One-parameter family of solutions for one class of Hammerstein nonlinear equations on a half-line
type: Item
instance of: scholarly article
title: One-parameter family of solutions for one class of Hammerstein nonlinear equations on a half-line
zbMATH Open document ID: 1186.45006
DOI: 10.1134/S1064562409060222
published in: Doklady Mathematics
publication date: 2010-01-29T00:00:00Z
review text: The existence of continuously many positive solutions with at most linear growth of the Hammerstein equation \[ \varphi(x)=\int_0^\infty K(x,t)Y(\varphi(t))dt \] on the half-line \((0,\infty)\) is proved where \(K(x,t)\geq K_0(x-t)\) is in a certain specified sense close to a convolution kernel \(K_0\), \(K_0\) being the Laplace transform of a measure with certain properties, and the nonlinearity \(Y\) has the rather particular form \(Y(u)=u-\omega(u)\) with \(\omega\geq0\) being decreasing for large arguments. Also results about factorizing the corresponding linear integral operator are obtained. The proof is in a sense constructive, using a variant of successive approximation with starting value depending on a scalar parameter; for different parameters, different solutions are obtained.
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45M20
Mathematics Subject Classification ID: 45L05
Mathematics Subject Classification ID: 47H30
zbMATH DE Number: 5664307
zbMATH Keywords: Hammerstein equation
zbMATH Keywords: unbounded interval
zbMATH Keywords: positive solution
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: infinitely many solutions
zbMATH Keywords: Wiener-Hopf integral operator
zbMATH Keywords: Laplace transform
author: Kh. Khachatryan
MaRDI profile type: Publication
cites work: The Wiener-Hopf equation whose kernel is a probability density
cites work: Q3684614
description: scientific article; zbMATH DE number 5664307