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Nonresonance problems for differential inclusions in separable Banach spaces

Authors:	Zouhua Ding Athanassios G Kartsatos

Institution:	Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700 ; Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Abstract:	Let be a real separable Banach space. The boundary value problem $\begin{equation}\begin {split} % &x' \in A(t)x+F(t,x),~t\in \mathcal {R}_+, &Ux = a, \end {split} % \tag {(B)} % \end{equation*}$ is studied on the infinite interval $R_+=0,\infty ).$ Here, the closed and densely defined linear operator $A(t):X\supset D(A)\to X,~t\in \mathcal {R}_+,$ generates an evolution operator The function $F:\mathcal {R}_+\times X\to 2^X$ is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either is bounded and is compact for or is compact and is equicontinuous. The mapping $U:C_b(\mathcal {R}_+,X)\to X$ is a bounded linear operator and $a\in X$ is fixed. The nonresonance problem is solved by using Ma's fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in $C_b(\mathcal {R}_+,X).$

Keywords:	Boundary value problem on an infinite interval differential inclusion upper semicontinuous function compact evolution operator

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