Table Of ContentMonotone Flows and Rapid
Convergence for Nonlinear Partial
Differential Equations
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Monotone Flows and Rapid
Convergence for Nonlinear Partial
Differential Equations
V. Lakshmikantham
and
s.
Koksal
Boca Raton London New York
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Taylor & Francis Group, an informa business
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© 2003 by V. Lakshmikantham and S. Koksal
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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
