Nonresonance problems for differential inclusions in separable Banach spaces

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 $C_b(\mathcal {R}_+,X).$