Topological characterization of the solution set of Fredholm equations with \(f\)-compactly contractive perturbations and its applications
type: Item
instance of: scholarly article
title: Topological characterization of the solution set of Fredholm equations with \(f\)-compactly contractive perturbations and its applications
zbMATH Open document ID: 1014.47034
author: Z. Kukharski
author: V. G. Zvyagin
author: V. T. Dmitrienko
published in: Russian Mathematics
publication date: 2002-06-09T00:00:00Z
review text: This article deals with the degree theory for completely continuous and \(f\)-compactly restricting perturbations \(f-g\) (in particular, \(f\)-condensing) of \(C^r\)-smooth mappings \(f\) which are defined on open subsets \(X\) of a real Banach space \(E\), Fredholm on the zero set of \(f-g\), and take values in a real Banach space \(F\). The authors introduce and study the index \(\text{ind}_2 (f-g,X,0)\) of the zero set of \(f-g\) on \(X\) with values in the ring of Rholin-Thom nonoriented bordisms and show that this index has the usual ``degree'' properties: the non-triviality of the zero set, if this index is nonzero, the invariance under homotopies from a natural class of deformations, and so on. As applications, the authors prove the solvability in the space \(C^2([0,1],\mathbb R^n)\) of the following nonlinear second-order boundary value problem \[ a_0\ddot x^m(t)+ a_1\ddot x^{m-1}(t)+\cdots+ a_m=G\bigl(t, x(t),\dot x(t),\ddot x(t)\bigr), \quad x(0)=c_0,\;x(1)=c_1. \] The main difference of the theory presented in this article from the previous is that the authors assume the Fredholm property of \(f\) only at points of some set containing zeros of \(f-g\).
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 58B15
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1752613
zbMATH Keywords: completely continuous and compactly restricting perturbations
zbMATH Keywords: Rohlin-Thom bordisms
zbMATH Keywords: degree theory
zbMATH Keywords: solvability
zbMATH Keywords: nonlinear second-order boundary value problem
zbMATH Keywords: Fredholm property
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1752613