A Primer of Nonlinear AnalysisCambridge University Press, 09.03.1995 - 180 Seiten This is an introduction to nonlinear functional analysis, in particular to those methods based on differential calculus in Banach spaces. It is in two parts; the first deals with the geometry of Banach spaces and includes a discussion of local and global inversion theorems for differential mappings. In the second part, the authors are more concerned with bifurcation theory, including the Hopf bifurcation. They include plenty of motivational and illustrative applications, which indeed provide much of the justification of nonlinear analysis. In particular, they discuss bifurcation problems arising from such areas as mechanics and fluid dynamics. |
Inhalt
0 Preliminaries and notation | 1 |
1 Differential calculus | 9 |
2 Continuity and differentiability of Nemitski operators | 15 |
3 Higher derivatives | 23 |
4 Partial derivatives Taylors formula | 26 |
2 Local inversion theorems | 30 |
2 The Implicit Function Theorem | 36 |
3 A stability property of orbits | 38 |
4 Bifurcation from the simple eigenvalue | 91 |
5 A bifurcation theorem from a multiple eigenvalue | 100 |
Appendix | 105 |
6 Bifurcation problems | 107 |
2 The Bénard problem | 112 |
3 Small oscillations for secondorder dynamical systems | 119 |
4 Water waves | 123 |
5 Periodic solutions of a semilinear hyperbolic equation | 130 |
3 Global inversion theorems | 45 |
2 Global inversion with singularities | 55 |
Appendix | 60 |
4 Semilinear Dirichlet problems | 61 |
1 Problems at resonance | 62 |
2 Problems with asymmetric nonlinearities | 71 |
5 Bifurcation results | 79 |
2 Some elementary examples | 82 |
3 The Lyapunov Schmidt reduction | 89 |
7 Bifurcation of periodic solutions | 136 |
2 Nonlinear oscillations of autonomous systems | 139 |
3 The Lyapunov Centre Theorem | 145 |
4 The restricted threebody problem | 153 |
Problems | 160 |
| 165 | |
| 170 | |
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assumption Banach spaces bifurcating branch bifurcation point boundary conditions bounded C¹(R C²(R codimension compact constant continuous map corresponding deduce defined definition denote dF(u Dirichlet Dirichlet problem discuss eigenvalue equation example exists F is differentiable f satisfies Fréchet Fredholm Alternative given Global Inversion Theorem hence Hölder spaces holds true Implicit Function Theorem infer integral Inv(X inverts F Ker(L Lagrangian points Lemma Let F Let us consider linear map locally invertible map F Moreover neighbourhood Nemitski operators nonlinear nontrivial solutions notation parameter partial derivative periodic solutions point for F preceding precisely Proof of Theorem Proposition prove Remark resp respect Section sequence simple eigenvalue singular point Sobolev spaces subspace super-solution Suppose T-periodic Theorem 4.1 topological trivial solution unique solution yields