Asymptotics of nonlinearities and operator equations
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 nonnegative kernels, etc. The book is a valuable addition to the English language mathematics literature. I recommend it as a research reference for advanced graduate students and practitioners in the fields of operator theory, differential and integral equations, and control theory.
second-order nonlinear differential equations
Dirichlet problems for elliptic partial differential equations
integral equations with nonnegative kernels
scientific article; zbMATH DE number 784259