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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 784816, 12 pages
http://dx.doi.org/10.1155/2013/784816
Research Article

Nonlocal Problem for Fractional Evolution Equations of Mixed Type with the Measure of Noncompactness

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 8 December 2012; Revised 14 March 2013; Accepted 18 March 2013

Academic Editor: Xinan Hao

Copyright © 2013 Pengyu Chen and Yongxiang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A general class of semilinear fractional evolution equations of mixed type with nonlocal conditions on infinite dimensional Banach spaces is concerned. Under more general conditions, the existence of mild solutions and positive mild solutions is obtained by utilizing a new estimation technique of the measure of noncompactness and a new fixed point theorem with respect to convex-power condensing operator.

1. Introduction

In this paper, we use a new estimation technique of the measure of noncompactness and fixed point theorem with respect to convex-power condensing operator to discuss the existence of mild solutions and positive mild solutions for nonlocal problem of fractional evolution equations (NPFEE) of mixed type with noncompact semigroup in Banach space : where is the Caputo fractional derivative of order ; , is a closed linear operator and generates a uniformly bounded -semigroup in , is a Carathéodory type function, , is a constant, mapping from some space of functions to be specified later, and are integral operators with integral kernels , , and , .

In recent years, fractional calculus has attracted many physicists, mathematicians, and engineers, and notable contributions have been made to both theory and applications of fractional differential equations. It has been found that the differential equations involving fractional derivatives in time are more realistic to describe many phenomena in practical cases than those of integer order in time. For more details about fractional calculus and fractional differential equations we refer to the books by Miller and Ross [1], Podlubny [2], and Kilbas et al. [3] and the papers by Eidelman and Kochubei [4], Lakshmikantham and Vatsala [5], Agarwal et al. [6], Darwish and Ntouyas [710], and Darwish et al. [11]. One of the branches of fractional calculus is the theory of fractional evolution equations. Since fractional order semilinear evolution equations are abstract formulations for many problems arising in engineering and physics, fractional evolution equations have attracted increasing attention in recent years; see [1226] and the references therein.

The study of abstract nonlocal Cauchy problem was initiated by Byszewski and Lakshmikantham [27]. Since it is demonstrated that the nonlocal problems have better effects in applications than the traditional Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained; see [1732] and the references therein for more comments and citations. In the past few years, the existence, uniqueness, and some other properties of mild solutions to nonlocal problem of fractional evolution equations (1) with have been extensively studied by using Banach contraction mapping principal, Schauder’s fixed point theorem and Krasnoselskii’s fixed point theorem, when is a compact semigroup. For more details on the basic theory of nonlocal problem for fractional evolution equations, one can see the papers of Diagana et al. [17], Wang et al. [18], Li et al. [19], Zhou and Jiao [20], Wang et al. [21], Wang et al. [22], Wang et al. [23], Chang et al. [24], Balachandran and Park [25], and Balachandran and Trujillo [26]. However, for the case that the semigroup is noncompact, there are very few papers studied nonlocal problem of fractional evolution equations; that only Wang et al. [14] discussed the existence of mild solutions for nonlocal problem of fractional evolution equations under the situation that is an analytic semigroup of uniformly bounded linear operators.

It is well known that the famous Sadovskii's fixed point theorem is an important tool to study various differential equations and integral equations on infinite dimensional Banach spaces. Early on, Lakshmikantham and Leela [33] studied the following initial value problem (IVP) of ordinary differential equation in Banach space : and they proved that if for any , is uniformly continuous on and satisfies the noncompactness measure condition where , is a positive constant, denotes the Kuratowski measure of noncompactness in , then IVP (3) has a global solution provided that satisfies the condition In fact, there are a large amount of authors who studied ordinary differential equations in Banach spaces similar to (3) by using Sadovskii's fixed point theorem and hypothesis analogous to (4); they also required that the constants satisfy a strong inequality similar to (5). For more details on this fact, we refer to Guo [34], Liu et al. [35] and Liu et al. [36].

It is easy to see that the inequality (5) is a strong restrictive condition, and it is difficult to be satisfied in applications. In order to remove the strong restriction on the constant , Sun and Zhang [37] generalized the definition of condensing operator to convex-power condensing operator. And based on the definition of this new kind of operator, they established a new fixed point theorem with respect to convex-power condensing operator which generalizes the famous Schauder's fixed point theorem and Sadovskii's fixed point theorem. As an application, they investigated the existence of global mild solutions and positive mild solutions for the initial value problem of evolution equations in : they assume that generate a equicontinuous -semigroup; the nonlinear term is uniformly continuous on and satisfies a suitable noncompactness measure condition similar to (4). Recently, Shi et al. [38] developed the IVP (6) to the case that the nonlinear term is and obtained the existence of global mild solutions and positive mild solutions by using the new fixed point theorem with respect to convex-power condensing operator established in [37], but they also require that the nonlinear term is uniformly continuous on .

We observed that, in [3338], the authors all demand that the nonlinear term is uniformly continuous; this is a very strong assumption. As a matter of fact, if is Lipschitz continuous on with respect to , then the condition (4) is satisfied, but may not be necessarily uniformly continuous on .

Motivated by the above mentioned aspects, in this paper we studied the existence of mild solutions and positive mild solutions for the NPFEE (1) by utilizing a new fixed point theorem with respect to convex-power condensing operator due to Sun and Zhang [37] (see Lemma 8). Furthermore, we deleted the assumption that is uniformly continuous by using a new estimation technique of the measure of noncompactness (see Lemma 7).

2. Preliminaries

In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper.

Let be a real Banach space with the norm . We denote by the Banach space of all continuous -value functions on interval with the supnorm , and by the Banach space of all -value Bochner integrable functions defined on with the norm .

Definition 1 (see [3]). The fractional integral of order with the lower limit zero for a function is defined as where is the Euler gamma function.

Definition 2 (see [3]). The Caputo fractional derivative of order with the lower limit zero for a function is defined as where the function has absolutely continuous derivatives up to order .

If is an abstract function with values in , then the integrals which appear in Definitions 1 and 2 are taken in Bochner's sense.

For , define two operators and by where is the function of Wright type defined on which satisfies

Let , where stands for the Banach space of all linear and bounded operators in . The following lemma follows from the results in [12, 13, 20].

Lemma 3. The operators and have the following properties.(1)For any fixed , and are linear and bounded operators; that is, for any , (2)For every , and are continuous functions from into .(3)The operators and are strongly continuous, which means that, for all and , one has

Definition 4. A function is said to be a mild solution of the NPFEE (1) if it satisfies

Next, we recall some properties of the measure of noncompactness that will be used in the proof of our main results. Since no confusion may occur, we denote by the Kuratowski measure of noncompactness on both the bounded sets of and . For the details of the definition and properties of the measure of noncompactness, we refer to the monographs [39, 40]. For any and , set . If is bounded, then is bounded in and .

Lemma 5 (see [39]). Let be a Banach space; let be bounded and equicontinuous. Then is continuous on , and

Lemma 6 (see [41]). Let be a Banach space; let be a bounded and countable set. Then is Lebesgue integral on , and

Lemma 7 (see [32, 42]). Let be a Banach space; let be bounded. Then there exists a countable set , such that .

Lemma 8 (fixed point theorem with respect to convex-power condensing operator (see [37])). Let be a Banach space; let be bounded, closed, and convex. Suppose is a continuous operator and is bounded. For any and , set If there exist and positive integer such that for any bounded and nonprecompact subset , then has at least one fixed point in .

Lemma 9. For and , one has

3. Existence of Mild Solutions

In this section, we discuss the existence of mild solutions for the NPFEE (1). We first make the following hypotheses. (H1) The function satisfies the Carathéodory type conditions; that is, is strongly measurable for all , and is continuous for a.e. .(H2) For some , there exist constants , and functions such that for a.e. and all satisfying , (H3) There exist positive constants such that for any bounded and countable sets and a.e. : (H4) The nonlocal function is continuous and compact, and there exist a constant and a nondecreasing continuous function such that, for some and all ,

Theorem 10. Let be a real Banach space; let be a closed linear operator, and generate an equicontinuous -semigroup of uniformly bounded operators in . Assume that the hypotheses (H1), (H2), (H3) and (H4) are satisfied, then the NPFEE (1) have at least one mild solution in provided that

Proof. Consider the operator defined by By direct calculation, we know that is well defined. From Definition 4, it is easy to see that the mild solution of the NPFEE (1) is equivalent to the fixed point of the operator . In the following, we will prove has a fixed point by applying the fixed point theorem with respect to convex-power condensing operator.
Firstly, we prove that there exists a positive constant , such that . If this is not true, then for each , there would exist and such that . It follows from Lemma 3(1), the hypotheses (H2) and (H4) and Hölder inequality that Dividing both sides of (25) by and taking the lower limit as , we get which contradicts (23).
Secondly, we prove that is continuous in . To this end, let be a sequence such that in . By the continuity of and the Carathéodory continuity of , we have From the hypothesis (H2), we get for each Using the fact that the function is Lebesgue integrable for , by (27) and (28) and the Lebesgue dominated convergence theorem, we get that Therefore, we know that which means that is continuous.
Now, we demonstrate that the operator is equicontinuous. For any and , we get that where It is obvious that Therefore, we only need to check tend to independently of when .
For , by Lemma 3(3), as .
For , by the hypothesis (H2), Lemma 3(1), and Hölder inequality, we have For , by the hypothesis (H2), Lemmas 3(1), and 9 and Hölder inequality, we get that For , , it is easy to see that . For and small enough, by the hypothesis (H2), Lemma 3 and the equicontinuity of , we know that As a result, tends to zero independently of as , which means that is equicontinuous.
Let , where means the closure of convex hull. Then it is easy to verify that maps into itself and is equicontinuous. Now, we are in the position to prove that is a convex-power condensing operator. Take ; we will prove that there exists a positive integer such that for any bounded and nonprecompact subset For any and , by (17) and the equicontinuity of , we get that is also equicontinuous. Therefore, we know from Lemma 5 that By Lemma 7, there exists a countable set , such that Thanks to the fact that we have where , . Therefore, by (24), (39), and (41), Lemma 6 and the hypotheses (H3) and (H4), we get that
Again by Lemma 7, there exists a countable set , such that Therefore, by (24), (41), and (43), Lemma 6, and the hypotheses (H3) and (H4), we have that where is the Beta function.
Suppose that Then by Lemma 7, there exists a countable set , such that From (24), (41), and (46), using Lemma 6, and the hypotheses (H3) and (H4), we get that Hence, by the method of mathematical induction, for any positive integer and , we have Consequently, from (38) and (48), we have By the well-known Stirling's formula, we know that Thus, Therefore, there exists a large enough positive integer such that Hence, we get that Thus, is a convex-power condensing operator. It follows from Lemma 8 that has at least one fixed point , which is just a mild solution of the NPFEE (1). This completes the proof of Theorem 10.

If we replace the hypotheses (H2) and (H4) by the following hypotheses: there exist a function , and a nondecreasing continuous function such that for all and a.e. ; the nonlocal function is continuous and compact; there exist constants and such that for all , ;

we have the following existence result.

Theorem 11. Let be a real Banach space; let be a closed linear operator and generate an equicontinuous -semigroup of uniformly bounded operators in . Assume that the hypotheses (H1), (H2)′, (H3), and (H4)′ are satisfied, then the NPFEE (1) has at least one mild solution in provided that there exists a constant with

Proof. From the proof of Theorem 10, we know that the mild solution of the NPFEE (1) is equivalent to the fixed point of the operator defined by (24). For any , by (24), (55), and the hypotheses and , we have which implies . By adopting completely similar method with the proof of Theorem 10, we can prove that the NPFEE (1) have at least one mild solution in . This completes the proof of Theorem 11.

Similar with Theorem 11, we have the following result.

Corollary 12. Let be a real Banach space; let be a closed linear operator and generate an equicontinuous -semigroup of uniformly bounded operators in . Assume that the hypotheses (H1), (H2)′, (H3) and (H4)′ are satisfied, then the NPFEE (1) have at least one mild solution in provided that

4. Existence of Positive Mild Solutions

In this section, we discuss the existence of positive mild solutions for the NPFEE (1). we first introduce some notations and definitions which will be used in this section.

Let be an ordered Banach space with the norm and let be a positive cone in which defines a partial ordering in by if and only if , where denotes the zero element of . is said to be normal if there exists a positive constant such that implies ; the infimum of all with the property above is called the normal constant of . For more definitions and details of the cone , we refer to the monographs [40, 43].

Definition 13. A -semigroup in is called to be positive, if order inequality holds for each , and .

For more details of the properties of positive -semigroup, we refer to the monograph [44] and the paper [45].

Lemma 14. If is a positive -semigroup in , then and are also positive operators.

Proof. Since the semigroup and the function defined by (10) are positive, by (9) we can easily conclude that the operators and are also positive. This completes the proof.

Here, we will obtain positive mild solutions under the following assumptions.(A1) The function satisfies the Carathéodory type conditions.(A2) There exist a constant and a function such that, for all and a.e. , .(A3) There exist positive constants such that for any bounded and countable sets and a.e. , (A4) The nonlocal function is continuous and compact, and there exist constants and such that for all , , where is the normal constant of the positive cone .

Theorem 15. Let be an ordered Banach space, be a normal positive cone with normal constant , be a closed linear operator, and generate a positive and equicontinuous -semigroup of uniformly bounded operators in . Assume that the assumptions (A1), (A2), (A3), and (A4) are satisfied, then the NPFEE (1) have at least one positive mild solution in .

Proof. From the proof of Theorem 10, we know that the positive mild solution of the NPFEE (1) is equivalent to the fixed point of operator defined by (24) in . We choose big enough such that Then for any , by (24) and the assumptions (A2) and (A4), we have that By the normality of the cone , (59), (60), the assumption (A4), and Hölder inequality, we get Thus, . Let . Similar to the proof of Theorem 10, we can prove that is a convex-power condensing operator. It follows from Lemma 8 that has at least one fixed point , which is just a positive mild solution of the NPFEE (1). This completes the proof of Theorem 15.

Theorem 16. Let be an ordered Banach space, be a normal positive cone with normal constant , be a closed linear operator and generate a positive and equicontinuous -semigroup of uniformly bounded operators in . Assume that the assumptions (A1), (A3) and the following assumptions: There exist nonnegative continuous functions , and abstract continuous function such that The nonlocal function is continuous and compact, and there exists a constant such that, for any , ,are satisfied, then the NPFEE (1) have at least one positive mild solution in .

Proof. From the proof of Theorem 10, we know that the positive mild solution of the NPFEE (1) is equivalent to the fixed point of operator defined by (24) in . Let We can prove that , where denotes the spectral radius of bounded linear operator. In fact, for any , by (63), we get that where , . Further, By the method of mathematical induction, for any positive integer and , we obtain that Therefore, we have Thus, combining (50) we get that Let . From [46] we know that there exists an equivalent norm in such that where denotes the operator norm of with respect to the norm .
Let and . Choose For any , by (24) and the assumptions and , we have that By the normality of the cone , (69), (70), and (71), we have Thus, we have proved that . Let . Similar to the proof of Theorem 10, we can prove that is a convex-power condensing operator. It follows from Lemma 8 that has at least one fixed point , which is just a positive mild solution of the NPFEE (1). This completes the proof of Theorem 16.

Remark 17. Positive operator semigroups are widely appearing in heat conduction equations, neutron transport equations, reaction diffusion equations, and so on [47]. Therefore, using Theorems 15 and 16 to these partial differential equations are very convenient.

Remark 18. Analytic semigroup and differentiable semigroup are equicontinuous semigroup [48]. In the application of partial differential equations, such as parabolic equations and strongly damped wave equations, the corresponding solution semigroup is analytic semigroup. Therefore, the results obtained in this paper have broad applicability.

Acknowledgments

This research was supported by NNSF of China (11261053 and 11061031), NSF of Gansu Province (1208RJZA129), and Project of NWNU-LKQN-11-3.

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