Fredholm mappings and the generalized boundary value problem
type: Item
instance of: scholarly article
title: Fredholm mappings and the generalized boundary value problem
zbMATH Open document ID: 0818.34014
published in: Differential and Integral Equations
publication date: 1995-08-22T00:00:00Z
review text: The author proves several assertions concerning various properties (e.g. generic properties, surjectivity, bifurcation problem) of the mapping \(F: X\to Y\) (\(X\) and \(Y\) are Banach spaces) which is the sum of a linear bounded Fredholm mapping of index zero and of a completely continuous mapping. These results are applied to boundary value problems of the form \(A(x)+ f(t, x, x',\dots, x^{(m)})= q(t)\), \(a\leq t\leq b\), \(\ell_ j(x)= 0\), \(j= 1,\dots, n\), where \(-\infty< a< b< \infty\), \(0\leq m\leq n- 1\), \(f: [a,b]\times \mathbb{R}^{m+ 1}\to \mathbb{R}\) is continuous, \(q\in C([a, b], \mathbb{R})\), \(A\) is an \(n\)th order linear differential expression with continuous coefficients and \(\ell_ j\) are linear continuous functionals on \(C^{n- 1}([a, b],\mathbb{R})\). In particular, the author proves that if a certain a priori estimate is satisfied then the given problem possesses bifurcation points. Furthermore, he proves that the existence of at most finitely many solutions is generic for the given type of boundary value problems. In the cases when the corresponding linearized homogeneous problem is positive or self-adjoint some more results are obtained, as well.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 716204
zbMATH Keywords: surjectivity
zbMATH Keywords: bifurcation problem
zbMATH Keywords: Fredholm mapping
zbMATH Keywords: completely continuous mapping
zbMATH Keywords: boundary value problems
author: Valter Šeda
reviewed by: Milan Tvrdý
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 716204