On the singular impulsive functional integral equations involving nonlocal conditions
type: Item
instance of: scholarly article
title: On the singular impulsive functional integral equations involving nonlocal conditions
zbMATH Open document ID: 1270.45011
author: Rong-Nian Wang
author: Yan Wang
author: De-Han Chen
published in: Osaka Journal of Mathematics
publication date: 2012-11-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1350306592
review text: The authors consider a singular impulsive functional integral equation of the form \[ u(t)= T(t) H(u)+{1\over \Gamma(\alpha)} \int^t_0 (t-s)^{\alpha-1} T(t-s) F(s,u(s))\,ds \] for \(t\in[0, T]\), but \(t\) not equal to \(t_i\), and \(\Delta u(t_i)= I_i(u(t_i))\), \(i= 1,\dots,n\). Here \(T(t)\) is a compact semigroup on the Banach space \(X\), \(0<\alpha <1\), \(H\) is a nonlocal operator mapping \(\text{PC}([0,T]; X)\) into \(X\). (PC = piecewise continuous) and \(I_i: X\to X\), \(\Delta u(t_i)= u(t_i^+)- u(t_i^-)\), \(i= 1,\dots,n\). Under certain growth conditions on \(H\), \(I_i\) and \(F\) (including a Lipschitz condition on \(H\) but not on \(I_i\)), the authors prove the existence of a solution. The proof is based on Schauders fixed point theorem.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 6110921
zbMATH Keywords: Volterra equations
zbMATH Keywords: nonlocal initial conditions
zbMATH Keywords: singular impulsive functional integral equation
zbMATH Keywords: Banach space
zbMATH Keywords: Schauders fixed point theorem
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6110921