In what follows, we recall some definitions, notations, and results that we need in the sequel. Throughout this paper, is a Banach space, and , , are closed linear operators defined on a common domain which is dense in ; the notations and represent the resolvent set of the operator and the domain of endowed with the graph norm, respectively. For , we fix , and we represent by the norm of in . Let and be Banach spaces. In this paper, the notation stands for the Banach space of bounded linear operators from into endowed with the uniform operator topology, and we abbreviate this notation to when . Furthermore, for appropriate functions , the notation denotes the Laplace transform of . The notation stands for the closed ball with center at and radius in . On the other hand, for a bounded function and , the notation is given by
and we simplify this notation to when no confusion about the space arises.
We will define the phase space axiomatically, using ideas and notations developed in [19]. More precisely, will denote the vector space of functions defined from into endowed with a seminorm denoted , and such that the following axioms hold.

(A)
If , , is continuous on and , then for every the following conditions hold:

(i)
is in ,

(ii)
,

(iii)
,
where is a constant; , is continuous, is locally bounded, and are independent of .

(A1)
For the function in (A), the function is continuous from into .

(B)
The space is complete.
Example 2.1 (the phase space ).
Let , and let be a nonnegative measurable function which satisfies the conditions (g5), (g6) in the terminology of [19]. Briefly, this means that is locally integrable and there exists a nonnegative, locally bounded function on , such that , for all and , where is a set with Lebesgue measure zero. The space consists of all classes of functions , such that is continuous on , Lebesguemeasurable, and is Lebesgue integrable on . The seminorm in is defined by
The space satisfies axioms (A), (A1), (B). Moreover, when and , we can take , , and , for (see [19, Theorem ] for details).
For additional details concerning phase space we refer the reader to [19].
To obtain our results, we assume that the integrodifferential abstract Cauchy problem
has an associated resolvent operator of bounded linear operators on .
Definition 2.2.
A one parameter family of bounded linear operators on is called a resolvent operator of (2.3)(2.4) if the following conditions are verified.

(a)
The function is strongly continuous and for all and .

(b)
For , , and
for every .
The existence of a resolvent operator for problem (2.3)(2.4) was studied in [20]. In this paper, we have considered the following conditions.

(P1)
The operator is a closed linear operator with dense in . There is positive constants , such that
where , , for some , and for all .

(P2)
For all , is a closed linear operator, , and is strongly measurable on for each . There exists (the notation stands for the set of all locally integrable functions from into ) such that exists for and for all and . Moreover, the operatorvalued function has an analytical extension (still denoted by ) to such that for all , and , as .

(P3)
There exists a subspace dense in and positive constant , such that , , for every and all .
In the sequel, for and ,
for , , are the paths
and oriented counterclockwise. In addition, are the sets
We now define the operator family by
The following result has been established in [20, Theorem 2.1].
Theorem 2.3.
Assume that conditions (P1)–(P3) are fulfilled, then there is a unique resolvent operator for problem (2.3)(2.4).
In what follows, we always assume that the conditions (P1)–(P3) are verified.
We consider now the nonhomogeneous problem.
In the rest of this section, we discuss existence and regularity of solutions of
where and . In the sequel, is the operator function defined by (2.11). We begin by introducing the following concept of classical solution.
Definition 2.4.
A function , is called a classical solution of (2.12)(2.13) on if ; the condition (2.13) holds and (2.12) is verified on .
Definition 2.5.
Let , we define the family by
for each .
The proof of the next result is in [20]. For reader's convenience, we will give the proof.
Lemma 2.6.
If the function is exponentially bounded in , then is exponentially bounded in .
Proof.
If there are constants, such that , we obtain
where .
The next result follows from Lemma 2.6. We will omit the proof.
Lemma 2.7.
If the function is exponentially bounded in , then is exponentially bounded in .
We now establish a variation of constants formula for the solutions of (2.12)(2.13). The proof of the next result is in [20]. For reader's convenience, we will give the proof.
Theorem 2.8.
Let . Assume that and is a classical solution of (2.12)(2.13) on , then
Proof.
The Cauchy problem (2.12)(2.13) is equivalent to the Volterra equation
and the resolvent equation (2.6) is equivalent to
To prove (2.16), we notice that
Therefore,
We obtain
It is clear from the preceding definition that is a solution of problem (2.3)(2.4) on for .
Definition 2.9.
Let . A function is called a mild solution of (2.12)(2.13) if
The proof of the next result is in [20]. For reader's convenience, we will give the proof.
Theorem 2.10.
Let and . If , then the mild solution of (2.12)(2.13) is a classical solution.
Proof.
To begin with, we study the case in which . Let be the mild solution of (2.12)(2.13) and assume that . It is easy to see that and
where is given by . From [7, Lemma 3.12], we obtain that is a classical solution and satisfies
Moreover, from (2.24) and taking into account that for all and , we deduce the existence of constants , (which are independent from ) such that
Now, we assume that . Let be a sequence in such that in . From [7, Lemma 3.12], we know that , , is a classical solution of (2.12)(2.13) with in the place of . By using the estimate (2.25), we deduce the existence of functions , such that in and in . These facts, jointly with our assumptions on , permit to conclude that in . On the other hand,
we obtain
Now, by making on
we conclude that is a classical solution of (2.12)(2.13). The proof is finished.
The proof of the next result is in [20]. For reader's convenience, we will give the proof.
Theorem 2.11.
Let and . If , then the mild solution of (2.12)(2.13) is a classical solution.
Proof.
Let , there is on such that on and on . Put proceeding as in the proof of Theorem 2.10. It follows that is a classical solution of (2.12)(2.13). Moreover, from [7, Lemma 3.13], we obtain
from which we deduce the existence of positive constants (independent from ) such that
With similar arguments as in the proof of Theorem 2.10, we conclude that is a classical solution of (2.12)(2.13). We omit additional details. The proof is completed.
To establish our existence results, we need the following Lemma.
Lemma 2.12.
Let . If is compact for some , then and are compact for all .
Proof.
From the resolvent identity, it follows that is compact for every . We have from [20, Lemma 2.2] that is a compact operator for ; therefore, is a compact operator for .
From [20, Lemma 2.5], we have, is uniformly continuous for , for any fixed, we can select points , such that if , we obtain , for all and .
Therefore, for all , we have that
Noting now that
from (2.31), we find that
Thus,
where is compact and , then we observe that as . This permits us to conclude that is relatively compact in . This proves that is a compact operator for all .
For completeness, we include the following wellknown result.
Theorem 2.13 (LeraySchauder alternative, [21, Theorem ]).
Let be a closed convex subset of a Banach space with . Let be a completely continuous map. Then, has a fixed point in or the set is unbounded.