We study the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in a separable Banach space . Combining the techniques of operator semigroup, noncompactness measures, and the fixed point theory, we obtain new existence of mild solution without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the semigroup generated by is compact. An application of the abstract result is also given.

1. Introduction

In this paper, we denote that is a positive constant and assume that a family of closed linear operators satisfies the following.(A1) The domain of is dense in a Banach space and independent of . (A2) The operator exists in (the Banach space of all linear and bounded operators on ) for any with Re and (A3) There exist constants and such that

Under condition (A2), each operator , , generates an analytic semigroup , , and there exists a constant such that where , ([1]).

Moreover, , which follows from condition (A2), where and is a positive constant independent of both and .

This paper is concerned with existence result for nonautonomous fractional integrodifferential equations with nonlocal conditions in a separable Banach space : where , , is a family of closed linear operators in and satisfies (A1)–(A3), with and , are given functions to be specified later. The fractional derivative is understood here in the Riemann-Liouville sense.

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The field of the application of fractional calculus is very broad. We can see it in the study of the memorial materials, earthquake analysis, robots, electric fractal network, fractional sine oscillator, electrolysis chemical, fractional capacitance theory, electrode electrolyte interface description, fractal theory, especially in the dynamic process description of porous structure, fractional controller design, vibration control of viscoelastic system and pliable structure objects, fractional biological neurons, and probability theory. For details, see the monographs of Kilbas et al. [2], Kiryakova [3], Lakshmikantham and Vatsala [4], Miller and Ross [5], Samko et al. [6] and Podlubny [7], and the references therein. Some recent contributions to the theory of fractional differential equations can be seen in [820] and the references therein. Among the previous researches, most of researchers focus on the case that the differential operators (possibly unbounded) in the main parts are independent of time . However, when treating some parabolic evolution problems, it is usually assumed that the partial differential operators depend on time (i.e., it is the case of the problems under considerations being nonautonomous), since this class of operators appears frequently in the applications (see [21] and the references therein).

Moreover, since the work of Byszewski [22], the nonlocal Cauchy problems have been investigated in many papers (cf., e.g., [1315, 20, 2325] and the references therein). The nonlocal conditions give a better description in applications than standard ones, and the Cauchy problem with nonlocal initial condition can be applied in physics with better effect than the classical Cauchy problem with traditional initial conditions. The existence of mild solutions of nonautonomous fractional evolution equations with nonlocal conditions of the form (4) is an untreated original topic, which in fact is the main motivation of the present paper.

In this paper, using a pair of evolution families and associated with the semigroup , we give a reasonable concept of solution to problem (4) in Section 2. Moreover, in general, the semigroup generated by is not compact, so we obtain the main result based on the theory of measures of noncompactness and the condensing maps. These techniques are often used to deal with abstract integer order differential equations but rarely used in abstract fractional order differential equations(e.g., [820] and the references therein). We will study (4) under suitable hypotheses based on a special noncompactness measure and the properties of fixed points set of condensing operators [26, 27] and establish a new existence result for (4) without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the semigroup generated by is compact (see Theorem 14). As one can see, our result is obtained under assumptions weaker than those required previously in the similar literature. The result is new even for the case of (autonomous). Moreover, an example is given to show an application of the abstract result.

2. Preliminaries

Throughout this paper, we set , a compact interval in . We denote by a separable Banach space with norm , 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 .

We set

Next, we recall the definition of the Riemann-Liouville integral.

Definition 1 (see [6]). The fractional (arbitrary) order integral of the function of order is defined by where is the Gamma function. Moreover, , for all .

Remark 2. We have (1) [6].

(2) Obviously, for , it follows from Definition 1 that where is a beta function.

Definition 3 (see [7]). The Riemann-Liouville derivative of order with the lower limit zero for a function can be written as
Based on the work in [12], we give the following definition of the operator family .

Definition 4. Let be a probability density function defined on such that its Laplace transform is given by We define operator families by the semigroup associated with as follows:

By using the family , we denote and construct the family by

Lemma 5 (see [12]). The operator-valued functions and are continuous in uniform topology in the variables , , where , , for any . Clearly,

Moreover, we have

A mild solution of (4) can be defined as follows.

Definition 6. A function satisfying the equation is called a mild solution of (4), where

We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [26, 27]) which are used later in this paper.

Definition 7. 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 .

As an example of the MNC, we may consider the Hausdorff MNC:

We know that is monotone, nonsingular, invariant with respect to union with compact sets, algebraically semiadditive, and regular. This means that (i)for any with , , (ii)for any , , , (iii)for every relatively compact set , , , (iv)for each , , (v) is equivalent to the relative compactness of .

In Section 3, we will establish an existence result to the problem (4) using the following assertion about -estimates for a multivalued integral (Theorem 4.2.3 of [27]).

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

Proposition 8. For an integrable, integrably bounded multifunction where is a separable Banach space, let where . Then for all .

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

Definition 9. 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 application of the topological degree theory for condensing maps (see, e.g., [26, 27]) yields the following fixed point principle which will be used later.

Theorem 10. Let be a bounded convex closed subset of and a -condensing map. Then fix is nonempty.

3. Main Result

We need the hypotheses as follows.(H1)Function satisfies that is measurable for all and is continuous for a.e. , and there exists a function such that for almost all . (H2)There exists a function such that, for any bounded set , where . (H3)The function is completely continuous and there exists a positive constant such that

Define the operator as follows: It is clear that the operator is well defined. For some MNC , we will show that the operator is -condensing on every bounded subset of . To this end, we divided the proof into three propositions.

Proposition 11. The operator is continuous.

Proof. Let be a sequence such that   in as . Since satisfies (H1), for almost every and , we have For , we can prove that is continuous. In fact, noting (H1) and in , we know that there exists such that for sufficiently large. Therefore, we get In view of (15) and (16), we obtain
Therefore, the fact (by (A2)), (H3), and the Lebesgue dominated convergence theorem ensure that Therefore, we deduce that

Proposition 12. The operator transforms bounded sets into equicontinuous ones.

Proof. For any , we set . Let and . Then where For , It follows from Lemma 5, (19), and (H3) that as . For , from (15) and (19), Similarly, , as and . For , from (15) and (19), we have Similarly, , as .
So, the set is equicontinuous.

Proposition 13. The operator is -condensing.

Proof. Noting that, for any , we have so, we can take the appropriate such that
For every bounded subset , we consider the measure of noncompactness in the space with values in the cone of the following way: where is the module of equicontinuity of given by
Let be a nonempty, bounded set such that
For any , we set
We consider the multifunction , Obviously, is integrable, and from (16) and (H1) it follows that is integrably bounded. Moreover, noting (H2), we have the following estimate for a.e. : Applying Proposition 8, we have Set Then from (49), Furthermore
Therefore, combining with (40), we have
For any , we set and consider the multifunction : Obviously, is integrable, and from (15) and (H1) it follows that is integrably bounded. Moreover, for a.e. , we can obtain by the similar technique used in (49)–(53); then
Now, from (53) and (57), can be chosen so that where . Then from (45), we have .
Further, from Proposition 12 we know that and integrating with (45) one yields . Hence . The regularity property of implies the relative compactness of . Now, it follows from Definition 9 that is -condensing.

Theorem 14. Assume that (H1), (H2), and (H3) are satisfied; then problem (4) has at least one mild solution on the interval .

Proof. Let us introduce in the space the equivalent norm defined as 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 .
Combining with (H1)–(H3), Remark 2(2), and (15), (16) and (19), we have Therefore where .
Dividing both sides of (62) by and taking , we have This contradicts (42). Hence for some positive number , . From Proposition 13 it follows that is -condensing and we apply Theorem 10 to complete the proof.

4. Example

In this section, set , and we consider the following integrodifferential problem: where is the Riemann-Liouville fractional partial derivative of order , , , is a constant to be specified later. Also , are continuous functions and there exists a positive constant such that is a continuous function and is uniformly Hölder continuous in ; that is, there exist and such that Define by Then generates an analytic semigroup satisfying assumptions (A1)–(A3) ([28]).

For , , we set where and Now .

Then (64) can be reformulated as the abstract (4).

Moreover, where .

For , we have Therefore, for any bounded set , we have where . Now, Theorem 14 implies that the problem (64) has at least a mild solution.

5. Conclusion

This paper deals with the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in an abstract space. Sufficient conditions for the existence of mild solution are derived with the help of the fixed point theorem for condensing maps. An example is provided to illustrate the obtained result.


The author is grateful to the referees for their valuable suggestions. This work was partly supported by the NSF of China (11201413), the NSF of Yunnan Province (2009ZC054M, 2013FB034), the Educational Commission of Yunnan Province (2012Z010), and the Foundation of Key Program of Yunnan Normal University.