A measure of weak noncompactness in \(L^1(\mathbb{R}^N)\) and applications
type: Item
instance of: scholarly article
title: A measure of weak noncompactness in \(L^1(\mathbb{R}^N)\) and applications
author: Bilal Boulfoul
author: Smail Djebali
published in: Mediterranean Journal of Mathematics
publication date: 2022-03-14T00:00:00Z
review text: In this article a new measure of weak noncompactness in the Banach space \(L^1(\mathbb{R}^N)\) is proposed. It is a generalization of the Banas-Knap measure of weak noncompactness, which in turn is a generalization of the De Blasi measure of weak noncompactness. Then, the multidimensional nonlinear functional integral equation \[ u(t,x)=f(t,x,u(t,x))+ g\left( t,x, \int_0^t \int_{\Omega}\, k(t,x,s,y,u(s,y)) dyds \right),\ t\ge 0,\, x\in \Omega. \] is considered. The existence of solutions to this equation is provided using a recent fixed point theorem of Krasnosel'skii type combined with the new previously introduced measure of weak noncompactness. The above equation easily includes the following Volterra-Fredholm integral equation as a special case \[ u(t,x)=f(t,x,u(t,x))+ \int_0^t \int_{\Omega}\, k(t,x,s,y,u(s,y)) dyds ,\ t\ge 0,\, x\in \Omega. \] Further, also the mixed Volterra-Fredholm integral equation \[ u(t,x)=\phi(x)+\frac1\eta \left( \int_0^t \big( v(s,x)-u(s,x) \big) ds + \int_0^t \int_{\Omega}\, h(s,x,y,u(s,y)) dyds \right) \] can be seen as a particular case of the multidimensional nonlinear functional integral equation under investigation. This last equation is obtained by integration from the integro-differential equation of parabolic type deriving from the neural modeling fields \[ \eta\frac{\partial}{\partial t}u(t,x)+u(t,x)=v(t,x) + \int_{\Omega} h(t,x,y,u(t,y)) dy,\ t\in [0,T],\, x\in \Omega, \] with \(u(0,x)=\phi(x),\ x\in \Omega\).
reviewed by: Paola Rubbioni
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 47H08
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 97I70
zbMATH DE Number: 7488620
zbMATH Keywords: measure of weak noncompactness
zbMATH Keywords: Krasnosel'skii fixed point theorem
zbMATH Keywords: nonlinear contraction
zbMATH Keywords: functional integral equation
zbMATH Keywords: Carathéodory condition
MaRDI profile type: Publication
full work available at URL: s00009-022-01991-x
OpenAlex ID: W4212864930
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DOI: 10.1007/S00009-022-01991-X
description: scientific article; zbMATH DE number 7488620