 19.1. Since  3/2 d2 π λ + 1 e 1 4α1 eu , A1 (λ1 + u) = α1  3/2 d2 π λ + 2 e 2 4α2 elu , A2 (λ2 + lu) = α2 then condition (A) is satisfied. Equation (19.45) will have the nontrivial solution for some λ1 if    

d2 d2  α2 3/2 q1 λ1 + 14 α1 − α2 1 2 e . λ2 = ln − α1 q2 In this case, boundary value problem (19.49) is the Neumann problem for the equation of type u = σg(α1 , d1 )(eu − elu ),  ∂  u = 0, (∇u, d1 ) = 0, ∂n Γ

(19.53)

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Toward General Theory of Differential-Operator and Kinetic Models

where 

λ1

σ=e ,

g(α1 , d1 ) = ag1

π α1

3/2

d2 1

e 4α1 .

Problem (19.53) can have nontrivial solutions u(σ) → 0 for σ → σ ∗ only in the neighborhood of eigenvalues σ ∗ of the problem u = σ ∗ g(α1 , d1 )(1 − l2 )u,  ∂  = 0, (∇u, d1 ) = 0. ∂n Γ After obtaining eigenvalues σ ∗ (bifurcation points of (19.53) and having built solutions u(σ) → 0 for σ → σ ∗ , we find solutions of the VM system (19.1)–(19.5), (19.38) by formulas (19.46). Case 2. l2 = α2 q2 m1 /α1 q1 m2 and to find λ1 , λ2 , ϕ, ψ, we need to use Theorem 19.2. Moreover,  Ai (λi + u) =

π αi

3/2

d2

λi + 4αi

e

i

eu .

Therefore, condition (A) is satisfied. Also, (B) is reduced to (B  ): 

  d21 (1 − l2 ) e 4α1 (1 − l1 ) − q1 α1 c2  3/2   d2 π (d1 , d2 )(l2 − 1) λ + 2 . e 2 4α2 4α2 (l1 − 1) − = q2 α2 α2 c2 π α1

3/2

d2

λ1 + 4α1

1

Since q1 q2 < 0, then from (B  ), one can define eλ2 if parameters α1 , α2 , d1 , d2 , l1 , l2 satisfy the inequality  τ−

d21

w α1



w τ − (d1 , d2 ) α2

 > 0,

The Family of Steady-State Solutions of Vlasov–Maxwell System

307

2 where τ = 4α1 (1−l1 ) and w = 1−l c2 . The solution of equations (19.51) is the well-known Liouville equation:

u = σh(α1 , α2 , d1 , d2 )eu ,  2α1 q1 (l2 − l1 ) ∂  e0 , (∇u, d) = 0, =  ∂n Γ m1 (l2 − 1)  3/2 d2 1 π λ e 4α1 . σ = e , h(α1 , α2 , d1 , d2 ) = b α1

(19.54)

Thus, in the case of distributions of exponential form (19.52), the solution of (19.1)–(19.5), (19.38) is reduced to the linear Neumann problem for l2 = l1 and to the problem of bifurcation point for equation of sinh-Gordon type or to the Neumann problem for the Liouville equation (19.54), where σ ∈ R+ an arbitrary constant for l2 = l1 , and to additional restriction on parameters α1 , α2 , d1 , d2 , l1 , l2 included in (19.52). 19.1.2.

Solutions with norm

Let us examine the construction solutions with respect to the norm definition. With respect to normalization condition (19.6) and taking into account relation (19.23), free parameters α1 , α2 , d1 , λ, l must satisfy the conditions a Φˆ 1 e 4πq1 b Φˆ 2 e 4πq2

 

Ω2

Ω2

eu(x) dx = 1, (19.55) lu(x)

e

dx = 1,

ˆ 2 = ln | − λ a |. Therefore, ˆ 1 = ln λ, Φ where Φ b aλ 4πq1

 Ω2

eu(x) dx = 1,

−

aλ 4πq2

 Ω2

elu(x) dx = 1.

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Toward General Theory of Differential-Operator and Kinetic Models

Moreover, function u(x) satisfies equation (19.34), which takes the form   1 1 u = λa − (19.56) (eu − elu ). x1 2α1 cy1 Hence, one needs to solve (19.56) with condition (19.55). Thus, we obtain the solutions with norm. Excluding aλ from (19.55), we obtain the equation    q1 eu(x) q2 elu(x) 1 1  , (19.57) + − u = 4π lu u(x) dx x1 2α1 c1 y1 Ω2 e dx Ω2 e with condition  Ω2

  q1 elu(x) + q2 eu(x) dx = 0.

(19.58)

Thus, the normed solution u(x) leads to the problem (19.55)–(19.56) or to the problem (19.57)–(19.58). If solution u∗ (x) is found satisfying (19.57) and equation    1 q1 1 , eu − elu  − u = 4π u(x) dx x1 2α1 cy1 Ω1 e then condition (19.58) will also be satisfied for this solution, giving us a normalized solution. 1 Let l2 = αα21 qq21 m m2 . Then l2 = l1 and det G = 0. In this case, we use Lemma 19.2 for the study of (19.26)–(19.27). Since in Lemma 19.2 ˆ of (19.31)–(19.32), then we we are interested in the real solutions Φ need to find conditions, with the fulfillment of which systems     x  1  1   (19.59) G   =  , x2  1     1 x     1 (19.60) G   = −  1 x2 

The Family of Steady-State Solutions of Vlasov–Maxwell System

309

have the positive solutions x1 > 0, x2 > 0. If aij elements of matrix G, then G

−1

1 = det G

 −a12  . a11 

 a  22  −a21

Moreover, det G < 0. Therefore, one of the systems has the positive solution if D = (a22 − a12 )(a11 − a21 ) > 0. Since  l1 − 1 k(1 − l2 ) , =b + x1 2α2 cy1   1 − l1 l2 − 1 =a + , x1 2α1 cy1 

a22 − a12 a11 − a21 where ab < 0, l1 < 0, then 

1 + |l1 | k2 (l2 − 1) + D = |ab| x1 2α1 cy1



 l2 − 1 1 + |l1 | + . x1 2α1 cy1

Moreover D > 0 if 1 − l2 1 + |l1 | > max x1 2cy1



1 k , α1 α2



or 1 − l2 1 + |l1 | < min x1 2cy1



 1 k , . α1 α2

Substituting values l1 , x1 , y1 into the above inequalities, we obtain the new ones   l2 4α1 c2 (α1 q1 m2 + α2 |q2 |m1 ) > max 1 − l2 , (1 − l2 ) , l1 q1 d21 m2   (19.61) l2 4α1 c2 (α1 q1 m2 + α2 |q2 |m1 ) < min 1 − l2 , (1 − l2 ) . l1 q1 d21 m2

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Toward General Theory of Differential-Operator and Kinetic Models

Introducing notation L(αi , qi , mi , d1 ) =

4α1 c2 (α1 q1 m2 + α2 |q2 |m1 ), q1 d21 m2

we write (19.61) in the form   l2 , L(αi , qi , mi , d1 ) > max 1 − l2 , (1 − l2 ) l1   l2 L(αi , qi , mi , d1 ) < min 1 − l2 , (1 − l2 ) . l1

(19.62) (19.63)

We construct the straight line 1 − l2 on plane yOl2 and parabola y = (1 − l2 ) ll21 , where l1 is a given constant. If we fix αi , qi , mi , d1 , then l1 and value L(αi , qi , mi , d1 ) in (19.62), (19.63) will become specific constants. Let us draw straight line y = 1 and let us denote as α, β, γ abscissas of intersection points with parabola. Then for l2 ∈ (−∞, α), inequality (19.63) will be satisfied, a22 − a12 < 0, a11 − a21 < 0 and for l2 (β, γ), inequality (19.62) with a22 − a12 > 0, a11 − a21 > 0. Since det G < 0, then system (19.59) has the positive solution for l2 ∈ (−∞, α), and for l2 ∈ (β, γ), system (19.60) has the positive solution. ˆ 2 , we build vector ˆ 1 and Φ Having defined Φ    Φ  ˆ 1 + u Φ= , ˆ 2 + u Φ ˆ 1 and where u satisfies the Liouville equation u = eu if Φ ˆ 2 correspond to the solution of system (19.59) or the equation Φ ˆ 1 and Φ ˆ 2 correspond to the solution of (19.60). For u = −eu if Φ determining the functions ϕ1 and ψ1 , we use Lemma 19.1. Knowing Φ, we substitute it in the right-hand side of the first equation of system (19.26). One can obtain the linear equation m1 ˆ ˆ ϕ1 (x) = (aeΦ1 + beΦ2 )eu . 2α1 q1

(19.64)

The Family of Steady-State Solutions of Vlasov–Maxwell System

311

Let, for the definition, function u(x) satisfy equation u(x) = eu(x) , then for ˆ

Θ=

ˆ

2(aeΦ1 + beΦ2 )α1 q1 , m1

ˆ 1 + Θu(x). function ϕ1 (x) = Θu(x) satisfies (19.64). Thus, ϕ1 (x) = Φ Since Φ1 = ϕ1 + ψ1 , then ψ1 = (1 − Θ)u(x). We demonstrate that here functions fi (v, x) do not satisfy normalization condition (19.6) not with those values of the free parameters if q2 = −q1 . Indeed, it is necessary for normalizing that a Φˆ 1 b ˆ 1 e = eΦ2 =  , u(x) q1 q2 dx Ω2 e where

 ˆ    eΦ1  1     G  ˆ  =  . eΦ2  1

Therefore,     1 1 k 1 ˆ1 ˆ Φ − − e +b eΦ2 = 1, a x1 2α1 cy1 x1 2α2 cy1     l2 kl2 l1 l1 ˆ1 ˆ Φ − − e +b eΦ2 = 1, a x1 2α1 cy1 x1 2α2 cy1 or the same     a Φˆ 1 b Φˆ 2 q1 q2 kq2 q1 − e + − e = 1, x1 2α1 cy1 q1 x1 2α2 cy1 q2     q1 l2 q2 l1 q2 kl2 a Φˆ 1 b Φˆ 2 q1 l1 − e + − e = 1. x1 2α1 cy1 q1 x1 2α2 cy1 q2 Since

ˆ1 a Φ q1 e



q1

=

ˆ2 b Φ q2 e ,

then it is necessary that

1 − l2 1 − l1 − x1 2α2 cy1



 + q2

1 − l2 1 − l1 −k x1 2α2 cy1

 = 0.

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Toward General Theory of Differential-Operator and Kinetic Models

If q2 = −q1 , then it is necessary that α11 = αk2 . Since k = αα21 ll21 , then l2 1 α1 = α1 l1 , i.e., l1 = l2 . However, in our case, l1 = l2 providing us with contradiction. So, while l1 = l2 , the normalizing condition (19.6) is fulfilled for q2 = −q1 .

Chapter 20

Boundary Value Problems for the Vlasov–Maxwell System 20.1.

Introduction

At present, the investigation of the Vlasov equation goes in two different directions. The first direction is related to the existence theorems for the Cauchy problem and uses a priori estimation techniques as the basis for research. The second one implements the reduction of the initial problem to a simplified one introducing a set of distribution functions (ansatz), followed by reconstruction of the characteristics for electromagnetic fields in an evident form. This is a rather restrictive approach, since the distribution function has a special form. On the other hand, it allows to solve a problem in an explicit form, which is important for applications. The statement and investigation of the boundary value problem for the Vlasov equation are very difficult and have been considered in simplified cases only (see [19, 39, 63]). Reducing it to the boundary value problem for a system of nonlinear elliptic equations allows us to show solvability in some cases. Doing the same for the initial statement of the problem is not that simple. Nevertheless, both directions are related in terms of special structures used for studying kinetic equations. For example: • energy integral is applied in both cases for obtaining energy estimations in existence theorems and for construction of Lyapunov functionals;

313

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Toward General Theory of Differential-Operator and Kinetic Models

• virial identities in stability and instability analyses in special classes of solutions of the Vlasov equation. It is known that the solution of Vlasov equation (see [172, 173]) is an arbitrary function of first integrals of the characteristic system (until now, their smoothness remains a complicated unsolved problem), defining the trajectory of a particle motion in the electromagnetic field   1 qi (20.1) E(r, t) + V × B(r, t) , r˙ = V, V˙ = mi c 



where r = (x, y, z) ∈ Ω2 ⊆ R3 and V = (Vx , Vy , Vz ) ∈ Ω1 ⊂ R3 are, 

respectively, the position and velocity of a particle, E = (Ex , Ey , Ez ) 

is a tension of electrical field, B = (Bx , By , Bz ) is the magnetic induction and mi , qi are mass and charge of a particle of ith kind. For the N -component distribution function, the classical Vlasov–Maxwell (VM) system has the form   qi 1 E + V × B · ∇V fi = 0, ∂t fi + V · ∇r fi + mi c i = 1, . . . , N,

(20.2)

∂t E = c curl B − j,

(20.3)

div E = ρ,

(20.4)

∂t B = −c curl E,

(20.5)

div B = 0.

(20.6)

The charge and current densities are defined as follows: ρ(r, t) = 4π

n 

 qi

Ω1

i=1

j(r, t) = 4π

n  i=1

fi dV, (20.7)

 qi

Ω1

fi V dV.

Boundary Value Problems for the Vlasov–Maxwell System

315

We impose the specular reflection condition on the boundary for the distribution function fi (t, r, v) = fi (t, r, v − 2(vNΩ (r))NΩ (r)),

t ≥ 0,

r ∈ ∂Ω,

v ∈ Ω,

where NΩ (r) is a normal vector to the boundary surface. In applied problems, the impact of magnetic field is often neglected. This limit system is known as the Vlasov–Poisson (VP) one, where the Maxwell equations degenerate to the Poisson equation  n  qi fi dV, (20.8) ϕ = 4π i=1

Ω1

where ϕ(r, t) is a scalar potential of the electric field. In the general case, the distribution function may be represented in the form fi = fi (Hi1 , Hi2 , . . . , Hil ),

i = 1, . . . , N,

where Hil is the first integral (which is constant along the characteristics of the equation) for (20.1). In reality, it is not easy to select a structure of the distribution function which is connected with electromagnetic potentials aiming to transform the initial system into a simplified form. Hence, in practice, one is usually restricted to the integrals of energy Hi = −ci |V |2 +ϕ(r, t) or Hi0 = −di |V |2 +ϕ(r) as in the stationary problem case (see [172, 173]). At the same time, an introduction of the ansatz  ailmkj V1m V2k V3j Hil = ϕil + (V, dil ) + (Ail V, V ) + m+k+j=3



generalizes the form of the distribution function. Here, V = (V1 , V2 , V3 ) and (Ail V, V ) is a quadratic form; the following ones are forms of higher degrees. In this case, matrices Ail and coefficients ailmkj should be connected with the system (20.2)–(20.6) converting the first integrals for the characteristic system (20.1) into Hil . The first statement of the existence problem of classical solutions for the one-dimensional Vlasov equation has been given by Irodanskij [81], and the existence of generalized (weak) solutions for the twodimensional problem has been proved by Arsen’ev [9].

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Toward General Theory of Differential-Operator and Kinetic Models

The results of [122, 46, 12, 79, 168, 42, 175, 176, 17, 126] are devoted to the existence of solutions for (20.2) and (20.8), and the results of [37, 38, 60, 56–59, 70, 71, 136, 63, 129] concern its generalization to the VM system (20.2)–(20.6). Some rigorous results obtained by [63, 19, 39, 22, 170, 171, 11, 26, 64, 43, 126, 30, 3] are related to the analysis of the systems (20.2)–(20.6) and (20.2)–(20.8) in the bounded domains with boundary conditions. We have to mention that techniques used to prove the existence of solutions of Cauchy problem for the VM and VP systems for (x ∈ R3 , v ∈ R3 ) have limited applicability in bounded domains. Hence, the necessity to study VM and VP systems with boundary conditions is valid. That is why before presenting our own results, we have to outline some already published results on VM and VP systems in bounded domains. 20.1.1.

Existence and properties of the solutions of the Vlasov–Maxwell and Vlasov–Poisson systems in bounded domains

In the case of spherical symmetry, rather complete results were obtained by Batt, Faltenbacher and Horst [15]. In [14], a family of “local isotropic” solutions of nonstationary problem of the VP system for distribution function   (U − Ar)2 , f (t, r, V ) = Φ W (t, r) + 2 U (t, r) = W (t, r) + Φ : R → [0, ∞),

(Ar)2 , 2

t ∈ R,

r ∈ D ⊂ R3 , v ∈ R3 ,

W : R3 → R

was constructed. Here, U is the potential and A is the antisymmetric 3 × 3-matrix. Under these assumptions, the VP system is reduced to the Dirichlet boundary value problem for the nonlinear elliptic equation  1 Φ(W + |v|2 )dv, w = (w1 , w2 , w3 ) ∈ R. W + 2|w| = 4π 2 R3

Boundary Value Problems for the Vlasov–Maxwell System

317

The existence of the solution for the named problem is proved using the lower–upper solution method. The stationary solutions of n-component VP system for distribution function depending on the integral of energy fi (E) were studied by Vedenyapin [171]. He proved the existence of solution for the Dirichlet problem u(r)|∂D = u0 (r),    n  1 2 mk |v| + qk u dv, qk gk ψ(u) = 4π 2 R3

−u(r) = ψ(u),

(20.9)

k=1

d ψ(u) ≥ 0. where an arbitrary function ψ satisfies the condition (i) du Here u(r) is the scalar potential, gk (·) are the nonnegative continuously differentiable functions, D ⊂ R3 is the domain with a smooth enough boundary, and u0 (r) is the potential given on the boundary. If r ∈ D ⊂ Rp , v ∈ Rp , then the boundary value problem (20.9) has a unique solution for arbitrary nonnegative functions gk (here, readers may refer to [170]). Rein [130] proved the existence of solution of (20.9) by the variational method under condition (i). In [11], Batt and Fabian studied a transformation of the stationary VP system into (20.9) in the general case, considering distribution functions depending on energy fi (E) and on the sum of energy and momentum fi (E + P ). Using a lower–upper solution method, they proved the existence of the solutions (20.9) under condition (i). Therefore, the condition (i) became a primary condition to prove the existence theorems for the problem (20.9). The general weak global solution of the VP system has been presented in [174]. Dolbeault [43] proved the existence and uniqueness of Maxwellian solutions

f (t, x, v) =

−|v|2 1 ρ(x)e 2T , (2πT )N/2

(x, v) ∈ Ω × RN

using variational methods. A new direction in the study of the VP system is connected with the free boundary problems for semiconductor modeling. Caffarelli et al. [30] considered a semilinear elliptic integro-differential equation

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Toward General Theory of Differential-Operator and Kinetic Models

with Neumann boundary condition φ = q(n − p − C) in ∂φ = 0 on ∂ν

Ω, (20.10)

∂Ω,

where local densities of electrons n(x) and holes p(x) in insulated semiconductor are given by Boltzmann–Maxwellian statistics N exp(qφ(x)/(kT )) , n(x) =  Ω exp(qφ/(kT ))dx

P exp(−qφ(x)/(kT )) p(x) =  . Ω exp(−qφ)/(kT ))dx

and C(x) is the given background, x ∈ Ω. The bounded domain is Ω ⊂ Rd . Using variational problem statement, they proved the existence and uniqueness of the solutions and showed that the limit potential is a solution of the free boundary problem. Concerning a study of nonlocal problem (20.10), we recommend the paper by Maslov [111]. 20.1.2.

Existence and properties of solutions of the VM system in the bounded domains  Changing velocity v by its relativistic analog vˆ = v/ 1 + |v|2 , we have to face another complicated problem, since the classical VM system is not invariant in the sense of Galilei and Lorentz. Adding boundary conditions E(t, x) × NΩ (x) = 0,

B(t, x)NΩ (x) = 0,

t ≥ 0,

x ∈ ∂Ω

to the system (20.2)–(20.7), we obtain a different problem statement. Here, NΩ is the vector of the unit normal to ∂Ω and reflection condition fk (t, x, v) = fk (t, x, v˜(x, v)),

t ≥ 0,

x ∈ ∂Ω,

v ∈ R3 ,

(20.11)

where v˜ : R3 → R3 is the bijective mapping for x ∈ ∂Ω. One of the most known reflection mechanisms is a specular reflection condition

Boundary Value Problems for the Vlasov–Maxwell System

319

of the form v˜(x, v) = v − 2(vNΩ (x))NΩ (x),

x ∈ ∂Ω,

v ∈ R3

or reflection condition v˜(x, v) = −v,

x ∈ ∂Ω,

v ∈ R3 .

At present, only a few number of papers study the VM system in bounded domains. For the first time, the boundary value problem for one-dimensional VM system has been considered by [37]. In [134], stationary classical solutions (f1 , . . . , fn , E, B) for the VM system of the special form (Rudykh–Sidorov–Sinitsyn ansatz (RSS)) fk (x, v) = ψk (−αk v 2 + μ1k U1 (x), vd + μ2k U2 (x)), E(x) =

1 ∇U1 (x), α1 q1

B(x) = −

1 (d × ∇U2 (x)) q1 d2

were constructed. Here, functions ψk : R2 → [0, ∞) and parameters d ∈ R3 \{0}, αk > 0, μik = 0 (k ∈ {1, . . . , n}, i ∈ {1, 2}) are given. Functions U1 , U2 have to be defined. This approach (RSS ansatz) is closely connected with the paper by Degond [38]. Batt and Fabian [11] applied the RSS ansatz technique for the VM system with distribution functions ψ(E), ψ(E, F ), ψ(E, F, P ), where functions E(x, v), F (x, v) and P (x, v) are the first integrals of the Vlasov equation (20.2). Braasch in his own thesis [26] extended RSS results to the relativistic VM system. 20.2.

Collisionless Kinetic Models (Classical and Relativistic Vlasov–Maxwell Systems)

In this area, the existence theorems (and global stability) of renormalized solutions in the bounded domains (when trace is defined in the boundary) were proved by [115]. Ben Abdallah and Dolbeault

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Toward General Theory of Differential-Operator and Kinetic Models

[24] also developed the entropic methods in bounded domains for a qualitative study of behavior of global solutions of the VP system. Regularity theorems of weak solutions on the basis of scalar conservation laws and averaging lemmas were proved by Jabin and Perthame [83]. Jabin [82] also obtained the local existence theorems of weak solutions of the VP system in the bounded domains. For modeling of ionic beams, Ambroso et al. [4] proposed some new kinetic models with a source. 20.3.

Quantum Models: Wigner–Poisson and Schr¨ odinger–Poisson Systems

Ben Abdallah, Degond and Markowich [21] considered the ChildLangmuir regime for stationary Schr¨ odinger equation. Authors developed a semiclassical analysis for quantum kinetic equations with passage in the limit h → 0 to classical Vlasov equation with special boundary ”transition” conditions from quantum zone to classical. New results were obtained for Boltzmann–Poisson, Euler–Poisson, Wigner–Poisson–Fokker–Planck systems (such as the existence and uniqueness of the solutions, hydrodynamic limits, solutions with a minimum energy and dispersion properties). 20.4.

Mixed Quantum-Classical Kinetic Systems

Vlasov–Schr¨odinger (VS) and Boltzmann–Schr¨ odinger systems for the one-dimensional stationary case are considered in [20]. Nonstationary problems for the VS system with boundary “transition” conditions from classical zone (Vlasov equation) to quantum (Schr¨ odinger equation) were studied by Ben Abdallah, Degond and Gamba [20]. We study the special classes of stationary and nonstationary solutions of the VM system. When constructed, such solutions lead us to the systems of nonlocal semilinear elliptic equations with boundary conditions. Applying the lower–upper solution method, the

Boundary Value Problems for the Vlasov–Maxwell System

321

existence theorems for solutions of the semilinear nonlocal elliptic boundary value problem under the corresponding restrictions upon a distribution function are obtained. It was shown that under certain conditions upon the electromagnetic field, the boundary conditions and specular reflection condition for distribution function are satisfied.

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Chapter 21

Stationary Solutions of Vlasov–Maxwell System In this section, we consider the system   ∂ qi 1 ∂ fi (r, V ) + fi (r, V ) = 0, E+ V ×B · V · ∂r mi c ∂V

(21.1)

rot E = 0,

(21.2)

div B = 0,  n  qk fk (r, V )dV, div E = 4π

(21.3)

k=1

(21.4)

Ω1

 n 4π  qk V fk (r, V )dV. rot B = c Ω1

(21.5)

k=1

Here, fi (r, V ) is a distribution function of the particle of ith kind, 



r = (x, y, z) ∈ Ω2 and V = (Vx , Vy , Vz ) ∈ Ω1 ⊂ R3 are coordinate and velocity of particle, respectively, E and B are electric field strength and magnetic induction, respectively, mi and qi are mass and charge of particle of ith kind, respectively. We consider stationary distributions of the form  fi (r, V ) = fi (−αi |V |2 + ϕi , V · di + ψi ) = fˆi (R, G)

(21.6)

and corresponding self-consistent electromagnetic fields E and B satisfying (21.2)–(21.5).

323

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Toward General Theory of Differential-Operator and Kinetic Models

Condition (i). We assume that fˆi (R, G) are fixed differentiable functions of own arguments, αi ∈ R+ and di ∈ R3 are free parameters, |di | = 0, ϕi = c1i + li ϕ, ψi = c2i + ki ψ, where c1i and c2i are constants, and for all ϕi , ψi , the integrals   fi dV, V fi dV R3

R3

are converged. Unknown functions ϕi (r), ψi (r) have to be defined in such a manner that system (21.1)–(21.5) will satisfy the relation (E(r), di ) = 0, i = 1, . . . , N . The last condition is necessary for solvability of (21.1) in a class (21.6) for ∂ fˆi /∂R|v=0 = 0. 21.1.

Problem Reduction to the System of Nonlinear Elliptic Equations

We construct the system of equations to define the set of functions ϕi , ψi . By substituting (21.6) into (21.1) and equating the coefficients at ∂ fˆi /∂R and ∂ fˆi /∂G to zero, we obtain mi ∇ϕi , 2αi qi mi c ∇ψi , B(r) × di = − qi E(r) =

(21.7) (21.8)

(E(r), di ) = 0. Here, ϕi , ψi are arbitrary functions satisfying conditions (∇ϕi , di ) = 0,

i = 1, . . . , N,

(∇ψi , di ) = 0.

(21.9) (21.10)

Vector B is B(r) =

mi c λi (r) di − [di × ∇ψi ] 2 , 2 di qi di

(21.11)

where function λi (r) = (B, di ) has to be defined. Having defined ϕi , ψi such that system (21.2)–(21.5) is to be satisfied, one can find unknown functions fi , E, B by formulas (21.6), (21.7), and (21.11).

Stationary Solutions of Vlasov–Maxwell System

325

Unknown vectors ∇ϕi , ∇ψi are linear-dependent by virtue of (21.7) and (21.8). Then, we consider ϕi , ψi in the form ϕi = c1i + li ϕ,

ψi = c2i + ki ψ,

(21.12)

where c1i and c2i are constants. Because of (21.7) and (21.8), parameters li , ki are connected by the following relations: m1 αi qi , i = 1, . . . , N, (21.13) li = α1 q1 mi ki

qi k1 d1 = di . m1 mi

(21.14)

From (21.4) with (21.7), one obtains the system  n 8παi qi  qk fk (r, V )dV. ϕi = mi Ω1 k=1

Since div[di × ∇ψi ] = 0, then substituting (21.11) into (21.3), one has (∇λi (r), di ) = 0.

(21.15)

Taking into account (21.11), from (21.5), we obtain the system of linear algebraic equations for ∇λi  n mi c 4π 2  d ∇λi × di = di ψi + qk V fk dV. (21.16) qi c i Ω1 k=1

To solve (21.16), it is necessary and sufficient (according to Fredholm theorem [167]) that ψi must be the solution of the following equation:  n 4πqi  qk (V, di )fk dV. ψi = − mi c2 Ω1 k=1

Furthermore, vector Ci (r)di + di × J(r) is a general solution of (21.16) with  n  4π  qk V fk dV, J= c Ω1 k=1

(21.17)

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Toward General Theory of Differential-Operator and Kinetic Models

where Ci is an arbitrary function. Taking into account (21.12)– (21.14), it is easy to show that functions ϕ, ψ satisfy the system  n 8παq  ϕ = qk fk dV, m Ω1

(21.18)

 n 4πq  ψ = − 2 qk (V, d)fk dV, mc Ω1

(21.19)

k=1

k=1









with α = α1 , q = q1 , m = m1 , d = d1 . Lemma 21.1. Vector di × J(r) is potential and unique solution of (21.16) satisfying condition (21.15). Proof. Since ψ satisfies (21.19), then (21.17) is a general solution of (21.16). Due to (21.15), one can put Ci ≡ 0. Therefore, di × J is a unique solution of (21.15)–(21.16). We show that di ×J is a potential. In fact, rot[di × J] = −(di , ∇)J + d(∇, J), where (∇, J) ≡ 0,

(di , ∇)J = (di , ∇)rotB = rot(di , ∇)B.

Due to (21.11),  (di , ∇)B = (di , ∇) =

λi mi c di − [di × ∇ψi ] 2 2 di qi di



mi c di (∇λi , di ) − 2 di qi d2i × [di × ∇(di , ∇ψi )],

(∇λi , di ) = 0,

(∇ψi , di ) = 0.

Hence, rot[di × J] ≡ 0, di × J = ∇λi , and the lemma is proved. Corollary 21.1. ∇λi (r) = [di × J(r)].



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Stationary Solutions of Vlasov–Maxwell System

Lemma 21.2. Let b(x) = (b1 (x), b2 (x), b3 (x)), x ∈ R3 , ∂bj ∂bi = , ∂xj ∂xi

i, j = 1, 2, 3.

Then b(x) = ∇λ(x), where  1 (b(τ x), x)dτ + const. λ(x) = 0

The proof is developed by straight calculation. Corollary 21.2.    1 d di λi = 2 β + (d × J(τ x), x)dτ , d d2i 0

i = 1, . . . , N,

β = const. (21.20)

The result follows from Lemma 21.2, Corollary 21.1 and (21.14). We are searching for the solutions (21.18) and (21.19) satisfying orthogonality conditions (21.9) and (21.10). Assuming d1i = 0, i = 1, 2, 3, we consider solutions in the form ϕ = ϕ(ξ, η), ψ = ψ(ξ, η),     d211 x z y y + 2 . − − ξ= d12 d13 d12 d11 + d212 d11   x y |d1 |d11 d12  , d1 = (d11 , d12 , d13 ). (21.21) − η= 2 2 d13 (d11 + d12 ) d11 d12 Moreover, the problem is reduced to the study of nonlinear (semilinear) elliptic equations  n  qk fk dV, (21.22) ϕ = μ k=1

ψ = ν

n  k=1

qk

Ω1

 Ω1

(V, d)fk dV,

(21.23)

where 

d = d1 , μ=

8παq , mw(d)

ν=−

· =

∂2· ∂2 + , ∂ξ 2 ∂η 2

4πq , mc2 w(d)

w(d) =

d2 . d13 (d211 + d212 )

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Toward General Theory of Differential-Operator and Kinetic Models

We note that every solution (21.22)–(21.23) due to (21.21) satisfies the orthogonality conditions (21.9)–(21.10). From preceding result, we have the following theorem. Theorem 21.1. Let the distribution function have the form (21.6). Then the electromagnetic field {E, B} is defined by formulas m ∇ϕ, 2αq    1 mc d (d × J(rˆ r ), rˆ)dτ − [d × ∇ψ(ˆ r )] 2 , B(ˆ r) = 2 β + d qd 0 E(ˆ r) =



r ) and ψ(ˆ r ) satisfy where rˆ = (ξ, η), β = const, and functions ϕ(ˆ system (21.22) and (21.23). Let us introduce scalar and vector potentials U (r), A(r): E(r) = −∇U (r),

B(r) = rot A.

Then due to (21.7), (21.11) and (21.20), field {E, B} is defined via potentials {U, A} by formulas U =−

m ϕ, 2αq

A=

mc ψd + A1 (r), qd2

where (A1 , d) = 0. Unknown potentials U, A can be defined in a subspace D of enough smooth functions on the set Ω ⊂ R3 with a smooth boundary ∂Ω and moreover to satisfy conditions (∇U, d) = 0,

(∇(A, d), d) = 0

and on the boundary U |∂Ω2 = u0 (r),

(A, d)|∂Ω2 = u1 (r).

(21.24)

329

Stationary Solutions of Vlasov–Maxwell System

Corollary 21.3. Let the distribution function be (21.6). Then the VM system (21.1)–(21.5) with boundary conditions (21.24) has a solution fi = fi (−αi |V |2 + c1i + li ϕ∗ (r), di V + c2i + ki ψ ∗ (r)), where ki , li satisfy (21.13) and (21.14), m ∇ϕ∗ (r), 2αq    1 mc d ∗ (d × J (τ r), r)dτ − [d × ∇ψ ∗ (r)] 2 , B= 2 β+ d qd 0  n 4π  qk V f dv. J ∗ (r) = c Ω1 E=

k=1

Functions ϕ∗ , ψ ∗ belong to D and are defined from system (21.22)– (21.23) with boundary conditions

21.2.

2αq u0 (r), m q u(r). = mc

ϕ|∂Ω2 = −

(21.25)

ψ|∂Ω2

(21.26)

System Reductions

Lemma 21.3. If f (V + d, r) = f (−V + d, r),

d ∈ R3 ,

then the following equality holds: j = d · ρ,

(21.27)

  where j = Ω1 V f dV is a vector of current density and ρ = Ω1 f dV is the charge density.

330

Toward General Theory of Differential-Operator and Kinetic Models

Proof. Using the change of variables method in integral of the form Vi = ξi + di (i = 1, 2, 3), one obtains

 Ω1

V f dV

 Vi f (V, r)dV = J1 + J2 + J3 , where  J1 = di

f (ξ + d, r)dξ,

Ω1 ∞ ∞ ∞

 J2 + J3 =

0



0 0



0

ξi f (ξi + d, r)dξ



0

0

+ −∞

−∞

−∞

ξi f (ξi + d, r)dξ.

It is easy to show that J3 = −J2 and equation (21.27) follows.



Taking into account Lemma 21.3, equations (21.22) and (21.23) can be transformed to the form ϕ = μ

n 

qi Ai ,

(21.28)

i=1 n νd2  ki qi Ai , ψ = 2α li

(21.29)

i=1

 where Ai = Ω1 fi dV, i = 1, . . . , N . Let (ξ, η) ∈ Ω, where Ω is a bounded domain in R2 with a smooth boundary ∂Ω. We set a value of scalar potential on the boundary ∂Ω as follows: ϕ(ξ, η) ∂Ω = A(ξ, η).

(21.30)

Consider two cases, when (21.28) and (21.29) are reduced to one equation.

Stationary Solutions of Vlasov–Maxwell System

331

Case 1. li = ki , i = 1, . . . , N . Lemma 21.4. If li = ki and u∗ satisfies the equation u = a(d, α)

n 

qi Ai (γi + li u)

(21.31)

k=1

with γi = c1i + c2i , u = ϕ + ψ,

i = 1, . . . , N, a(d, α) = 2πq

4α2 c2 − d2 , mc2 αw(d)

then system (21.28)–(21.29) possesses a solution ϕ = Θ(d, α)u∗ + ϕ0 ,

ψ = (1 − Θ(d, α))u∗ − ϕ0 ,

where Θ(d, α) = 4α2 c2 /(4α2 c2 − d2 ), 4α2 c2 = d2 . Knowing some solution u∗ of equation (21.31) is being obtained under the conditions of Lemma 21.4 and the value of potential on the boundary ϕ ∂Ω = A(ξ, η), one finds ϕ0 by means of solution of the linear problem ϕ0 = 0, ϕ0 ∂Ω = A(ξ, η) − Θu∗ ∂Ω .

(21.32)

Hence, in the first case, we transformed the problem to a solution of “solving” equation (21.31) and linear Dirichlet problem (21.32). One has the following result. Theorem 21.2. Let ki = li , i = 1, . . . , N . Then the VM system (21.1)–(21.5) with boundary condition (21.30) has a solution fi = fi (−αi |V |2 + V di + γi + li u∗ (ξ, η)), E=

m (Θ(d, α)∇u∗ (ξ, η) + ∇ϕ0 ), 2αq

332

Toward General Theory of Differential-Operator and Kinetic Models

d B= 2 d

   1 β+ (d × J(τ rˆ), rˆ)dτ 0

− [d × (∇(1 − Θ(d, α))u∗ (ξ, η) − ϕ0 )]

mc . qd2

where u∗ (ξ, η) is a function satisfying “solving” equation (21.31), 

γi , βi = const, rˆ = (ξ, η) and ϕ0 (ξ, η) is a harmonic function defined from the linear problem (21.32). 



Case 2. l2 = · · · = ln = l, k2 = · · · = kn = k, l = k. We note that for N = 2, cases 1 and 2 exhaust all possible connections between parameters li and ki . We construct solution ϕ, ψ of (21.28) and (21.29) satisfying the condition ϕ + ψ = lϕ + kψ. 

Suppose functions fi = fi (−αi |V |2 + V di + ϕi + ψi ) such that the following condition holds. Condition (A). There are constants γi , i = 1, . . . , N such that ΘqA1 (γ1 + u) + τ

n 

qi Ai (γi + u) = 0

i=2

for Θ = 4α2 c2 (1 − l) + d2 (k − 1),

k τ = 4α2 c2 (1 − l) + d2 (k − 1) . l

We remark that the corresponding distribution function satisfies the condition of Lemma 21.3. 



Lemma 21.5. Let l2 = l3 = · · · = ln = l, k2 = k3 = · · · = kn = k, l = k. We assume that condition (A) holds. Then system (21.28)– (21.29) possesses a solution ϕ=

k−1 ∗ u , k−l

ψ=

1−l ∗ u , k−l

333

Stationary Solutions of Vlasov–Maxwell System

where u∗ satisfies the equation u =  =

1 , c2

h=

h A1 (γ1 + u), a(α, l) + b(d, k, l)

d2 (k − l)2 8παq 2 , mw(d)

a = 4α2 (1 − l)l,

(21.33) b = d2 (k − 1)k.

Proof. By changing ϕ = lu, ψ = ku, the system is reduced to (21.33) due to (A). Hence ϕ=

k−1 u, k−l

ψ=

1−l u. k−l



From Lemma 21.5, we have the following theorem. 

Theorem 21.3. Let α2 q2 /m2 = · · · = αn qn /mn , k2 = · · · = kn = k. n qn , 1} and condition (A) holds. Then the VM system Let k ∈ / { αm n (21.1)–(21.5) with boundary condition (21.30) on scalar potential ϕ has a solution fi = fi (−αi |V |2 + V di + γi + u∗ ), m(k − 1) ∇u∗ , 2αq(k − l)    1 cm(1 − l) d . (d × J(τ rˆ), rˆ)dτ − [d × ∇u∗ ] 2 B= 2 β+ d qd (k − l) 0 E=

Here, u∗ satisfies (21.33) with condition u∗ |∂Ω =

k−1 m A(ξ, η), k − l 2αq

(21.34)



where β and γi are constants and rˆ = (ξ, η). The problem (21.33)–(21.34) at  → 0 possesses solution u∗ = u0 + O(), where u0 is a harmonic function satisfying condition (21.34). One can show the existence of other solutions of (21.33) and (21.34) using the parameter continuation method and results of the Branching theory [169].

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Chapter 22

Existence of Solutions for the Boundary Value Problem We realize the form of distribution function. Let fi = exp(−αi |V |2 + V di + li ϕ + ki ψ).

(22.1)

Distributions (22.1) have meaning in applications. By substituting (22.1) into (21.28) and (21.29), taking into account (21.12)–(21.14) and (21.27), we arrive at the system   3/2  n  d2i π qi exp γi + exp(li ϕ + ki ψ), ϕ = μ αi 4αi k=1   3/2  n d2i ki d2 ν  π qi exp γi + ψ = exp(li ϕ + ki ψ) . 2α αi 4αi li k=1 (22.2) By introducing the normalization condition    fi dV dx = 1, i = 1, . . . , N ; Ω ∈ R2 ; x = (ξ, η), Ω

R3

we transform (22.2) to the form ϕ = μ

n 



qi exp(li ϕ + ki ψ)

Ω

i=1

ψ =

d2 ν 2α

n  i=1

ki qi exp(li ϕ + ki ψ) li

exp(li ϕ + ki ψ)dx

−1

,

 Ω

exp(li ϕ + ki ψ)dx

(22.3)

−1

.

(22.4) Consider the general case when it is not possible to transform (22.4) to one equation. Without loss of generality, we can consider that 335

336

Toward General Theory of Differential-Operator and Kinetic Models 

l2 = k2 ; q = q1 . Let q1 < 0 and qi > 0, i = 2, . . . , N . By introducing new variables u1 = ϕ + ψ,

ui = −(li ϕ + ki ψ), i = 2, . . . , N

and using them with boundary conditions (21.25)–(21.26), one obtains the system − ui =

n 

Cij Aj ,

i = 1, . . . , N,

(22.5)

j=1

where



A1 = eu1

Ω

eu1 dx

−1



 ,

Aj = e−uj

j = 2, . . . , N,

Ω

e−uj dx

−1

,

 αi 1 8π (d1 , di ) · |qi |qj 1 − 2 2 Zi Zj , Zi = , Cij = w(d1 ) mi αi 2d1 c ui = u0i , x ∈ ∂Ω, i = 1, . . . , N.

(22.6)

It is easy to check that solultions (22.4) and (22.5) are equivalent in the sense that solution (22.5) defines complete solution of (22.3) and (22.4). In fact, ϕ, ψ are defined via u1 , u2 , because l2 , k2 and ui are linear-dependent for i = 3, . . . , N . Here, we assume that u0i ∈ C 2+α , ∂Ω ∈ C 2+α , α ∈ (0, 1). We give auxiliary results. Lemma 22.1. Let n 

⎛ Cij > 0,

j=1

⎝

n 

⎞ Cij < 0⎠ .

j=1

Then Fi (u) = ⎛

n 

Cij Aj (u) ≥ 0,

j=1

⎝Fi (u) =

n  j=1

ui ≥ min u0i , ∂Ω

⎞ Cij Aj (u) ≤ 0, ui ≤ max u0i ⎠ . ∂Ω

337

Existence of Solutions for the Boundary Value Problem

 Proof. It is easy to see that Ω Fi (u)dx = nj=1 Cij > 0. Moreover, the set Ω+ = {x ∈ Ω : Fi (u(x)) > 0} is nonempty. We denote connected components by Ω− , i.e., maximum (by inclusion) connected subspace Ω− = {x ∈ Ω : Fi (u(x)) < 0}, and we show that Ω− = ∅. Hence, on the one hand, Fi (u(x)) = 0, where x ∈ ∂Ω, and on the other hand, −ui (x) = Fi (u(x)) < 0,

x ∈ Ω.

Thus, ui is bounded in Ω and it reaches its maximum on ∂Ω = ¯ Ω\Ω, i.e., maxx∈Ω¯ u(x) = u(x0 ), x0 ∈ ∂Ω. However, since function Fi (u) decreases for fixed ( Ω e−uj dx)−1 , then one obtains Fi (u(x)) > ¯ that contradicts the definition of the set Ω− . Fi (u(x0 )) = 0, x ∈ Ω  By the analogy case, nj=1 Cij < 0 is considered. Here, readers may refer to [94].  Lemma 22.2 ([61]). Let max(u − v)(x) = (u − v)(x0 ) > 0. Ω

Then u(x0 )



e

−u(x0 )



e

Ω

u(x)

e

Ω

−u(x)

e

dx dx

−1 −1

v(x0 )



>e

−v(x0 )

0. ¯ Ω

Existence of Solutions for the Boundary Value Problem

339

Evidently, x0 , due to (22.8) cannot belong to the boundary ∂Ω. Then due to the maximum principle, one has a contradiction as follows: 0 ≤ −(vk − uk )(x0 ) n

 ew1 (x0 ) e−wj (x0 ) ep(u1 )(x0 ) + − Ck1 p(u ) Ckj −vj ≤ Ck1 v 1 1 dx dx Ω e dx Ωe Ωe j=2

−

n 

(uk − vk )(x0 ) e−p(uj )(x0 ) Ckj −p(u ) + 2 (x ) < 0. j dx 1 + u e k 0 Ω j=2

Thus, vi ≤ ui . By analogy, the proof of inequality ui ≤ wi is given. We assume that there exist a number l ∈ {1, 2, . . . , N } and the ¯ such that there are two solutions u1 , u2 of (22.5)–(22.6), point y0 ∈ Ω 1 2 ui ≡ ui , i = l, u1l (y0 ) > u2l (y0 ). Using Lemma 22.2, we arrive at again to contradiction: 0 ≤ −(u1l − u2l )(y0 ) < 0, which proves uniqueness. We construct an upper solution and a lower solution of (22.5)–  (22.6). Let nj=1 Cij ≥ 0, i = 1, . . . , N . Then from Lemma 22.1, we have ui ≥ 0. At first, we construct an upper solution in the form: vi ≡ 0, − wi =

n  j=2



Cij |Ci1 | − w , −w j 1 dx Ωe Ω e dx

wi |∂Ω = max u0i ≡ w0

(22.11) (22.12)

i,∂Ω

with x = (ξ, η) ∈ Ω ⊂ R2 . From (22.8), it follows that wi must satisfy inequalities n 

Cij e−wj − |Ci1 |ew1 ≥ 0,

i = 1, . . . , N.

(22.13)

j=2

Consider the auxiliary problem −g = 1,

g|∂Ω = w0 .

We assume that domain Ω is contained in a strip 0 < x1 < r and one introduces the function g(x) = w0 + er − ex1 . It is easy to show that (q − g) = −ex1 + 1 < 0

in Ω,

340

Toward General Theory of Differential-Operator and Kinetic Models

q − g = er − ex1 ≥ 0 on ∂Ω. Therefore, according to the maximum ¯ and principle (see [55]), q − g ≥ 0, if x ∈ Ω 

w0 ≤ g(x) ≤ w0 + er − 1 = M.

(22.14)

We denote the right part in (22.11) by zi = const ≥ 0. Then from (22.11) and (22.14), we obtain wi ≤ M zi , wi = zi g(x) and (22.11)– (22.12) is equivalent to the following finite-dimensional algebraic system: zi =

n 



j=2

Cij |Ci1 |  − zg = Li (z). −z g j 1 dx dx Ωe Ωe

Let us introduce the norm |z| = max1≤i≤N |zi |. Then, due to (22.14), we obtain the following chain of inequalities:     n  |Ci1 |  Cij 

−z g −

|L(z)| ≤ max  z1 g dx  1≤i≤n  e j dx  Ωe j=2 Ω ⎧ ⎫ n ⎨ ⎬  1 max Cij eM zj − |Ci1 |e−M z1 ≤ ⎭ |Ω| 1≤i≤N ⎩ j=2

≤

⎧ n ⎨

1 max |Ω| 1≤i≤N ⎩

Cij eM |z| − |Ci1 |e−M |z|

j=2

⎫ ⎬ ⎭

where |Ω| = mes Ω, Ω ⊂ R2 .  Lemma 22.3. Let nj=1 Cij > 0. Introduce notations n 



Cij = ai ,

|Ci1 | = bi ,

j=2

min

1≤i≤N

,

(22.15) 

ai = α2 > 1. bi

Let the following inequalities hold: αai −

1 |Ω| bi ≤ ln α, α M

i = 1, . . . , N.

(22.16)

1 ln α, and functions Then equation Lz = z has a solution zi ≤ M vi ≡ 0, wi = zi g(x) are lower and upper solutions of the problem (22.5)–(22.6).

Existence of Solutions for the Boundary Value Problem

Proof.

341

Let |z| = R. From (22.13) we have ai e−M R − bi E M R ≥ 0

1 1 ln α. By substituting a maximum value R = M ln α with R ≤ M in (22.15), it is easy to check that (22.16) gives estimation |L(z)| ≤ |z|, and the existence of the fixed point Lz = z follows from the  Brayer theorem (see [78]). n Let now j=1 Cij ≤ 0, i = 1, . . . , N . By analogy with preceding result, we obtain the following result.  Lemma 22.4. Let nj=1 Cij ≤ 0, β 2 = min1≤i≤N (bi /ai ) > 1 and the following inequalities hold:

|Ω| bi − βai ≤ ln β, β M

i = 1, . . . , N.

(22.17)

Then functions vi = −zi g(x), wi = 0 are lower and upper solutions of (22.5)–(22.6) with zi = −Li (−z). It follows from Theorem 22.1 and smoothness

−u of−1the function Fi (u) under the fixed functional coefficients ( Ω e j dx) that there exists a constant M (v, w) > 0 such that ∂u∂ j Fi ≥ −M with i, j = 1, . . . , N . ¯ n → C(Ω) ¯ n defined by formulas Moreover, the mapping G : C(Ω) Gi u = Fi + M ui will be monotonic increasing in ui because of monotonicity of coefficients. We set operator T1 : z = T1 z, − zi + M zi = Gi u > 0,

zi |∂Ω = u0i .

(22.18)

Due to the maximum principle, zi > 0 (u0i > 0). Thus, operator T1 is positive and monotonic. Moreover, T1 is completely continuous which is proved in the same way as that for operator T . It is evident v ≤ T1 v, T1 w ≤ w. We note that a cone of nonnegative functions is ¯ Therefore, due to uniqueness (Theorem 22.1), we normal in C(Ω). can apply the classical theory of monotone operator (see [91]) for problem (22.18) and obtain the following result. Theorem 22.2. Operator T1 has a unique fixed point u = T1 u, vi ≤ ui ≤ wi , where for any y0 : vi ≤ y0i ≤ wi , successive approximations yn+1 = T1 yn are uniformly converged to u.

342

Toward General Theory of Differential-Operator and Kinetic Models

Corollary 22.1. We define successive approximations in the following way: u0i = 0, + M un+1 = Fi (un ) + M uni , −un+1 i i |∂Ω=u0i , i = 1, 2, n = 0, 1, . . . ; un+1 i   m1 qk αk m2 unk = (z2 − zk )un1 + (zk − z1 )un2 , mk (z2 − z1 ) |q1 |α1 q2 α2 k = 3, . . . , n. Then {uni }, i = 1, . . . , n, are monotone and uniformly converged to solution (22.5)–(22.6). Remark 22.1. In the case n = 1, boundary value problem (22.5)– (22.6) was considered in [61, 94]. 22.1.

Existence of Solution for Nonlocal Boundary Value Problem

Consider plasma in domain Ω ⊂ R2 with a smooth boundary ∂Ω ∈ C 1 consisting of N kinds of charged particles. It is assumed that particles interact among themselves only by means of own charges q1 , . . . , qn ∈ R\{0}. Every particle of ith kind is described by the distribution function fi = fi (x, v, t) ≥ 0, where t ≥ 0 is the time, x ∈ Ω is the position and v ∈ R3 is the velocity. Plasma motion is described by the classical VM system (21.1)–(21.5) with boundary conditions (21.24). We impose the reflection condition (20.11) for the distribution function. In this section, we study stationary solutions (f1 , . . . , fn , E, B) of the VM system of special form fi = fˆi (−αi v 2 + c1i + li ϕ(x), vdi + c2i + ki ψ(x)), (22.19) m ∇ϕ, (22.20) E(x) = 2αq cm (22.21) B(x) = − 2 (d × ∇ψ), qd

Existence of Solutions for the Boundary Value Problem

343

where functions fˆi : R2 → [0, ∞[ and parameters d ∈ R3 \{0}, αi > 0, c1i , c2i , li , ki (see formulas of connection (21.12)–(21.14)) are given, and functions ϕ, ψ have to be defined. Earlier, using the lower– upper solution method, the existence theorem of classical solutions of boundary value problem (21.28)–(21.30) is proved for the distribution function, fi = eϕ+ψ . Under proof of the existence theorem (Theorem 22.1), we essentially apply the monotonic property of the right parts of (22.5). In the general case of the distribution function (22.19), the system (21.28)–(21.29) does not possess good monotonic properties and therefore we cannot apply techniques of lower and upper solutions for nonlinear elliptic system in a cone developed by [2]. Therefore, we show the existence of solutions of the boundary value problem (21.28)–(21.29) and (21.25)–(21.26) by the method of lower–upper solutions without monotonic conditions. We note that the approach (22.19)–(22.21) is connected with papers of [11, 38]. In these papers, integrals E, F (x, v) and P (x, v) of the Vlasov equation are introduced and solutions of the VM system for distribution function (i = 1 — particles of single kind) of the form fˆ(E), fˆ(E, F ) or fˆ(E, F, P ) are considered. The case of the distribution function of fˆ(E, P ) and particles of various kinds (species i = 1, . . . , N ) in these papers is not considered. Thus, we consider the boundary value problem (21.28)–(21.29) and (21.25)–(21.26). Let q < 0 (electrons) and qi > 0 (positive ions), i = 2, . . . , N . Then the problem (21.28)–(21.29) takes the form   n  8παq qA1 − |qi |Ai = h1 , (22.22) ϕ = mw(d) i=2   n  ki 4πq d2 qA1 − |qi |Ai = h2 , (22.23) −ψ = mc2 w(d) 2α li i=2

where Ai = Ω fi dv, i = 1, . . . , N , and fi is ansatz (22.19). Remark 22.2. In case ki = li , system (22.22)–(22.23) is transformed to one equation and we may use Theorem 22.1. Theorem 22.3 ([112]). Let Ω ⊂ Rn be the bounded domain with ¯ × Rn → Rn satisfy boundary ∂Ω ∈ C 2,μ for some μ ∈]0, 1[. Let h : Ω

344

Toward General Theory of Differential-Operator and Kinetic Models

the following smoothness conditions: for all r > 0, there exists Cr > 0 ¯ and for all y, y1 , y2 ≤ r: such that for all x, x1 , x2 ∈ Ω (I) there exist the inequalities |h(x1 , y) − h(x2 , y)| ≤ Cr |x1 − x2 |μ , |h(x, y1 ) − h(x, y2 )| ≤ Cr |y1 − y2 |. (II) there exists ordered couple (u, w) of lower v and upper w ¯ n , v ≤ w(x) in Ω, ¯ v≤0≤w solutions, i.e., v, w ∈ C 2 (Ω)n ∩C 1 (Ω) on ∂Ω, ∀x ∈ Ω : ∀z ∈ Rn ,

v(x) ≤ z ≤ w(x),

zk = vk (x) : vk (x) ≥ hk (x, z) and ∀x ∈ Ω : ∀z ∈ Rn ,

v(x) ≤ z ≤ w(x),

zk = wk : wk (x) ≤ hk (x, z) for all k ∈ {1, . . . , N }. The vector inequality v(x) ≤ z ≤ w(x) means a component-wise comparison. ¯ n of the problem Then there exists a solution u ∈ C 2,μ (Ω) u = h(·, u(·)) u=0

in Ω

on ∂Ω

¯ such that v ≤ u ≤ w in Ω. Since the right parts in (22.22)–(22.23) are nonlocal, then we give sufficient conditions on function fˆi to apply the McKenna–Walter theorem. Lemma 22.5. Let α > 0 and fˆ : R2 → [0, ∞[ satisfy the following conditions: (1) fˆ ∈ C 1 (R2 ); (2) fˆ and fˆ are bounded, there exists R0 ∈ R such that supp(fˆ) ⊂ [R0 , ∞[×R.

Existence of Solutions for the Boundary Value Problem

345

Then function hα,fˆ : R2 → R2 , given via ⎛ ⎞ 2αq  4πq ⎜ ⎟ 2 hα,fˆ(u) = ⎝ 1 ki ⎠ fˆ(−αv + u1 , vd + u2 )dv, mw(d) R3 − c2 li is continuously differentiable and there are R, C1 , C2 such that     3/2 0 C1 (u1 + R)+ ≤ hα,fˆ(u) ≤ −C2 (u1 + R)2+ C2 (u1 + R)2+ for any function u ∈ R2 . Proof.

Moving on to the spherical system of coordinates v1 = ρ sin Θ cos ϕ,

we obtain h1α,fˆ(u) =

8παq 2 mc2 w(d)

 R3

v2 = ρ sin Θ sin ϕ,

v3 = ρ cos Θ,

fˆ(−αv 2 + u1 , vd + u2 )dv ∞ π



= P (q, α, d, m)

0

0



2π 0

fˆ(−αϕ2 + u1 , ϕk(ρ, Θ) + u2 )

× sin(Θ)ϕ2 dρΘdϕ     P (q, α, d, m) u1 π 2π ˆ −1 k(ρ, Θ) (u1 − s) + u2 ) f (s, α = α2 0 −∞ 0  × sin(Θ)(u1 − s) (u1 − s)dρdΘds   P (q, α, d, m) u1 K1 (s, u1 − s, u2 )(u1 − s) (u1 − s)ds, = 2 α −∞ where k(ρ, Θ) = d1 cos(ρ) sin(Θ) + d2 sin(ρ) sin(Θ) + d3 cos(Θ) and

 K1 (s, t, ϕ) =

π 0



2π 0

√ fˆ(s, α−1 k(ρ, Θ) t + ϕ) sin(Θ)dρdΘ.

Similar expressions are satisfied for h2α,fˆ and K2 (s, t, ϕ). Due to condition (2), kernels K1 , K2 are bounded and by applying Lebesgue

346

Toward General Theory of Differential-Operator and Kinetic Models

theorem on dominant convergence, it is easy to prove that hα,fˆ ∈  C 1 (R2 )2 . Theorem 22.4. Let Ω ⊂ R2 be a two-dimensional domain with boundary ∂Ω ∈ C 2,μ , μ ∈]0, 1[. Let fˆ1 , . . . , fˆn : R2 → [0, ∞[ satisfy conditions (1) and (2) of Lemma 22.5. Then the problem (21.28)– ¯ (21.29) and (21.25)–(21.26) possesses a smooth solution ϕ ∈ C 2 (Ω), 2 1 ¯ Moreover, the distribution function fn ∈ C (Ω ¯ × R3 ) ψ ∈ C (Ω).

generates the classical stationary solution (f1 , . . . , fn , E, B) of the VM system of the form (22.19)–(22.21) in Ω.

Proof. Consider the system (22.22)–(22.23). The right parts in it may change sign depending on relations  (A1) qA1 − ni=2 |qi |Ai = G(q, A) > 0. Hence qA1 >

n 

|qi |Ai >

i=2

(A2) qA1 −

n

i=2 |qi |Ai

T− |qi |Ai .

i=2

= G1 (q, A) < 0. Hence,

qA1
0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ G1 0

i 1

+

+

G1 0},

l− = min{|li | |li < 0}

and l = min(l+ , l− ). We define a lower solution and an upper solution in Ω ⎞ ⎛ −l+ ⎟ ⎜ 2  n ⎟  v=⎜ |l | ⎠ ⎝ −1 i  c2i |G| 1 + R2 l i=1

and

⎛

⎞ l− ⎜ ⎟ 2  n ⎟,  w=⎜ | |l ⎝ ⎠ i 2 −1 − c2i |G| 1 + R l i=1

and on the boundary vi ≤ u0i ,

wi ≥ u10i , x ∈ ∂Ω

with v = (v1 , v2 ) , w = (w1 , w2 ) . Assuming that the right parts h1 (·), h2 (·) of (22.22), (22.23) are invariant under the transition on the constant vector, we can change the last conditions on the following ones: vi ≤ 0,

wi ≥ 0, x ∈ ∂Ω.

Moreover, operator −1 is defined with respect to zero boundary ¯ conditions and v ≤ 0 ≤ w in Ω. Due to the above given estimation for hf and conditions (A1) and (A2), we obtain v1 = 0 ≥ h1f (v1 , z2 ),

z2 ∈ R,

w1 = 0 ≤ h1f (w1 , z2 ),

z2 ∈ R,

v2 ≥

n  i=1

c2i |G|(li z1 + R)2+ ≥ h2f (z1 , v2 ),

z1 ∈ [v1 , w1 ]

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Toward General Theory of Differential-Operator and Kinetic Models

and w2 ≤ −

n  i=1

c2i |G|(li z1 + R)2+ ≤ h2f (z1 , w2 ),

z1 ∈ [v1 , w1 ].

¯ U = (ϕ, ψ) of (22.22))– Thus, the existence of solutions U ∈ C 2,μ (Ω),  (22.23) follows from McKenna–Walter theorem. Remark 22.3. Existence of stationary solutions for the relativistic VM system has been proved in dissertation [26] as using RSS [134] ansatz.

Chapter 23

Nonstationary Solutions of the Vlasov–Maxwell System 23.1.

Reduction of the Vlasov–Maxwell System to Nonlinear Wave Equation

Let us consider the nonstationary VM system (20.2)–(20.6) for N -component distribution function with additional condition    n  1 qi2 (23.1) E + [V × B] · ∇V fi dV = 0. m i R3 c i=1

We consider distribution functions of the form fi = fi (−αi |V |2 + V di + Fi (r, t)),

di ∈ R3 ,

αi ∈ [0, ∞)

(23.2)

and corresponding fields E(r, t), B(r, t) satisfying equations (20.2)– (20.6) and (23.1). If functions Fi (r, t), vectors di and vector functions E, B are connected among themselves by relations qi ∂Fi + (E, di ) = 0, ∂t mi ∇Fi −

(23.3)

2αi qi qi E+ [B × di ] = 0, mi mi c

i = 1, . . . , N,

(23.4)

then functions (23.2) satisfy (20.2) and one can have the following equations: 1 ∂Fi + (∇Fi , di ) = 0, ∂t 2αi 1 ∂fi (∇fi , di ) = 0. + ∂t 2αi 349

(23.5)

350

Toward General Theory of Differential-Operator and Kinetic Models

By introducing auxiliary vectors Ki = (Kix (r, t), Kiy (r, t), Kiz (r, t)), we transform (23.4) into the system ∇Fi −

2αi qi E = Ki , mi

(23.6)

qi [B × di ] = −Ki . mi c

(23.7)

We note that equation (23.7) is solvable with respect to vector B if and only if (Ki , di ) = 0.

(23.8)

We define functions Fi (r, t) and vectors Ki (r, t) in the form Fi = λi + li U (r, t),

Ki = ki K(r, t),

where λi , ki and li are constants, and l1 = k1 = 1. Then from (23.6) and (23.7), it follows that E(r, t) =

mi (li ∇U − ki K), 2αi qi

B(r, t) =

ki mi c γ di + [K × di ] , 2 di qi d2i

where γi (r, t) = (B, di ) remain arbitrary functions. Let li = ki =

m1 αi qi , α1 q1 mi

αi d1 = α1 di , αi γ1 = α1 γi , i = 1, . . . , N. Then E(r, t) =

m (∇U − K), 2αq

B(r, t) =

mc γ + [K × d] 2 , 2 d qd

(23.9) (23.10)

where the following notations are introduced: 

m = m1 ,







α = α1 , d = d1 , γ = γ1 .

Nonstationary Solutions of the Vlasov–Maxwell System

351

Moreover, K ⊥ d. Due to (23.3) and (23.8), the function U (r, t) satisfies the following linear equation: 2α

∂U + (∇U, d) = 0. ∂t

(23.11)

Having defined U, K such that the Maxwell equations (20.2)–(20.5) satisfy for the distribution function fi = fi (−αi |V |2 + V di + λi + li U (r, t)),

(23.12)

we can find unknown functions fi , E, B using (23.9), (23.10) and (23.12). Lemma 23.1. Densities of charge ρ and current j defined by formulas     n n qi fi dV, j(r, t) = 4π qi V fi dV, ρ(r, t) = 4π R3 i=1

R3 i=1

are connected among themselves by the following relation: 1 dρ + rot Q(r) + ∇ϕ0 (r), ϕ0 (r) = 0. (23.13) 2α An equality (23.13) follows directly from the continuity equation j=

∂ρ +∇×j =0 ∂t and 1 ∂ρ + (d, ∇ρ) = 0, ∂t 2α which is corollary of (23.5). By substituting (23.9) and (23.10) into (20.3) are (20.5), one obtains  n 8παq  qi fi dV, (23.14) U = div K + m R3 i=1

(d, ∇γ) +

mc (d, rot K) = 0. q

(23.15)

352

Toward General Theory of Differential-Operator and Kinetic Models

Due to Lemma 23.1 and taking into account rot Q(r) + ∇ϕ0 = 0 (which can always be assured by calibrating),   d V fi dV = fi dV. 2α R3 R3 Thus, after substituting the (23.9) and (23.10) into (20.2), we obtain relation ∇γ × d =

md2 ∂ (∇U − K) 2αcq ∂t  n mc 2πd2  d rot[K × d]. qi fi dV − + αc q R3

(23.16)

i=1

Having used Fredholm alternative, we set the function U (r, t), and from the condition that its solution ∇γ is a gradient of function γ(r, t), we find K(r, t) as a function of U . Thus, from the solvability condition of (23.16) with respect to (22.17), one obtains  n 2πqd2  ∂2U = qi fi dV + c2 div K. ∂t2 αm 3 R i=1

Due to (23.14), this equality is transformed into  n  2πq 2 ∂2U 2 2 2 = c U + qi fi dV. (d − 4α c ) ∂t2 αm R3

(23.17)

i=1

In the following, we apply (23.17) for solvability (23.16). If function U satisfies (23.17), then (23.16) is satisfied and moreover    md2 ∂ 1  mc ν rot[K × d] + (∇U − K) = F, ∇γ = 2 d + d × − d q 2αcq ∂t d2 (23.18) where ν(r, t) = (∇γ, d) is kept arbitrary. It follows from (23.18) that the vector field F(r, t) must be irrotational. Since U satisfies (23.11), we define K in a class of vectors satisfying the condition 2α

∂K + (d · ∇)K = 0. ∂t

(23.19)

Nonstationary Solutions of the Vlasov–Maxwell System

353

Then d × rot[K × d] = −2α[d × ∂K/∂t] and (23.18) transforms into    ν m 2 2 2 ∂K 2 ∂ ∇γ = 2 d + d × (4α c − d ) +d ∇U . d ∂t ∂t 2αcqd2 Up to arbitrary function b(U ) and arbitrary vector-function a(r), one can put K(r, t) =

d2

d2 (∇U + b(U )d + a(r)). − 4α2 c2

(23.20)

ν . d2

(23.21)

Then ∇γ = If b(U ) = −

1 (∇U, d), d2

a(r) = ∇ϕ0 (r),

where ∇ϕ ⊥ d, then (23.20) satisfies (23.19). Proof is obtained by directly substituting (23.20) into (23.19) taking into account (23.11). Thus, vector   1 d2 ∇U − 2 (∇U, d)d + ∇ϕ0 (r) , (23.22) K(r, t) = 2 d − 4α2 c2 d where ∇ϕ ⊥ d satisfies condition (23.19). Moreover, it is evident that K ⊥ d. If ϕ0 (r) = 0, then for any U (r, t) satisfying (23.17), vector function (23.22) satisfies (23.14) that can be showed by substituting (23.22) into (23.14). We show that in (23.21) ν ≡ 0. In fact, (d, rot(∇U, d)) = (d, ∇(∇U, d) × d) ≡ 0 for arbitrary U, (d, rot K) = 0 and due to (23.15) d ⊥ ∇γ. But then from (23.21), ν ≡ 0. Therefore, ∇γ = 0 and γ is a constant. It remains to be shown that functions (23.9) and (23.10), where U (r, t) satisfies (23.17) and K(r, t) are expressed via U and ϕ0 by

Toward General Theory of Differential-Operator and Kinetic Models

354

formula (23.22), satisfy (20.4). By substituting (23.9) and (23.10) into (20.4), we obtain the chain of equalities as    1 ∂K 1 ×d − rotK d2 ∂t 2α   ∂ 1 m [∇U × d] + rot(∇U, d)d) = q(d2 − 4α2 c2 ) ∂t 2α    m ∂U 1 = ∇ + rot(∇U, d) × d = 0. q(d2 − 4α2 c2 ) ∂t 2α m q



Remark 23.1. If (23.13) holds, then functions γ = const, ∇γ = d × rot Q. Hence, we have the following theorem. Theorem 23.1. Let fi (s) be arbitrary differential functions; moreover,  fi (−|V |2 +T )dV < ∞, T ∈ (−∞, +∞), ai ∈ [0, ∞), di ∈ R3 , R3





αi d = αdi , α = α1 , d = d1 . Then every solution U (r, t) of hyperbolic equation (23.17) with condition (23.11) corresponds to solution of the system (20.1)–(20.5) of the form fi = fi (−αi |V |2 + V di + λi + li U (r, t)),

(23.23)

B=

mc γ [∇(U + ϕ0 (r)) × d], d+ d2 q(d2 − 4α2 c2 )

E=

m {∇(4α2 c2 U + d2 ϕ0 (r) − (∇U, d)d)}, 2αq(4α2 c2 − d2 )

(23.24)

where ϕ0 (r) is an arbitrary function satisfying ϕ0 = 0, ∇ϕ0 ⊥ d.

Nonstationary Solutions of the Vlasov–Maxwell System

355

Corollary 23.1. In the stationary case, equation (23.17) is transformed to the form  n  2πq 2 2 2 U (r) = (4α c − d ) qi fi dV (23.25) αmc2 R3 i=1

with condition (∇U, d) = 0.

(23.26)

Remark 23.2. If fi = es , S = −αi |V |2 + V di + λi + li U, li = then



 R3

fi dV =

π αi

αi mqi , αmi q

3/2 exp{d2i /4αi + λi + li U }.

In that case, the “resolving” equation (23.17) is written as 2πq 2 ∂2U (d − 4α2 c2 )π 3/2 = c2 U + 2 ∂t αm n  × qi (αi )−3/2 exp{d2i /4αi + λi + li U }. i=1

Due to paper [109], for case N = 2 (two-component system), this equation is transformed into ∂2U = c2 U + λb(eU − elU ), l ∈ R− , λ ∈ R+ , ∂t2 b=

(23.27)

2πq 2 π 3/2 2 2 (d − 4α2 c2 )ed /4α . αm α

Due to l = −1, equation (23.27) is a wave sinh-Gordon equation ∂2U = c2 U + 2λb sinh U. ∂t2

(23.28)

Toward General Theory of Differential-Operator and Kinetic Models

356

Remark 23.3. Due to conditions of Theorem 23.1, a scalar Φ and the vector A potentials are defined by formulas Φ=

2αq(d2

m {4α2 c2 U (r, t) + d2 ϕ0 }, − 2α2 c2 )

mc d{U (r, t) + ϕ0 } + Θ(r), A= 2 q(d − 4α2 c2 )

(23.29)

where Θ(r) =

γ  (d2 z, d3 x, d1 y) + ∇p(r), d = (d1 , d2 , d3 ) 2 d

(23.30)

and p(r) is an arbitrary harmonic function. Since function U (r, t) satisfies (23.11), then potentials Φ, A are connected among themselves by the Lorentz calibration 1 ∂Φ + div A = 0. c ∂t For analysis of (23.28), we direct a constant vector d ∈ R3 along 

axis Z, i.e., we assume that d = (0, 0, dz ). Moreover, the solution U (x, y, z, t) for (23.11) has the form  U =U

d t . x, y, z − 2α

(23.31)

Solution (23.31) describes the wave-spreading velocity running in a positive direction along axis Z with a constant velocity d/2α, where d/2α < c. By substitution of ξ = z − (d/2α)t, we reduce (23.28) to ∂2U (4α2 c2 − d2 ) ∂ 2 U ∂2U + + = 2λp sinh U, ∂x2 ∂y 2 4α2 c2 ∂ξ 2

(23.32)

where 

p=

2πq 2 π 3/2 (4α2 c2 − d2 ) exp(d2 /4α) > 0, αmc2 α

λ ∈ R+ .

Nonstationary Solutions of the Vlasov–Maxwell System

357

Moreover, introducing a new variable η = (4α2 c2 /(4α2 c2 − d2 ))1/2 ξ, we transform (23.32) into ∂2U ∂2U ∂U  + + 2 = 2λp sinh U, U = U (x, y, η). ∂x2 ∂y 2 ∂η

(23.33)

It is easy to reconstruct some solutions of (23.33) by the Hirota method [66]. 23.2.

Existence of Nonstationary Solutions of the Vlasov–Maxwell System in the Bounded Domain

Here, we consider the classical solutions (f1 , . . . , fn , E, B) of the VM system of special form (23.23)–(23.24), which we write in the following form: fi (x, v, t) = fˆi (−αi v 2 + vdi + li U (x, t)), E(x, t) =

(23.34)

2 2

m 4α c ∇U (x, t) + ∂t U (x, t)d , 2 2 2 2αq(4α c − d ) (23.35)

B(x, t) = −

mc ∇U (x, t) × d, − d2 )

q(4α2 c2

(23.36)

where functions fˆi : R → [0, ∞[ and vector d ∈ R3 \{0} are given ¯ → R has to be defined. Assuming that and function U : [0, ∞[×Ω 1 ∂Ω ∈ C , we add the VM system by the boundary conditions for the electromagnetic field E(x, t) × nΩ (x) = 0,

B(x, t)nΩ (x) = 0,

t ≥ 0, x ∈ ∂Ω, (23.37)

and the specular reflection condition for the distribution function on the boundary fi (t, x, v) = fi (t, x, v − 2(vnΩ (x))nΩ (x)), where nΩ is the unit normal vector of ∂Ω.

t ≥ 0, x ∈ ∂Ω, v ∈ R3 , (23.38)

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Toward General Theory of Differential-Operator and Kinetic Models

To prove the existence of classical solutions of (20.2)–(20.6) and (23.34)–(23.38), we apply the method of lower–upper solutions developed for nonlinear elliptic systems. In contrast to the stationary problem, the nonstationary one is more complicated because we need to add the equation of first order (23.11) to nonlinear wave equation (23.17). Hence, the problem is not “strongly” elliptic and one needs further development in the lower–upper solution method. Lemma 23.2. Let Ω ⊂ Rn be the bounded domain with boundary ¯ and h ∈ C 0,1 (Ω ¯ × R) such that ∂Ω ∈ C 2,α , α ∈]0, 1[. Let u0 ∈ C 2,α (Ω) loc h(x, ·) is a monotonic increasing function for every x ∈ Ω. Then the boundary value problem u = h(·, u(·)) u = u0

in Ω

on ∂Ω

(23.39)

¯ possesses a unique solution u ∈ C 2.α (Ω). Proof. Due to monotonicity of h, it is easy to check that there exist p1 , p2 ∈ C 0,α (Ω) such that p2 (x) ≤ 0 ≤ p1 (x) and  ≤ p1 (x) for s ≤ 0, h(x, s) ≥ p2 (x) for s ≥ 0 ¯ Let u01 = min(u0 , 0) and u02 = max(u0 , 0). Let uk ∈ for all x ∈ Ω. 2,α ¯ C (Ω) be the solution of linear boundary value problem for k ∈ (1, 2)  uk = pk in Ω, uk = u0k

on ∂Ω.

¯ it follows that u1 is Due to maximum principle u1 ≤ 0 ≤ u2 in Ω, a lower solution and u2 is an upper solution for (23.39). Then from the theorem of existence it follows that (23.39) has a unique solution ¯  u ∈ C 2,α (Ω). Remark 23.4. Lemma 23.2 is a well-known statement and does not require additional comments. We only remark that the condition of monotonicity of function h(x, ·) for the VP system was first applied by Vedenyapin [170].

Nonstationary Solutions of the Vlasov–Maxwell System

359

Introduce the following conditions on function fˆ : R → [0, ∞[: fˆ ∈ C 1 (R); for all u ∈ R, f ∈ L1 (u, ∞); f is measurable function and f (s) ≤ Ce−s for almost everywhere (a.e.) s ∈ R; (f4) f is decreasing, f (0) = 0 and there exists μ ≥ 0 such that for all s ≤ 0, f (s) ≤ C|s|μ . (f1) (f2) (f3)

Lemma 23.3 (Braasch [26]). Let function f : R → [0, ∞[ be given and  f (v 2 + vd + u), u ∈ R. hf (u) = c R3

Then the following claims hold. (1) Assume conditions (f2) and (f3). Then hf : continuous and nonnegative, hf (u) =

c1 |d|

 1

∞  |d|s2 −|d|s2

R → R is

sf (s + t + u)dtds

for all u ∈ R. (2) Assume condition (f3). Let ψ : R → [0, ∞[ be a measurable function and ψ ≤ f (a.e.). Then hψ ≤ hf . (3) Assume condition (f4) and |d| < 1. Then, the following conditions (f2), (f3), hf (continuously differentiable) and hf (u) ≤ Ce−u for all u ∈ R are satisfied. (4) Assume condition (f4) and |d| < 1. Then from (f4), it follows that hf is a decreasing function and |hf (u)| ≤ C|u|μ for all u ∈ R, where C = C(μ, |d|). Lemma 23.4. Let Ω ∈ R2 with a smooth boundary ∂Ω ∈ C 1 . Let fˆ1 , . . . , fˆn : R → [0, ∞[ satisfy conditions (f1)–(f3) and |d| < 1. Let

360

Toward General Theory of Differential-Operator and Kinetic Models

¯ × R → R be given by hf : Ω n

hf (x, U ) = −

 2πq qi (4α2 c2 − d2 ) αm



i=1

R3

fˆi (−αv 2 + vdi + li U (x, t))dv,

¯ is a solution of boundary problem and we assume U ∈ C 2 (Ω) ⎧ 2  ∂ U ⎨ − c2 U = hf (·, U ) in Ω, LU = (23.40) ∂t2 ⎩ U =0 on ∂Ω. We define ˜ (x + td), U (x, t) = U 

K(x, t) = −

t ≥ 0,

¯ x ∈ Ω,

d2 (∇U (x, t) − |d|−2 ∂t U (x, t)d), 4α2 c2 − d2

¯ t ≥ 0, x ∈ Ω,

K ∈ C 1 ([0, ∞[×Ω)3 and E, B by means of (23.35) and (23.36). Then (f1 , . . . , fn , E, B) is a classical solution of the VM system in Ω and it satisfies boundary conditions (23.37) and (23.38). Proof. Due to Lemma 23.4, hf is a continuous and continuously differentiable function. The function U satisfies equation (23.11). Therefore, it follows from Theorem 23.1 that f1 , . . . , fn is a solution of the Vlasov equation, and E, B are solutions of the Maxwell system. Since U vanishes on ∂Ω, then from definition U and translation invariance Ω in d, we obtain that U and ∂t U vanish on [0, ∞[×∂Ω. Hence, ∇U × nΩ = K × nΩ = 0 on [0, ∞[×∂Ω. From the last one, one can obtain E(x, t) × nΩ (x) = (K(x, t) − ∇U (x, t)) × nΩ (x) = 0 and B(x, t) × nΩ (x) = |d|−2 (nΩ (x) × K(x, t))d = 0 at t ≥ 0 and x ∈ ∂Ω. Therefore, the boundary conditions (23.37) are  satisfied.

Nonstationary Solutions of the Vlasov–Maxwell System

361

Theorem 23.2. Let Ω ⊂ R3 . Let f1 , . . . , fn : R → [0, ∞[ satisfy condition (f1) and be (pointwisely) less than the corresponding functions ψ1 , . . . , ψn : R → [0, ∞[ satisfying condition (f4) with ˜ ∈ C 2 (Ω) μ > 0. We suppose that |d| < 1 and there exists function U C such that ˜ (x + td), U (x, t) = U

t ≥ 0,

x ∈ Ω.

Then (23.35) in Lemma 23.4 possesses a smooth solution and f1 , . . . , fn generates the classical solution (f1 , . . . , fn , E, B) of the VM system in Ω of the form (23.34)–(23.36). Proof. Since elliptic operator L in (23.40) has constant coefficients, then by linear change of coordinates, it is possible to transform to 

Laplace operator L = . We introduce notations F = (f1 , . . . , fn ) and write the right part hF of (23.40) as hF (x, U ) = −c1 (c2 − d2 )

n 

qi hfi (li U (x)),

i=1

where functions hf1 , . . . , hfn are defined in Lemma 23.4. From Lemmas 23.3 and 23.4, we obtain ⎧   ⎪ ˜ (x)) = ≥ −c1 (c2 − d2 ) Ci |qi |hψi (|li |U h1 (x, U ), ⎪ ⎪ ⎨ qi >0 hF (x, U )  ⎪  ⎪ ˜ (x)) = ⎪ Ci |qi |(−|li |U h2 (x, U ), ⎩ ≤ c1 (c2 − d2 ) qi 0 and for all W0 ∈ S, there exists ˆ 0 − W0  < δ, then the norm δ = δ(, T ) such that when the norm W W (r, V, t) − W0  <  for 0 < t < T , where 0 < T < ∞. The equilibrium configuration, which we tested on stability, represents the charged electron–ion bundle with nonrelativistic movement of particles confined in a cylinder with a finite radius and retained by the magnetic field (see [38]). Moreover, (d, n) = 0.

(24.3)

In cylindrical geometry (ρ, Θ, Z) the boundary conditions (24.1), (24.2) along with (23.26) and (24.3) are concretized   ∂U  ∂U  = 0, = 0. ∂Θ ∂Ω2 ∂Z ∂Ω2 Let Gi (fi ) be smooth functions and   Gi (fi )dV < ∞, Ω1

Ω1

fi dV < ∞.

Then the following first integrals by (24.1)–(24.2) are valid T =

1 8π

 Ω2

{E 2 + B 2 }dr +

N   i=1

 Ω1

Ω2

1 mi V 2 fi drdV , 2

(24.4)

365

Linear Stability of the Stationary Solutions of the VM System

F1 =

N   i=1

Ω1

 Ω2

Pi fi drdV ,

F2 = (d, P ),   N   1 Pi fi drdV + E × Bdr, P = 4πc Ω2 Ω1 Ω2

(24.5) Pi = mi V.

i=1



Definition 24.2 ([69]). The stationary solution W0 = (f0i , E0 , B0 ) for system (20.2)–(20.6) is called formally stable if there exists the Lyapunov functional L, which possesses an isolated minimum in a stationary point W0 . If the second variation of functional L is strongly positive, then 

W0 = (f0i , E0 , B0 ) is an isolated minimum. Following Chetaev’s method, we introduce the Lyapunov functional in the form of a bundle of the first integrals L − L0 = T + F1 + λF2 ,

(24.6)

where L0 is a functional value which is calculated along the nonperturbed (stationary W0 ) state of the system and λ is an auxiliary parameter (Lagrange’s coefficient). Calculate the first variation of functional (24.6) on variables fi , E, B. In addition, we restrict our consideration to the twocomponent (N = 2) system of (20.2)–(20.6). The first variation of energy integral (24.4), in the point of equilibrium, has the type 1 δT = 4π

 Ω2

{E0 δE + B0 δB}dr +

2   i=1

Ω1

 Ω2

mi 2 V δfi drdV . 2 (24.7)

Reduce the first subintegral expression in a functional (24.7) using the connection of fields E, B with potentials ϕ and A 1 ∂A , E0 = −∇ϕ0 , c ∂t B = rot A, B0 = rot A0

E = −∇ϕ −

(24.8) (24.9)

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Toward General Theory of Differential-Operator and Kinetic Models

and gauge condition

1 ∂Φ c ∂t +div A

= 0. After transformations, we have

   

1 1 ∂2 δT = ϕ0 2 2 δϕ − δϕ − A0 δA dr 4π Ω2 c ∂t − ϕ0 (δE, n) dS ∂Ω2

+

∂Ω2

(δA × B0 )ndS +

2   i=1

 Ω1

Ω2

1 mi V 2 δfi dr dV . 2

Similar calculations of the first variation for the momentum integral (24.5), taking into account (24.8)–(24.9) and gauge condition, give d (d, δP ) = − 4πc

 Ω2

 ∇ϕ0 ∇δA + ϕ0 δA + A0 δϕ

 1 ∂ [d × δA]rot A0 dr + c ∂t

{ϕ0 [n × δB] − ϕ0 ∇δA + [n × B0 ]δϕ} dS + ∂Ω2

+d

2   i=1

 Ω1

Ω2

Pi δfi drdV

and the expression 

 Ω2

 1∂ [d × δA] rot A0 dr (d, ∇ϕ0 )∇δA + c ∂t

vanishes through the use of (23.26) and (23.29). In addition, introduce the first integrals of the type  Φ(ϕ)dr, F3 = F4 =

2   i=1

 Ω1

Ω2

(24.10)

Ω2

Ψi (x, y, z, t)fi drdV ,

(24.11)

367

Linear Stability of the Stationary Solutions of the VM System

where Φ(ϕ) is a twice differentiable function of its argument and Ψi satisfies the equation 1 ∂Ψi + (∇Ψi , d) = 0 ∂t 2α

(24.12)

where, by the terminology of Moiseev, Sagdeev, Tur, Yanovsky [116], the functions Ψi are Lagrangian invariants. We show that the functionals (24.10) and (24.11) are really first integrals of (20.2)–(20.6) in view of the symmetry problem along d. Differentiate (24.10) and (24.11) with time. Since the function Φ(ϕ) satisfies the equation ∂Φ 1 + (d, ∇Φ) = 0, ∂t 2α then we obtain    1 ∂Φ d dr = − Φdr = (d, ∇Φ)dr dt Ω2 2α Ω2 Ω2 ∂t 1 Φ(n, d)dS = 0. =− 2α ∂Ω2 Further, 2

d  dt i=1



 Ω1

Ω2

2  

Ψi fi drdV



qi ∂ Ψi fi − Ψi ∇r (V fi ) − ∇V ∂t m i Ω1 Ω2 i=1  

1 · Ψi E + [V × B] drdV c  

  2  1 ∂Ψi + (∇Ψi , d) fi drdV = ∂t 2α Ω1 Ω2 i=1

 Ψi V fi dV dS −

=

∂Ω2

qi − mi



Ω1



∂Ω1

Ω2

Ψi



 1 E + [V × B] fi dr dS1 = 0. c

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Remark 24.1. The integral (24.10) allows a generalization of the type  Φ(ϕ, Ψ1 , . . . , Ψn )dr, F3 = Ω2

where the functions Ψn are Lagrangian invariants of (24.12). Consider a final structure of the Lyapunov functional (24.10) and (24.11) ˆ = L − L0 = T + F1 + λF2 + F3 + F4 . L

(24.13)

After preliminary calculations, the first variation of functional (24.13) has the form ˆ i , δE, δB) δL(δf    2   mi 2  V + Gi (f0i ) + dλPi + Ψ0i + qi ϕ0 δfi drdV = 2 i=1 Ω1 Ω2    d 1  A0 δA + λ (ϕ0 δA + A0 δϕ) − Φ (ϕ0 )δϕ dr − 4π 4πc Ω2 1 {ϕ0 (δE, n) − [δA × B]n}dS − 4π ∂Ω2 d {ϕ0 [n × δB] − ϕ0 ∇δA + [n × B0 ]δϕ}dS −λ 4πc ∂Ω2 where λ = −1/2α. Assuming ϕ0 |∂Ω2 = 0

(24.14)

and taking into account (23.24) and (23.29), we have ˆ= δL

 d 1 2  mi V + Gi (f0i ) − Pi + Ψ0i + qi ϕ0 δfi drdV 2 2α i=1 Ω1 Ω2    d2  ϕ0 δϕdr Φ (ϕ0 ) + (24.15) + 16πα2 c2 Ω2 2  





Linear Stability of the Stationary Solutions of the VM System

369

with condition  δϕ0 ∂Ω2 = 0. From equality to zero of the first variation (24.15) of functional (24.13), we have the equation of the equilibrium states 1 d Pi + Ψ0i + qi ϕ0 = 0, Gi (f0i ) + mi V 2 − 2 2α Φ (ϕ0 ) +

d2 ϕ0 = 0 16πα2 c2

(24.16) (24.17)

with condition (24.14). The correlation (24.16) permits us to concretize a structure of the distribution functions for which we can show stability. Introduce the notation   d 1  2 mi V − Pi + Ψ0i + qi ϕ0 . (24.18) Gi (fi0 ) = −H, H = 2 2α A choice of the functions Gi (fi0 ) of type (24.18) implies 

f0i (r, V ) = f0i (H). Then by putting Gi (f0i ) =

1 f0i βi , ln βi γ

γ>0

(24.19)

we concretize the distribution functions f0i = γ exp(−βi H).

(24.20)

1 {f0i ln f0i − f0i − f0i ln γ}. βi

(24.21)

From (24.19), we have Gi (f0i ) =

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By analyzing the equations of equilibrium states (24.16) and (24.17) and stationary equations (23.25) and (23.26), we define the values of parameter βi in (24.20) and functions Ψ0i βi =

2αi , mi

Ψ0i = −

qi (d, A0 ). 2αc

(24.22)

Moreover, by (23.29) and (24.17), we obtain

U0 |∂Ω2

d2 mc U0 + Φ (U0 ) = 0, 8παq(d2 − 4α2 c2 )   d2 0 = − 2 2 ϕ  , ϕ0 = 0, ϕ0 |∂Ω2 = b(r), 4α c ∂Ω2

(24.23)

where b(r) is a given function. The harmonic function ϕ0 in (24.23) may be discussed as a given external field on the boundary ∂Ω2 . Further, it is easy to write the sufficient conditions of positive definiteness of the initial Lyapunov functional (24.13) taking into account (24.21) and (24.22) |d| < c; ln f0i ≥ 1 + ln |γ|, 2α   2   d 1 2 mi V − Pi + Ψi (r) > 0, 2 2α i=1   Ω1 a(V )f (r, V )dV , a =  Ω1 f (r, V )dV

(24.24) (24.25)

Since d/2α = V = V − VT , where VT is a mean of random heat velocity, then (24.25) becomes 2   1 i=1

2

mi V 2 −



 2 d m 0 qi d2 Pi ϕ − ϕ . 0 2α 4α2 c2 2αq

(24.26)

i=1

The condition of (24.26) places a restriction on the value of fields in the system; moreover, d2 /c2 1.

371

Linear Stability of the Stationary Solutions of the VM System

The second variation of the functional (24.13) has the form   1 1 2ˆ 2 2 δ L= {(δE) + (δB) }dr + (d, δB × δE)dr 4π Ω2 4απc Ω2   2    2 Gi (f0i )(δfi ) dV dr + Φ (ϕ0 )(δϕ)2 dr. + i=1

Ω2

Ω1

Ω2

(24.27) It is easy to show that, taking into account (24.1)–(24.3), the second variation of the Lyapunov functional (24.27) is an integral of a linearized VM system. Sufficient conditions for the stability of equilibrium solutions (24.16)–(24.17) can be obtained from the condition of the positive definiteness of the subintegral expression ˆ in the in formula (24.27). For the positive definiteness of δ2 L neighborhood of a stationary state, it is sufficient that we have the following conditions: |d| < c; 2α

Gi (f0i ) > 0;

Φ (ϕ0 ) > 0,

or by (24.16) |d| < c, 2α

∂f0i (H) < 0, ∂H

Φ (ϕ0 ) > 0,

i = 1, 2.

Using the stability theorem [156], we obtain the sufficient conditions of stability for the stationary (equilibrium) solutions (24.16), (24.17) by measure ρ. As a measure by which the stability is studied, we choose the quantity ρ = δE2L2 (Ω2 ) + δB2L2 (Ω2 ) +

2  i=1

δfi L2 (Ω1 ×Ω2 ) + δϕ2L2 (Ω2 ) . (24.28)

Let the potential ϕ0 and a solution ϕ0 of the linear boundary value problem (24.23) satisfy the inequalities (24.24) and (24.26); then for the stability of stationary (equilibrium) solutions (24.16) and (24.17)

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Toward General Theory of Differential-Operator and Kinetic Models

by measure (24.28), it is sufficient that the following conditions are satisfied: |d| < c, 0 < Ci ≤ Gi (f0i ) ≤ bi , bi > 0, Ci < bi , 2α 0 < l1 ≤ Φ (ϕ0 ) ≤ l2 , l2 > 0, l1 < l2 , Ci , bi , l1 , l2 = const, or the conditions |d| < c, 2α 1 ∂f0i (H) 1 ≤ − < 0, ≤ − Ci ∂H bi 0 < l1 ≤ Φ (ϕ0 ) ≤ l2 .

Chapter 25

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System The problem of bifurcation analysis of the VM system, formulated for the first time by Vlasov, proved to be very complicated against the general background of the progress of the bifurcation theory in other directions and it remains open at this moment. There exist only separate results. One simple theorem about the point of bifurcation is proposed in [150], and another one is proven in [149] for the stationary VM system. The goal of this chapter is to prove the general existence theorems for the bifurcation points of stationary VM system with the given boundary conditions to the potentials of electromagnetic field and on the density of charge and current. For studying the bifurcation points of the VM system, the results of the branching theory of [165, 169] and the index theory [36, 93] are used. Let us note that the methods, adapted in [149, 150], were not sufficient for examining the situation in general, as we elaborate in the following. Let us consider many-component plasma consisting of electrons and positively charged ions of various kinds, described by manyparticle distribution function of the form fi = fi (r, v), i = 1, . . . , N . Plasma is located in domain D ⊂ R3 with a smooth boundary. Particles interact among themselves only by means of their own charges; we neglect collisions between particles.

373

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Toward General Theory of Differential-Operator and Kinetic Models

The behavior of plasma is described as follows (classical version of stationary VM system [172]):   1 v · ∂r fi + qi /mi E + v × B · ∂v fi = 0, c r ∈ D ⊂ R3 ,

i = 1, . . . , N,

(25.1)

curl E = 0, div B = 0,

div E = 4π

N 

 qk

k=1



R3

fk (r, v)dv = ρ, (25.2)

 N 4π   qk vfk (r, v)dv = j, curl B = c R3 k=1

where ρ(r) and j(r) are the densities of charge and current, and E(r) and B(r) are electric and magnetic fields, respectively. We look for the solution E, B, f of the VM system (25.1), (25.2) for r ∈ D ⊂ R3 with boundary conditions on potentials and densities U |∂D = u01 , ρ|∂D = 0,

(A, d)|∂D = u02 ,

(25.3)

j|∂D = 0,

(25.4)

where E = −∂r U , B = curl A, and U and A are scalar and vector potentials, respectively. We call a trivial solution E 0 , B 0 , f 0 if ρ0 = 0 and j 0 = 0 inside domain D. Here, we study the case of the distribution functions of the special form [134]  fi (r, v) = λfˆi (−αi v 2 + ϕi (r), v · di + ψi (r)) = λfˆi (R, G),

ϕi : R3 → R, λ ∈ R+ ,

ψi : R3 → R, 

αi ∈ R+ = [0, ∞),

r ∈ D ⊆ R3 , di ∈ R3 ,

v ∈ R3 , (25.5)

i = 1, . . . , N,

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System

375

where functions ϕi and ψi , which generate appropriate electromagnetic field (E, B), should be found. We are interested in the dependence of unknown functions ϕi and ψi on parameter λ in distribution function (25.5). First, we consider λ which does not depend on physical parameters αi and di used in (25.5). For example, in case αi = αi (λ), di = di (λ), the distribution function    m 3/2 −m|v|2 f (r, v) = + d · v + ϕ(r) · exp 2πkT 2kT provides a dependence α = α(λ), where λ = (m/(2πkT ))3/2 , α = −m/(2kT ), k is the Boltzmann constant, and T is the temperature of electrons. In this case, parameter λ has a dimension of the temperature. Definition 25.1. A point λ0 is called the bifurcation point of VM system with conditions (25.3) and (25.4) if in any neighborhood of vector (λ0 , E 0 , B 0 , f 0 ) correspond to trivial solution with ρ0 = 0, j 0 = 0 in domain D: there exists a vector (λ, E, B, f ), satisfying (25.1)–(25.4), for which E − E 0  + B − B 0  + f − f 0  > 0. Let ϕ0i and ψi0 be constants with corresponding densities ρ0 and j 0 , induced in medium by distributions fi , which are equal to zero in D for ϕ0i , ψi0 . Then N   k=1

R3

qk fˆk dv = 0,

N   3 k=1 R

qk v fˆk dv = 0

for ϕi = ϕ0i and ψi = ψi0 , k = 1, . . . , N . Let di = σi d1 , i = 1, . . . , n and let σi be a constant. Vector di ∈ R3 and parameter αi ∈ R+ characterize the chaotic heat motion of particles of the kind i. We examine the case, when di and αi are different, i.e., nonisothermic plasma as most frequently being encountered in the applications.

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Toward General Theory of Differential-Operator and Kinetic Models

Then VM system possesses a trivial solution fi0 = λfˆi (−αi v 2 + ϕ0i , v · di + ψi0 ), B 0 = βd1 ,

E 0 = 0,

β = const

for any λ. Our aim is to construct nontrivial solutions of the stationary VM system. We obtain conditions implying the existence of points λ∗ ∈ R+ (bifurcation points), possessing a neighborhood, where VM system has nontrivial solutions in domain D ⊂ R3 . For these solutions, we have ρ|D = 0, j|D = 0, but ρ|∂D = 0, j|∂D = 0. It is assumed that the scalar and vector potentials of the desired electromagnetic field are given at the boundary of the domain. The branching equation (BEq) was derived by Vainberg and Trenogin in seminal books [167, 169]. We proved that for the sufficiently general case of fi = fˆi (a(−αi v 2 + ϕi ) + b(v · di + ψi )), where a and b are constants, BEq is the potential equation. On this basis, the asymptotics are constructed for the nontrivial branches of solutions in a neighborhood of the bifurcation point. Let us note that the problem of bifurcation points in the theory of collisionless plasma without allowance for magnetic field was studied in [65, 67, 68]. Apparently, the problem of bifurcation points for the general VM system has not been considered earlier. This chapter is organized as follows. First, we prove two existence theorems of the bifurcation points for the nonlinear operator equation in Banach space, generalizing results on bifurcation points in [165]. The method of proof of these theorems uses index theory of vector fields [36, 93] and this makes it possible to study not only points but also bifurcation surfaces with the minimum limitations to the equation. Then we compose the problem of bifurcation point of the VM system to the problem of bifurcation point of semilinear elliptic system. The last one is considered as operator equation in Banach space. The boundary value problem and the problem on bifurcation points were formulated, and a spectrum of the problem for linearized system is studied. Then, the BEq is constructed. Finally, the existence theorem for bifurcation points is proved based

377

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System

on the analysis of BEq, and the asymptotics of nontrivial branches is constructed for the solutions of the VM system. 25.1.

Bifurcation of Solutions of Nonlinear Equations in Banach Spaces

Let E1 and E2 be Banach spaces and Υ be normed space. Let us consider equation Bx = R(x, ),

(25.6)

where B : D ⊂ E1 → E2 is a closed linear operator with dense domain of definition in E1 . Operator R(x, ) with the values into E2 is defined, continuous and continuously differentiable in Frechet sense over x in the neighborhood Ω = {x ∈ E1 ,  ∈ Υ : x < r,  < ρ}. Assume R(0, ) = 0 and Rx (0, 0) = 0. Let the operator B be a Fredholm operator. Let us introduce basis {ϕi }N 1 in subspace N (B), ∗ ), and also systems {γ }N ∈ E ∗ , {z }N ∈ E , basis {ψi }N in N (B i 1 i 1 2 1 1 which are biorthogonal to these bases. Definition 25.2. A point 0 is called the bifurcation point of (25.6) if in any neighborhood of point x = 0, 0 , there is a couple (x, ) with x = 0, satisfying equation (25.6). It is known [169] that the problem of bifurcation point of equation (25.6) is equivalent to the problem of bifurcation point of finitedimensional system L(ξ, ) = 0,

(25.7)

where ξ ∈ RN , L : RN × Υ → RN . We call (25.7) the branching equation (BEq). We write (25.6) as the system ˜ = R(x, ) + Bx

N 

ξs zs ,

(25.8)

s = 1, . . . , n,

(25.9)

s=1

ξs = x, γs ,

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Toward General Theory of Differential-Operator and Kinetic Models

 ˜ def where B = B+ N s=1 ·, γs zs has it inverse bounded. Equation (25.8) has a unique small solution x=

N 

ξs ϕs + U (ξ, )

(25.10)

s=1

for ξ → 0,  → 0. Substitution of (25.10) into (25.9) gives formulas for coordinates of vector function L : RN × Υ → RN : N   (25.11) ξs ϕs + U (ξ, ),  , ψk . Lk (ξ, ) = R s=1

Here, the derivatives

∂Lk

def = Rx (0, )(I − ΓRx (0, ))−1 ϕi , ψk = aik ()

∂ξi ξ=0 are continuous in the neighborhood of point  = 0, ΓRx (0, ) < 1. Introducing the set Ω = { | det[aik ()] = 0} containing the point  = 0, we are able to formulate the following condition. Condition (A). We assume that there exists the set S, in a neighborhood of point 0 ∈ Ω, which possesses Jordan continuum S = S+ ∪ S− , 0 ∈ ∂S+ ∩ ∂S− . Moreover, there is continuous mapping (t), t ∈ [−1, 1] such that  : [−1, 0) → S− ,  : (0, 1] → S+ , 1 (0) = 0 , det[aik ((t))]N i=i,k=1 = α(t), where α(t) : [−1, 1] → R is a continuous function becoming zero only in t = 0. Theorem 25.1. Let condition (A) hold and α(t) be a monotone increasing function. Then 0 is the bifurcation point of bifurcation equation (25.6). Proof. Let us take arbitrarily small r > 0 and δ > 0. Let us consider the continuous vector field def

H(ξ, Θ) = L(ξ, ((2Θ − 1)δ)) : RN × R → RN , given for ξ, Θ ∈ M , where M = {ξ, Θ| ξ = r, 0 ≤ Θ ≤ 1}. Case 1. If there exists a pair (ξ ∗ , Θ∗ ) ∈ M with H(ξ ∗ , Θ∗ ) = 0, then by above definition, 0 is the bifurcation point.

379

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System

Case 2. We assume that H(ξ, Θ) = 0 for all (ξ, Θ) ∈ M and hence, 0 is not the bifurcation point. Then vector fields H(ξ, 0) and H(ξ, 1) are homotopic on the sphere ξ = r. Hence, their rotations [91] coincide: J(H(ξ, 0), ξ = r) = J(H(ξ, 1), ξ = r).

(25.12)

Since vector fields H(ξ, 0) and H(ξ, 1) and their linearization def

L− 1 (ξ) =

def L+ 1 (ξ) =

N 

aik ((−δ))ξk |N i=1 ,

k=1 N 

aik ((+δ))ξk |N i=1

k=1

are nondegenerate on the sphere ξ = r, then due to smallness of r > 0 fields, (H(ξ, 0) and H(ξ, 1)) are homotopic in their linear parts + L− 1 and L1 (ξ). Thus, J(H(ξ, 0), ξ = r) = J(L− 1 (ξ), ξ = r),

(25.13)

J(H(ξ, 1), ξ = r) = J(L+ 1 (ξ), ξ = r).

(25.14)

Since linear fields L± 1 (ξ) are nondegenerate according to the theorem of Kronecker index, the following equalities hold: J(L− 1 (ξ), ξ = r) = sign α(−δ), J(L+ 1 (ξ), ξ = r) = sign α(+δ) Since α(−δ) < 0, α(+δ) > 0, then due to (25.13) and (25.14), relation (25.12) is not valid. Hence, we found a couple (ξ ∗ , Θ∗ ) ∈ M , for which  H(ξ ∗ , Θ∗ ) = 0 and ξ0 is the bifurcation point. Remark 25.1. If conditions of Theorem 25.1 are satisfied for all  ∈ Ω0 ⊂ Ω, then Ω0 is the bifurcation set of equation (25.6). Moreover, if Ω0 is a connected set and every point is contained in their own neighborhood homeomorphism with some domain from RN , then Ω0 is called the n-dimensional bifurcation manifold.

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For example, taking Υ = Rn+1 , n ≥ 1, we have Ω0 as a bifurcation

set of (25.6) and it contains the point  = 0, while ∇ det[aik ()]  = 0. The generalization of this result (see [165]) follows from Theorem 25.1, with Υ = R, and also other known strengthening of Krasnosel’skii theorem about the bifurcation point of the odd multiplicity [91]. Stronger results in the theory of bifurcation points are obtained for (25.6) with the potential BEq in ξ when L(ξ, ) = gradξ U (ξ, ).

(25.15)

k N This condition is satisfied if matrix [ ∂L ∂ξi ]i,k=1 is symmetrical. By means of the differentiation of superposition, we find from (25.11) that N    ∂U ∂Lk = Rx ξs ϕs + U (ξ, ),  ϕi + , ψk , (25.16) ξi ∂ξi

s=1

where according to (25.8) and (25.10) ϕi +

∂U = (I − ΓRx )−1 ϕi . ∂ξi

(25.17)

Operator I − ΓRx is continuously invertible because ΓRx  < 1 for the sufficiently small norm ξ and . By substituting (25.17) into (25.16), we obtain equalities   ∂Lk = Rx (I − ΓRx )−1 , ψk , ∂ξi

i, k = 1, . . . , n.

The following assertion occurs. Lemma 25.1. In order for BEq (25.7) to be potential, it is sufficient that matrix Ξ = [ Rx (ΓRx )m ϕi , ψk ]N i,k=1 is symmetrical for all (x, ) in the neighborhood of point (0, 0). Corollary 25.1. Let all matrices [ Rx (ΓRx )m ϕi , ψk ]N i,k=1 ,

m = 0, 1, 2, . . .

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System

381

be symmetrical in any neighborhood of point (0, 0). Then BEq (25.7) is potential. Corollary 25.2. Let E1 = E2 = H and H be the Hilbert space. If operator B is symmetrical in D, and operator Rx (x, ) is symmetrical for all (x, ) in neighborhood of point (0, 0) in D, then BEq is potential. We assume that BEq (25.7) is potential. Then it follows from the proof of Lemma 25.1 that corresponding potential U in (25.15) takes the form N 1  ai,k ()ξi ξk + w(ξ, ), U (ξ, ) = 2 i,k=1

where w(ξ, ) = O(|ξ|2 ) for ξ → 0. Theorem 25.2. Let BEq (25.7) be potential. Let Condition (A) hold. Moreover, let symmetrical matrix [aik ((t))] possess ν1 positive eigenvalues for t > 0 and ν2 positive eigenvalues for t < 0, ν1 = ν2 . Then 0 will be the bifurcation point of equation (25.6). Proof. Let us have arbitrarily small δ > 0 and consider function U (ξ, ((2Θ − 1)δ)) defined into Θ ∈ [0, 1] in neighborhood of critical point ξ = 0. Case 1. If there exists Θ∗ [0, 1] such that ξ = 0 is nonisolated critical point of function U (ξ, ((2η ∗ − 1)δ)), then due to Definition 24.1, 0 is the bifurcation point. Case 2. We assume that point ξ = 0 is the isolated critical point of function U (ξ, ((2Θ − 1)δ)) for all Θ ∈ [0, 1], where (t) is a continuous function from Condition (A). Then with for all Θ ∈ [0, 1] for this function, the Conley index [36] KΘ of critical point ξ = 0 is determined. Let us recall that   2  ∂ U (ξ, ((2Θ − 1)δ))   = α((2Θ − 1)δ). det    ∂ξi ∂ξk =0

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Toward General Theory of Differential-Operator and Kinetic Models

Since α((2Θ − 1)δ) = 0 for Θ = 12 , then critical point ξ = 0 for Θ = 12 is nondegenerated. Thus, the Conley index KΘ with certain Θ = 12 necessary is equal to the number of positive eigenvalues of the corresponding Hessian according to the definition (see [36, p. 6]). Therefore, KΘ = ν1 , K1 = ν2 , where ν1 = ν2 due to conditions of Theorem 25.2, hence KΘ = K1 . We assume that 0 is not the bifurcation point. Then ∇ξ U (ξ, ((2Θ − 1δ)) = 0 with 0 < ξ ≤ r, where r > 0 sufficiently small number, Θ ∈ [0, 1]. In view of the homotopic invariance of the Conley index (see [36, Theorem 4, p. 52]), KΘ is a constant for Θ ∈ [0, 1] and K0 = K1 . Hence, in the second case, we will always find the pair (ξ ∗ , Θ∗ ) satisfying equation ∇ξ U (ξ, ((2Θ − 1)δ)) = 0 for arbitrarily small r > 0, δ > 0, where  0 < ξ ∗  ≤ r, Θ∗ ∈ [0, 1]. Thus, 0 is the bifurcation point. Remark 25.2. Another proof of Theorem 25.2 for case Υ = R, ν+ = n, ν− = 0 with the application of the Rolle theorem was given in [166]. Remark 25.3. Theorems 25.1 and 25.2 (see Remark 25.1) make it possible to build not only the points of bifurcation but also bifurcation sets, surfaces and curves of bifurcation. Corollary 25.3. Let Υ = R and BEq be potential. Furthermore, let [aik ()]N i,k=1 be the positively defined matrix for  ∈ (0, r) and negatively defined for  ∈ (−r, 0). Then  = 0 is the bifurcation point of equation (25.6). Let us consider connection of eigenvalues of matrix [aik ()] with eigenvalues of operator B − Rx (0, ). Lemma 25.2. Let E1 = E2 = E,  ∈ R, and ν = 0 be isolated Fredholm point of operator function B − νI. Then k

sign () = (−1) sign

k  i

νi () = sign

N 

μi (),

i

where k is the root number of operator B and {μ}N 1 are the eigenvalues of matrix [aik ()], () = det[aik ()].

383

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System

Proof. Since {μi }N are eigenvalues of matrix [aik ()], then 1 N () = (). Thus, it suffices to prove equality () = i μi (−1)k ki νi (). Since zero is an isolated Fredholm point of operator function B − νI, then operators B and B ∗ have the corresponding complete Jordan system [169] (s)

ϕi

(1)

= (Γ)(s−1) ϕi ,

i = 1, . . . , n,

(s)

ψi

(1)

= (Γ∗ )(s−1) ψi ,

s = 1, . . . , Pi ,

(25.18)

where (P ) ϕi i , ψj

= δij ,

(P ) ϕi , ψj j

= δij ,

i, j = 1, . . . , n,

N 

Pi = k.

i=1

Let us recall that (1) 

ϕi

(P )

(1) 

(Pi )

= ϕi = Γϕi i , ψi = ψi = Γ∗ ψi −1

N  (Pi ) Pi ) ·, ψi ϕi , Γ= B+

, (25.19)

1

where k = l1 + · · · + ln is the root number of operator B − Rx (0, ). Small eigenvalues ν() of operator B − Rx (0, ) satisfy the following BEq [169]: 

L(ν, ) = det| Rx (0, ) + νI)(I − ΓRx (0, ) − νΓ)−1 ϕi , ψj |ni,j=1 = 0. (25.20) Due to the Weierstrass theorem (see [169, p. 66]) and relations (25.18) and (25.19), equation (25.20) can be transformed into L(ν, ) ≡ (ν k + Hk−1 ()ν k−1 + · · · + H0 ())Ω(, ν) = 0 in the neighborhood of zero, where Hk−1 (), . . . , H0 () = () continuous functions , Ω(0, 0) = 0, H0 (0) = 0. Hence, operator B − Rx (0, ) has k ≥ n small eigenvalues νi (), i = 1, . . . , n, which we can find from equation

Then

k i

ν k + Hk−1 ()ν k−1 + · · · + () = 0. νi () = ()(−1)−1 .

384

Toward General Theory of Differential-Operator and Kinetic Models

Let  ∈ R. Let us consider computation of asymptotics of eigenvalues μ() and ν(). Let us introduce the block presentation of matrix [aik ]N i,k=1 satisfying the following condition. l rik 0 l Condition (B). Let [aik ()]N i,k=1 = [Aik ()]i,k=1 ∼ [ Aik ]i,k=1 for  → 0, where [Aik ] are blocks of dimension [ni ×nk ], n1 +· · ·+nl = n, 

min(ri1 , . . . , ril ) = rii = ri and rik > ri (or for k < i), i = 1, . . . , l. l 0 Let 1 det[Aii ] = 0. Condition (B) denotes that matrix [aik ()]N i,k=1 allows the block presentation, which is “asymptotically  triangular” for  → 0. Lemma 25.3. Let Condition (B) hold. Then formulas n1 r1 +···+nl rl det[aik ()]N i,k=1 = 

l  1

 det|A0ii | + 0(1) ,

and μi = ri (Ci + 0(1)),

i = 1, . . . , l,

(25.21)

define the dominant terms for all n eigenvalues of matrix |aik ()|N i,k=1 , n i where μi , Ci ∈ R , and Ci is the vector of eigenvalues of matrix A0ii . Proof. Due to Condition (B) and property of linearity of determinant, we obtain

A0 + 0(1) 0(1) . . . . . . 0(1)

11

0

A21 + 0(1) A022 + 0(1) 0(1) . . . 0(1) n1 r1 +···+nl rl

det det[aik ()] = 

... ... ...



0 0

Al1 + 0(1) ... All + 0(1)

=

n1 r1 +···+nl rl

l  i

 det|A0ii |

+ 0(1) .

Substituting μ = ri c(), i = 1, . . . , l, into equation det|aik () − μδik |N i,k=1 = 0 and using the property of linearity of determinant,

385

Bifurcation of Stationary Solutions of the Vlasov–Maxwell System

we obtain the following equation: ⎧ i−1 ⎨ n1 r1 +···+ni−1 ri−1 +(ni +···+nl )ri det|A0jj |  ⎩ j=1

× det(A0ii − c()E)c()ni+1 +···+nl

⎫ ⎬ + ai () = 0, ⎭

i = 1, . . . , l, (25.22)

where ai () → 0 for  → 0. Hence, the coordinates of the dominant terms Ci in asymptotics (25.21) satisfy equations det|A0ii − cE| = 0, i = 1, . . . , l.  If k = n, then operator B − Rx (0, ), just as matrix [aik ()]N i,k=1 has n small eigenvalues. In this case, the following result holds. Corollary 25.4. Let the operator B not have I adjoint elements and assume Condition (B). Then formula νi = −ri (Ci + 0(1)),

i = 1, . . . , l,

(25.23)

defines all n small eigenvalues of operator B − Rx (0, ), where Ci ∈ Rn is the vector of eigenvalues of matrix A0ii , i = 1, . . . , l, n1 + · · · + nl = n. N Proof. In this case, due to Lemma 25.2, we have 1 Pi = n (root number k = n) and operator B − Rx (0, ) possesses n small  eigenvalues. Since l1 ni = n, A0ii is the square matrix, then (25.23) gives n eigenvalues, where dominant terms coincide with dominant terms in (25.21) with an accuracy to the sign. For computing eigenvalues ν of operator B − Rx (0, ), we transform (25.20) into ⎤N ⎡ ∞  (j) bik ν j ⎦ = 0, (25.24) L(ν, ) ≡ det ⎣aik () + j=1

i,k=1

where (j)

bik = [(I − ΓRx (0, ))−1 Γ]j−1 (I − ΓRx (0, ))−1 ϕi , γk .

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Toward General Theory of Differential-Operator and Kinetic Models

Substituting ν = −ri c() into (25.24) and taking into account the property of linearity of determinant, we obtain equation which differs from (25.22) only in terms of the error of calculation ai (). Then in conditions of Corollary 25.4, the dominant terms define all small eigenvalues of operator B − Rx (0, ) and matrices −[aik ()] are found from the same equations and hence are identical.  25.2.

Conclusions

1. Due to Lemma 25.3, we can change Condition (A) in Theorem 25.1 by the following condition. Condition (A∗ ). Let E1 = E2 = E, and ν = 0 be an isolated Fredholm point of operator function B − νI. Let in the neighborhood of the point 0 ∈ Ω there exist the set S, containing point 0 , which presents the continuum S = S+ ∪ S− . Furthermore, let us assume that



 

νi () · νi () < 0, 0 ∈ ∂S+ ∩ ∂S− ,

i

∈S+

i

∈S−

where {νi ()} are small eigenvalues of operator B − Rx (0, ). 2. If the dominant terms of asymptotics of small eigenvalues of operator B − Rx (0, ) and matrix [aik ()]N i,k=1 coincide, then one can apply eigenvalues of such operator in Theorem 25.2. Due to Corollary 25.4, it is possible if E1 = E2 = H. Then operators B and Rx (0, ) are symmetrical, and Condition (B) is satisfied. We note that Condition (B) is also realized in [151, 165] on the bifurcation point with potential BEq; moreover, r1 = · · · = rn = 1.

Chapter 26

Statement of the Boundary Value Problem and the Bifurcation Problem Let us present one preliminary result about the reduction of the VM system (25.1)–(25.2) with boundary conditions (25.3) and (25.4) to the quasilinear system of elliptic equations for the distribution function (25.5). Assume that the following condition is satisfied. Condition (C). fˆi (R, G) are given, differentiable functions in distribution (25.5); αi and di are constant parameters; |di | =  0, ϕi = c1i + li ϕ(r), ψi = c2i + ki ψ(r), c1i and c2i are constant values, parameters li and ki are connected by relations q1 m1 αi qi qi , ki d1 = di , k1 = l1 = 1, (26.1) li = α1 q1 mi m1 mi   and integrals R3 fˆi dv, R3 fˆi vdv converge for all ϕi , ψi . 





Let us introduce the following notations: m1 = m, α1 = α, q1 = q. Theorem 26.1. Let fi be defined in (25.5) and condition C hold. Also, assume that vector function (ϕ, ψ) is a solution of the system of equations  N  8παq , qk fk dv, μ = ϕ = μ m R3 k=1  N  4πq qk (v, d)fk dv, ν = − 2 , (26.2) ψ = ν mc R3 k=1

ϕ|∂D = −

2αq u01 , m 387

ψ|∂D =

q u02 mc

Toward General Theory of Differential-Operator and Kinetic Models

388

in subspace (∂r ϕi , di ) = 0,

(∂r ψi , di ) = 0,

i = 1, . . . , N.

(26.3)

Then the VM system (25.1)–(25.2) has a solution m ∂r ϕ, E= 2αq

d B= 2 d

   1 mc β+ (d × J(tr), r)dt − [d × ∂r ψ] 2 , qd 0 (26.4)

where  N 4π  qk vfk dv, J= c R3 

β = const.

k=1

Potentials U =−

m ϕ, 2αq

A=

mc ψd + A1 (r), qd2

(A1 , d) = 0,

(26.5)

satisfying condition (25.3), are defined over this solution. We introduce the notations   vfi dv, ρi = ji = R3

R3

fi dv,

i = 1, . . . , N

and introduce the following condition. Condition (D). There exist vectors βi ∈ R3 such that ji = βi ρi , i = 1, . . . , N . For example, Condition D is satisfied for distribution fi = fi (−αi v 2 + ϕi ) + b((di , v) + ψi ) for βi =

b 2αi a di ,

where a and b are constant values.

(26.6)

Statement of the Boundary Value Problem and the Bifurcation Problem

389

We assume that Condition D holds. Then system (26.2) is transformed into ϕ = λμ

N 

qi Ai ,

ψ = λν

N 

i=1

qi (βi , d)Ai ,

(26.7)

i=1

where 



Ai (li ϕ, ki ψ, αi , di ) = b βi = di , 2αi a

R3

b d2 ki (βi , d) = , a 2α li

fˆi dv, (di , d) d2 ki = . αi α li

(26.8)

In the case of normalized distribution functions, this system allows the following generalization. Let   fi dvdr = Ni , D

R3

where (26.9) fi = f1i (ϕ, ψ, v) + f2i (ϕ, ψ) · M,  and the integral R3 M (v)dv converges. Then, one has the integral identity       Ni − D R3 f1i dvdr  . fi dv = f1i (ϕ, ψ, v)dv + f2i (ϕ, ψ) · R3 R3 D f2i dr Therefore, for the functions ϕ and ψ occurring in the distribution fi in the case of normalized distributions functions fi , we obtain the following system of quasilinear integro-differential equations: ϕ = λμ  A˜i =

R3

N 

qi A˜i ,

ψ = λν

i=1

f1i dv + λf2i · 



N 

qi (βi , d)A˜i ,

i=1

ξi −

 f1i dvdr , D f2i dr

 

1 λ  D R3

where λ = N , N = (N1 , . . . , Nn ), ξi = Ni /N .

(26.10)

390

Toward General Theory of Differential-Operator and Kinetic Models

Remark 26.1. Apparently, distribution functions of the form (26.9) can be useful for the analysis of stationary solutions of the Boltzmann equation, since they allow one to simplify the collision integral by separating the variables r and v. If, in addition, we normalize the function f1i by   f1i dvdr = Ki , D

R3

then we can study distribution function with different numbers Ni and Ki . From now on, for simplicity, we consider the auxiliary vector d in (25.5) directed along the axis Z. Due to (26.3), we can take ϕ = ϕ(x, y), ψ = ψ(x, y), x, y ∈ D ⊂ R2 in system (26.7). Moreover, let N ≥ 3 and klii = const. Let D be a bounded domain in R2 with boundary ∂D of class C 2,α , α ∈]0, 1[. Boundary conditions (25.3) and (25.4) for local densities of charge and current provide equalities: N 

qk Ak (lk ϕ0 , kk ψ 0 , αi , di ) = 0,

k=1 N 

qk (βk , d)Ak (lk ϕ0 , kk ψ 0 , αi , di ) = 0

(26.11)

k=1

for all ∈ ι, where ι is a neighborhood of the point = 0 and ϕ0 = −

2αq u012 , m

Remark 26.2. If N = 2 and βi = d2 ki 2α li ,

ψ0 = di 2αi ,

q u02 . mc

(26.12)

then due to (26.11)–(26.12)

we obtain an alternative: either in (26.11) A1 = and (βi , d) = A2 = 0 or ki = li , i = 1, 2. In this case, and also for klii = const, system (26.7) is reduced to one equation and the bifurcation of the solutions is impossible.

Statement of the Boundary Value Problem and the Bifurcation Problem

391

Using (26.11)–(26.12) for the system (26.7) with boundary conditions ϕ|∂D = ϕ0 ,

ψ|∂D = ψ 0 ,

(26.13)

one obtains a trivial solution ϕ = ϕ0 , ψ = ψ 0 , for all λ ∈ R+ . Then due to Theorem 26.1 for any λ, the VM system with boundary conditions (25.3) and (25.4) has the trivial solution E0 =

m ∂r ϕ0 = 0, 2αq

B 0 = βd1 ,

r ∈ D ⊂ R2 ,

f 0 = λfˆi (−αi v 2 + c1i + li ϕ0 , (v, di ) + c2i + ki ψ 0 ). Under this condition, ρ and j vanish in domain D. Thus, our aim is to find λ0 providing nontrivial solution for the neighborhood system (26.7), (26.13). Then the corresponding densities ρ and j vanish in domain D, and point λ0 is the bifurcation point of the VM system (25.1)–(25.4). Let functions fi be analytical in (25.5). Using the Taylor series expansion,   ∞  ∂ ∂ i 1 (x − x0 ) + (y − y 0 ) A(x0 , y 0 ), A(x, y) = i! ∂x ∂y i≥0

and expressing the linear terms, we can rewrite system (26.7) in operator form (L0 − λL1 )u − λr(u) = 0. Here,

(26.14)



 0 , u = (ϕ − ϕ0 , ψ − ψ 0 ) , (26.15) L0 = 0 

s s N μks ∂A μls ∂A  ∂x ∂y qs L1 = ∂As s νls (βs , d) ∂A s=1 ∂x νks (βs , d) ∂y x=ls ϕ0 , y=ks ψ0

 μT1 μT2 = , (26.16) νT3 νT4

Toward General Theory of Differential-Operator and Kinetic Models

392

r(u) =

N ∞  

is (u)bs ,

(26.17)

i≥l s=1

where qs is = i! 



∂ ∂ ls u1 + ks u2 ∂x ∂y

i

As (ls ϕ0 , ks ψ 0 )

are ith-order homogeneous forms in u,   ∂ i1 +i2 As (x, y) =0 i i 1 2 ∂x ∂y x=ls ϕ0 , y=ks ψ0 with 2 ≤ i1 + i2 ≤ l − 1,



s = 1, . . . , N, l ≥ 2, bs = (μ, ν(βs , d)) .

The existence problem for a bifurcation point, λ0 , of the system (26.7), (26.13) also can be stated as the existence problem for a bifurcation point for the operator equation (26.14). ¯ and C 0,α (D) ¯ with Let us introduce the Banach spaces C 2,α (D) the norms  · 2,α and  · 0,α , respectively, and let W 2,2 (D) be the ordinary Sobolev L2 space in D. 

Let us introduce the Banach space E of vectors u = (u1 , u2 ) , where ui ∈ L2 (D), L2 is the real Hilbert space with inner product (·, ·) and the corresponding norm  · L2 (D). We define the domain  ˚ 2,2 (D) consists D(L0 ) as the set of vectors u = (u1 , u2 ) with ui ∈ W

of W 2,2 functions with zero trace on ∂D. Hence L0 : D ⊂ E → E is a linear self-adjoint operator. By virtue of the embedding ¯ W 2,2 (D) ⊂ C 0,α (D),

0 < α < 1,

(26.18)

the operator r : W 2,2 ⊂ E → E is analytic in a neighborhood of the origin. The operator L1 ∈ L(E → E) is linear and bounded. We keep same notations for matrix corresponding to operator L1 . By the embedding (26.18), any solution of equation (26.14) is a H¨ older function in D(L0 ). Moreover, since the coefficients of system (26.14) are constant, the vector r(u) is analytic, and ∂D ∈ C 2,α ; it follows

Statement of the Boundary Value Problem and the Bifurcation Problem

393

from well-known results of regularity theory for weak solutions of elliptic equations that the generalized solutions of equation (26.14) ˚ 2,2 (D) actually belong to C 2,α . Here, readers may also refer in W to [97]. Definition 26.1 ([167]). A point λ0 is called a bifurcation point of the problem (26.7), (26.13) if every neighborhood of the point (ϕ0 , ψ 0 , λ0 ) contains a point (ϕ, ψ, λ) satisfying system (26.7), (26.13) such that 0 ϕ − ϕ0 W ˚ 2,2 + ψ − ψ W ˚ 2,2 > 0, 2,2 (D). where  · W ˚ 2,2 is the norm in the space W

According to the Theorem 26.1 on the reduction of the VM system, the bifurcation points of problem (26.8), (26.13) will be referred to as the bifurcation points of the stationary VM system (25.1)–(25.4). Under the above assumptions about L0 and L1 , all 

singular points of the operator L(λ) = L0 −λL1 are Fredholm points. If N (L(λ0 )) = {0}, then by the Implicit Operator Theorem [167], for any δ > 0 there exists a neighborhood S of the point λ0 such that for all λ ∈ S the ball uE < δ contains only the trivial solution u = 0, so that λ0 is not a bifurcation point. Therefore, to find the bifurcation points it is necessary (but not sufficient) to find number λ0 such that N (L0 − λ0 L1 ) = {0}. The bifurcation points of the nonlinear equation (26.14) are necessarily spectral points of the linearized system (L0 − λL1 )u = 0.

(26.19)

To analyze the spectral problem (26.19) for physically acceptable parameter values, first we search for eigenvalues and eigenvectors of the matrix L1 in (26.19). To achieve it, we need several auxiliary assertions to be made. Let us introduce the conditions: (i) T1 < 0, (ii) (T1 T4 − T2 T3 ) > 0.

Toward General Theory of Differential-Operator and Kinetic Models

394

Lemma 26.1. If

 ∂ fˆk  > 0, ∂x x=lk ϕ0

then condition (i) is satisfied. 

Proof. We can assume that q = q1 < 0, qi > 0, i = 2, . . . , N and sign qi li = sign q. By the assumption of the lemma, we have  ∂Ai ∂ fˆi = dv > 0. ∂x R3 ∂x Therefore T1 < 0.



Let us introduce matrix Θ = Θij i,j=1,...,n = [0], Θij = qi qj (lj ki − kj li )(βi − βj , d). Lemma 26.2. If ∂Ai ∂Ai = , ∂x ∂y

i = 1, . . . , N,

N ≥ 3,

∂Ai >0 ∂x

and the matrix Θ is positive, then conditions (i) and (ii) are satisfied. For any ∂Ai /∂x, we have the identity     li ai ki ai ki (βi , d)ai − li (βi , d)ai T1 T4 − T2 T3 =

Proof.

=

N  i−1 

ai aj (lj ki − kj li )(βi − βj , d),

i=2 j=1

∂Ai . ∂x By virtue of this identity, for the expression T1 T4 − T2 T3 to be positive, it is sufficient that 

where ai = qi

Θij = qi qj (lj ki − kj li )(βi − βj , d) > 0,

i, j = 1, . . . , N,

N ≥ 3. (26.20)

Since the matrix Θ is positive, we have (26.20).



Statement of the Boundary Value Problem and the Bifurcation Problem

395

Remark 26.3. If βi = di /(2αi ), then (βi , d) =

d2 ki 2α li

(26.21)

and Θij =

d2 qi qj (lj ki − li kj )2 > 0, 2α li lj

i, j = 1, . . . , n,

because sign(qi /li ) = sign q. Remark 26.4. If N = 2 and βi = di /(2αi ), then by conditions (i), (ii) and (26.21), the following alternative occurs: either A1 = A2 = 0 under conditions (26.11), (26.12) or ki = li , i = 1, 2. For example if βi = i−1 N  

di 2αi ,

then (βi , d) =

= ai aj (lj ki − li kj )2 ·

i=2 j=1

d2 ki 2α li

and

d2 > 0. 2αli lj

Lemma 26.3. Let distribution function has the form (25.5) and di fi > 0. Then conditions D and (i), (ii) are satisfied for βi = ab 2α i and system (26.7) is transformed to potential type

ϕ a1  =λ 0 ψ

⎡ ∂V ⎤ 0 ⎣ ∂ϕ ⎦, ∂V a2

(26.22)

∂ψ

where V =

 N  qk k=1

lk

0

alk ϕ+bkk ψ

Ak (s)ds,

a1 = μ/a,

a2 =

νd2 . 2ab (26.23)

To prove it we just substitute (26.23) into (26.22)

Toward General Theory of Differential-Operator and Kinetic Models

396





Lemma 26.4. Let r = x ∈ R, v ∈ R2 , d = d2 . Then system (26.22) with potential (26.23) can be written as Hamiltonian system p˙ ϕ = −∂ϕ H,

ϕ˙ = ∂pϕ H,

p˙ ψ = −∂ψ H,

ψ˙ = ∂pψ H

with Hamiltonian H=−

p2ϕ p2ψ − + V (ϕ(x), ψ(x)). 2 2

Here V (ϕ, ψ) = λa1

 N  qk k=1

+ λa2

lk

alk ϕ

R2

0

 N  qk k=1

lk



bkk ψ

A(s, ψ)ds

 R2

0

A(ϕ, s)ds.

The proof follows from Lemma 2.2 of paper [64, p. 1152]. Lemma 26.5. Let conditions (i), (ii) are satisfied. Then the matrix L1 has one positive eigenvalue χ+ = μT1 + 0(1) and one negative eigenvalue χ− = η 

T1 T4 − T2 T3 + O( ), T1

η=

4π|q| >0 m

(26.24)

as = c12 → 0. Eigenvalues χ− generate eigenvectors of matrices L1 and L1 , respectively

T

− T12 c∗1 0 c1 + O( ), = = + O( ). ∗ c2 1 c2 0

Statement of the Boundary Value Problem and the Bifurcation Problem

Proof.

397

The characteristic equation of the matrix

μT2 μT1 − χ + ηT3

+ ηT4 − χ

has the form χ2 − χ(μT1 + ηT4 ) + ημ(T1 T4 − T2 T3 ) = 0. Since χ1,2 =

  1 μT1 + ηT4 ± (μT1 + ηT4 )2 − 4 ημ(T1 T4 − T2 T3 ) , 2

we obtain χ+ = μT1 + O(1),

χ− = η

T1 T4 − T2 T3 + O( ) T1

as → 0. Since μ < 0 and T1 < 0, it follows that χ+ > 0. It can be checked in a similar way that χ− < 0 by virtue of the inequalities η > 0, T1 T4 − T2 T3 > 0, T1 < 0. Solving the homogeneous systems (Ξ − χ− )c = 0,

(Ξ − χ− )c∗ = 0,

we can find the eigenvectors corresponding to χ− . Let us proceed to the computation of the bifurcation point λ0 . Assuming that λ = λ0 + in (26.14), consider the system (L0 − (λ0 + )L1 )u − (λ0 + )r(u) = 0

(26.25)

in a neighborhood of the point λ0 . Assume that either T2 = 0 and T3 = 0 or T2 = T3 = 0. To symmetrize the system for T2 = 0, T3 = 0, we multiply both sides of equation (26.25) by matrix   1 0  μT2 = 0. M= , where a ˜= νT3 0 a ˜ We rewrite (26.25) in the form Bu = B1 u + (λ0 + )R(u). Here B = M (L0 − λ0 L1 ),





R(u) = M r(u) = (r1 (u), r2 (u)),

(26.26)

398

Toward General Theory of Differential-Operator and Kinetic Models

B1 ∈ L(E → E) is a self-adjoint operator, since it is generated by the symmetric matrix B1 = M L1 , and B : D(L0 ) ⊂ E → E is a self-adjoint Fredholm operator.  Remark 26.5. If As (als ϕ + bks ψ), then ∂As = As b, ∂y

∂A = As a, ∂x

a ˜=

μb d2 b ds a, βs = . ν 2α a 2αs

In decomposition (26.17), is =

qs (i) A (als ϕ0 + bks ψ 0 )(als u1 + bks u2 )i . i! s

Therefore, ∂r1 /∂u2 = ∂r2 /∂u1 in this case, the matrix Ru (u) is symmetric for any u, and the operator Ru : E → E is self-adjoint for any u. ˜ = 1. If either T2 = 0 and T3 = 0 If T2 = T3 = 0, then one can set a or T3 = 0 and T2 = 0, then the problem cannot be symmetrized and the derivation of BEq must be performed directly for equation (26.25). Let us introduce the following condition: (iii) Let μ be an eigenvalue of the Dirichlet problem −e = μe,

e|∂D = 0,

and {e1 , . . . , en } be an orthonormal basis in the subspace of eigenfunctions. Let c− = (c1 , c2 ) be the eigenvector of the matrix L1 corresponding to the eigenvalue χ− < 0. Lemma 26.6. Let condition (iii) hold, and λ0 = −μ/χ− . Then dim N (B) = n and the system {ei }N i=1 , where ei = c− ei , is a basis in the subspace N (B).

Statement of the Boundary Value Problem and the Bifurcation Problem

399

Proof. Consider the matrix  whose columns are the eigenvectors of L1 corresponding to the eigenvalues χ− , χ+ . Then,   0 χ − , L0  = L0 , −1 L1  = 0 χ+ and substitution u = U reduces the equation Bu = 0 into M [L0 U − λ0 L1 U ] = M [(L0 U − λ0 −1 L1 U )] = 0. It follows that the linearized system (26.19) splits into two linear elliptic equations: U1 − λ0 χ− U1 = 0,

u1 |∂D = 0,

(26.27)

U2 − λ0 χ+ U2 = 0,

u2 |∂D = 0,

(26.28)

where λ0 χ− = −μ, λ0 χ+ > 0. By condition (iii), we have μ ∈ σ(−). Therefore, U1 =

N 

αi ei ,

αi = const,

U2 = 0,

i=1

and hence,

   c u   1−  1   = U =  c2− u2 

     N c1+  u1  c1−   αi ei . · =  c2+   0  c2−  i=1



Lemma 26.7. The operator B does not have B1 -adjoint elements. Since L1 c− = χ− c− , we obtain an equality         1 0 c  c     1−   1−  L c e , c e = χ

B1 ei , ek  =   1 − i − k e , e −  0 a c2−  i c2−  k ˜

Proof.

˜c22 )δik , = χ− (c21− + a

i, k = 1, . . . , n.

N 2 ˜c22 )N = 0 because Therefore, det B1 ei , ek N i,k=1 = χ− (c1− + a

χ− = 0,

c21i + a ˜c22− ≈ −2α

T2 2 c , T3

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Toward General Theory of Differential-Operator and Kinetic Models

and hence, according to the definition of generalized Jordan sequences [169], the operator B does not have B1 -adjoint elements. Without loss of generality, we can assume that the eigenvector ˜c22− ) = 1. Then c1− of the matrix L1 is chosen, so that χ− (c21− + a N the system of vectors {B1 ei }N i=1 is biorthogonal to {ei }i=1 . Hence, by Schmidt’s lemma [169], the operator ˘ =B+ B

N 

·, γi γi ,

i=1 

with γi = B1 ei , has a bounded inverse Γ ∈ L(E → E). Thus, Γ = Γ∗ ,

Γγi = ei .

(26.29) 

Remark 26.6. It follows from the proof of Lemma 26.4 that to construct the operator Γ, one can use the equation    Γ  1 0  −1 −1 Γ =   M ,  0 Γ2  where

 Γ1 =

 D

G1 (x, s)[·]ds,

Γ2 =

D

G2 (x, s)[·]ds,

where G1 (x, s) is the modified Green function of the Dirichlet problem (26.27) and G2 (x, s) is the Green function of the Dirichlet problem (26.28).

Chapter 27

Resolving Branching Equation Let us rewrite equation (26.26) in the form of the system  ˘ − B1 )u = (λ0 + )R(u) + (B ξi γi ,

(27.1)

i

ξi = u, γi , i = 1, . . . , n.

(27.2)

From (27.1) by the inverse operator theorem, we have −1

u = (λ0 + )(I − ΓB1 )

N

1  ΓR(u) + ξ i ei . 1−

(27.3)

i=1

Moreover, in virtue of (27.2) and (26.29), we must have λ0 +   (27.4) ξi + R(u), ei  = 0, 1− 1− where R(u) = Rl (u) + Rl+1 (u) + · · · . According to the implicit operator theorem, equation (27.3) has the unique solution u = u1 (ξe, ) + (λ0 + )(I − ΓB1 )−1 Γ{ul (ξe, ) + ul+1 (ξe, ) + · · · } (27.5) for sufficiently small  and |ξ|, where N

u1 (ξe, ) =

1  ξi ei , ul (ξe, ) = Rl (u1 (ξe, )), 1− i=1

ul+1 (ξe, ) = Rl+1 (u1 (ξe, )) ⎧ ⎪ ⎨0,  + ΓR2 (u1 (ξe, ))(λ0 + ) ⎪ ⎩ (I − ΓB1 )−1 Γu2 (ξe, ), 401

l > 2, l = 2,

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Toward General Theory of Differential-Operator and Kinetic Models

and so on. By substituting the solution (27.5) into (27.2), we obtain the desired bifurcation system (BEq):  ξ + L(ξ, ) = 0, 1−

(27.6)

where L = (L1 , . . . , LN ),   λ0 +  1 Rl (ξe), ei  + Rl+1 (ξe), ei  Li = (1 − )l+1 1− ⎫ ⎧ l > 2⎬ ⎨ 0, + ri (ξ, ), +  λ + ⎩ 0 R2 (ξe)(I − ΓB1 )−1 ΓR2 (ξe), ei , l = 2 ⎭ 4 (1 − ) ri = o(|ξ|l+1 ),

i = 1, . . . , n.

If L(ξ, ) = grad U (ξ, ), then BEq (27.6) is considered to be potential. In the potential case, matrix Lξ (ξ, ) is symmetric. Let fi = fi (ali ϕ + bki ψ), i = 1, . . . , N in (25.5). Then, by Remark 26.5, the matrices Ru (u) are symmetric for any u. Let us show that BEq (27.6) is potential if the matrix u(u) is symmetric for any u. Indeed, by (27.4), Li (ξ, ) =

λ0 +  R(u(ξe, )), ei , 1, . . . , n, (1 − )

where u(ξe, ) is defined by series (27.5). Therefore, the vector field L(ξ, ) is potential if and only if the matrix  N Ru ∂u(η, ) ej , ei ∂η

i,j=1



(η = ξe)

(27.7)

is symmetric. By virtue of (27.3) and the inverse operator theorem, we have the operator identity −1 1 ∂u(η, )  = I − (λ0 + )(I − ΓB1 )−1 ΓRu (u(η, )) , ∂η 1−

Resolving Branching Equation

403

which is a sufficiently small neighborhood of the point ξ = 0,  = 0. Since B1 , Γ and Ru are self-adjoint operators, it follows that 

∂u Ru ∂η

∗

= [I − (λ0 + )Ru (u(η, ))]−1

1 . 1−

Therefore, 

∂u Ru ∂η

∗

= [I − (λ0 + )Ru Γ(I − B1 Γ)−1 ]−1 Ru

1 ∂u = Ru 1− ∂η

by virtue of the operator identity [I − (λ0 + )Ru Γ(I − B1 Γ)−1 ]−1 Ru = Ru [I − (λ0 + )(I − ΓB1 )−1 ΓRu ]−1 . Hence, the operator Ru

∂u : E→E ∂η

in the matrix (27.7) is self-adjoint. Therefore, the matrix (27.7) is symmetric and L(ξ, ) = grad U (ξ, ). The foregoing implies the following assertion. Theorem 27.1. Let conditions (26.11), (26.12), and (i)–(iii) be satisfied and λ0 = −μ/χ− . Then the number of solutions of equation (26.25) such that u → 0 as for λ → λ0 coincides with the number of small solutions ξ → 0 as  → 0 of BEq (27.6). If Ai = Ai (akli ϕ + bki ψ), i = 1, . . . , N in system (26.2)–(26.3), (26.7), (26.13), and a, b are constants, then BEq is potential, that is, L(ξ, ) = ∇ξ U (ξ, ),

Toward General Theory of Differential-Operator and Kinetic Models

404

U : RN × R → R, where N  λ0 +  u(ξ, ) = − Rl (ξe), ei ξi (l + 1)(1 − )l+1 i=1

N

 λ0 +  Rl+1 (ξe), ei ξi l+2 (l + 2)(1 − ) i=1 ⎧ ⎪ 0, ⎪ ⎨ N − (λ0 + )   ⎪ R2 (ξe)(I − ΓB1 )−1 ΓR2 (ξe), ei ξi , ⎪ ⎩ (1 − )4 −

i=1

l+1

+ o(|ξ| 27.1.

⎫ l > 2⎪ ⎪ ⎬ l = 2⎪ ⎪ ⎭

).

(27.8)

The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions

Branching Equation (27.6) is the desired Lyapunov–Schmidt BEq (here, readers may refer to the book by Vainberg and Trenogin [169]) for the bifurcation point of the boundary value problem (26.7), (26.13). In the sequel, we need some properties of the real solutions and the structure of BEq 

L(ξ, ) =

 ξ + Ll (ξ, ) + o(|ξ|l ) = 0. 1−

(27.9)

We state these results from [165] in the form of two lemmas. Lemma 27.1. (1) Let n be odd, or  j (2) Let l be even and N j=1 |Ll (ξ, 0)| = 0 for ξ = 0, or (3) Let L(ξ, ) = ∇W (ξ, ). Then in every neighborhood of the point ξ = 0,  = 0, there exists a pair (ξ ∗ , ∗ ), ξ ∗ = 0 satisfying (27.9).

Resolving Branching Equation

405

Proof. If the point ξ = 0 is a nonisolated singular point of the vector field L(ξ, 0), then in every neighborhood of the point ξ = 0,  = 0, there exists a pair (ξ ∗ , 0) such that ξ = 0 satisfies (27.9) and the lemma is true. Assume ξ = 0 is an isolated point of the vector field L(ξ, 0). Let us consider three cases. Case 1. Let n be odd. Let us take the neighborhood |ξ| ≤ r, || ≤  and introduce the vector field Φ(ξ, t) =

(2t − 1) ξ + L(ξ, (2t − 1)). 1 − (2t − 1)

If Φ(ξ, t) = 0 for t ∈ [0, 1] and |ξ| = r, then the degree of the map   Φ , S(0, r) Jt = J Φ of the boundary of the sphere |ξ| = r into the unit sphere is well defined [36], and hence, Jt is the same integer for each t ∈ [0, 1]. But, J0 = (−1)N , J1 = 1N . Hence, Jt = const. Therefore, for all r > 0 and  > 0, there exist t∗ ∈ [0, 1] and ξ ∗ , |ξ ∗ | = r such that Φ(ξ ∗ , t∗ ) = 0. The corresponding pair (ξ ∗ , (2t∗ − 1)) satisfies system (27.9). Case 2. Let l be even and let N  j=1

|Ljl (ξ, 0)| = 0.

In this case, the field L(ξ, ) is homotopic to Ll (ξ, 0) on the sphere S(0, r) for || < δ with sufficiently small δ. Hence,     Ll (ξ, 0) L(ξ, ) , S(0, r) = J , S(0, r) J L Ll is even number because l is even. Let us fix an ∗ ∈ (−δ, δ) and introduce the Kronecker index [93] γ0 of the isolated singular point ξ = 0 of the field L(ξ, ∗ ),

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Toward General Theory of Differential-Operator and Kinetic Models

γ0 = (sign ∗ )N . By the Kronecker theorem [93],    L(ξ, ∗ ) J , S(0, r) = γi . L

(27.10)

i

Since the left-hand side of equation (27.10) is even, and γ0 is odd, it follows that along with the point ξ = 0, the sphere S(0, r) contains another singular point ξ ∗ = 0 of the field L(ξ, ∗ ). The pair (ξ ∗ , ∗ ) satisfies system (27.9). Case 3. Let L(ξ, ) = grad W (ξ, ), where   ξi2 + U (ξ, ), W (ξ, ) = 2(1 − )

|ξ| < δ,

δ > 0.

i

If the point ξ = 0 is a nonisolated critical point of the potential W (ξ, 0) (of the potential W (ξ, ) for 0 < || < δ), then the lemma is true. Let the point ξ = 0 be an isolated critical point of the potential W (ξ, 0) and of the potential W (ξ, ) for 0 < || < δ. Then the Morse– Conley indices (see [36]) are defined for the critical point ξ = 0 of the potential W (ξ, 0) and of the potential W (ξ, ) for 0 < || < δ, where δ is sufficiently small. By the homotopic invariance property of the Morse–Conley index (see [36], Theorem 1.4, p. 67), these indices are equal. But for 0 < || < δ, the point ξ = 0 is a nondegenerate critical point because ∂2 N w(ξ, ) = = 0. det ∂ξi ∂ξj (1 − )N =0 Therefore the Morse–Conley index is equal to the number of negative eigenvalues of the corresponding Hessian  δ . ik 1 −  i,k=1,...,n But then this index is zero for  > 0, and for  < 0 it is equal to n. Therefore, the point ξ = 0 cannot be an isolated critical point of the potentials W (ξ, 0) and W (ξ, ) for 0 < || < δ. Hence, ξ = 0 is a  bifurcation point of the BEq L(ξ, ) = 0.

Resolving Branching Equation

407

Lemma 27.2. (1) Let l be even, and let the system ξ + Ll (ξ, 0) = 0

(27.11)

have a simple real solution ξ 0 =  0. Then in a neighborhood of the point  = 0, system (27.9) has a real-valued solution of the form ξ = (ξ 0 + o(1))1/(l−1) .

(27.12)

(2) Let l be odd and let system (27.11) or the system − ξ + Ll (ξ, 0) = 0

(27.13)

have a simple real solution ξ 0 = 0. Then in the half-neighborhood  > 0 ( < 0), there exist two solutions of the form ξ = (±ξ 0 + o(1))||1/(l−1) .

(27.14)

(3) Let Ll (ξ, 0) = grad U (ξ) and let ξ 0 be an isolated extremum of the function U (ξ) on the sphere |ξ| = 1, U (ξ 0 ) = 0. Then there exists a solution of the form ξ = (c + o(1))||1/(l−1) , where ⎧ 1/(l−1) sign  ⎪ ⎪ ξ0, ⎪ ⎨ (l + 1)U (ξ 0 ) c= 1/(l−1)  ⎪ ⎪ 1 ⎪ ⎩± ξ 0 , U (ξ 0 ) > 0, (l + 1)U (ξ 0 )

(27.15)

if l is even, if l is odd. (27.16)

Proof. Under conditions (1) and (2) we consider the solutions of equation (27.11) in the form ξ = η()1/(l−1) . To define η(0), we obtain two systems: one for  > 0 and other for  < 0, that is systems (27.11) and (27.13). If l is even, then

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Toward General Theory of Differential-Operator and Kinetic Models

the substitution ξ = −ξ transforms equation (27.11) into equation (27.13). If l is odd, this substitution does not change equations. Therefore, in the case of simple real solutions ξ 0 , the existence of solutions of the form (27.12), (27.14) follows from the implicit function theorem. (3) Let Ll (ξ, 0) = grad U (ξ). Then we consider the desired solutions in the form  1/(l−1)  , ξ= 2q where |η| = 1 and q() is a scalar parameter satisfying the condition q(0) > 0 for odd l. For q and η, we obtain the system 2qη + Ll (η, 0) + Θ(η, q, ) = 0, |η| = 1, where Θ = o(1) as  → 0. Therefore, 2q(0)η(0) + Ll (η(0), 0) = 0 and |η(0)| = 1. Since by assumption, ξ 0 is an isolated extremum of the function U (ξ) on the sphere |ξ| = 1 and U (ξ 0 ) = 0, we set q(0) = (l + 1)U (ξ 0 ), η(0) = ξ 0 . Consider the perturbed vector field   ∂U + Θi , |η| = 1 . Φ (η, q) = 2qηi + ∂ηi Let S be the sphere of radius  > 0 centered at the point (q(0), η 0 ) in Rn+1 . Let us introduce the degree of the map: ⎧ ⎪ if (q(0), ξ 0 ) is an arg of the ⎪ +1 ⎪  ⎪  ⎨ min q(0)|ξ|2 + U (ξ, 0), Φ , S = J ⎪ Φ (−1)n+1 if (q(0), ξ 0 ) is an arg of the ⎪ ⎪ ⎪ ⎩ max q(0)|ξ|2 + U (ξ, 0). Since this degree is nonzero, it follows that the vector field Φ (η, q) = 0 has a singular point in a neighborhood of the point  (q(0), 0 ) for || < δ with sufficiently small δ. With the help of Theorem 27.1 and Lemmas 27.1 and 27.2, it is now possible to prove the following results on the bifurcation point for the problem (26.7), (26.13).

Resolving Branching Equation

409

Theorem 27.2. Let conditions (26.11), (26.12) and (i)–(iii) with λ0 = −μ/χ− as well as one of the following three conditions be satisfied: (1) n is odd;  (2) l is even and N i=1 |Rl (ξe), ei | = 0 for ξ = 0; (3) fi = fi (a(−αi v 2 + ϕi ) + b(vdi + ψi )), i = 1, . . . , N and a, b are constants. Then λ0 is a bifurcation point of the boundary value problem (26.7), (26.13). Proof. By assumptions (1)–(3) of the theorem, the assumptions of Lemmas 27.1 and 27.2 are satisfied for BEq (27.6) of the boundary value problem (26.7), (26.13). Equation (27.5) establishes a one-toone correspondence between the desired solutions of the boundary value problem and small solutions of BEq (27.6). Therefore, the  validity of Theorem 27.2 follows from these lemmas. Corollary 27.1. Let the potentials of the electromagnetic field satisfy conditions (25.3) and let the assumptions of Theorem 27.2 be satisfied. Then λ0 is a bifurcation point of the VM system (25.1)– (25.2). Example 27.1. Let the distribution functions of the VM system have the form [134] fi = λ exp(−αi v 2 + (di , v) + γi + li ϕ(r) + ki ψ(r)), and N  i=1 N  i=1

−3/2 qi αi

−3/2 qi αi



d2 exp γi + i 4αi 

d2 exp γi + i 4αi

 = 0, 

ki = 0. li

Conditions (26.1) and (26.12) for βi = di /(2αi ) and assumptions (1)–(3) of Theorem 27.2 are satisfied. Thus, BEq (27.6) is potential. If μ is an eigenvalue of the Dirichlet problem (iii), then, by

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Toward General Theory of Differential-Operator and Kinetic Models

Corollary 27.1, λ0 = −μ/χ is a bifurcation point of the VM system with conditions (25.3) and (25.4), where u10 = u20 = 0. Thus,       T1 = ai li , T2 = ai ki , T3 = ai li , 

T4 =



k2  ai i , ai = qi li



π αi

3/2

  d2i exp γi + , 4αi

 1 χ− = (νT4 − μT1 − (νT4 − μT1 )2 + 4νμ[T1 T4 − T2 T3 ]). 2 Theorem 27.3. Let conditions (26.11), (26.12), and (i)–(iii) with λ0 = −μ/χ− as well as the conditions of one of the three statements of Lemma 27.2 be satisfied. If (1) is satisfied, then the boundary value problem (26.7), (26.13) has the solution  N  ξi0 ei + o(1) |λ − λ0 |1/(l−1) . u= i=1

If (2) is satisfied, then there exist two solutions:  N   ξi0 ei + o(1) |λ − λ0 |1/(l−1) , u± = ± i=1

defined in the half-neighborhood λ > λ0 (λ < λ0 ), provided that xi0 satisfies system (27.11) (ξ 0 satisfies system (27.13)). If (3) is satisfied, then  N  ci ei + o(1) |λ − λ0 |1/(l−1) , (27.17) u= i=1

where the vector c is defined by equation (27.16). The proof follows from Lemmas 27.1, 27.2 and equation (27.5). Corollary 27.2. Let the potentials of the electromagnetic field satisfy condition (25.3) and let the assumptions of Theorem (27.3) be

Resolving Branching Equation

411

satisfied. Then the VM system (25.1)–(25.2) with conditions (25.3)– (25.4) has the solution (26.4), where ϕ=−

2αq u10 + u1 (r, λ), m

ψ=

q u02 + u2 (r, λ), mc

(27.18)

the functions u1 , u2 → 0 are defined by Theorem 27.3 as λ → λ0 . Let us consider more detailed distribution functions of the form fi = fi (−αv 2 + c1i + li ϕ) + b(vdi + c2i + ki ψ)),

(27.19)

where li and ki are connected by the linear relations (26.1), the integral  fi dv = Ai (ali ϕ + bki ψ) R3

converges and ∂Ai (s)/∂s < 0 for all s. In this case, the conditions (26.11), (26.12), (i), (ii) and the assumptions of Theorem 27.3 are satisfied by Lemmas 27.1 and 27.2; hence according to Lemma 27.1, BEq (27.6) is potential. In Theorem 27.3, case (3) occurs. Therefore, the form of the functions u1 (r, λ), u2 (r, λ) in (27.18) can be specified, namely, in the case of the distribution (27.19), the vector u = (u1 , u2 ) in equation (27.18) can be given by equations (27.17) and (27.16). Thus, if we found that vector c in equation (27.16) corresponds to a nonisolated extremum of the corresponding potential, then some of its coordinates may be arbitrary points of some sphere S ⊂ Rk , where k ≤ n (see [151, 165]). Then problem (26.25) will have a solution depending on free parameters. This case is possible if the domain D is symmetric and problem (26.25) has a spherical symmetry. Thus, the free parameters remaining in the solution have a group meaning. Let us show that this is just the situation that arises in our problem in the case of a circular cylinder. Let us introduce the following condition: (iv) D = {x ∈ R2 | x21 + x22 = 1} and the matrix R (u) is symmetric for any u.

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Toward General Theory of Differential-Operator and Kinetic Models

Let us pass to the polar coordinates x1 ρ cos Θ, x2 = ρ sin Θ in system (26.25). Then =

1 ∂ 1 ∂2 ∂2 + , u|ρ=1 = 0. + ∂ρ2 ρ ∂ρ ρ2 ∂Θ2

Condition (iii) now makes sense. According to it, 2

μ ∈ {μ(s) σ , s = 0, 1, . . . ,

σ = 1, 2, . . .},

(s)

where μσ are the zeros of the Bessel function Js (u). If μ = (s )2 μσ00 , s0 ≥ 1, then dim N (B) = 2; the vectors (s0 ) 0) c− Js0 (μ(s σ0 ρ)cos s0 Θ, c− Js0 (μσ0 ρ)sin s0 Θ

form a basis in the subspace N (B). By Lemma 27.1, BEq (27.6) is potential on the whole. Moreover, BEq (27.6) allows the group O(2) and by [102], the theorem has the form ξi |ξ|−1 L(|ξ|, ) = 0,

(27.20)

where L(|ξ|, ) =

∞ 

L2i+1 ()|ξ|2i+1 ,

i=0

l1 () =

 1−

is analytical. Let us note that in this case, the forms (27.18) which are even in ξ must be lacking on the left-hand sides of BEq (27.6), since L(|ξ|, ) is odd in ξ. Remark 27.1. In view of (27.20), the potential U of BEq in (27.8) has the form  U =− 0

|ξ|

L(s, )ds +

2 + ξ2 1 ξ12 2 . 1− 2

Resolving Branching Equation

413

Therefore,  U ||ξ|=1 = −

1

L(s, )ds +

0

1 . 2(1 − )

Let L2i+1 () ≡ 0, i = 1, 2, . . . , m − 1,

l2m+1 () = 0

in (27.20). Assuming that ξ1 = r cos α, ξ2 = r sin α, let us reduce the system (27.20) to the single equation  + L2m+1 ()r 2m + O(r 2m+2 ) = 0. 1−

(27.21)

Note that L2m+1 (0) = λ0 R2m+1 (ξe), ej ξj−1 |ξ|−2m , j = 1, 2 for all ξ if R2 u = · · · = R2m (u) = 0.



Remark 27.2. If R2 (u) = 0, then for all ξ, we have L3 (0) = λ0 ξj−1 |ξ|−2 R3 (ξe) + R2 (ξe), ej ,

j = 1, 2.

From equation (27.21), we find two solutions:  r1,2 = ±

2m

− L2m+1 (0)

+ O(||1/2m ),

which are real for L2m+1 (0) < 0. The two solutions          ξ1  cos α −     + O(||1/2 )     = ± 2m  sin α   ξ2  L2m+1 (0)

(27.22)

of BEq correspond to these solutions, where the parameter α ∈ R corresponding to the group O(2) remains arbitrary. Substituting (27.22) into (27.5) and the vector (27.5) into (27.18), we obtain two

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Toward General Theory of Differential-Operator and Kinetic Models

solutions: 

ϕ±

⎡

⎤ 2αq u01 ⎥ ⎢− = ⎣ qm ⎦ u02 mc   −T2 /T1 0) js0 (mu(s ± σ0 ρ) cos s0 (Θ − α) 1 $ λ − λ0 + 0(|λ − λ0 |1/(2m) ). × 2m − L2m+1 (0)



ψ±

For (λ − λ0 )L2m+1 (0) < 0, to these solutions, there correspond two real solutions fi± , E ± , B ± of the stationary VM system (25.1)–(25.2) with boundary conditions (25.3)–(25.4) determined by equations (26.4). In conclusion, we note that instead of condition (26.11), it sufficies to impose the following condition: (v) The potentials U and A of the electromagnetic field (E, B) satisfy conditions (25.3) and hence, N 

qi Ai (li ϕ0 , ki ψ 0 ) = 0,

i=1

where tions

ϕ0

N 

qi (βi , d)Ai (li ϕ0 , ki ψ 0 ) = 0,

i=1

and

ψ0

are harmonic functions with the boundary condi-

2αq q u01 (r), ψ 0 |∂D = u02 (r). m mc Moreover, if distribution functions has the form ϕ0 |∂D = −

fˆi = fi (−αv 2 + ϕi ) + b(vdi + ψi )), then the results are obtained by analogy in the case of nonconstant values u01 and u02 . The procedure presented can be used for constructing the nontrivial solutions of the integro-differential system (26.10). Therefore, analogous results also occur, in the problem of the point of bifurcation of the VM system with normalized distribution function.

Chapter 28

Numerical Modeling of the Limit Problem for the Magnetically Noninsulated Diode 28.1.

Introduction

This chapter is aimed at studying the stationary self-consistent problem of magnetic insulation under space charge limitation via the asymptotics of the Vlasov–Maxwell system. This approach has been introduced by Langmuir and Compton [98] and recently developed by Ben Abdallah et al. [19, 22] to analyze the space charge limited operation of a vacuum diode. In a dimensionless form of the Vlasov– Poisson system, the ratio of the typical particle velocity at the cathode to that reached at the anode appears as a small parameter [22]. The associated perturbation analysis provides a mathematical framework to the results of Langmuir and Compton [98], stating that the current flowing through the diode cannot exceed a certain value called the Child–Langmuir current. This chapter is concerned with an extension of this approach, based on the Child–Langmuir asymptotics to magnetized flows [19]. In particular, the value of the space charge limited current is determined when the magnetic field is small (noninsulated diode). Since the arising model could not be solved analytically, it is very important to discover its properties in noninsulated and nearly insulated cases first. For a better understanding of the discussed mathematical problem, we provide an extended list of bibliographic references: [19,22,23, 103, 113, 126, 133]. To establish the correspondence of the numerical

415

416

Toward General Theory of Differential-Operator and Kinetic Models

modeling results with rising physical effects in vacuum diode first, we need to introduce the description of how it really works. The other related important thing is a brief discussion of the physical processes giving rise to the diode current fluctuations. A better understanding of these properties is very essential for examining the current instabilities in the nearly magnetic insulated diode. The excellent description of these processes found in [88] is reproduced here. 28.2.

Description of Vacuum Diode

The vacuum diode consists of a hot cathode surrounded by a metal anode inside an evacuated enclosure. At sufficiently high temperatures, electrons are emitted from the cathode and are attracted to the positive anode. Electrons moving from the cathode to the anode constitute a current; they do so when the anode is positive with respect to the cathode. When the anode is negative with respect to the cathode, electrons are repelled by the anode and the reverse current is almost zero (due to the tail of the Maxwellian distribution of the electrons, it is greater than zero). The space between the anode and the cathode is evacuated, so that electrons may move between the electrodes unimpeded by collisions with gas molecules. If Vf = 0 and no emission occurs, the diode may be regarded as a parallel plate capacitor, whose potential difference is Vp . In this case, the potential distribution in the cathode–plate space is represented by a straight line which joins the points corresponding to cathode potential Vk = 0 and the plate potential Vp . When the filament voltage rises, the electrons leaving the cathode gang up in the interelectrode space as a cloud called space charge. This charge alters the potential distribution. Since the electrons making up the space charge are negative, the potential in the cathode–plate space goes up, though all points remain at positive potential. The vector of the electric field is directed from the plate to the cathode, so all the electrons escaping from the cathode make for the plate. In this case, the plate current equals the emission current. One could

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say the all electrons are being sucked away from the cathode by the anode. This region is known as the emission-limited region. As the filament voltage is increased, emission increases, and so does the space charge. Electrons having low initial velocities are driven back to the cathode by the negative space charge due to the electrons. The density of the electron cloud near the cathode increases to the point where it forms a negative potential region whose minimum, Vmin , is usually within a few hundredth or tenths of a millimeter of the cathode surface. Thus, there is a high retarding electric field near the cathode (0 < x < xmin ); the vector is directed away from the cathode to the plate. To overcome this field, an initial velocity the cathode should exceed a certain value v0 of the electrons leaving e Vmin . determined by Vmin , v0 > 2 m If the electron is below this value, the electron will not be able to overcome the potential barrier. It will slow down to a stop, and the field will push it back to the cathode. Accordingly, the retarding field region (from 0 to xmin ) contains not only electrons traveling away from the cathode but also those falling back toward the cathode. At a constant filament voltage, a dynamic equilibrium sets in, so that the number of electrons reaching the plate and the number falling back to the cathode are equal to the number of electrons emitted by the cathode. Therefore, plate current is smaller than emission current, or the cathode produces more electrons than the anode can. 28.3.

Shot Noise in a Diode

We know that the minimum signal observable in an electronic circuit is set by the level of electrical noise in the system. This is caused by a random fluctuation of voltage and current or electromagnetic fields. Shot noise in a diode is the random fluctuation in a diode current, I, due to the discrete nature of electronic charge. Noise causes the signal to fluctuate around a given value. The average of these fluctuations are zero due to their randomness. But the root mean square of the fluctuation is measurable. Perhaps, a discussion of the various types of electrical noise giving rise to degradation of the observed signal would be in order.

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Electrical noise may persist even after the input signal has been removed from the electronic circuit. This implies the existence of a basic limit below which signals are no longer distinguishable. The signal-to-noise ratio quantifies the observability of an output signal. Hence, a measure whether satisfactory amplification can be obtained is given by this ratio. Therefore, in order to ensure the maximum observability of an amplified weak signal, we must ensure that the noise power introduced by the circuit devices and components should be as small as possible. External sources of noise can produce electrical interference in circuits. This may be done by electromagnetic radiation. Examples of this would be narrow-frequency band sources, such as radio transmitters, local oscillators and power-supply cables and also broadband sources, such as lightning and fluorescent lamps. Another means by which electrical noise may be induced in an electronic circuit from an external source is electromagnetic induction. Since magnetic fields arise from alternating current, then by electromagnetic induction, corresponding noise signals may be induced into other circuits or different parts of the same electronic system. In order to reduce such effects, we take care of the positioning of critical circuit components to take advantage of the short range of such magnetic fields. Such effects can be greatly reduced by electrostatic screening (i.e., placing the entire circuit, or at least the sensitive portions of it, inside a closed metal box and connecting the box to earth potential). It is important that the total electrostatic screening for a system is earthed at one point only — this ensures that no large-area circuit earth loops can exist, in which signal may again be induced by electromagnetic induction. The main types of internal sources of noise present in electronic devices are thermal noise and shot noise. Thermal noise results due to the random motion of the current carriers in a metal or semiconductor, which increases with temperature. Thermal noise arises from the random motion of electrons in materials due to their thermal energy of 3kT /2 and therefore occurs even in the absence of an applied electric field. Shot noise results due to the random flow of electrons in an electric current and also due to the particle nature of electric charge.

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The current flow in a vacuum diode is due to the emission of electrons from the cathode which they travel to the anode. Each electron carries a discrete amount of charge and produces a small current pulse. The average anode current, Ia , is the summation of all the current pulses. The emission of electrons is a random process, depending on the surface condition of the cathode, shape of electrodes, and potential between the electrodes. This gives rise to random fluctuations in the number of electrons emitted and so the diode current contains a time-varying component. Since each electron arriving at the anode is like a “shot”, the fluctuating current gives rise to a mean-square shot noise current i2s . 28.4.

Description of the Mathematical Model

We consider a plane diode consisting of two perfectly conducting electrodes, a cathode (X = 0) and an anode (X = L) supposed to be infinite planes, parallel to (Y ; Z). The electrons, with charge −e and mass m, are emitted at the cathode and submitted to an applied electromagnetic field Eext = Eext X; Bext = Bext Z such that Eext ≤ 0 and Bext ≥ 0. Such an electromagnetic field does not act on the PZ component of the particle momentum. Hence, we consider a situation where this component vanishes, leading to a confinement of electrons to the plane Z = 0. The relationship between momentum and velocity is then given by the relativistic relations  ⎧ |P|2 ⎪ P ⎨V(P) = , γ = 1 + 2 2, γm m c (28.1) ⎪ ⎩ 2 2 2 V = (VX , VY ), P = (PX , PY ), |P| = PX + PY . which can also be written as V(P) = ∇P E(P),

(28.2)

where E is the relativistic kinetic energy, E(P) = mc2 (γ − 1),

(28.3)

and c is the speed of light. Moreover, we assume that the electron distribution function F does not depend on Y and that the flow is

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Toward General Theory of Differential-Operator and Kinetic Models

stationary and collisionless. The injection profile G(PX , PY ) at the cathode is assumed to be given, whereas no electron is injected at the anode. The system is then described by the so-called 1.5-dimensional Vlasov–Maxwell model:  dΦ ∂F ∂F dA ∂F VX + eVX +e − VY = 0, (28.4) ∂X dX ∂PX dX ∂PY e d2 Φ = N (X), 2 dX 0 d2 A = −μ0 JY (X), dX 2

X ∈ (0, L), X ∈ (0, L),

(28.5) (28.6)

subject to the following boundary conditions: F (0, PX , PY ) = G(PX , PY ), F (L, PX , PY ) = 0, Φ(0) = 0, A(0) = 0,

PX > 0,

PX < 0,

(28.7) (28.8)

Φ(L) = ΦL = −LEext ,

(28.9)

A(L) = AL = LBext .

(28.10)

In this system, the macroscopic quantities, namely, the particle densities N, X and Y are the components of the current densities JX , JY . In the above equations, 0 and μ0 are, respectively, the vacuum permittivity and permeability. The boundary conditions are justified by the fact that the electric field E = −dΦ/dX and the magnetic field B = −dA/dX are exactly equal to the external fields when self-consistent effects are ignored (N = JY = 0). The 1.5-dimensional model (28.4)–(28.10) ignores the selfconsistent magnetic field due to JX , which would introduce twodimensional effects, and is only an approximation of the complete stationary Vlasov–Maxwell system. In this chapter, we are especially interested in the case when the applied magnetic field is not strong enough to insulate the diode, JX does not vanish and our model can be viewed as an approximation of the Maxwell equations. In order to get a better insight into the behavior of the diode, we write the model in dimensionless variables in the spirit of [40].

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We first introduce the following units, respectively, for position, velocity, momentum, electrostatic potential, vector potential, particle density, current and distribution function: ¯ = L, X 2 ¯ = mc , Φ e

P¯ = mc, E = mc2 , ¯ ¯ ¯ = 0 Φ , J¯ = −ecN ¯ , F¯ = N , N ¯2 xX P¯ 2

V¯ = c,

mc A¯ = , e

and the corresponding dimensionless variables P X x = ¯ , p = ¯ = (px , py ), X P  V E p , ε = ¯ = 1 + p2 − 1, v = (vx , vy ) = ¯ =  V E 1 + p2 Φ φ = ¯, Φ

A a = ¯, A

N n= ¯, N

J j = ¯, J

F f = ¯. F

The next step is to express that particle emission at the cathode occurs in the Child–Langmuir regime: in such a situation, the thermal velocity VG is much smaller than the typical drift velocity supposed to be of the order of the speed of light c. Letting  = VG /c, we assume that 1 px py  , , px > 0, f (0, px , py ) = g (px , py ) = 3 g    where g is a given profile. The scaling factor 3 ensures that the incoming current remains finite independent of , whereas the dependence on p expresses the fact that electrons are emitted at the cathode with a very small velocity. For a detailed discussion of the scaling, we refer to [40]. The dimensionless system reads   dϕ ∂f  da ∂f  da ∂f  + − vy + vx = 0, vx (28.11) ∂x dx dx ∂px dx ∂py (x, px , py ) ∈ (0, 1) × R2 , d2 ϕ = n (x), dx2 d2 a = jy (x), dx2

x ∈ (0, 1),

(28.12)

x ∈ (0, 1).

(28.13)

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Toward General Theory of Differential-Operator and Kinetic Models

Here, n (x) is a particle density and jy (x) is a current density in Y direction. The initial and boundary conditions are also transformed: 1 px py  f  (0, px , py ) = g (px , py ) = 3 g , , px > 0, (28.14)    (28.15) f  (1, px , py ) = 0, px < 0, ϕ (0) = 0,

ϕ (1) = ϕL ,

(28.16)

a (0) = 0,

a (1) = ϕL .

(28.17)

Omitting the complete derivation of the limit system, when  → 0, we need to introduce some notions and notations used ahead. Definition 28.1. We define Θ(x) = (1 + ϕ(x))2 − 1 − a2 (x) as an effective potential. It is readily seen that electrons do not enter the diode unless the effective potential Θ is nonnegative in the vicinity of the cathode. Therefore, we always have Θ (0) ≥ 0. The limiting case Θ (0) = 0 is the space charge limited or the Child–Langmuir regime. In view of (28.16) and (28.17) (it still holds in the limit  → 0), this condition is equivalent to the standard Child–Langmuir condition dϕ dx (0) = 0. Let ΘL be the value of Θ at the anode ΘL = (1 + ϕL ) − 1 − a2L . If ΘL < 0, electrons cannot reach the anode x = 1, they are reflected by the magnetic forces back to the cathode and the diode is considered to be magnetically insulated. This enables us to define the Hull cutoff magnetic field, which is the relativistic version of the critical field introduced in [76] in the nonrelativistic case: = ϕ2L + 2ϕL . aH L The diode is magnetically insulated if aL > aH L and is not insulated H if aL < aL . In dimensional variables, the Hull cut-off magnetic field is given by  1 2mc2 H ΦL . Φ2L + B = Lc e Thus, our primary goal is a study of noninsulated, or nearly insulated diodes, which means Bext < B H . The complete derivation of the

Numerical Modeling of the Limit Problem

423

model is given in [22], while we need only its formal expressions d2 ϕ 1 + ϕ(x) , (x) = jx  2 dx (1 + ϕ(x))2 − 1 − a2 (x)

(28.18)

a(x) d2 a , (x) = jx  2 dx (1 + ϕ(x))2 − 1 − a2 (x)

(28.19)

with corresponding Cauchy and boundary conditions ϕ(0) = 0,

ϕ(1) = ϕL ,

dϕ (0) = 0, dx a(0) = 0,

a(1) = aL .

(28.20) (28.21) (28.22)

Let us recall that the unknowns are the electrostatic potential ϕ, the magnetic potential a and the current jx (which does not depend on x). It is to be noted that the whole construction of this model depends heavily on the assumption that the effective potential is positive. Actually, Θ could vanish at some points in the diode, leading to closed trajectories and trapped particles. Apart from heuristic discussions, some analytical remarks could also be made about the parametric dependences jx , β. In particular, they are (1 + ϕ(x))a (x) − ϕ (x)a(x) = β,  2jx Θ(x) − (ϕ (x))2 + (a (x))2 = β 2 .

(28.23) (28.24)

The analyses of these equations were made in [22], but the proposed approach do not provide any information to be immediately used in numerical computations. Nevertheless, this relation could be treated as an auxiliary method for verification of any jx , β range. Vector (jx , β) is hereinafter usually referred to as a parameter vector, depending on the boundary condition of the problem (28.18) and (28.19). Since the analysis of the couple of arbitrarily  chosen boundary conditions ϕL , aL is not very useful, we refer to Θ(x) and

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Toward General Theory of Differential-Operator and Kinetic Models

√

ΘL especially as a distance measure. The quantities (ϕL , aL ) and √ (ϕL , ΘL ) are algebraically equivalent on R+ to define the boundary conditions, thus we evaluate a sets of equally distant point and refer √ to them as (z, ΘL ), z = ϕL . Keeping in mind the above remarks, we devote the section to the analysis of the solution trajectories, their relation with the lower and upper estimations obtained by Sinitsyn and better solution approximations. Then we consider some known methods and describe newly developed numerical methods for solving the posed limit problem. Also, some issues of the numerical method stability in application to our problem are discussed. In the following section, we introduce the results of numerical experiments, describing the properties of the parameter vector for different “distances” ΘL . The numerical experiments have shown that the character of the parameter curves highly depends on the quantity ΘL , being unique for ΘL ≥ 1 and bifurcating for ΘL < 1. The accurate numerical modeling gave us a number of different parameter vectors √ that comply with the unique given boundary condition (ϕL , ΘL ). Due to high computation times and numerical sensitivity of the problem, it is difficult to evaluate the bifurcating solutions for Θ ≈ 1 and we fixed our attention on nearly insulated diode behavior for ΘL 1. Section 28.5 describes in brief the obtained results. 28.5.

Solution Trajectory, Upper and Lower Solutions

Finally, the limit model of magnetical noninsulation diode is described by the system of two second-order ordinary differential equations (28.18) and (28.19) with conditions (28.20)–(28.22). 28.5.1.

Existence of semitrivial solutions for the problem

Let us introduce the definition of cone in a Banach space X.

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425

Definition 28.2. Let X be a Banach space. A nonempty convex closed set P ⊂ X is called a cone if it satisfies the conditions: (i) x ∈ P, λ ≥ 0 implies λx ∈ P ; (ii) x ∈ P , −x ∈ P implies x = O, where O denotes zero element of X. Here, ≤ is the order in X induced by P , i.e., x ≤ y if and only if y − x is an element of P . We also assume that the cone P is normal in X, i.e., order intervals are norm-bounded. ¯ u = v = 0}, we introduce the norm In X ≡ {(u, v) : u, v ∈ C1 (Ω), |U |X = |u|C 1 + |v|C 1 and the norm |U |X = |u|∞ + |v|∞ in C, where U = (u, v). Here, a cone P is given by P = {(u, v) ∈ X : u ≥ 0, v ≥ 0 for all x ∈ Ω}. So, if u = 0 and v = 0 belong to P , then −u and −v do not belong. We work with classical spaces on the intervals ¯ with norm u ∞ = max{|u(x)| : I¯ = [a, b], Iˆ =]a, b], I = (a, b) : C(I) 1  ¯ ¯ x ∈ I}; C (I) = u ∞ + u ∞ ; Cloc (I), which contains all functions that are locally absolutely continuous in I. We introduce a space Cloc (I) because the limit problem is singular for ϕ = 0. The order ≤ in cone P is understood in the weak sense, i.e., y is increasing if a ≤ b implies y(a) ≤ y(b) and y is decreasing if a ≤ b implies y(a) ≥ y(b). Theorem 28.1 (Comparison principle in cone). Let ¯ ∩ Cloc (I). y ∈ C(I) The function f is defined on I × R. Let f (x, y) be increasing in y function. Then v  − f (x, v) ≥ w − f (x, w)

in mean on I,

(28.25)

¯ v(a) ≤ w(a), v(b) ≤ w(b) implies v ≤ w on I. For the convenience of defining an ordering relation in cone P , we make a transformation for the problem (28.18)–(28.19). Let F (ϕ, a) and G(ϕ, a) be defined as in Definition 28.3. Then through the

426

Toward General Theory of Differential-Operator and Kinetic Models

transformation ϕ = −u, the limit problem is reduced to the form −

1−u d2 u  = jx  = F˜ (jx , u, a), 2 2 dx2 (1 − u) − 1 − a

u(0) = 0,

u(1) = ϕL ,

a d2 a  ˜ = jx  = G(j x , u, a), 2 dx (1 − u)2 − 1 − a2 a(0) = 0,

(28.26)

a(1) = aL .

We note that all solutions of the initial problem, as well the problem (28.26), are symmetric with respect to the transformation of sign for the magnetic potential a : (ϕ, a) = (ϕ, −a) or the same (u, a) = (u, −a). Thus, we must search only positive solutions ϕ > 0, a > 0 in cone P or only negative ones: ϕ < 0, a < 0. Thanks to the symmetry of problem, it is equivalent and does not yield the extension of the types of sign-defined solutions of the problem (28.26)–(28.22) (respectively, (28.26)). Once more, we note that the introduction of negative electrostatic potential in problem (28.26) is connected with more convenient relation between order in cone and positiveness of the Green function for operator −u that we use in the following. Definition 28.3. A pair [(ϕ0 , a0 ), (ϕ0 , a0 )] is called (a) sub–super solution of the problem (28.18)–(28.22) relative to P if the following conditions are satisfied:

¯ (ϕ0 , a0 ) ∈ Cloc (I) ∩ C(¯(I) × Cloc (I) ∩ C(I), (28.27) ¯ × Cloc (I) ∩ C(I), ¯ (ϕ0 , a0 ) ∈ Cloc (I) ∩ C(I) ϕ0 − jx 

1 + ϕ0 (1 + ϕ0

(ϕ0 ) − jx  in I

)2



− 1 − a2

= F (ϕ0 , a) ≤ 0 in I,

1 + ϕ0 (1 +

ϕ0 )2

−1−

for all a ∈ [a0 , a0 ];



a2

= F (ϕ0 , a) ≥ 0

Numerical Modeling of the Limit Problem

a0 − jx 

a0 (1 +

(a0 ) − jx  in I

ϕ)2



−1−

a20

= G(ϕ, a0 ) ≤ 0

a0 (1 +

ϕ)2

427

in I,



−1−

(a0 )2

= G(ϕ, a0 ) ≥ 0

for all ϕ ∈ [ϕ0 , ϕ0 ];

ϕ0 ≤ ϕ0 , a0 ≤ a0 in I, and on the boundary ϕ0 (0) ≤ 0 ≤ ϕ0 (0),

ϕ0 (1) ≤ ϕL ≤ ϕ0 (1),

a0 (0) ≤ 0 ≤ a0 (0),

a0 (1) ≤ aL ≤ a0 (1);

(b) sub–sub solution of the problem (28.18)–(28.22) relative to P if a condition (28.27) is satisfied and ϕ0 − F (jx , ϕ0 , a0 ) ≤ 0 in I, a0 − G(jx , ϕ0 , a0 ) ≤ 0

in I,

(28.28)

and on the boundary, ϕ0 (0) ≤ 0,

ϕ0 (1) ≤ ϕL ,

a0 (0) ≤ 0,

a0 (1) ≤ aL .

(28.29)

Remark 28.1. In Definition 27.3, we take the expressions with square roots by modulus of effective potential Θ(·). By analogy with (28.28) and (28.29), we may introduce the definition of super–super solution in cone. Definition 28.4. We shall call the functions Φ(x, xai , jx ), Φ1 (x, xϕj , jx ) semitrivial solutions of the problem (28.18)–(28.22) if Φ(x, xai , jx ) is a solution of the scalar boundary value problem: ϕ = F (jx , ϕ, xai ) = jx  ϕ(0) = 0,

ϕ(1) = ϕL ,

1+ϕ , (1 + ϕ)2 − 1 − (xai )2 (28.30)

and Φ1 (x, xϕj , jx ) is a solution of the scalar boundary value problem: a , a = G(jx , xϕj , a) = jx (1 + xϕj )2 − 1 − a2 a(0) = 0,

a(1) = aL ,

(28.31)

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Toward General Theory of Differential-Operator and Kinetic Models

where xai , i = 1, 2, 3, and xϕj , j = 1, 2, are, respectively, the indicators of semitrivial solutions Φ(x, xai , jx ) and Φ1 (x, xϕj , jx ) defined in the following ways: • • • •

xa1 = 0 if a(x) = 0; xa2 = a0 if a = a0 is the upper solution of the problem (28.31); xϕ1 = ϕ0 if ϕ = ϕ0 is the upper solution of the problem (28.31); xϕ2 = ϕ0 if ϕ = ϕ0 is the lower solution of the problem (28.30).

From Definition 27.4, we obtain the following types of scalar boundary value problems for semitrivial (in the sense of Definition 27.4) solutions: ϕ = F (ϕ, 0) = jx  ϕ(0) = 0,

1+ϕ (1 + ϕ)2 − 1

,

ϕ(1) = ϕL ,

ϕ = F (ϕ, a0 ) = jx 

(A1 ) 1+ϕ (1 + ϕ)2 − 1 − (a0 )2

,

ϕ(0) = 0, ϕ(1) = ϕL , (A2 )

ϕ = F (ϕ, a0 ) = jx 

1+ϕ (1 + ϕ)2 − 1 − (a0 )2

,

ϕ(0) = 0, ϕ(1) = ϕL , (A3 )

a = G(ϕ0 , a) = jx 

a (1 +

ϕ0 )2

− 1 − a2

,

a(0) = 0, a(1) = aL , (A4 )

a = G(ϕ0 , a) = jx 

a (1 + ϕ0 )2 − 1 − a2

,

a(0) = 0, a(1) = aL . (A5 )

We find the solutions of problems (A1 ), (A2 ), (A3 ) for ϕ0 < ϕ0 , where ϕ0 (xa1 ) and ϕ0 (xa2 ) are, respectively, lower and upper solutions of problem (A1 ). The solution (ϕ, a) of the initial problem should belong to the interval ϕ ∈ Φ(ϕ, 0) ∩ Φ(ϕ, a0 ) ∩ Φ(ϕ, a0 ), a ∈ Φ1 (ϕ0 , a) ∩ Φ1 (ϕ0 , a).

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429

Moreover, the ordering of the lower and upper solutions of problems (A1 ), (A2 ), (A3 ) is satisfied: ϕ0 (xa1 ) < ϕ0 (xa2 ) < ϕ0 (xa2 ) < ϕ0 (xa1 ). We consider the solution of problems (A4 ) and (A5 ) for a0 < a0 . In this case, the following ordering of lower and upper solutions of problems (A4 ) and (A5 ) is satisfied: a0 (xϕ1 ) < a0 (xϕ2 ) < a0 (xϕ2 ) < a0 (xϕ1 ). We go over to the direct study of the problem (28.30), which includes the cases (A1 ), (A2 ), (A3 ). Let us consider the boundary value problem (28.30) with F (x, ϕ) : (0, 1] × (0, ∞) → (0, ∞).

(B1 )

In condition (B1 ) for F (x, ϕ), we dropped index ai , considering a general case of nonlinear dependence F of x. We shall assume that F is a Carath´eodory function, i.e., F (·, s)

measurable for all s ∈ R,

(B2 )

F (x, ·)

is continuous almost every where for x ∈]0, 1],

(B3 )

and the following conditions hold:  1 s(1 − s)F ds < ∞,

(B4 )

0

∂F/∂ϕ > 0, i.e., F is increasing in ϕ.

(B5 )

There exist γ(x) ∈ L1 (]0, 1]) and α ∈ R, 0 < α < 1 such that |F (x, s)| ≤ γ(x)(1 + |s|−α ) for all (x, s) ∈]0, 1] × R.

(B6 )

We are interested in a positive classical solution of equation (28.30), i.e., ϕ > 0 in P for x ∈]0, 1] and ϕ ∈ C([0, 1) ∩ C 2 (]0, 1]). The problem (28.30) is singular, therefore, condition (B1 ) is not fulfilled on the interval ϕ ∈ (0, ∞) and in this connection, the well-known theorems on the existence of lower and upper solutions in cone P do not work. It follows from Theorem 28.1 that since F in (28.30) is increasing

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Toward General Theory of Differential-Operator and Kinetic Models

in ϕ, then ϕ < w for x ∈]0, 1], where ϕ and w satisfy the differential inequality (28.25). Theorem 28.2. Assume conditions (B2 )–(B6 ). Then there exists a positive solution ϕ ∈ C([0, 1]) ∩ C 2 (]0, 1]) of the boundary value problem (28.30). The application of monotone iteration techniques to equation (28.30) gives an existence of maximal solution ϕ(x, ¯ jx ) such that ¯ xj ) < w(x) ϕ(x, xj ) ≤ ϕ(x,

for x ∈]0, 1].

Proposition 28.1. Let 0 < c ≤ jx ≤ jxmax . Then (A1 ), ϕ = F (jx , ϕ, 0) = jx 

1+ϕ ϕ(2 + ϕ)

,

ϕ(0) = 0,

ϕ(1) = ϕL ,

has a lower positive solution

if

u0 = δ2 x4/3

(28.32)

 4δ3 ≥ 9jxmax (1 + δ2 )/ 2 + δ2

(28.33)

and an upper positive solution u0 = α + βx

(α, β > 0)

(28.34)

with ϕL ≥ δ2 ,

(28.35)

where δ is defined from (28.33).

 Remark 28.2. Square root is taken as |ϕ(2 + ϕ)| in the case of negative solutions. Here, u0 = −x is an upper solution, and u0 = −2 +  is a lower solution (0 <  < 1). Hence, (A1 ) has the negative solution only for 0 < ϕL < −2 because F (x, −2) = −∞. It follows from (28.33) and (28.35) that a value of current is limited by the value of electrostatic potential on the anode ϕL jx ≤ jxmax ≤ F(ϕL ).

(28.36)

Numerical Modeling of the Limit Problem

431

The analysis of lower and upper solutions (28.32) and (28.34) exhibits that for δ2 = ϕL > 2 and α = β ≤ 1 interval in x between lower and upper solutions is decreased, and for the large values of the potential ϕL diode makes on regime ϕL x4/3 . Proposition 28.2. Let 0 < c ≤ jx ≤ jxmax . Then (A4 ), a , a(0) = 0, a(1) = aL , a = G(jx , ϕ0 , a) = jx  (1 + ϕ0 )2 − 1 − a2 with a lower solution a0 = 0 and an upper solution a0 = u0 > 0, conditions (28.27) and (28.16), has a unique solution a(x, jx , c), which is positive; moreover,  0 ≤ aL ≤ ϕ0 (2 + ϕ). Remark 28.3. The problem (A5 ) is considered by analogy with 0 0 0 problem (A4 ), change  of an upper solution a = u to a lower a = u0 one and 0 ≤ aL ≤ ϕ0L (2 + ϕ0L ). Thus, we have the following main result. Theorem 28.3. Assume that conditions (B2 ), (B3 ) and (B6 ) and inequalities (28.14), (28.17) and aL ≤

jx j max F(ϕL ) ≤ x ≤ 2 2 2

are fulfilled. Then the problem (28.18)–(28.22) possesses a positive solution in cone P such that

z2 ∈ [0, ϕ0 ], ϕ0 ≥ jx F (ϕ0 , z2 ), (ϕ0 ) ≤ jx F (ϕ0 , z2 ),

a0 ≥ G(jx , z1 a0 ), (a0 ) ≤ G(jx , z1 , a0 ),

z2 ∈ [0, ϕ0 ], z1 ∈ [ϕ0 , ϕ0 ], z1 ∈ [ϕ0 , ϕ0 ],

where ϕ0 = δ2 x4/3 is a lower solution of problem (A1 ), ϕ0 = α+ βx (α, β > 0) is an upper solution of problem (A1 ) with condition ϕL ≥ δ2 , and a0 =  0 is a lower solution of problem (A4 ) with condition 0 ≤ aL ≤ ϕ0 (2 + ϕ0 ).

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28.5.2.

Toward General Theory of Differential-Operator and Kinetic Models

Analysis of the known upper and lower solutions

Up to this moment, the analytical solution of the ordinary differential equation (ODE) system defined by (28.18) and (28.19) with respect to the conditions (28.20)–(28.22) is unknown. The only known result partially describing the form of the solution trajectory was given in [45]. According to this, both solution trajectories are bounded by the upper and lower solutions: yUP (x) = kx + b,

k, b > 0,

yLOW (x) = c2 x4/3 .

(28.37) (28.38)

Using the boundary conditions (28.20) and (28.22), one can obtain quite good solution trajectory estimations c2 ϕL x4/3 ≤ ϕ(x) ≤ ϕL x,

0 ≤ a(x) ≤ aL x

defined on x ∈ [0, 1]. Here and everywhere, we assume boundary conditions ϕL , aL to be correctly defined, i.e., ΘL > 0. Looking forward and leaving the discussion of numerical solution methods in the following sections, here, we provide some numerical solution trajectory examples both for ϕ(x) and a(x) evaluated for different boundary conditions. The straightforward analysis of the trajectories shows that the lower solutions obtained in Section 28.5.1 could be made significantly better and the upper solutions are exactly ϕL x and aL x. The lower solution obviously could be written as yLOW (x) = y(1)xγ ,

γ > 1,

(28.39)

where y(1) is either ϕL or aL . The value of the parameter γ depends only on ϕL , aL and could be found numerically. 28.5.3.

First lower solution hypothesis

First, we suggested that (28.39) could be the exact solution for ϕ(x) and a(x). In this case, at each point of the numerical solution, we

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should have γ

ϕ(x ˆ i ) = ϕL xi ϕ ,

(28.40)

a ˆ(xi ) = aL xγi a .

Here and elsewhere, we denote by yˆ(x) a numerical solution of any ODE w.r.t. y(x). Hence, γϕ =

ˆ i) ln ϕ(x ϕL

ln xi

,

γa =

i) ln aˆ(x aL

ln xi

,

(28.41)

and the best way to verify this hypothesis is to evaluate the mean m and dispersion σ 2 for each boundary condition pair available. Using the same data of numerical integration with 50,000 segments and rejecting the initial and boundary points, we have a 49,999-number sample. The results of this statistical estimation are provided Table 28.1. As it will be explained in Section 28.7, the problem (28.18)–(28.19) is highly numerically sensitive, thus even a moderate diminishing of the integration step can remarkably improve the estimations for γ. Thus, we provide the same results calculated from the 75,000 segments of numerical integration. Omitting the initial and boundary integration points, we have a 74,999-number sample. The numbers presented in Table 28.2 are nearly the same and even slightly worse because the real numerical computational error in both cases is about 10−16 –10−17 . This is too close to the numerical tolerance of the 80-bit floating point hardware computations. Using hypothesis (28.39) and substituting in equations (28.18) and (28.19) with respect to the conditions (28.20)–(28.22), we obtain Table 28.1.

γ parameter estimation; 49,999 sample points.

ϕL

aL

mγϕ

σγ2ϕ

mγa

σγ2a

1.0 8.0 0.3

1.0 3.0 0.8

1.4099739532037706 1.5451754873805480 1.4838493294052769

5.285e–4 3.255e–3 2.937e–3

1.1059698785688596 1.3763413139333533 1.0522660035274544

1.3227e–3 1.006e–2 5.148e–4

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Table 28.2.

γ parameter estimation; 74,999 sample points.

ϕL

aL

mγϕ

σγϕ

mγa

σγ2a

1.0 8.0 0.3

1.0 3.0 0.8

1.4098055071829239 1.5449948650231333 1.4836787676313544

5.326e–4 3.267e–3 2.947e–3

1.1059247093400567 1.3762442069420176 1.0522339187018956

1.328e–3 1.0065e–2 5.155e–4

Table 28.3.

Hypothesis (28.39) γ parameter estimation.

ϕL

aL

γϕ

γa

1.0 8.0 0.3

1.0 3.0 0.8

1.436828731 1.658475999 1.693216882

1.292242432 1.644909010 1.268396508

two quadratic equations: γϕ2 − γϕ −

1 + ϕL ξL = 0, ϕL

γa2 − γa − ξL = 0, where ξL ≡

√jx . ΘL

Using the assumption γ > 1, we obtain   1 1 1 + 1 + 4ξL + , (28.42) γϕ = 2 ϕL   1 (28.43) γa = 1 + 1 + 4ξL . 2 It immediately follows that γϕ > γa , which complies with the results in Table 28.1. Substituting example data, we obtain Table 28.3. Obviously, being substituted in (28.40), the numbers from Table 28.3 represent functions passing below the numerical trajectory solution curve. 28.6.

Second Lower Solution Hypothesis

Comparing numbers from Tables 28.1 and 28.3, we can see that (28.40) lower solution passes below the real solutions trajectories.

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More importantly, the behavior of the lower solution for a(x) do not da correspond to the initial condition dx (0) = 0. Hence, we can suppose that the real solution curve (or at least its estimation) should be expressed in the form a ¯(x) = c1 x − c2 xγa .

(28.44)

Applying conditions (28.20)–(28.22) to equation (28.44), we obtain the system of equations ⎧ a ¯(0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a ¯(1) = c1 + c2 = aL , ⎪ ⎪ ⎨ d¯ a (0) = c1 , ⎪ ⎪ ⎪ dx ⎪ ⎪ ⎪ 2¯ ⎪ ⎪ ⎩d a (1) = −c2 γa (γa − 1). dx2 Using equation (28.19), we obtain ⎧ da ⎪ ⎪ ⎪c1 = dx (0), ⎪ ⎪ ⎪ ⎨ da (0) − aL , c2 = ⎪ dx ⎪ ⎪ ⎪ aL ⎪ ⎪ ξL = 0. ⎩γa2 − γa + da dx (0) − aL Calculating the positive solution on γa , we have a ¯(x) =

da (0)x − dx





da (0) − aL x dx

1 2

r 1+ 1+4ξL

! aL

da (0)−a L dx

.

(28.45)

Using the same experimental data, we found Table 28.4. To verify this hypothesis by means of statistical approach, we use the same idea as above. Assuming c1 and c2 to be known, we evaluate the mean and dispersion for γa in (28.44) to compare them with analytical data given in Table 28.3. Since the relation a ˆ(xi ) − c1 xi c2 could be negative due to the computational error of numerical integration, we consider only the valid points of the whole samples. −

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Table 28.4.

Hypothesis (28.44) parameter estimation.

ϕL

aL

c1

c2

γa

1.0 8.0 0.3

1.0 3.0 0.8

0.879738089874 1.72776197665 0.759092882499

−0.120261910125 −1.27223802334 −0.040907117500

2.341253235040985 2.158751926015232 3.128246828693846

Table 28.5 provides the truncated estimations for the parameter γa , rejecting lower significant digits without loss of statistical relevance. Comparing Columns 4 and 5, we can see that according to the numerical estimation, the function (28.44) will pass above its analytically estimated analog in the third example for ϕL = 0.3, aL = 0.8 and below in the two first ones. It means that such a remarkable difference of the estimations have to be studied more attentively. To make such an analysis, Table 28.5 provides the two last columns entitled A and N , representing the Euclidean norm of the vector difference (ˆ a(x) − a ¯(x)) evaluated for two different parameters γa : analytically estimated (see Table 28.4) and statistically estimated (see Table 28.5, all estimation digits were used in computations). In the third line, we see that an analytical estimation is better than the statistical one, but the mistake levels are close in all three examples. To verify this idea, we have to compare the values  da da  γa −1 (0) − (0) − aL γa = a ¯ (1) = c1 · 1 − c2 · γa · x dx dx for all three numerical experiments, where the third column is the numerically calculated derivative a ˆ (1). Defining these results proves that our estimations defined by formula (28.45) are consistent. But, leaving the initial point with the same tangent as a numerical solution, we arrived to the final one with a smaller value in two cases. This means that the curve (28.45) is definitely not a lower solution, but it could be assumed as quite a good approximation of a solution a(x).

Valid ϕL aL points

γa

mγa

σγ2a

Hypothesis 28.39 γa parameter estimation. A

N

1.0 1.0 48744 2.34125 2.36129 2.182e–3 22.54118 22.50162 8.0 3.0 49438 2.15875 2.23132 1.999e–3 245.63749 243.99216 0.3 0.8 47833 3.12825 2.67611 1.126e–2 6.98219 7.21990

Experimental 1 2 3

Hypothesis

Calculated

1.16130167600785401476 1.16218421557717 4.47420825989693827300 4.51242935767056 0.88706044309118177670 0.87651519455565

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Table 28.5.

437

438

Toward General Theory of Differential-Operator and Kinetic Models

28.7.

Numerical Methods

28.7.1.

Formal analysis of limit problem

The second-order ODE system (28.18) and (28.19) 1 + ϕ(x) d2 ϕ(x) , = jx  2 dx (1 + ϕ(x))2 − 1 − a2 (x) a(x) d2 a(x) = jx  2 dx (1 + ϕ(x))2 − 1 − a2 (x) is not completely defined by its initial and boundary conditions (28.20)–(28.22): ϕ(0) = 0, dϕ(0) = 0, dx a(0) = 0,

ϕ(1) = ϕL ,

a(1) = aL

since application of any numerical integration method for Cauchy problem (28.18) and (28.19) requires values dϕ = 0, dx da(0) =β a(0) = 0, dx to be given. Here, the last of the initial conditions as well as the “parameter”, constant jx , are unknown. Therefore, we need to define these to variables jx , β as unknown parameters. Depending on this, pair p = (jx , β), we will reach the different final trajectory points ϕ(1, jx , β) and a(1, jx , β). To remove this obstacle, we investigate the boundary problem   ϕ(1, jx , β) − ϕL = 0, (28.46) Im(jx , β) = a(1, jx , β) − aL ϕ(0) = 0,

being defined by a nonlinear function Im(jx , β). Using only all the equations, (28.18)–(28.22) and (28.46), the limit problem becomes completely defined. But, the new problem

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immediately arises, namely, the analytical solution of the system (28.18) and (28.19) is unknown. Then we cannot apply any of the already known methods for solving a vector system of nonlinear equations (28.46). For example, the application of the Newton method   ∂ Im −1 (xi ) Im (xi ) xi+1 = xi − ∂p Im requires the Jacobian J = ∂∂p to be analytically known. On the one hand, one can revise that system (28.18) and (28.19) is singular at x ≡ 0. A careful study of these equations shows that considering ϕ(2) (0) ≡ a(2) (0) = const for arbitrary constant value makes a system (28.18) and (28.19) consistent analytically. On the other hand, the only numerically accepted values are the real solution trajectory convexity numbers. It immediately follows that all explicit numerical ODE solution methods are not applicable at the starting point. The numerical experiments also show the equations (28.18) and (28.19) are highly sensitive to the value of numerical integration step h. Here, we give the table containing the Im values evaluated for different h = 1/n with the same p = (jx , β) given. For the numerical integration of the system (28.18) and (28.19), we use the standard implicit Euler second-order method

yi+1 = yi + hf (xi+1 , yi+1 ), along with the standard Newton method to solve this nonlinear equation. Comparing the rows (1, 4) and (3, 5) in Table 28.6, we see that a change of integration step size by 10 also changes the global integration error by the order of 10, not 100 times as it could be expected. Using any numerical analysis textbook, a reader could find a lot of examples with a second-order convergence speed that need even greater integration steps, about 10−2 . Then the unique acceptable integration method is Gear’s methods (see [53]).

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Table 28.6. n 1000 2500 5000 10,000 50,000

28.7.2.

Numerical integration errors.

ϕ

a

1.61425217900765E–0002 8.34380843012195E–0003 4.68455678253690E–0003 2.27729848639435E–0003 −5.55677931763120E–0004

−4.47533487147775E–0003 −2.46343215088107E–0003 −1.52983203144597E–0003 −9.16751253617249E–0004 −1.91669763913965E–0004

Gear’s method and problem numerical sensitivity

In this section, we provide only a very brief summary of the general results given in [53] and in later papers. It is well known that the classical Gear iterative methods of the M th consistency order are written as (k+1) Yi+1

=

(k)

M  k=1

(k)

(hbM −k fi+1−k − aM −k Yi+1−k ) + hbM fi+1 , (k)

(28.47)

where fi+1 = f (xi+1 , Yi+1 ) and the upper index k denotes an iteration number. The idea is to make some iterative approximations starting from the fixed, already evaluated solution points. For a better explanation of formula (28.47), we put three low consistency order methods in Table 28.7. One can see that the first consistency order method is equal to the simple-iteration implicit Euler method and it is just a matter of fact that an implicit Euler method is the simplest numerical method applicable for the solution of the stiff differential equations. Comparing the notions of the numerical method convergence order and consistency order for the implicit Euler method with respect to numbers given in Table 28.6, we see that a consistency order is kept for the problem (28.18)–(28.19). It seems to be a little strange, but Gear’s method for q = 2 proved to be more sensitive and the computational time for a desired error level was only slightly better compared with q = 1. Hence, we implemented

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Table 28.7.

441

Gear’s iterative method of qth consistency order.

Order q

Expression

1

Y (k+1) = Yi + hf (xi+1 , Yi+1 ) 1 (k+1) (k) Yi+1 = (4Yi − Yi−1 + 2hf (xi+1 , Yi+1 )) 3 1 (k+1) (k) Yi+1 = (18Yi − 9Yi−1 + 2Yi−2 + 6hf (xi+1 , Yi+1 )) 11

2 3

(k)

this method in the developed software, but never used it in the main modeling. The explanation of this fact lies in the convergence conditions (x,y) of Gear iterations (28.47). Let · be some matrix norm, ∂f ∂y function Jacobian. Then Gear iterations converge if and only if     ∂f  < 1. hbM (28.48)  ∂y  28.7.3.

Reduced problem statement

To apply any of the mentioned numerical methods and to study their convergence properties, we reduce the system of second-order ODEs (28.18)–(28.19) to the system of first-order ODEs: y1 = y2 , 1 + y1 , y2 = jx  (1 + y1 )2 − 1 − y32 y3 = y4 ,

(28.49)

y3 , y4 = jx  (1 + y1 )2 − 1 − y32 y¯0 = (y1 (0), y2 (0), y3 (0), y4 (0)) = (0, 0, 0, β). System (28.49) is the autonomous ODE system and has a smaller computational cost. Moreover, if we could overcome its “stiffness”, there are a number of special effective implicit numerical methods for solving this type of ODE systems.

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442

Hence, we have to estimate the Jacobian of system (28.49) near the singular initial point and for ΘL → 0. ∂f (x, y) ∂y ⎡

0

⎢ ⎢ 1 + y32 ⎢ ⎢−jx 3 ⎢ [(1 + y1 )2 − 1 − y32 ] 2 =⎢ ⎢ 0 ⎢ ⎢ ⎢ (1 + y1 )y3 ⎣ −jx 3 [(1 + y1 )2 − 1 − y32 ] 2

1 0

0 jx

0 0

jx

0

(1 + y1 )y3 3

[(1 + y1 )2 − 1 − y32 ] 2 0 (1 + y1 )2 − 1 3

[(1 + y1 )2 − 1 − y32 ] 2

⎤

⎥ ⎥ ⎥ 0⎥ ⎥ ⎥. 1⎥ ⎥ ⎥ ⎥ ⎦ 0 (28.50)

Here, nonzero elements could be greater and less than one, hence the use of the infinity matrix norm leads to additional difficulties in the analysis. Therefore, we use the Frobenius matrix norm in condition (28.48):  (1 + y32 )2 + 2(1 + y1 )2 y32 + ((1 + y1 )2 − 1)2 hbM 2 + jx2 3 [(1 + y1 )2 − 1 − y32 ] 2  [(1 + y1 )2 + y32 ]2 − 2[(1 + y1 )2 − 1 − y32 ] = hbM 2 + jx2 3 [(1 + y1 )2 − 1 − y32 ] 2      [(1 + y1 )2 + y32 ]2 jx2 = hbM 2 1 −  + j x 3 (1 + y1 )2 − 1 − y32 [(1 + y1 )2 − 1 − y32 ] 2 < 1.

(28.51)

Revising condition (28.51) in the initial point (y1 , y3 ) = (0, 0), we have that the expression under the sign of square root is infinite, and formally, Gear’s method of integration cannot converge. This expression is also large if ΘL → 0. This explains bad convergence √ properties for the numerical experiments with ΘL < 0.01. Moreover, even taking small h > 0 is not a solution because a larger number of computational steps increase the rounding error.

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In the case of the truly stiff (oscillating) problem, these local errors compensate each other. In our case, the solution is a smooth function posing the error accumulation. 28.7.4.

Steffensen’s derivative free self-convergent method

One of the most used derivative-free self-convergent methods for solving nonlinear functions and their systems is based on the so-called Steffensen’s estimation. A good discussion of this method and other suitable methods could be found in [163]. Temporarily assuming β and arbitrary parameter, we introduce the iterative function ϕ=x−

βf 2 (x) , f (x + βf (x)) − f (x)

where ϕ and β are formal parameters. If |1 + βf  (x)| < 1 and f (x)/f  (x) is small, the introduced function could be more exact than the classical Newton’s method. Taking β = −1/f  (x) makes this iterative function the third-order convergent. Moreover, one can check that f  (x) ≈

βf (x) . f (x + βf (x)) − f (x)

Supposing that β approximates −1/f  (x), we can introduce a formal two-step method ⎧ f (xi ) ⎪ ⎪ , xi+1 = xi − ⎪ ⎪ Γi ⎪ ⎪ ⎨ f (xi + βi f (xi ) − f (xi ) (28.52) , Γi = ⎪ βi f (xi ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩βi = − . Γi−1 To start the process, we have to provide two values: x0 and β0 . The choice of the starting point is problem-dependent, while the choice of β0 has an immediate impact on the iteration convergence

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444

or divergence. Usually, we choose β0 according to the following assumptions: (1) If the rough estimation of f  (x0 ) can be found, we assume β0 = 1 ; − f  (x 0) ) (2) β0 = − f (x0 +ff(x(x00))−f (x0 ) ;

(3) β0 = sign(f  (x0 )).

It is clear that for the “stiff” nonlinear functions with small local convergence domains, the right choice of initial parameter and starting point will be the clue. The situation becomes even worse for vector nonlinear functions. In this case, an algorithm (28.52) needs to approximate the Jacobian matrix. Let x = (x1 , . . . , xn ), f = (f1 (x), . . . , fn (x)), n > 1 be a problem dimension. Denote (k)

ti

(k)

= xi

(k) (k)

+ βij fi ,

(k) 

fi

= fi (x(k) ),

where the upper index is the iteration number and the lower index the element index. Then (k)

Jij = Jij (x(k) , t(k) ) (k)

=

(k)

(k)

(k)

(k)

(k)

fi (x1 , . . . , tj , . . . , xn ) − fi (x1 , . . . , xj , . . . , xn ) (k)

(k)

tj − xj

,

(28.53) (k)

where Jij is just an extension of the standard derivative difference estimation. Denote H (k) = [J (k) ]−1 . Then the generalization of the iterative function (28.52) is defined: ⎧ (k+1) = x(k) − H (k) f (k), x ⎪ ⎪ ⎪ ⎪ (k) (k−1) ⎪ ⎨βij = −Hij , i, j = 1, . . . , n, (28.54) n ⎪  ⎪ ⎪ (k) (k) (k) (k) ⎪ βij fj , i = 1, . . . , n. ⎪ ⎩ti = x + j=1

An iterative algorithm (28.54) can be easily implemented, but its numerical stability and accuracy depend on the choice of the matrix

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445

inversion algorithm and condition number of the Jacobian (28.53). In our case, n = 2, and we use the explicit computational expressions overriding additional internal computational errors. 28.7.5.

Advanced two-step derivative free self-convergent method

A standard Steffensens estimation discussed above is fast, has good numerical properties and seems to be enough. But in the process of numerical modeling, we found it acceptable only if a quantity  ΘL  0.1. (28.55) To operate some standard notations, other than ϕL , aL and ΘL , we need to denote a boundary condition definition domain for system (28.18)–(28.19) D = {ϕL , aL | (1 + ϕL )2 − 1 − a2L > 0}, ¯ = {ϕL , aL | (1 + ϕL ) − 1 − D 2

a2L

≥ 0}.

(28.56) (28.57)

¯ where domain D ¯ includes all the boundary A domain D ⊂ D, conditions related to the magnetically insulated diode. At the moment, we investigate the problem of noninsulated diode and the limitation (28.55) is not restrictive. Nevertheless, the singularity ΘL ≈ 0 makes impossible the modeling of the border cases and a numerical/analytical investigation of the problem √ properties in this critical state. For example, assuming ΘL = 0.01 and varying ϕL , we failed to obtain estimations of parameters jx , β even for ϕL > 12.3 using second-order Gear’s implicit and iterative methods and for ϕL > 10 using third-order Gear’s method. Hence, we examined the applicability of the two-stage method with pseudo-diagonal method developed by E. Dulov. This method description is not published yet, but it was verified in a number of well-known tests and practical problems including the threebody problem in the Differential-Algebraic equation statement.

446

Toward General Theory of Differential-Operator and Kinetic Models

It demonstrate high numerical stability along with the same speed of computations in this task. The derivation of the method is not hard but needs a lot of attention; we describe only the principle for this here. First of all, we assume matrix [βij ] to be diagonal (a vector of parameters). Hence, all the expressions (28.54) are significantly simplified. To obtain a convergence order, higher than the second one, we need to set some additional conditions for βi . Since the error of iterative algorithm (28.54) is an (n × n)-matrix, we cannot zero all its elements using only n parameters. So, we require diagonal elements to be zero, which gives the condition (k)

βi

=−

1 (k)

2Jii

.

(28.58)

The second algorithm step assumes the evaluation of better Jacobian estimation using the data available from the pseudo-diagonal version of (28.54) and convex approximation. For simplicity, here, we provide only the scalar form of the vector since vector generalization is straightforward. Let ρ(z − x)f  (x) = f (x + ρ(z − x)) − f (x),

ρ ∈ (0, 1]

(28.59)

with some additional point z and scalar ρ. This requirement makes the separate use of the convex approximation problematic, and usually, it is used as the second stage of some basic iterative algorithm. Hence, denoting f  (x) ≈ t(x) =

f (x + ρ(φ(x) − x)) − f (x) , ρ(φ(x) − x)

ρ ∈ (0, 1]

and supposing that a basic iterative function φ(x) is at least quadratically convergent, by (28.52) and (28.59), we obtain a number of different methods with guaranteed convergence orders ≥3, but according to requirements of practical convergence, the method of

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447

main interest is the two-stage method: φ(x) = x −

βf 2 (x) , f (x + βf (x)) − f (x)

t(x) =

f (x + ρ(φ(x) − x)) − f (x) , ρ(φ(x) − x)

u(x) =

f (φ(x)) , t(x)

¯ φ(x) = φ(x) + β=−

(28.60)

u(x)f (φ(x)) , f (φ(x) − u(x)) − f (φ(x))

u(x) . f (φ(x) − u(x))) − f (φ(x))

The merit of this two-step method is that at the second step, we use a better convex β approximation for calculating a resulting point. This two-stage method is more flexible, since we have one additional parameter involved: ρ. A direct analysis of (28.60) indicates that the error term is proportional to −

15(f (2) (α))3 ! (2) 3(f (α))2 ((1 − ρ) − βf  (α)(ρ + βf  (α))) 2(f  (α))5 " + 2f (2) (α)f  (α)(1 + βf  (α))(2 + βf  (α)) ,

where α is a solution point or the fixed point of the iterative function 1 ¯ in this equation gives zero, but φ(x). Substituting optimal β = − f  (α) it is easy to see that the first term in the square brackets depends both on β and ρ, and moreover, the error value involved in this term is minimal for ρ ≡ 1. A selection of ρ = 1 is meaningless, but it shows that taking ρ close to 1 makes two-stage algorithm converge faster. In our numerical experiments, we set ρ = 0.95. On the other hand, using ρ ≈ 1 imposes smaller integration step and increased computation time. Hence, we plan to make a set of additional experiments to find some “optimal” value of the convex approximation parameter ρ. As a conclusion, this two-stage method is twice more timeconsuming, but it allows one to extend the domain where we can

448

Toward General Theory of Differential-Operator and Kinetic Models

evaluate the critical domain estimations at least for ϕL = 80  12.5. The numerical modeling can be further continued, but the already obtained results allow one to make the qualified conclusions. 28.7.6.

Initial parameter approximation

Any of the introduced algorithms need some initial approximation to be given. It is clear that we have in general an implicit dependence g(ϕL , aL , jx , β) = 0 and one of the most important tasks of our numerical experiments was to build some kind of approximations jx = g1 (ϕL , aL ), β = g1 (ϕL , aL ).

(28.61)

Here, we assume that a reader could easily differ between parameter β as an initial condition of the ODE system and a parameter vector β0 for the one- or two-stage pseudo-diagonal derivative-free self-convergent method. Without making some good approximation (28.61), we cannot give any acceptable a priori estimation. Nevertheless, thousands of numerical runs for different ΘL showed that  1 1 ,− β0 = − 2ϕL 2aL is a good one. Hence, all our efforts were concentrated on the selection of the proper interpolation/extrapolation functions (28.61). 28.8.

Numerical Modeling

Recalling various remarks already made about the numerical stability of the solution, we have to logically divide all our experiments into blocks. First of all, we would like to get some information about parameter vector trajectories p = (jx , β) for distances ΘL ≥ 1.

28.8.1.

Steady solutions

This chapter is aimed only at a preliminary subject study, thus we made only one set of numerical experiments for ΘL = 1. As we

Numerical Modeling of the Limit Problem

Table 28.8.

449

jx and β parameter values; ΘL = 1.

ϕL

jx

β

0.45 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.5 3.0 4.0 5.0 6.0 7.0 8.0 9.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0

0.17864868540937 0.20404667713384 0.33864720780410 0.48182248854903 0.62941593031875 0.77957026817339 0.93066609035116 1.08188872124903 1.38117662430921 1.67486318624019 2.23898741598031 2.77064007114640 3.27110218726489 3.74297629896394 4.18905608370126 4.61195944040519 5.39796579347806 5.76436673093679 6.11532511808196 6.45219825697198 6.77617430735118 7.08829752910593 7.38948962373869 7.68056762187826 7.96225883273576 8.23521334223292 8.50001448436446 8.75718764266414

0.301433871969702 0.46737105858652 0.92943647499937 1.23100608581084 1.47230153366985 1.678247954551252 1.86001181745560 2.02350593270070 2.31111775388343 2.55883258218455 2.97439626139971 3.31638912417627 3.60768739672887 3.86166045267669 4.08690609535260 4.47437644974364 4.64282381562656 4.79821194565616 4.94242313464426 5.07695868731263 5.20303567114342 5.32165465243329 5.43364831404905 5.53971710886889 5.64045590084838 5.73637420356129 5.82791178241683 5.91545084189829

could expect, the errors of integration were smaller for small ϕ, about 10−17 , and grow quite fast. They became 10−12 only for ϕ = 80 (Tables 28.8 and 28.9). Remark 28.4. To ensure the correctness of the above results, we applied formulas (28.23) and (28.24) to some of the numerically evaluated solution curves with proper parameter vectors. The exact numerical differences vary, keeping 10−7 coincidence order sufficient for verification.

450

Toward General Theory of Differential-Operator and Kinetic Models

Table 28.9.

28.8.2.

jx and β parameter values; ΘL = 1.

ϕL

jx

β

23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0

9.00720767619584 9.25093999416217 9.48792823466859 9.71894291624990 9.94431109427948 10.1643328203516 10.3792840656884 10.5896457275944 11.5757743817479 12.4704804229310 13.2907790793456 14.0492090856849 15.4164568319502 16.6269099012908 17.7149884119297

5.99932565877550 6.08028976674331 6.15769586302146 6.23222252722460 6.30407601910147 6.37344121169832 6.44048444952802 6.50556737370551 6.80172674079113 7.05966669855577 7.28812444567524 7.49314 749947562 7.84921457963099 8.15164335637439 8.41407321211649

Partial conclusions

One of the main conclusions to be made is the fact that a statement of limit problem (28.18)–(28.22) does not comply with a Child– Langmuir regime that a current density jx is saturated in the case of noninsulated and nearly insulated diode. Moreover, the current density jx will infinitely grow if the voltage applied to the diode also grows. On the other hand, the experimental data show that jx grow faster for noninsulated diode than for a nearly insulated one. This could be seen as a preliminary numerical proof that a Child– Langmuir regime could be achieved only in a magnetically insulated diode. The second conclusion refers to the nearly insulated diode. The discovered properties described by a limit problem in mathematical statement fully comply with their physical expectations described in Section 28.1. Along with the first conclusion made, the obtained limit model is characterized as reliable one, which complies with the physical processes, underlying the thermovacuum diode with a plane cathode and anode.

Numerical Modeling of the Limit Problem

451

The third conclusion is the consistency of the proposed numerical methods for the problem (28.18)–(28.22). The only but serious deficiency raised by the model singularity is the difficulty in the translation of the dimensionless model into the physical space because the investigated boundary conditions are far away from the voltage and magnetic field values used either in semiconductor and home device manufacturing or in the modeling of the high-energy devices.

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Chapter 29

Open Problems In this chapter, we provide the list of open problems for the VP and VM equations, kindly handed over by Professor J¨ urgen Batt. Problem 1. One way to construct stationary spherically symmetric solutions of the VP system is described in [15]. A second method is the so-called inversion method: given a function h(u, r), such that the equation 1 2  (r u (r)) = h(u(r), r) r2 is solvable, that is, has a solution u = u(r), one defines ρ(r) = h(u(r), r) and has to solve the problem to find a function ϕ = ϕ(E, F ) with ϕ(E, F ) = 0 for E < a := u(∞) such that   2 ϕ(E, F ) 2π a 2r (E−u)  d(E, F ). (29.1) h(u, r) = r u 0 2r 2 (E − u) − F This equation then guarantees  2   v + u(|x|), x2 v 2 − (xv)2 dv, ϕ ρ(|x|) = 4π 2 R3 v2 + u(|x|) =: E, x2 v 2 − (xv)2 =: F, 2 that is, the above-defined ρ is the local density of the distribution function f := ϕ(E(x, v), F (x, v)).

453

454

Toward General Theory of Differential-Operator and Kinetic Models

The substitution ξ = a−u, η = 2r 2 (one-dimensional variables) gives (29.1) in the form     ξ  η(ξ−s1 ) √ 2 1 η 2 2π ϕ(a − s1 , s2 )  = √ ds2 ds1 h a − ξ, 2 π η 0 0 η(ξ − s1 ) − s2 or  ξ  η(ξ−s1 ) 1 f (s1 , s2 )  ds2 ds1 g(ξ, η) = √ π η 0 0 η(ξ − s1 ) − s2 in R+ × R+ with    √ η =: g(ξ, η), 2 2π 2 ϕ(a − s1 , s2 ) =: f (s1 , s2 ). h a − ξ, 2 The problem is, when g is given to find distribution f . The substitution s = s1 and t = s1 + sη2 give   1 x t f (s, y(t − s))ds √ dsdt, ξ =: x, η =: y. g(x, y) = π 0 0 x−t Abel’s integral equation  1 x ψ(t) √ dt, g(x, y) = π 0 x−t where  t f (s, y(t − s)) √ dt =: ψ(t) x−t 0 has the solution  x g(s, y) 1 d √ ds. ψ(x) = √ π dx 0 x−s The problem remains, given ψ, to solve (for f ) the equation  x f (s, y(x − s)ds ψ(x) = 0

or

 ψ(x) =

f (a)da 

1

= Rf (Θ(y),

1 + y2 where R is the Radon transform in   1 y , , Θ(y) =  1 + y2 1 + y2 Θ(y)a

notation in [120].

p(x, y)) 

1 1 + y2

xy , p(x, y) =  1 + y2

,

Open Problems

455

Problem 2. In connection with Problem 1 in [77], Hunter and Qian have extended the inversion method to cylindrically symmetric stationary solutions on a formal level. An embedding of their arguments in a rigorous mathematical frame is needed. Problem 3. A dificult open problem is the global existence of classical solutions to the Vlasov–Maxwell system for general sufficiently smooth initial data. Glassey and Schaffer have settled this existence problem for a 2D version of the VM system in 1998, for a 2 12 D version in 1997 and a 1 12 D version in 1990 [58]. In the latter paper, some regularity questions are still open. But, we still have dificulties in the general existence theorem for the classical initial value problem of the VP system (for a mathematical development of this theorem, see [131]). Problem 4. The existence theorem for classical global solutions to the initial value problem of the VP system have been extended to singular initial values. Many stationary solutions with spherical symmetry are examples of (trivially globally existing) singular solutions of the VP system, e.g., certain polytropes (corresponding to the M-solutions of the Emden–Fowler equation) and certain Camm models (corresponding to the M-solutions of the Matukuma equation). The semiexplicit spherically symmetric solutions constructed by Kurth [95] are singular nonstationary global solutions. Problem 5. The above-mentioned existence theorem have been extended to initial values with infinite mass. Instationary solutions with infinite mass have been constructed in [14]. The first effort to manage initial values with infinite mass is made in Ref. [32] (see also [31]). Problem 6. In the above-mentioned paper [14], the existence of time-periodic solutions was proven, and also the semiexplicit solutions of [95] are partially time-periodic. A general result for timeperiodic solutions is however still missing. The question is: Which initial values lead to time-periodic solutions? It might be helpful to look into the paper in Ref. [179].

456

Toward General Theory of Differential-Operator and Kinetic Models

Problem 7. In [12], it was proven that initial values with spherical symmetry and with bounded v-support lead to solutions with timeglobal bounded v-support. The growth of the v-support of a solution f of the VP system is estimated by the function hη (t) := sup{|v(0, t, x, v) − v| x, v ∈ R3 }. Horst showed that hη (t) = O(t1+δ ), δ > 0. Batt and Rein proved hη (t) = O(t2 ) in the x-periodic case. For cylindrically symmetric solutions, the estimate is not known. In general, the uniqueness of weak solutions to an initial value problem is open, and for the total energy one, only knows E(t) ≤ E(0) (instead of equality). The paper by Robert [132] seems to be a further attempt in this direction. Majda et al. [106] have proven the nonuniqueness of weak (rather singular) solutions. The exact border between nonuniqueness and uniqueness in the field of weak and strong solutions is not known. Horst and Hunze [72] have developed a concept to obtain weak solutions for the VP system, which has become a guideline to handle other cases (Vlasov–Poisson–Fokker– Planck, flat case of the VP system). Problem 8. In [13], an elementary proof was given for the approximability of the solutions of the unmodified VP system by the masspoint solutions of modified N -body problems when N → ∞ and the modification parameter δ → 0 in an appropriate way. However, this is not what the physicists call the “mean field limit”. For the modified VP system, the mean field limit has been proven in the classical paper of Braun and Hepp [27]: “The Vlasov dynamics and its fluctuations in the N1 limit of interesting classical particles.” Problem 9. This is same as for Problem 8, but for the Vlasov– Maxwell system. Horst has shown the global existence for the classical solution of the modified VM system, where modification means that the current density j is replaced by the convolution j(·, t) ∗ δ, where δ is a member of a canonical sequence of δ-functions. Problem 10. The “flat” VP system is obtained if x, v ∈ R3 are replaced by x, v ∈ R2 and in the integral form of Poisson’s equation, i.e., in the formula for the potential u, and in the definition of the

Open Problems

457

3 local density ρ, the integrals R2 over R are replaced by integrals 2 R2 over R (over all x or all v, respectively). The question is: Can one obtain the solutions of the flat VP systems from the solutions of the VP system (in 3D) by a suitable limit process? The “jump relations” (see [62, Chapters 6–8]) might play an important role. Dietz in her dissertation [41] (unpublished) has proven the existence of classical local solutions and weak global solutions. Problem 11. The unicity of classical solution of the VP system (or of related systems) is not fully understood. Are two C 1 -solutions f1 and f2 of the initial value problem with the same initial value f 0 ∈ Cc1 (R6 ) equal? Problem 12. Generalize the approach of [16] introduced for Emden– Fowler equation to other equations.

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Index

asymptotically stable solution, 107

formally stable solution, 365 Fredholm alternative, 352 Fredholm operator, 209 Fredholm point, 4 Fuchs–Frobenius solution, 85 functional perturbaton of the argument, 97

Banach–Steinhaus theorem, 37 Banach–Kantorovich space, 135 Bessel function, 412 bi-interlaced equation, 271 bi-interlacing, 266 bifurcation manifold, 380 bifurcation point, 262, 377 Boltzmann equation, 389 branch point, 175 branch surface, 180 branching, 83, 84, 262–264 branching equation, 377 Brayer theorem, 341

Gear method, 441, 442 generalized Jordan chain, GJC, 5, 179 generalized Jordan set, 6 generalized Schmidt lemma, 243 Green function, 400 Gronwall–Bellman inequality, 109 H¨ older function, 393 Hirota method, 357 homotopy principle, 177

Caratheodory function, 429 characteristic polynomial, 85, 87, 89, 94 Chetaev method, 365 Child–Langmuir asymptotics, 415 Child–Langmuir regime, 422 complete canonical GJS, 6 cone set, 424 Conley index, 382, 406 contractive operator, 91 convex majorants, 135

ill-posed problem, 15, 205, 243 implicit operator theorem, 225 interlaced equation, 213, 277 interwining operators, 266 isolated singular point, 4 Jordan chain, 89, 93 Jordan normal form, 85, 91

Dirichlet BVP, 298 Kantorovich majorant, 138 Karman problem, 203 Kovalevskaya theorem, 127 Krasnosel’skij theorem, 380 Kronecker theorem, 405

effective potential, 422 epimorphisms, 246 Euler method, 440 Euler operator, 85, 86 471

472

Toward General Theory of Differential-Operator and Kinetic Models

Lagrangian invariants, 367 Lavrentyev method, 9 Liouville equation, 307 Lyapunov majorants, 135 Lyapunov–Kantorovich convex majorants, 140 majorant equation, 138 Maxwellian distribution, 416 McKenna–Walter theorem, 348 method of successive approximations, 85, 87, 89 method of undetermined coefficients, 91, 92, 94 minimal branch, 241 neutral type perturbaton, 97 Newton diagram, 84, 86, 95, 217, 221, 263 non-degenerate field, 184 oscillations, 262 Picard theorem, 110 polyhedron, 218 potential type branching equation, 277 principal solution, 138 regular point, 3 regularization, 205 regularization equation, 206

regularizing equation, 207 root number, 6 Schauder theorem, 338 Schmidt operator, 4 Schmidt pseudo-resolvent, 214 sectorial neighborhood, 257 semitrivial solutions, 427 sinh-Gordon equation, 355 singular point, 3 skeleton chains, 120 skeleton-attached operator, 121 small solution, 83, 84, 88 solution uniformization, 213 spherical symmetry, 265 stabilizing operator, 32 Steffenson’s derivative, 445 Steffenson’s estimation, 443 sub–super solution, 426 supporting lines, 221 3-body problem, 446 (T, M )-interlacement, 283 Tikhonov regularization, 9 total generalized Jordan set, 6 Trenogin regularizer, 259 vector norm, 136 Vlasov equation, 313 Vlasov–Maxwell system, 289, 316 Vlasov–Poisson system, 315