A Topological Introduction to Nonlinear AnalysisSpringer Science & Business Media, 17 avr. 2013 - 146 pages Nonlinear analysis is a remarkable mixture of topology, analysis and applied mathematics. Mathematicians have good reason to become acquainted with this important, rapidly developing subject. But it is a BIG subject. You can feel it: just hold Eberhard Zeidler's Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems [Z} in your hand. It's heavy, as a 900 page book must be. Yet this is no encyclopedia; the preface accurately describes the " ... very careful selection of material ... " it contains. And what you are holding is only Part I of a five-part work. So how do you get started learning nonlinear analysis? Zeidler's book has a first page, and some people are quite comfortable beginning right there. For an alternative, the bibliography in [Z], which is 42 pages long, contains exposition as well as research results: monographs that explain portions of the subject to a variety of audiences. In particular, [D} covers much of the material of Zeidler's book. What makes this book different? The answer is in three parts: this book is (i) topological (ii) goal-oriented and (iii) a model of its subject. |
Table des matières
AscoliArzela Theory | 8 |
Brouwer Fixed Point Theory | 18 |
Equilibrium Heat Distribution | 29 |
Some Topological Background 43 | 42 |
LeraySchauder Degree | 57 |
Properties of the LeraySchauder Degree | 63 |
A Separation Theorem | 72 |
The Degree Calculation | 87 |
The KrasnoselskiiRabinowitz Bifurcation Theorem | 95 |
Nonlinear SturmLiouville Theory | 108 |
Euler Buckling | 118 |
A Singular Homology 135 | 133 |
| 144 | |
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Expressions et termes fréquents
Ascoli-Arzela theorem assume B₁ Banach space bifurcation point bifurcation theorem boundary condition bounded subset Brouwer degree Brouwer Fixed Point certainly choose closed subset compact linear operator compact map compact set compact space compact subset completely continuous contains continuous function convergent subsequence cross product defined definition denote differential equation disjoint e-net equicontinuous euclidean space Euler buckling problem exists Figure finite-dimensional fixed point theorem Frechet differentiable function f H₄(X Hn Sn Hn(Sn homology theory homomorphism homotopy homotopy property hypothesis identity map implies inclusion integer interval isomorphism Lemma Leray-Schauder degree Let f map f metric space nonlinear Sturm-Liouville eigenvalue nonzero normed linear space open subset Proof prove real number relatively compact Rn-m sequence solution Sturm-Liouville eigenvalue problem Sturm-Liouville problem subspace suppose tells totally bounded U₁ vector space write zeros λι μη