Impulsive problems for semilinear differential equations with nonlocal conditions
type: Item
instance of: scholarly article
title: Impulsive problems for semilinear differential equations with nonlocal conditions
zbMATH Open document ID: 1188.34073
author: Zhenbin Fan
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-01-08T00:00:00Z
review text: This paper deals with the impulsive differential equation with nonlocal conditions \[ \begin{aligned} u'(t) &= Au(t) + f(t, u(t)), 0\leq t\leq b, t\neq t_i,\\ u(0) &= u_0-g(u),\\ \Delta u(t_i) &= I_i(u(t_i)), i = 1, 2, \dots, p, 0 < t_1 < t_2 < \dots < t_p < b, \end{aligned} \] where \(A : D(A) \subseteq X \to X\) is the infinitesimal generator of a strongly continuous semigroup \(T (t), t \geq 0,\) \(X\) is a real Banach space, \(\Delta u(t_i) =u(t_i^+)-u(t_i^-),\) \( u(t_i^+), u(t_i^-)\) denotes the right and the left limit of \(u\) at \(t_i,\) respectively and \(f, g, I_i \) are appropriate continuous functions. By using the Hausdorff measure of noncompactness and fixed point techniques, the author proves the existence of a mild solution without the Lipschitz continuity of the mapping \(f,\) in the cases when (i) \(g\) and \(I_i\) are Lipschitz and the semigroup \(T (t), t > 0\), generated by the linear operator \(A\) is compact, and (ii) \(g\) is not Lipschitz and not compact.
reviewed by: Sotiris K. Ntouyas
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 5655116
zbMATH Keywords: nonlocal condition
zbMATH Keywords: impulsive differential equations
zbMATH Keywords: fixed point
zbMATH Keywords: \(C_0\)-semigroup
MaRDI profile type: Publication
full work available at URL: j.na.2009.07.049
OpenAlex ID: W2061733059
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DOI: 10.1016/J.NA.2009.07.049
description: scientific article; zbMATH DE number 5655116