Nonresonance problems for differential inclusions in separable Banach spaces

Abstract

Let X X be a real separable Banach space.  The boundary value problem (B) a m p ; x ′ ∈ A ( t ) x + F ( t , x ) ,   t ∈ R + , a m p ; U x = a , \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 , ∞ ) . R_+=[0,\infty ).  Here, the closed and densely defined linear operator A ( t ) : X ⊃ D ( A ) → X ,   t ∈ R + , A(t):X\supset D(A)\to X,~t\in \mathcal {R}_+, generates an evolution operator W ( t , s ) . W(t,s).  The function F : R + × X → 2 X 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 F is bounded and W ( t , s ) W(t,s) is compact for t > s , t > s, or F F is compact and W ( t , s ) W(t,s) is equicontinuous.  The mapping U : C b ( R + , X ) → X U:C_b(\mathcal {R}_+,X)\to X is a bounded linear operator and a ∈ X 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 ( R + , X ) . C_b(\mathcal {R}_+,X).