Nonlinear Functional Analysistopics. However, only a modest preliminary knowledge is needed. In the first chapter, where we introduce an important topological concept, the so-called topological degree for continuous maps from subsets ofRn into Rn, you need not know anything about functional analysis. Starting with Chapter 2, where infinite dimensions first appear, one should be familiar with the essential step of consider ing a sequence or a function of some sort as a point in the corresponding vector space of all such sequences or functions, whenever this abstraction is worthwhile. One should also work out the things which are proved in § 7 and accept certain basic principles of linear functional analysis quoted there for easier references, until they are applied in later chapters. In other words, even the 'completely linear' sections which we have included for your convenience serve only as a vehicle for progress in nonlinearity. Another point that makes the text introductory is the use of an essentially uniform mathematical language and way of thinking, one which is no doubt familiar from elementary lectures in analysis that did not worry much about its connections with algebra and topology. Of course we shall use some elementary topological concepts, which may be new, but in fact only a few remarks here and there pertain to algebraic or differential topological concepts and methods. |
Contents
| 1 | |
Borsuks Theorem | 21 |
Concluding Remarks | 27 |
Topological Degree in Infinite Dimensions | 35 |
Compact Maps | 55 |
Set Contractions | 68 |
Concluding Remarks | 87 |
Monotone and Accretive Operators | 95 |
Solutions in Cones | 217 |
Solutions in Cones | 238 |
Approximate Solutions | 256 |
AProper Maps and Galerkin for Differential Equations | 267 |
Multis | 278 |
Multis and Compactness | 299 |
Extremal Problems | 319 |
Extrema Under Constraints | 332 |
Monotone Operators on Banach Spaces | 111 |
Accretive Operators | 123 |
Concluding Remarks | 133 |
Implicit Functions and Problems at Resonance | 146 |
Problems at Resonance | 172 |
Fixed Point Theory | 186 |
Fixed Point Theorems Involving Compactness | 203 |
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A-proper A₁ accretive B₁ bifurcation point bounded sets choose closed convex compact cone consider continuous conv convergent convex functional convex set D₁ defined differential equations eigenvalue example Exercise exists F₁ F₂ finite finite-dimensional Fredholm operators function given Hence Hilbert space Hint homeomorphism hypermaximal implicit function theorem implies K₁ Let F let us prove linear Lipschitz lower semicontinuous M₁ maximal monotone metric neighbourhood nonexpansive nonlinear norm Notice open bounded P₁ properties Proposition real Banach space reflexive result satisfies semicontinuous semigroup Stanislaw Jerzy Lec subset subspace Suppose T₁ topological uniformly convex w-periodic x₁ y₁ zero λ₁ λε λη λο χο Ω₁