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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonresonance problems
for differential inclusions
in separable Banach spaces


Authors: Zouhua Ding and Athanassios G. Kartsatos
Journal: Proc. Amer. Math. Soc. 124 (1996), 2357-2365
MSC (1991): Primary 34A60
MathSciNet review: 1340383
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Abstract: Let $X$ 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 $W(t,s).$ 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 $F$ is bounded and $W(t,s)$ is compact for $t > s,$ or $F$ is compact and $W(t,s)$ 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 $C_b(\mathcal {R}_+,X).$


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Additional Information

Zouhua Ding
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: ding@chuma.usf.edu

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: hermes@gauss.math.usf.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03439-9
PII: S 0002-9939(96)03439-9
Keywords: Boundary value problem on an infinite interval, differential inclusion, upper semicontinuous function, compact evolution operator
Received by editor(s): December 16, 1994
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society