, Volume 20, Issue 3, pp 477-497
Date: 17 Feb 2012

Global Solutions for Nonlinear Delay Evolution Inclusions with Nonlocal Initial Conditions

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Abstract

We prove a sufficient condition for the existence of global C 0-solutions for a class of nonlinear functional differential evolution equation of the form $$ \left\{\begin{array}{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \end{array}\right. $$ where X is a real Banach space, A generates a nonlinear compact semigroup having an exponential decay, \(F:\mathbb{R}_+\times X\times C([\,-\tau,0\,];\overline{D(A)})\rightsquigarrow X\) is a nonempty, convex, weakly compact valued and almost strongly-weakly u.s.c. multi-function with linear growth and the nonlocal function \(g:C_{b}([\,-\tau,+\infty);\overline{D(A)})\to C([\,-\tau,0\,];\overline{D(A)})\) is nonexpansive.