Existence of solutions of a class of abstract second order nonlinear integrodifferential equations
type: Item
instance of: scholarly article
title: Existence of solutions of a class of abstract second order nonlinear integrodifferential equations
zbMATH Open document ID: 1014.45008
DOI: 10.1155/S1048953302000138
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2002-12-10T00:00:00Z
full work available at URL: https://eudml.org/doc/50094
review text: The following nonlinear second order integrodifferential equation is investigated \[ Kx''(t)= Ax(t)+f\left(t,x(t), \int^t_0 g\bigl(t,x (s), x'(s)\bigr)ds, x'(t)\right), \;t\in J=[0,T] \tag{1} \] together with the initial conditions \(x(0)=x_0\), \(x'(0)=y_0\). Here \(K\) and \(A\) are linear operators with domains contained in a Banach space \(X\) and with values in a Banach space \(Y\) and the functions \(g:J\times J\times X\times X\to X\) and \(f:J \times X\times X\times X\to Y\) being continuous and sublinear. Moreover, the operator \(K\) is continuously invertible with \(AK^{-1}\) generating a strongly continuous cosine family of bounded linear operators on \(X\). Under some extra assumptions an existence theorem for the problem (1) is proved with help of the Schaefer fixed point theorem.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1844035
zbMATH Keywords: nonlinear second order integrodifferential equation
zbMATH Keywords: Banach space
zbMATH Keywords: existence
zbMATH Keywords: Schaefer fixed point theorem
author: Krishnan Balachandran
author: Jong Yeoul Park
reviewed by: Józef Banaś
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1844035