A coincidence degree for bifurcation problems
type: Item
instance of: scholarly article
title: A coincidence degree for bifurcation problems
zbMATH Open document ID: 1028.47046
DOI: 10.1016/S0362-546X(02)00308-5
author: Cristinel Mortici
author: Silviu Sburlan
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-05-04T00:00:00Z
review text: This article deals with coincidence degree for pairs \((L, A)\) of operators in a Hilbert space \(H\); the first operator is assumed to be Leray-Schauder type, the second one compact and monotone. This degree is defined by the equation \[ d_\lambda\bigl( (L,A),D\bigr)= d_{LS}\bigl(I-(I+ \lambda A)^{-1}(I-L), D,0\bigr) \] and has the properties of the classical topological degree; it can be used in order to study bifurcation phenomena, for equations of the type \[ Lu+\lambda Au+F(\lambda,u)+ G(\lambda,u)=0 \] with \(F( \lambda,u)\) and \(G(\lambda,u)\) satisfying the conditions \(\|F(\lambda,u) \|\leq k\|u\|\) for some \(k>0\) and \(\|G(\lambda,u)\|= o(\|u \|)\) for \(u\to 0\) uniformly in \(\lambda\); in particular, analogues to Krasnosel'skiĭ and Rabinowitz theorems about bifurcation are given. At the end of the article, using the Browder-Ton theorem about imbedding reflexive separable Banach spaces into Hilbert spaces, the main results are enlarged to vector fields of monotone type defined from a Banach space into its dual.
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 1903500
zbMATH Keywords: bifurcation problems
zbMATH Keywords: pairs of operators
zbMATH Keywords: coincidence degree
zbMATH Keywords: Hilbert space
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
cites work: Convergence of approximants by regularization for solutions of nonlinear functional equations in Banach spaces
cites work: Q3857336
full work available at URL: s0362-546x(02)00308-5
OpenAlex ID: W1980708304
description: scientific article; zbMATH DE number 1903500