This paper is concerned with the existence of mild solutions for the fractional integrodifferential equations with finite delay and almost sectorial operators in a separable Banach space . We obtain existence theorem for mild solutions to the above-mentioned equations, by means of measure of noncompactness and the resolvent operators associated with almost sectorial operators. As an application, the existence of mild solutions for some integrodifferential equation is obtained.

1. Introduction

Fractional differential and integrodifferential equations have received increasing attention during recent years and have been studied extensively (see, e.g., [115] and references therein) since they are playing an increasingly important role in engineering, physics, electrolysis chemical, fractional biological neurons, condensate physics, statistical mechanics, and so on.

Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (see, e.g., [3, 4, 9, 10, 14, 1620] and references therein).

We mention that much of the previous research on the fractional equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, a compact semigroup, or an analytic semigroup, or is a Hille-Yosida operator (see, e.g., [14, 810, 1214]). However, as presented in [15, Examples 1.1 and 1.2], for which the resolvent operators do not satisfy the required estimate to be a sectorial operator. Von Wahl in [21] first introduced examples of almost sectorial operators which are not sectorial. Recently, the study of evolution equations involving almost sectorial operators has been investigated extensively. However, much less is known about the fractional evolution equations with almost sectorial operators (see [15] and the references therein).

In this paper, we are concerned with the following fractional integrodifferential equations: where . The fractional derivative is understood here in the Caputo sense. is a separable Banach space. is an almost sectorial operator to be introduced later. Here (, where denotes the space of all continuous functions from to .

For any continuous function defined on and any , we denote by the element of defined by .

Our paper is organized as follows. In Section 2, we give out some preliminaries about fractional order operator, measure of noncompactness, and almost sectorial operators. The existence result will be established in Section 3. In Section 4, an example is given to show the application of the abstract result.

2. Preliminaries

Throughout this paper, we denote by a separable Banach space with norm . For a linear operator , we denote by the domain of , by the resolvent set of , and by , the resolvent of . Moreover, we denote by the Banach space of all linear and bounded operators on and by the space of all -valued continuous functions on with the supremum norm as follows: Moreover, we abbreviate with for any .

Let us recall the following known definitions. For more details, see [7, 11].

Definition 2.1 (see [11]). The fractional integral of order with the lower limit zero for a function is defined as provided that the right side is pointwise defined on , where is the gamma function.

Definition 2.2 (see [11]). The Riemann-Liouville derivative of order with the lower limit zero for a function can be written as

Definition 2.3 (see [11]). The Caputo derivative of order for a function can be written as

Remark 2.4. (1) If , then
(2) the Caputo derivative of a constant is equal to zero.

We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [22, 23]).

Definition 2.5. Let be a Banach space, the family of all nonempty subsets of , a partially ordered set and . If for every : then we say that is a measure of noncompactness (MNC) in .
A MNC is called: (i) monotone if , implies ; (ii) nonsingular if for every ; (iii) invariant with respect to union with compact sets if for every relatively compact set , . If is a cone in a normed space, we say that the MNC is; (iv) algebraically semiadditive if for each ; (v) regular if is equivalent to the relative compactness of ; (vi) real if is with the natural order.

As an example of the MNC possessing all these properties, we may consider the Hausdorff MNC:

Now, let be a multifunction. It is called: (i) integrable, if it admits a Bochner integrable selection for a.e. ; (ii) integrably bounded, if there exists a function such that

We present the following assertion about -estimates for a multivalued integral [23, Theorem ].

Proposition 2.6. For an integrable, integrably bounded multifunction , where is a separable Banach space, let where . Then, for all .

Let be a Banach space, a monotone nonsingular MNC in .

Definition 2.7. A continuous map is called condensing with respect to a MNC (or -condensing) if for every bounded set which is not relatively compact, we have
The following fixed point principle (see, e.g., [22, 23]) will be used later.

Theorem 2.8. Let be a bounded convex closed subset of and a -condensing map. Then, is nonempty.

Theorem 2.9. Let be a bounded open neighborhood of zero and a -condensing map satisfying the boundary condition: for all and . Then, is a nonempty compact set.

To prove the main result, we will need the following generalization of Gronwall's lemma for singular kernels [24, Lemma ].

Lemma 2.10. Let be continuous functions. If is nondecreasing and there are constants and such that then there exists a constant such that

Next, we recall the knowledge of almost sectorial operator, for more details, we refer to [25, 26].

Let and with be the open sector: and its closure, that is,

Let us recall the following definition.

Definition 2.11. Let and . By , we denote the family of all linear closed operators which satisfy:(1);(2) for every , there exists a constant such that

A linear operator will be called an almost sectorial operator on if .

Remark 2.12. Let , then the definition implies that .

Remark 2.13 (see [15]). From [26], note in particular that if , then generates a semigroup with a singular behavior at in a sense, called semigroup of growth . Moreover, the semigroup is analytic in an open sector of the complex plane , but the strong continuity fails at for data which are not sufficiently smooth.

We denote the semigroup associated with by . For , forms an analytic semigroup of growth order , where , the integral contour is oriented counter-clockwise ([15, 26]).

We have the following proposition on [26, Theorem 3.9].

Proposition 2.14. Let with and . Then the following properties remain true:(i) is analytic in and (ii) the functional equation for all holds;(iii) there exists a constant such that (iv) if , then .

Clearly, we note that the condition (ii) of the Proposition 2.14 does not satisfy for or .

The relation between the resolvent operators of and the semigroup is characterized by.

Proposition 2.15 (see [26, Theorem 3.13]). Let with and . Then for every with , one has

Based on the work in [15], we define operator families and by where with is a function of Wright-type (cf. e.g., [15]) as follows:

We collect some basic properties on . For more details, we refer to ([7, 11, 15, 25]).

Proposition 2.16. For , , the following results hold: (1), ; (2); (3).

Theorem 2.17 (see [15, Theorem ]). For each fixed , and are linear and bounded operators on . Moreover, for all , , ,

Theorem 2.18 (see [15, Theorem 3.2]). For , and are continuous in the uniform operator topology. Moreover, for every , the continuity is uniform on .

Remark 2.19 (see [15, Theorem 3.4]). Let . Then for all ,

Next, we will present the definition of mild solution of problem (1.1).

According to Definitions 2.12.3, we can rewrite problem (1.1) in the equivalent integral equation: provided that the integral in (2.25) exists, where

Set formally applying the Laplace transform to (2.25), we get then thus provided that the integral in (2.30) exists.

Then, using Proposition 2.16, we have Similarly, we have Thus, from (2.30)–(2.32), we obtain

We invert the last Laplace transform to obtain

Then from the above induction, when with , we can give the following definition of the mild solution of (1.1).

Definition 2.20. A continuous function satisfying the equation: is called a mild solution of (1.1).

Remark 2.21 (see [15], Remark 4.1). (1) In general, since the operator is singular at , solutions to problem (1.1) are assumed to have the same kind of singularity at as the operator .
(2) When with , it follows from Remark 2.19 that the mild solution is continuous at .

3. Main Result

Throughout this section, let be an operator in the class and . Moreover, we require the following assumptions: (Hf) (1) satisfies is measurable for all and is continuous for a.e. , and there exists a positive function () such that for almost all ; (2) there exists a nondecreasing function such that for any bounded set : (Hg) (1) and is continuous for a.e. , and for each , the function is measurable. Moreover, there exists a function with such that for almost all ; (2) for any bounded set , there exists a function such that where .

Theorem 3.1. Let with , . Assume that (Hf) and (Hg) are satisfied. Then for every with , the mild solution set of problem (1.1) is a nonempty compact subset of the space , provided that where and .

Proof. We define the operator in the following way: It is clear that the operator is well defined.
We define Let . It is easy to see that satisfies and if and only if satisfies and .
Let be an operator defined by for and
Clearly the operator has a fixed point is equivalent to having one.
We define . Next we will prove that has a fixed point on .
Let be a sequence such that in as . Since satisfies (Hf)(1) and satisfies (Hg)(1), for almost every and , we get
Noting that Therefore, and
Since in , it follows that there exists such that for sufficiently large. Moreover, noting that , we have
Similarly, It follows from the Lebesgue's Dominated Convergence Theorem that and from (2.23), we have Therefore, we obtain that then we see that is continuous.
Let us consider the MNC in the space with values in the cone of the following way: for every bounded subset , where is the module of equicontinuity of given by where is a constant chosen so that Noting that for any , we have so, we can take the appropriate to satisfy (3.22) and (3.23).
Next, we show that the operator is -condensing on every bounded subset of .
Let be a nonempty, bounded set for which Noting that and (3.25), we can see that .
Next, we estimate . For any , we set We consider the multifunction : Obviously, is integrable, and from (2.23), (Hf)(1), and (3.13), it follows that is integrably bounded. Moreover, noting that (Hf)(2), we have the following estimate for a.e. : Applying Proposition 2.6, we have Therefore, from (3.22), we have
Similarly, if we set then we can see that the multifunction , is integrable, and from (2.23), (Hg)(1), and (3.14), it follows that is integrably bounded. Moreover, noting that (Hg)(2), Proposition 2.6, and (3.23), we have the following estimate for a.e. :
Now, from (3.31) and (3.34), can be chosen so that where .
From (3.25), we have . Next, we will prove that .
Let , such that and , noting that (Hf)(1) and (Hg)(1), we obtain
Using (2.23), (3.13), and (3.14), we have Clearly, tends to zero as . Similarly, for , we have
For ,  for small enough, noting that (2.23), (3.13), and (3.14), we have it follows from Theorem 2.18 that tends to zero as and .
For the case when , we can see
Thus, the set is equicontinuous, then . From (3.25), we get that . Hence .
The regularity property of implies the relative compactness of . Now, it follows from Definition 2.7 that is -condensing.
Consider the set Next, we show that there exists some such that . Suppose on the contrary that for each there exist and some such that .
Noting the Hölder inequality, we have
By (2.23), (Hf)(1), (Hg)(1), and (3.42), we have where .
Then, Dividing both sides of (3.44) by , and taking , we have This contradicts (3.5). Hence for some positive number , . According to Theorem 2.8, problem (1.1) has at least one mild solution.
Next, for , we consider the following one-parameter family of maps: We will demonstrate that the fixed point set of the family : is a priori bounded. In fact, let , for , we have where
We denote that . Let such that . Then, by (3.48), we can see By Lemma 2.10, there exists a constant such that Hence, .
Now, we consider a closed ball as follows: We take the radius large enough to contain the set inside itself. Moreover, from the proof above-mentioned, is -condensing, and it remains to apply Theorem 2.9.

4. Application

Example 4.1. Let