Existence of nonlinear neutral integrodifferential equations of Sobolev type with nonlocal condition
type: Item
instance of: scholarly article
zbMATH Open document ID: 1387.47034
author: P. Anukokila
author: B. Radhakrishnan
author: M. Balasubramanian
publication date: 2014-11-04T00:00:00Z
full work available at URL: 1.pdf
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K32
zbMATH DE Number: 6365589
zbMATH Keywords: existence
zbMATH Keywords: neutral differential equation
zbMATH Keywords: impulsive differential equation
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: fixed point theorem
MaRDI profile type: Publication
title: Existence of nonlinear neutral integrodifferential equations of Sobolev type with nonlocal condition
published in: Dynamics of Continuous, Discrete \& Impulsive Systems. Series A. Mathematical Analysis
review text: The purpose of this paper is to prove the existence of mild solutions for a nonlinear impulsive neutral delay integrodifferential equation of Sobolev type with time varying delays using semigroup theory and the fixed point approach. More precisely, it is used the Mönch type fixed point theorem and the Hausdorff measure of noncompactness. Since operators of Mönch type generalize compact operators, the obtained result is a generalization of other results from the literature. As an application of the abstract result, in the last part of this work the authors establish conditions which ensure the existence of a mild solution for the partial differential equation NEWLINE\[NEWLINE\begin{aligned} \frac{\partial }{\partial t}\left[z\left(t,x\right)-z_{xx}\left(t,x\right)+\int_{-\infty}^{t} a_1\left(s-t\right)z_t\left(s,t\right) ds\right]-z_{xx}\left(t,x\right) \\ =\rho\left(t,z\left(t,x\right)\right)+\int_{0}^{t}\sigma\left(t,s,z\left(t,x\right)\right)ds, \, 0\leq x\leq \pi,\\ z\left(t,0\right)=z\left(t,\pi\right)=0, \, t\geq 0,\\ z\left(0,x\right) + \sum_{k=1}^{p}z\left(t_k,x\right) = z_0\left(x\right), 0<t_1 <t_2 < \ldots < t_p<a; \, x\in \left[0,\pi\right],\\ \Delta z|_{t=t_i} = I_i\left(z\left(x\right)\right)=\left(\gamma_i \left(z\left(x\right)\right)+t_i\right)^{-1}, \, z\in X, \, 1\geq i \geq p. \end{aligned}NEWLINE\]
reviewed by: Andrei Horvat-Marc
description: scientific article; zbMATH DE number 6365589