Mild solutions of non-autonomous second order problems with nonlocal initial conditions

Abstract

Artículo de publicación ISIIn this paper we establish the existence of mild solutions for a non-autonomous abstract semi-linear second order differential equation submitted to nonlocal initial conditions. Our approach relies on the existence of an evolution operator for the corresponding linear equation and the properties of the Hausdorff measure of non-compactness.Partially supported by: CONICYT under grant FONDECYT 1130144 and DICYT-USACH, FONDECYT 1110090 and MECESUP PUC 0711

An existence theorem for a non-autonomous second order nonlocal multivalued problem

“…Moreover, as in [8], a map S : J ×J → L(X), where L(X) denote the space of all bounded linear operators in X with the norm . L(X) , is said to be a "f undamental operator" if {S(t, s)} t,s∈J is a fundamental system.…”

“…L(X) , is said to be a "f undamental operator" if {S(t, s)} t,s∈J is a fundamental system. Moreover, as in [8], we introduce, for each (t, s) ∈ J × J the operator…”

“…As in [8], we will use the identification between the functions defined on T and the 2π−periodic functions from R to C. Next, put the Sobolev space…”

“…Recently in [8] H.R. Henríquez, V. Poblete, J.C. Pozo have studied the existence of mild solutions for a nonlocal problem governed by the following non-autonomous wave equation…”

“…Recently the existence of nonlocal mild solutions in Banach space has been investigated for semilinear non-autonomous second order differential equations in [9], in [7] and in [8], while there exists an extensive literature for the autonomous case (see, for example, [6], [11], [12] , [13] and [14]). The note is organized in the following way.…”