Nonlinear Hammerstein integral equations via local linking and mountain pass
type: Item
instance of: scholarly article
title: Nonlinear Hammerstein integral equations via local linking and mountain pass
zbMATH Open document ID: 1237.45006
author: Donal O'Regan
publication date: 2012-02-09T00:00:00Z
review text: The author considers nonlinear Hammerstein type integral equations \[ u(x) = \int_{\Omega} k(x,y) f(y,u(y)) dy,\;\;\; x\in \Omega, \tag{1} \] where \(\Omega\) is a bounded closed subset of \({\mathbb R}^n\), \(k(\cdot,\cdot)\) is a continuous symmetric kernel and \(K\) is the corresponding linear integral operator, \(N_f\) is a nonlinear superposition (Nemytsky) operator generated by a continuous function \(f: \Omega\times {\mathbb R} \rightarrow {\mathbb R}\). The solvability of the equation (1) in \(C(\Omega)\) of the equation (1) is reduced to the solvability of the following operator equation in \(L^2(\Omega)\) \[ v = K^{\frac{1}{2}} N_f K^{\frac{1}{2}} v. \tag{2} \] The solvability of the equation ({2}) is studied on the basis of the following notion of local linking. Definition. Let \(W = (W, \|\cdot\|)\) be a real Banach space and let \(W = W_1 \oplus W_2\) be a direct sum decomposition of \(W\) with \(W_1\) being finite dimensional. Assume that the origin is a critical point of a real valued function \(\Phi\) defined on \(W\) (\(\Phi \in C^1(W, {\mathbb{R}})\)) and \(\Phi(0) = 0\). It is said that \(\Phi\) has a local linking near the origin if for some \(r > 0\) we have \[ \Phi(u_1) \leq 0,\quad u_1\in W_1,\quad \|u_1\| \leq r, \] \[ \Phi(u_2) > 0,\quad u_2\in W_2,\quad 0 < \|u_2\| \leq r. \] A number of unique and multiple solvability results for integral equation ({1}) is obtained by using the approach developed by the author earlier on the basis of the above notion.
reviewed by: Sergei V. Rogosin
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 47H30
zbMATH DE Number: 6004391
zbMATH Keywords: nonlinear Hammerstein integral equation
zbMATH Keywords: local linking
zbMATH Keywords: Morse theory
zbMATH Keywords: critical points
zbMATH Keywords: critical value
MaRDI profile type: Publication
full work available at URL: s12215-011-0056-0
OpenAlex ID: W2009383722
cites work: A new approach to the Morse-Conley theory and some applications
cites work: Infinite dimensional Morse theory and multiple solution problems
cites work: Applications of local linking to critical point theory
cites work: Existence of solutions to nonlinear Hammerstein integral equations and applications
cites work: Q3560017
cites work: Q4820378
cites work: Q3749777
DOI: 10.1007/S12215-011-0056-0
published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 6004391