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Differentiable Operators and Nonlinear Equations : Operator Theory, Advances and Applications - Victor Khatskevich

Differentiable Operators and Nonlinear Equations

Operator Theory, Advances and Applications

Hardcover Published: 1st December 1993
ISBN: 9783764329297
Number Of Pages: 284

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The need to study holomorphic mappings in infinite dimensional spaces, in all likelihood, arose for the first time in connection with the development of nonlinear analysis. A systematic study of integral equations with an analytic nonlinear part was started at the end of the 19th and the beginning of the 20th centuries by A. Liapunov, E. Schmidt, A. Nekrasov and others. Their research work was directed towards the theory of nonlinear waves and used mainly the undetermined coefficients and the majorant power series methods, which subsequently have been refined and developed. Parallel with these achievements, the theory of functions of one or several complex variables was gradually enriched with more significant and subtle results. The present book is a first step towards establishing a bridge between nonlinear analysis, nonlinear operator equations and the theory of holomorphic mappings on Banach spaces. The work concludes with a brief exposition of the theory of spaces with indefinite metrics, and some relevant applications of the holomorphic mappings theory in this setting. In order to make this book accessible not only to specialists but also to students and engineers, the authors give a complete account of definitions and proofs, and also present relevant prerequisites from functional analysis and topology. Contents: Preliminaries . Differential calculus in normed spaces . Integration in normed spaces . Holomorphic (analytic) operators and vector-functions on complex Banach spaces . Linear operators . Nonlinear equations with differentiable operators . Nonlinear equations with holomorphic operators . Banach manifolds . Non-regular solutions of nonlinear equations . Operators on spaces with indefinite metric . References . List of Symbols . Subject Index

Introduction. Part 1 Preliminaries: Sets and relations; Topological spaces; Convergence. Directedness; Metric spaces; Spaces of mappings; Linear topological spaces; Normed spaces; Linear operators and functionals; Conjugate space. Conjugate operator; Weak topology and reflexivity; Hilbert spaces. Part 2 Differential calculus in normed spaces: The derivate and the differential of a nonlinear operator; Lagrange formula and Lipschitz condition; Examples of Frechet differentiable operators; Lemmas about differentiable operators; Partial derivatives; Multilinear operators. Duality. Homogeneous forms; Higher order derivatives; Complete continuity of operators and of their derivatives. Part 3 Integration in normed spaces: Riemann - Stieltjes integrals of vector-functions; Pettis integral and the connection with Riemann - Stilties integral; Antiderivatives of vector-functions; Integral representations; Integrals of operators in Banach spaces. Part 4 Holomorphic (analytic) operators and vector-functions on complex Banach spaces: Differentiability in complex and real sense - Cauchy - Riemann conditions; The p-topology and holomorphy; Cauchy integral theorems and their consequences; Uniqueness theorems and maximum principles; Schwartz Lemma and its generalizations; Uniformly bounded families of p-holomorphic (holomorphic) operators; Montel property. Part 5 Linear operators: The spectrum and the resolvent of a linear operator; Spectral radius; Resolvent and spectrum of the adjoint operator; The spectrum of a completely continuous operator; Normally solvable operators; Noether and Fredholm operators; Projections. Splitable operators; Invariant subspaces. Part 6 Nonlinear equations with differentiable operators: Fixed points. Banach principle; Non-expansive operators; Fixed points for differentiable operators; Some applications of fixed point principle; Implicite and inverse operators. Connection with fixed points. Part 8 Nonlinear equations with holomorphic operators: s-fixed points for holomorphic operators; A converse of Banach principle; Criterions for the existence of an s-fixed point and its extension with respect to a parameter; Regular fixed points. Geometric criterions; Apriori estimates and the extension of an s-solution to the boundary of the domain; Local inversion of holomorphic operators and a posteriori error estimates; Single-valued small solutions in some degenerate cases. Part 8 Banach manifolds: Basic definitions; Smooth mappings; Submanifolds; Complex manifolds and Stein manifolds; Part 9 Non-regular solutions of nonlinear equations: Ramification of solutions. Statement of the problem; Equations of ramification; Equations of ramification for an analytic operator; The problem of the coefficients; The description of the set of fixed points for an analytic operator. Part 10 Operators on spaces with indefinite metric: Spaces with indefinite metric; Angle operators; Plus-operators; Symmetric

ISBN: 9783764329297
ISBN-10: 3764329297
Series: Operator Theory, Advances and Applications
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 284
Published: 1st December 1993
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.39 x 15.6  x 1.75
Weight (kg): 0.59