Uniqueness of solutions to an Abel type nonlinear integral equation on the half line
type: Item
instance of: scholarly article
title: Uniqueness of solutions to an Abel type nonlinear integral equation on the half line
zbMATH Open document ID: 1263.45003
author: Wojciech Mydlarczyk
published in: Central European Journal of Mathematics
publication date: 2013-02-18T00:00:00Z
review text: The author studies the integral equation \[ u(x)=\int_{-\infty}^{x}(x-s)^{\alpha-1}g(u(s))\;ds\text{, }-\infty<x\text{, }0<\alpha, \] and obtains that any two nontrivial solutions of this equation are translations of each other; note that in this paper it is assumed that the constant \(\alpha\) appearing above satisfies the inequality \(1<\alpha<4\). More precisely, the author proves that if \(u_1\) and \(u_2\) are nontrivial solutions of the above integral equation, then it holds that \(u_2(x)=u_1(x+c)\), for some constant \(c\in\mathbb{R}\) and each \(x>-\infty\). In order to prove this result, the author assumes that the function \(g\;: \;[0,+\infty)\rightarrow[0,+\infty)\) satisfies \(g(0)=0\), \(g(u)>0\) for all \(u>0\), and \(g\) is nondecreasing; furthermore, it is supposed that \(g\) is continuous on \([0,+\infty)\). It is also worth noting that in addition to the aforementioned assumptions imposed on \(g\) it is also supposed that \(g\) satisfies the inequality \[ \int_0^{\delta}\left(\frac{s}{g(s)}\right)^{\frac{1}{\alpha}}\frac{ds}{s}=+\infty \] for some \(\delta>0\). With all these assumptions in hand, the previously mentioned uniqueness-type result is obtained. This paper is clearly written and could be of interest to researchers interested in integral equations generally and, more specifically, in both the existence and uniqueness of solution to integral equations.
reviewed by: Christopher Goodrich
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45M20
zbMATH DE Number: 6137111
zbMATH Keywords: nonlinear Volterra integral equations
zbMATH Keywords: integral inequalities
zbMATH Keywords: nontrivial solutions
zbMATH Keywords: uniqueness of solution
zbMATH Keywords: generalized Osgood condition
MaRDI profile type: Publication
full work available at URL: s11533-012-0104-9
OpenAlex ID: W2011380129
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DOI: 10.2478/S11533-012-0104-9
description: scientific article; zbMATH DE number 6137111