Nonlinear Functional AnalysisSpringer Science & Business Media, 2013 M11 11 - 450 páginas topics. 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. |
Contenido
1 | |
12 | |
Concluding Remarks | 27 |
8 | 33 |
Compact Maps | 55 |
7 | 62 |
Set Contractions | 68 |
Concluding Remarks | 87 |
Approximate Solutions | 256 |
AProper Maps and Galerkin for Differential Equations | 267 |
Exercises | 275 |
Multis | 278 |
13 | 287 |
16 | 298 |
Multis and Compactness | 299 |
Extremal Problems | 319 |
Borsuks Theorem | 90 |
Monotone and Accretive Operators | 95 |
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 |
Solutions in Cones | 217 |
Solutions in Cones | 238 |
Exercises | 329 |
Extrema Under Constraints | 332 |
17 | 340 |
Critical Points of Functionals | 349 |
Bifurcation | 378 |
Global Bifurcation | 398 |
Further Topics in Bifurcation Theory | 411 |
Epilogue | 426 |
445 | |
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A-proper accretive B₁ bifurcation bounded sets chapter choose closed convex condition cone continuous conv convergent convex functional convex set D₁ defined differential equations eigenvalue example Exercise exists F₁ F₂ finite finite-dimensional Fix F fixed point theorem Fredholm operators function given Hence Hilbert space Hint homeomorphism hyperaccretive hypermaximal implies K₁ Let F let us prove Let X linear Lipschitz lower semicontinuous maximal monotone metric multis neighbourhood nonexpansive nonlinear norm Notice open bounded operators proof to Theorem properties Proposition real Banach space reflexive result satisfies semicontinuous semigroup Stanislaw Jerzy Lec strictly convex subsets subspace Suppose T₁ topological uniformly convex unique solution weakly x₁ xn+1 y₁ zero