Approximation-solvability of Nonlinear Functional and Differential Equations

by Wolodymyr V. Petryshyn

Overview

This reference/text develops a constructive theory of solvability on linear and nonlinear abstract and differential equations - involving A-proper operator equations in separable Banach spaces, and treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.;Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional appoximations, this book: offers an elementary introductions to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.;There are also over 900 display equations, and an appendix that contains basic theorems from real function theory and measure/integration theory.

Product Details

ISBN-13:	9780367402570
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Chapman & Hall/CRC Pure and Applied Mathematics , #171
Pages:	392
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Wolodymyr V. Petryshyn is a Professor of Mathematics at Rutgers University, New Brunswick, New Jersey. The author or coauthor of more than 100 professional papers and a member of the American Mathematical Society and the Shevchenko Society, he is the recipient of the M. Krylov Award (1992) of the Ukrainian Academy of Sciences, the highest award the Academy can bestow on a research mathematician, for originating and developing the theory of A-proper mappings and its application to the solvability of ordinary and partial differential equations. He was cited by the Academy for his outstanding contributions to the growth of linear and nonlinear functional analysis and elected by the Presidium of the Academy in 1992 as a foreign full member. Dr. Petryshyn has lectured extensively in Israel, Ukraine, Russia, China, Germany, and the United States. He received the B.A. (1953), M.S. (1954), and Ph.D. (1961) degrees in mathematics from Columbia University, New York, New York.

Table of Contents

Preface v

I Solvability of Equations Involving A-Proper and Pseudo-A-Proper Mappings 1

1 Definitions and some facts about A-proper mappings and the approximation-solvability of equation (1.1) 2

2 Further examples of A-proper mappings 20

3 A-properness and the constructive solvability of some ordinary and partial differential equations 34

4 Existence theorems and pseudo-A-proper mappings 49

II Equations Involving Linear A-Proper Mappings 55

1 Approximation-solvability and bounded A-proper mappings 56

2 Equations involving unbounded linear operators 64

3 Generalized Fredholm alternative 78

4 Constructive solvability of linear elliptic boundary value problems 86

5 Stability of projective methods and A-properness 95

III Fixed-Point and Surjectivity Theorems for P_γ-Compact and A-Proper-Type Maps 100

1 Outline of the Brouwer degree theory and some of its consequences 101

2 Constructive fixed-point theorems for P_γ-compact maps 106

3 Constructive surjectivity theorems for P_γ-compact vector fields 121

4 Approximation-solvability and solvability of equations involving maps of A-proper type 127

5 Solvability of equations involving semilinear weakly A-proper mappings 160

IV Generalized Degree for A-Proper Mappings and Applications 188

1 Definitions of topological degrees for compact and strict set-contractive vector fields and some of their properties 189

2 Generalized degree for A-proper mappings 196

3 Single-valued generalized degree for compact and k-ball-contractive perturbations of firmly monotone and accretive maps 212

4 Some applications of the generalized degree 219

5 Calculation of generalized degree 237

6 Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications 250

V Solvability of PDEs and ODEs and Bifurcation Problems 265

1 Solvability of partial differential equations 265

2 Solvability of ordinary differential equations 288

3 Approximation-solvability of periodic boundary value problems 304

4 Variational BPs and ABPs for PDEs and ODEs 314

Appendix

1 Upper and lower bounds and limits of functions 323

2 Finite element methods for ODEs and elliptic PDEs 324

3 Lebesgue measure, integral, and some important convergence theorems 330

4 Lebesgue spaces L_p (Q) 334

5 Weak and weak* convergence 336

6 Sobolev spaces W^m_p (Q), W^m_p(Q) 336

References 339

Notation 363

Index 367