Asymptotics of nonlinearities and operator equations
type: Item
instance of: scholarly article
title: Asymptotics of nonlinearities and operator equations
zbMATH Open document ID: 0863.47047
published in: Operator Theory: Advances and Applications
publication date: 1995-08-13T00:00:00Z
review text: This is an English translation by Mircea Martin of the Russian edition of the book, originally published by Nauka, Moscow, in 1992. The monograph is divided into five chapters entitled: Norm estimates for solutions of integral-functional inequalities (Chapter 1), Two-sided estimates for nonlinearities (Chapter 2), The use of arguments of leading eigenvalues (Chapter 3), Weak nonlinearities (Chapter 4), and One-sided estimates for nonlinearities (Chapter 5). The main object of investigation is the nonlinear equation \[ x(t)=Af(t,x(t)),\quad t\in\Omega,\tag{E} \] where \(\Omega\) is a finite measure space, \(f:\Omega\times\mathbb{R}^n\to \mathbb{R}^n\) is a Carathéodory function satisfying suitable growth conditions, and \(A\) is a linear operator from a certain function space over \(\Omega\) into another (often from \(L^p(\Omega)\) into itself, \(1\leq p\leq\infty\)). Various abstract existence theorems for (E) are proved by Schauder's fixed point theorem and topological degree methods, or by variational techniques. In many instances \(A\) is assumed to be a completely continuous operator in \(L^2(\Omega)\), while \(f\) satisfies two-sided or one-sided growth restrictions (Chapters 2 and 5). In the latter case, \(A\) is also subject to positivity or strict positivity conditions. In Chapter 3, the author considers the case when \(A\) is normal and its leading eigenvalues have nonzero imaginary parts, while in Chapter 4 he studies a slight modification of (E), namely \[ x(t)= A(x(t)+\varphi(t,x(t)))+b(t), \] where \(x:\Omega\to\mathbb{R}\), \(A\) is a normal completely continuous operator on \(L^2(\Omega)\), \(b\in L^2(\Omega)\), and \(\varphi\) is a so-called weak nonlinearity. The general theory is applied to the discussion of two-point boundary value problems for second-order nonlinear differential equations, Dirichlet problems for elliptic partial differential equations, forced oscillations in control systems and systems with delay, integral equations with nonnegat…
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 784259
zbMATH Keywords: two-sided estimates for nonlinearities
zbMATH Keywords: one-sided estimates
zbMATH Keywords: integral-functional inequalities
zbMATH Keywords: leading eigenvalues
zbMATH Keywords: Carathéodory function
zbMATH Keywords: Schauder's fixed point theorem
zbMATH Keywords: topological degree
zbMATH Keywords: variational techniques
zbMATH Keywords: completely continuous operator
zbMATH Keywords: strict positivity
zbMATH Keywords: weak nonlinearity
zbMATH Keywords: two-point boundary value problems
zbMATH Keywords: second-order nonlinear differential equations
zbMATH Keywords: Dirichlet problems for elliptic partial differential equations
zbMATH Keywords: forced oscillations in control systems
zbMATH Keywords: systems with delay
zbMATH Keywords: integral equations with nonnegative kernels
author: Alexander Krasnosel'skij
reviewed by: Sergiu Aizicovici
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784259