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).