Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations SERIES IN MATHEMATICAL ANALYSIS AND APPLICATIONS Series in Mathematical Analysis and Applications (SIMAA) is edited by Ravi P. Agarwal, Florida Institute of Technology, USA and Donal O'Regan, National University of Ireland, Galway, Ireland. The series is aimed at reporting on new developments in mathematical analysis and applications of a high standard and of current interest. Each volume in the series is devoted to a topic in analysis that has been applied, or is potentially applicable, to the solutions of scientific, engineering and social problems. Volume 1 Method of Variation of Parameters for Dynamic Systems V. Lakshmikantham and S.G. Deo Volume 2 Integral and Integrodifferential Equations: Theory, Methods and Applications edited by Ravi P. 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British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library ofC ongress Cataloging in Publication Data A catalog record for this book has been requested Contents Preface ix Part A 1 1 Elliptic Equations 3 1.1 Introduction.................................................................................... 3 1.2 Monotone Iterates: A Preview..................................................... 4 1.3 Monotone Iterative Technique..................................................... 9 1.4 Generalized Quasilinearization .................................................. 22 1.5 Weakly Coupled Mixed Monotone Systems ............................ 30 1.6 Elliptic Systems in Unbounded Domains.................................. 34 1.7 Monotone Iterative Technique (MIT) for Systems in Unbounded Domains.......................................................................................... 39 1.8 Notes and Comments.................................................................... 42 2 Parabolic Equations 45 2.1 Introduction.................................................................................... 45 2.2 Comparison Theorems................................................................. 46 2.3 Monotone Iterative Technique..................................................... 49 2.4 Generalized Quasilinearization .................................................. 55 2.5 Monotone Flows and Mixed Monotone Systems...................... 62 2.6 Generalized Comparison Results (GCRs) for Weakly Coupled Systems.......................................................................................... 68 2.7 Stability and Vector Lyapunov Functions ............................... 72 2.8 Notes and Comments.................................................................... 77 3 Impulsive Parabolic Equations 79 3.1 Introduction.................................................................................... 79 3.2 Comparison Results for Impulsive Parabolic Systems (IPSs) . 79 3.3 Coupled Lower and Upper Solutions........................................ 90 vi CONTENTS 3.4 Generalized Quasilinearization ............................................... 97 3.5 Population Dynamics with Impulses..................................... 108 3.6 Notes and Comments................................................................. 112 4 Hyperbolic Equations 113 4.1 Introduction................................................................................. 113 4.2 Variation of Parameters (VP) and Comparison Results . . . . 113 4.3 Monotone Iterative Technique.................................................. 126 4.4 The Method of Generalized Quasilinearization...................... 131 4.5 Notes and Comments................................................................. 137 Part B 139 5 Elliptic Equations 141 5.1 Introduction................................................................................. 141 5.2 Comparison Result.................................................................... 142 5.3 Monotone Iterative Technique (MIT): Semilinear Problems . 146 5.4 Monotone Iterative Technique (MIT): Quasilinear Problems . 154 5.5 Monotone Iterative Technique (MIT): Degenerate Problems . 164 5.6 Generalized Quasilinearization (GQ): Semilinear Problems . . 174 5.7 Generalized Quasilinearization (GQ): Quasilinear Problem . . 182 5.8 Generalized Quasilinearization (GQ): Degenerate Problems . 189 5.9 Notes and Comments................................................................. 196 6 Parabolic Equations 199 6.1 Introduction................................................................................. 199 6.2 Monotone Iterative Technique.................................................. 200 6.3 Generalized Quasilinearization ............................................... . 209 6.4 Nonlocal Problems: Existence and Comparison Results . . . . 224 6.5 Generalized Quasilinearization (GQ): Nonlocal Problems . . . 236 6.6 Quasilinear Problems: Existence and Comparison Results . . 246 6.7 Generalized Quasilinearization (GQ): Quasilinear Problems . 254 6.8 Notes and Comments................................................................. . 268 7 Hyperbolic Equations 269 7.1 Introduction................................................................................. 269 7.2 Notation and Comparison Results ........................................ 269 7.3 Monotone Iterative Technique.................................................. 275 7.4 Generalized Quasilinearization ............................................... 281 7.5 Notes and Comments .............................................................. . 292 CONTENTS vii Appendix A 293 A.l Sobolev Spaces........................................................... ................293 A.2 Elliptic Equations ..................................................... ................297 A.3 Parabolic Equations.................................................. ................300 A.4 Impulsive Differential Equations ............................ ................305 A.5 Hyperbolic Equations .............................................. ................306 Bibliography 309 Index 317 Preface An interesting and fruitful technique for proving existence results for non­ linear problems is the method of lower and upper solutions. This method coupled with the monotone iterative technique manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive exis­ tence results in a closed set, generated by the lower and upper solutions. The lower and upper solutions serve as rough bounds, which can be improved by monotone iterative procedures. Moreover, the iteration schemes can also be employed for the investigation of qualitative properties of solutions. The ideas embedded in these techniques have proved to be of immense value and have played a crucial role in unifying a wide variety of nonlinear problems. Another fruitful idea of Chaplygin is to obtain approximate solutions of nonlinear problems which are not only monotone but also converge rapidly to the solution. Here strict lower and upper solutions and the assumption of convexity are used for nonlinear initial value problems (IVPs). The method of quasilinearization developed by Bellman and Kalaba, on the other hand, uses the convexity assumption and provides a lower bounding monotone sequence that converges to the assumed unique solution once the initial ap­ proximation is chosen in an adroit fashion. If we utilize the technique of lower and upper solutions combined with the method of quasilinearization and employ the idea of Newton and Fourier, it is possible to construct con­ currently lower and upper bounding monotone sequences whose elements are the solutions of the corresponding linear problems. Of course, both sequences converge rapidly to the solution. Furthermore, this unification provides a framework to enlarge the class of nonlinear problems consider­ ably to which the method is applicable. For example, it is not necessary to impose the usual convexity assumption on the nonlinear function involved, since one can allow much weaker assumptions. In fact, several possibilities can be investigated with this unified methodology and consequently this technique is known as generalized quasilinearization. Moreover, these ideas are extended, refined and generalized to various other types of nonlinear