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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 916543, 19 pages
http://dx.doi.org/10.1155/2012/916543
Research Article

On Nonlinear Neutral Fractional Integrodifferential Inclusions with Infinite Delay

1School of Mathematics, Yunnan Normal University, Kunming 650092, China
2Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 12 February 2012; Accepted 25 February 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Fang Li et al. 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

Of concern is a class of nonlinear neutral fractional integrodifferential inclusions with infinite delay in Banach spaces. A theorem about the existence of mild solutions to the fractional integrodifferential inclusions is obtained based on Martelli’s fixed point theorem. An example is given to illustrate the existence result.

1. Introduction

As have been seen, the field of the application of fractional calculus is very broad. For instance, 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. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional-order. The main feature of fractional order differential equation is containing the noninteger order derivative. It can effectively describe the memory and transmissibility of many natural phenomena. These differential equations have been studied by many researchers (cf., e.g., [111] and references therein).

As an generalization of differential equations, differential inclusions have also been investigated (cf., e.g., [1, 7, 12, 13] and references therein). Moreover, equations with delay are often more useful to describe concrete systems than those without delay. So the study of these equations has been attracted so much attention (cf., e.g., [1, 4, 8, 12, 1421] and references therein).

In this paper, we pay our attention to the investigation of the existence of mild solutions to the following fractional integrodifferential inclusions of neutral type with infinite delay in a Banach space 𝑋: 𝐷𝑞𝑥(𝑡)𝑔𝑡,𝑥𝑡𝐴𝑥(𝑡)𝑔𝑡,𝑥𝑡+𝑡0𝐾(𝑡,𝑠)𝐹𝑠,𝑥(𝑠),𝑥𝑠[],𝑥𝑑𝑠,𝑡0,𝑇0=𝜙𝒫,(1.1) where 0<𝑞<1, the fractional derivative is understood in the Caputo sense ([2], see Definition 2.3 in Section 2),𝒫 is an admissible phase space, 𝑥𝑡(,0]𝑋 defined by𝑥𝑡],(𝜃)=𝑥(𝑡+𝜃),for𝜃(,0(1.2)𝑇>0, 𝑔[0,𝑇]×𝒫𝑋, 𝐴 generates a compact and uniformly bounded semigroup 𝑆() on 𝑋 which implies that there exists 𝑀1 such that𝑆(𝑡)𝑀,𝑡0,(1.3)𝐾[0,𝑇]×[0,𝑇]𝐑, 𝜙 belongs to 𝒫 with𝜙(0)=0,(1.4) and 𝐹 is a multivalued map to be specified later.

2. Preliminaries

Throughout this paper, 𝑋 is a Banach space with norm , 𝐿(𝑋) is the Banach space of all linear continuous operators on 𝑋, 𝐽=[0,𝑇], and 𝐶(𝐽,𝑋) (𝐶([0,),𝑋)) is the space of all 𝑋-valued continuous functions on 𝐽([0,)).

Moreover, we abbreviate 𝑢𝐿1(𝐽,𝐑+)as 𝑢𝐿1, for any 𝑢𝐿1(𝐽,𝐑+).

We use the notation 𝔅(𝑋) to denote the family of all nonempty subsets of 𝑋. Let 𝔅bd(𝑋),𝔅cl(𝑋),𝔅cp(𝑋),𝔅cv(𝑋),and𝔅cp,cv(𝑋) denote, respectively, the family of all nonempty bounded, closed, compact, convex, and compact-convex subsets of 𝑋.

See the following definition about admissible phase space according to [8, 1421].

Definition 2.1. A linear space 𝒫 consisting of functions from 𝐑 into 𝑋 with norm 𝒫 is called an admissible phase space if 𝒫 has the following properties.(H1)For any 𝑡0𝑅 and 𝑎>0, if 𝑥(,𝑡0+𝑎]𝑋 is continuous on [𝑡0,𝑡0+𝑎] and 𝑥𝑡0𝒫, then 𝑥𝑡𝒫, 𝑥𝑡 is continuous in 𝑡[𝑡0,𝑡0+𝑎], and𝑥(𝑡)𝐶𝑥𝑡𝒫,(2.1) for a positive constant 𝐶. (H2)There exists a continuous function 𝐶1(𝑡)>0 and a locally bounded function 𝐶2(𝑡)0 in 𝑡0 such that𝑥𝑡𝒫𝐶1𝑡𝑡0max𝑠[𝑡0,𝑡]𝑥(𝑠)+𝐶2𝑡𝑡0𝑥𝑡0𝒫(2.2)     for 𝑡[𝑡0,𝑡0+𝑎] and 𝑥 as in (H1).(H3)The space (𝒫,𝒫) is complete.

Remark 2.2. (H1) is equivalent to that for any 𝑡0𝑅 and 𝑎>0, if 𝑥(,𝑡0+𝑎]𝑋 is continuous on [𝑡0,𝑡0+𝑎] and 𝑥𝑡0𝒫, then 𝑥𝑡𝒫, 𝑥𝑡 is continuous in 𝑡[𝑡0,𝑡0+𝑎], and 𝜙(0)𝐶𝜙𝒫,𝜙𝒫(2.3) for a positive constant 𝐶.

Now we recall some very basic concepts in the fractional calculus theory. For more details see, for example, [2, 9, 11].

We set for 𝛽0, 1𝑔{𝛽}(𝑡)=Γ𝑡(𝛽)𝛽1,𝑡>0,0,𝑡0,(2.4) and 𝑔0(𝑡)=0, where Γ() is the Gamma function.

Definition 2.3. Let 𝑓𝐿1(0,;𝑋) and 𝛼0. Then the expression 𝐼𝛼1𝑓(𝑡)=(𝑔{𝛼}𝑓)(𝑡)=Γ(𝛼)𝑡0(𝑡𝑠)𝛼1𝑓(𝑠)𝑑𝑠,𝑡>0,𝛼>0(2.5) with 𝐼0𝑓(𝑡)=𝑓(𝑡) is called Riemann-Liouville integral of order 𝛼 of 𝑓.

Definition 2.4. Let 𝑓(𝑡)𝐶𝑚1([0,);𝑋), 𝑔{𝑚𝛼}𝑓𝑊𝑚,1(𝐼,𝑋) (𝑚,0𝑚1<𝛼<𝑚). The Caputo fractional derivative of order 𝛼 of 𝑓 is defined by 𝐷𝛼𝑓(𝑡)=𝐷𝑚𝐼𝑚𝛼𝑓(𝑡)𝑚1𝑖=0𝑓(𝑖)(0)𝑔𝑖+1,(𝑡)(2.6) where 𝐷𝑚=𝑑𝑚/𝑑𝑡𝑚.

The following concepts are also very basic, which will be used later.

A multivalued map 𝐺𝑋𝔅(𝑋) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥𝑋. 𝐺 is bounded on bounded sets if𝐺(𝐵)=𝑥𝐵𝐺(𝑥)(2.7) is bounded in 𝑋 for all 𝐵𝔅bd(𝑋), that is,sup𝑥𝐵{sup{𝑦𝑦𝐺(𝑥)}}<.(2.8)

A multivalued map 𝐺𝐽𝔅cl(𝑋)is said to be measurable if for each 𝑥𝑋 the function 𝑌𝐽𝐑 defined by𝑌(𝑡)=𝑑(𝑥,𝐺(𝑡))=inf{𝑥𝑧𝑧𝐺(𝑡)}(2.9) is measurable.

If for each 𝑥𝑋, the set 𝐺(𝑥) is a nonempty, closed subset of 𝑋, and for each open set 𝐵 of 𝑋 containing 𝐺(𝑥), there exists an open neighborhood 𝑉 of 𝑥 such that 𝐺(𝑉)𝐵, then 𝐺 is called upper semicontinuous (u.s.c.) on 𝑋.

If for every 𝐵𝔅bd(𝑋), 𝐺(𝐵) is relatively compact, then 𝐺 is said to be completely continuous.

If the multivalued map 𝐺 is completely continuous with nonempty compact values, then 𝐺 is u.s.c. if and only if 𝐺 has a closed graph, that is,𝑥𝑛𝑥,𝑦𝑛𝑦,𝑦𝑛𝑥𝐺𝑛imply𝑦𝑥𝐺.(2.10) We say that 𝐺 has a fixed point if there is some 𝑥𝑋 such that 𝑥𝐺(𝑥).

For more details on multivalued maps we refer to the book by Deimling [22].

The following is the multivalued version of the fixed-point theorem due to Martelli [23].

Lemma 2.5. Let 𝑋 be a Banach space and let 𝑁𝑋𝔅cp,cv(𝑋) be an upper semicontinuous is bounded; then 𝑁 has a fixed point and completely continuous multivalued map. If the set Ω={𝑦𝑋𝜆𝑦𝑁𝑦forsome𝜆>1}(2.11) is bounded, then 𝑁 has a fixed point.

Following Liang and Xiao [14, 15], let 𝒫[0,𝑇] be the set defined by𝒫[0,𝑇]=]𝑥(,𝑇𝑋𝑥|𝐽𝐶(𝐽,𝑋),𝑥0.𝒫(2.12)

Let 𝑇 be the norm of 𝒫[0,𝑇] defined by𝑦𝑇=𝑦0𝒫+max{𝑦(𝑠)0𝑠𝑇},𝑦𝒫[0,𝑇].(2.13)

Based on the work in [7, 11], we set𝑄(𝑡)=0𝜉𝑞(𝜎)𝑆(𝑡𝑞𝜎)𝑑𝜎,𝑅(𝑡)=𝑞0𝜎𝑡𝑞1𝜉𝑞(𝜎)𝑆(𝑡𝑞𝜎)𝑑𝜎,(2.14) and 𝜉𝑞 is a probability density function defined on (0,)(see [7]) such that𝜉𝑞1(𝜎)=𝑞𝜎11/𝑞𝜛𝑞𝜎1/𝑞0,(2.15) where𝜛𝑞(1𝜎)=𝜋𝑛=1(1)𝑛1𝜎𝑞𝑛1Γ(𝑛𝑞+1)𝑛!sin(𝑛𝜋𝑞),𝜎(0,).(2.16)

Remark 2.6. It is not difficult to verify that for 𝑣[0,1], 0𝜎𝑣𝜉𝑞(𝜎)𝑑𝜎=0𝜎𝑞𝑣𝜛𝑞(𝜎)𝑑𝜎=Γ(1+𝑣).Γ(1+𝑞𝑣)(2.17) Then, we can see 𝑅(𝑡)𝑞𝑀𝑡Γ(1+𝑞)𝑞1,𝑡>0.(2.18)

We define the mild solution to problem (1.1) as follows.

Definition 2.7. A function 𝑥𝒫[0,𝑇] satisfying the equation ],𝑥(𝑡)=𝜙(𝑡),𝑡(,0𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑥𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,𝑡𝐽,(2.19) is called a mild solution of problem (1.1), where 𝑓𝑆𝐹,𝑥=𝑓𝐿1(𝐽,𝑋)𝑓(𝑡)𝐹𝑡,𝑥(𝑡),𝑥𝑡.fora.e.𝑡𝐽(2.20)

Remark 2.8. (1) Since we only consider the following case: 𝜙(0)=0,(2.21) we define the mild solution to problem (1.1) in the way as mentioned before.
(2) For general 𝜙(0), one can define the mild solution to problem (1.1) similarly and obtain the same conclusion by the similar arguments given in this paper. So we only pay attention to the essential case: 𝜙(0)=0.(2.22)

3. Results and Proofs

We will require the following assumptions.(A1)𝐹𝐽×𝑋×𝒫𝔅cp,cv(𝑋);  (𝑡,𝑣,𝑤)𝐹(𝑡,𝑣,𝑤) is measurable with respect to 𝑡 for each (𝑣,𝑤)𝑋×𝒫; for every 𝑡𝐽, the map 𝐹(𝑡,,)𝑋×𝒫𝔅cp,cv(𝑋) is u.s.c., and the set𝑆𝐹,𝑣=𝑓𝐿1(𝐽,𝑋)𝑓(𝑡)𝐹𝑡,𝑣(𝑡),𝑣𝑡fora.e.𝑡𝐽(3.1)

is nonempty.(A2) There exist two functions 𝜇𝑖𝐿1(𝐽,𝐑+)(𝑖=1,2) such that𝐹(𝑡,𝑣,𝑤)=sup{𝑓𝑓𝐹(𝑡,𝑣,𝑤)}𝜇1(𝑡)𝑣+𝜇2(𝑡)𝑤𝒫,(𝑡,𝑣,𝑤)𝐽×𝑋×𝒫.(3.2)(A3)There exist positive constants 𝑎 and𝑏 such that𝑔𝑡,𝜑𝑎𝜑𝒫+𝑏,for𝑡𝐽,𝜑𝒫.(3.3)(A4)For each 𝑡𝐽, 𝐾(𝑡,) is measurable on [0,𝑡] and𝐾||𝐾||(𝑡)=esssup(𝑡,𝑠),0𝑠𝑡(3.4)

is bounded on 𝐽. The map 𝑡𝐾(𝑡,) is continuous from 𝐽 to 𝐿(𝐽,𝐑).

The following lemma will be used in the proof of our main result.

Lemma 3.1 (see [24]). Let 𝐼 be a compact real interval and let 𝐸 be a Banach space. Let 𝐹 be a multivalued map satisfying hypothesis (A1) and let Υ be a linear continuous mapping from 𝐿1(𝐼,𝐸)𝐶(𝐼,𝐸). Then, Υ𝑆𝐹𝐶(𝐼,𝐸)𝔅cp,cv(𝐶(𝐼,𝐸)),𝑥Υ𝑆𝐹𝑆(𝑥)=Υ𝐹,𝑥(3.5) is a closed graph operator in 𝐶(𝐼,𝐸)×𝐶(𝐼,𝐸).

To prove the main result, we consider the multivalued map 𝒩𝒫[0,𝑇]𝔅(𝒫[0,𝑇]) defined by 𝒩(𝑥)(𝑡)=𝜌𝒫[0,𝑇]],𝜌(𝑡)=𝜙(𝑡),𝑡(,0𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑥𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,𝑡𝐽,,(3.6) where 𝑓𝑆𝐹,𝑥.

It is clear that the fixed points of 𝒩 are mild solutions to problem (1.1).

For 𝜙𝒫, we define the function ],𝑦(𝑡)=𝜙(𝑡),𝑡(,00,𝑡𝐽,(3.7) then 𝑦𝒫[0,𝑇].

Set 𝑥(𝑡)=𝑢(𝑡)+𝑦(𝑡), 𝑡(,𝑇].

It is obvious that 𝑥 satisfies (2.19) if and only if 𝑢 satisfies 𝑢0=0 and for 𝑡𝐽, 𝑢(𝑡)=𝑄(𝑡)g(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,(3.8) where 𝑓𝑆𝐹,𝑢=𝑓𝐿1(𝐽,𝑋)𝑓(𝑡)𝐹𝑡,𝑢(𝑡)+𝑦(𝑡),𝑢𝑡+𝑦𝑡.fora.e.𝑡𝐽(3.9) Let 𝒫0[0,𝑇]=𝑢𝒫[0,𝑇]𝑢0.=0(3.10) For any 𝑢𝒫0[0,𝑇], 𝑢𝑇=𝑢0𝒫+max{𝑢(𝑠)0𝑠𝑇}=max{𝑢(𝑠)0𝑠𝑇}.(3.11) Thus (𝒫0[0,𝑇],𝑇) is a Banach space.

Set 𝐵𝑟=𝑢𝒫0[0,𝑇]𝑢𝑇𝑟,for𝑟0.(3.12) For 𝑢𝐵𝑟, from Definition 2.1, we have 𝑢𝑡+𝑦𝑡𝒫𝑢𝑡𝒫+𝑦𝑡𝒫𝐶1(𝑡)max0𝜏𝑡𝑢(𝜏)+𝐶2𝑢(𝑡)0𝒫+𝐶1(𝑡)max0𝜏𝑡𝑦(𝜏)+𝐶2𝑦(𝑡)0𝒫𝐶1𝑟+𝐶2𝜙𝒫=𝑟,(3.13) where 𝐶𝑖=sup𝑡𝐽𝐶𝑖(𝑡)(𝑖=1,2).(3.14) Define the operator 𝒩𝒫0[0,𝑇]𝒫𝔅0[0,𝑇](3.15) by 𝒩(𝑢)(𝑡)=𝒫0[0,𝑇](𝑡)=𝑄(t)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0,𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,𝑡𝐽(3.16) where 𝑓𝑆𝐹,𝑢.

We can see that if 𝒩 has a fixed point in 𝒫0[0,𝑇], then 𝒩 has a fixed point in 𝒫[0,𝑇] which is a mild solution of problem (1.1).

Assume the following.(A5) The function 𝑔𝐽×𝒫𝑋 is completely continuous, and for every bounded set 𝐵𝒫0[0,𝑇], the set {𝑡𝑔(𝑡,𝑢𝑡)𝑢𝐵} is equicontinuous in 𝑋.

Then we can deduce that 𝒩 has a fixed point under the assumptions (A1)–(A5). For this purpose, we will show that the multivalued operator 𝒩 is completely continuous, u.s.c. with convex values. The proof of this conclusion will be given by proving the following six propositions.

Proposition 3.2. 𝒩𝑢 is convex for each 𝑢𝒫0[0,𝑇].

Proof. For 1(𝑡),2(𝑡)𝒩𝑢, there exist 𝑓1,𝑓2𝑆𝐹,𝑢 such that for each 𝑡𝐽 we have 𝑖(𝑡)=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓𝑖(𝜏)𝑑𝜏𝑑𝑠,𝑖=1,2.(3.17) Let 𝛽[0,1]. Then for each 𝑡𝐽, we get 𝛽1(𝑡)+(1𝛽)2(𝑡)=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝛽𝑓1(𝜏)+(1𝛽)𝑓2(𝜏)𝑑𝜏𝑑𝑠.(3.18) Since 𝐹 has convex values, 𝑆𝐹,𝑢 is convex, we see that 𝛽1(𝑡)+(1𝛽)2(𝑡)𝒩𝑢.(3.19)

Proposition 3.3. 𝒩 maps bounded sets into bounded sets in 𝒫0[0,𝑇].

Proof. Let 𝑢𝐵𝑟. If 𝒩𝑢, then there exists 𝑓𝑆𝐹,𝑢 such that (𝑡)=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,for𝑡𝐽.(3.20) In view of (A3) and (3.13), 𝑔𝑡,𝑢𝑡+𝑦𝑡𝑎𝑟+𝑏.(3.21) Hence from (A2), (A3), and (3.13), it follows that 𝑔(𝑡)𝑄(𝑡)𝑔(0,𝜙)+𝑡,𝑢𝑡+𝑦𝑡+𝑞𝑀𝐾Γ(1+𝑞)𝑡0(𝑡𝑠)𝑞1𝑠0𝜇1(𝜏)𝑢(𝜏)+𝑦(𝜏)+𝜇2𝑢(𝜏)𝜏+𝑦𝜏𝒫𝑑𝜏𝑑𝑠𝑀𝑎𝜙𝒫+𝑏+𝑎𝑟+𝑏+𝑞𝑀𝐾𝑟𝜇Γ(1+𝑞)1𝐿1+𝑟𝜇2𝐿1𝑡0(𝑡𝑠)𝑞1𝑑𝑠𝑀𝑎𝜙𝒫+𝑏+𝑎𝑟+𝑏+𝑀𝑇𝑞𝐾𝑟𝜇Γ(1+𝑞)1𝐿1+𝑟𝜇2𝐿1=𝜔,(3.22) where 𝐾=sup𝑡𝐽𝐾(𝑡).(3.23) Therefore, for each 𝒩(𝐵𝑟), we have 𝑇𝜔.(3.24)

Proposition 3.4. 𝒩 maps bounded sets into equicontinuous sets in 𝒫0[0,𝑇].

Proof. Let 𝒩𝑢 for 𝑢𝐵𝑟, and let 0<𝑡2<𝑡1𝑇. Then we have 𝑡1𝑡2𝑄𝑡1𝑡𝑄2𝑔𝑡𝑔(0,𝜙)+1,𝑢𝑡1+𝑦𝑡1𝑡𝑔2,𝑢𝑡2+𝑦𝑡2+𝑡10𝑠0𝑅𝑡1𝑠𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠𝑡20𝑠0𝑅𝑡2𝑠𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠=𝐼1+𝐼2+𝐼3.(3.25) It follows from the continuity of 𝑆(𝑡) in the uniform operator topology for 𝑡>0 that 𝐼1tendsto0,as𝑡2𝑡1.(3.26) The equicontinuity of 𝑔 ensures that 𝐼2tendsto0,as𝑡2𝑡1.(3.27) For 𝐼3, we obtain 𝐼3𝐾𝑡20𝑠0𝑅𝑡1𝑡𝑠𝑅2(𝑠𝑓𝜏)𝑑𝜏𝑑𝑠+𝐾𝑡1𝑡2𝑠0𝑅𝑡1(𝑠𝑓𝜏)𝑑𝜏𝑑𝑠𝐾𝑟𝑡20𝑅𝑡1𝑡𝑠𝑅2𝑠𝑑𝑠+𝑞𝑀Γ(1+𝑞)𝑡1𝑡2𝑡1𝑠𝑞1𝑑𝑠𝑞𝑟𝐾𝑡200𝜎𝑡1𝑠𝑞1𝑡2𝑠𝑞1𝜉𝑞𝑡(𝜎)𝑆1𝑠𝑞𝜎𝑑𝜎𝑑𝑠+𝑞𝑟𝐾𝑡200𝜎𝑡2𝑠𝑞1𝜉𝑞(𝑆𝑡𝜎)1𝑠𝑞𝜎𝑡𝑆2𝑠𝑞𝜎+𝑑𝜎𝑑𝑠𝑞𝑀𝑟𝐾Γ(1+𝑞)𝑡1𝑡2𝑡1𝑠𝑞1𝑑𝑠𝑞𝑀𝑟𝐾Γ(1+𝑞)𝑡20|||𝑡1𝑠𝑞1𝑡2𝑠𝑞1|||𝑑𝑠+𝑞𝑟𝐾𝑡200𝜎𝑡2𝑠𝑞1𝜉𝑞𝑆𝑡(𝜎)1𝑠𝑞𝜎𝑡𝑆2𝑠𝑞𝜎+𝑑𝜎𝑑𝑠𝑀𝑟𝐾𝑡Γ(1+𝑞)1𝑡2𝑞,(3.28) where 𝑟𝜇=𝑟1𝐿1𝜇+𝑟2𝐿1.(3.29) Clearly, the first term and third term on the right-hand side of (3.28) tend to 0 as 𝑡2𝑡1. The second term on the right-hand side of (3.28) tends to 0 as 𝑡2𝑡1 as a consequence of the continuity of 𝑆(𝑡) in the uniform operator topology for 𝑡>0.
Thus the set{𝒩𝑢𝑢𝐵𝑟} is equicontinuous.

Proposition 3.5. (𝒩𝐵𝑟)(𝑡) is relatively compact for each 𝑡𝐽, where 𝒩𝐵𝑟𝒩𝐵(𝑡)=(𝑡)𝑟.(3.30)

Proof. Fix 𝑡(0,𝑇]. For arbitrary 0<𝜀<𝑡 and arbitrary 𝛿>0, write 𝜀,𝛿(𝑡)=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑞0𝑡𝜀(𝑡𝑠)𝑞1𝛿𝜎𝜉𝑞(𝜎)𝑆((𝑡𝑠)𝑞𝜎)𝑠0𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝜎𝑑𝑠=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑞𝑆(𝜀𝑞𝛿)0𝑡𝜀(𝑡𝑠)𝑞1𝛿𝜎𝜉𝑞(𝜎)𝑆((𝑡𝑠)𝑞𝜎𝜀𝑞𝛿)𝑠0𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝜎𝑑𝑠,(3.31) where 𝑓𝑆𝐹,𝑢. Since 𝑆(𝑡) is compact for each 𝑡(0,𝑇] and (A5), the set 𝑈𝜀,𝛿=𝜀,𝛿𝒩𝐵(𝑡)𝑟(3.32) is relatively compact. Moreover, (𝑡)𝜀,𝛿(𝑡)𝑞0𝑡𝜀(𝑡𝑠)𝑞1𝛿0𝜎𝜉𝑞(𝜎)𝑆((𝑡𝑠)𝑞𝜎)𝑠0𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝜎𝑑𝑠+𝑞𝑡𝑡𝜀(𝑡𝑠)𝑞10𝜎𝜉𝑞(𝜎)𝑆((𝑡𝑠)𝑞𝜎)𝑠0𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝜎𝑑𝑠𝑀𝐾𝑟𝑇𝑞𝛿0𝜎𝜉𝑞(𝜎)𝑑𝜎+𝑀𝑟𝐾𝜀𝑞,Γ(1+𝑞)(3.33) which implies that (𝒩𝐵𝑟)(𝑡) is relatively compact.

Now, it follows from Propositions 3.33.5 and the Ascoli-Arzela theorem that𝒩𝒫0[0,𝑇]𝒫𝔅0[0,𝑇](3.34)

is completely continuous.

Proposition 3.6. 𝒩 has a closed graph.

Proof. Suppose that 𝑢𝑛𝑢,𝑛𝒩𝑢𝑛with𝑛.(3.35) We claim that 𝒩𝑢.(3.36) In fact, the assumption 𝑛𝒩𝑢𝑛 implies that there exists 𝑓𝑛𝑆𝐹,𝑢𝑛 such that 𝑛(𝑡)=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑛𝑡+𝑦𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓𝑛(𝜏)𝑑𝜏𝑑𝑠,𝑡𝐽.(3.37) We will show that there exists 𝑓𝑆𝐹,𝑢 such that (𝑡)=𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,𝑡𝐽.(3.38) Obviously, as 𝑛, we have 𝑛(𝑡)+𝑄(𝑡)𝑔(0,𝜙)𝑔𝑡,𝑢𝑛𝑡+𝑦𝑡(𝑡)+𝑄(𝑡)𝑔(0,𝜙)𝑔𝑡,𝑢𝑡+𝑦𝑡0.(3.39) Consider the following linear continuous operator: Υ𝐿1(𝐽,𝑋)𝐶(𝐽,𝑋),𝑓Υ(𝑓)(𝑡)=𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠.(3.40) By virtue of Lemma 3.1, we know that Υ𝑆𝐹 is a closed graph operator. Moreover, we get 𝑛(𝑡)+𝑄(𝑡)𝑔(0,𝜙)𝑔𝑡,𝑢𝑛𝑡+𝑦𝑡𝑆Υ𝐹,𝑢𝑛.(3.41) Since 𝑢𝑛𝑢 and 𝑛, it follows from Lemma 3.1 that (𝑡)+𝑄(𝑡)𝑔(0,𝜙)𝑔𝑡,𝑢𝑡+𝑦𝑡=𝑡0𝑠0𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠,(3.42) for some 𝑓𝑆𝐹,𝑢.

Now, we can conclude that 𝒩 is a completely continuous multivalued map, u.s.c. with convex values. Next, we give the existence result of problem (1.1).

Theorem 3.7. Assume that (A1)–(A5) are satisfied; then there exists a mild solution of (1.1) on (,𝑇] provided that 𝑎𝐶1<1.

Proof. Define Ω=𝑢𝒫0[0,𝑇]𝜆𝑢𝒩𝑢,forsome𝜆>1.(3.43) Then, according to the previous propositions and discussions, we see that we only need to prove that the set Ω is bounded.
Take 𝑢Ω. Then there exists 𝑓𝑆𝐹,𝑢 such that𝑢(𝑡)=𝜆1𝑄(𝑡)𝑔(0,𝜙)+𝑔𝑡,𝑢𝑡+𝑦𝑡+𝑡0𝑠0.𝑅(𝑡𝑠)𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠(3.44) It follows from Definition 2.1 and (A2) that 𝑎𝑢(𝑡)<𝑀𝜙𝒫𝐶+𝑏+𝑎1max0𝜏𝑡𝑢(𝜏)+𝐶2𝜙𝒫++𝑏𝑞𝑀Γ(1+𝑞)𝑡0(𝑡𝑠)𝑞1𝑠0||||𝐾(𝑠,𝜏)𝑓(𝜏)𝑑𝜏𝑑𝑠𝑀1+𝑎𝐶1max0𝜏𝑡+𝑢(𝜏)𝑞𝑀𝐾Γ(1+𝑞)𝑡0(𝑡𝑠)𝑞1𝑠0𝜇1(𝜏)𝑢(𝜏)+𝑦(𝜏)+𝜇2𝑢(𝜏)𝜏+𝑦𝜏𝒫𝑑𝜏𝑑𝑠𝑀1+𝑎𝐶1max0𝜏𝑡+𝑢(𝜏)𝑞𝑀𝐾Γ(1+𝑞)𝑡0(𝑡𝑠)𝑞1𝑠0𝜇1+(𝜏)𝑢(𝜏)𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝑞1𝑠0𝜇2𝑢(𝜏)𝜏𝒫𝑑𝜏𝑑𝑠+𝐶2𝑡0(𝑡𝑠)𝑞1𝑠0𝜇2(𝜏)𝜙𝒫𝑑𝜏𝑑𝑠𝜃1+𝑎𝐶1max0𝜏𝑡𝑢(𝜏)+𝑞𝑀𝐾𝜇1𝐿1Γ(1+𝑞)𝑡0(𝑡𝑠)𝑞1max0𝜏𝑠+𝑢(𝜏)𝑑𝑠𝑞𝑀𝐾𝜇2𝐿1𝐶1Γ(1+𝑞)𝑡0(𝑡𝑠)𝑞1max0𝜏𝑠𝑢(𝜏)𝑑𝑠=𝜃1+𝑎𝐶1max0𝜏𝑡𝑢(𝜏)+𝜃2𝑡0(𝑡𝑠)𝑞1max0𝜏𝑠𝑢(𝜏)𝑑𝑠,(3.45) where 𝑀1=𝑀𝑎𝜙𝒫+𝑏+𝑎𝐶2𝜙𝒫𝜃+𝑏,1=𝑀1+𝑀𝐾𝜇Γ(1+𝑞)2𝐿1𝐶2𝑇𝑞𝜙𝒫,𝜃2=𝑞𝑀𝐾𝜇1𝐿1+𝐶1𝜇2𝐿1.Γ(1+𝑞)(3.46) Denote 𝜅(𝑡)=max0𝑠𝑡𝑢(𝑠),(3.47) and let ̃𝑡[0,𝑡] such that ̃𝜅(𝑡)=𝑢(𝑡). Then, by (3.45), we get 𝜅(𝑡)𝜃1+𝑎𝐶1𝜅(𝑡)+𝜃2𝑡0(𝑡𝑠)𝑞1𝜅(𝑠)𝑑𝑠.(3.48) Furthermore, 𝜃𝜅(𝑡)11𝑎𝐶1+𝜃21𝑎𝐶1𝑡0(𝑡𝑠)𝑞1𝜅(𝑠)𝑑𝑠.(3.49) It is known from [25, Lemma  7.1.1] that, for any continuous functions 𝑣,𝑤𝐽[0,+), if 𝑤() is nondecreasing and there are constants 𝑎>0 and 0<𝛼<1 such that 𝑣(𝑡)𝑤(𝑡)+𝑎𝑡0(𝑡𝑠)𝛼𝑣(𝑠)𝑑𝑠,(3.50) then there exists a constant 𝑘=𝑘(𝛼) such that 𝑣(𝑡)𝑤(𝑡)+𝑘𝑎𝑡0(𝑡𝑠)𝛼𝑤(𝑠)𝑑𝑠,foreach𝑡𝐽.(3.51) By virtue of this general fact and (3.49), we see that there exists a constant ̃̃𝑘=𝑘(𝑞) such that 𝜃𝜅(𝑡)11𝑎𝐶1+̃𝑘𝜃21𝑎𝐶1𝑡0(𝑡𝑠)𝑞1𝜃11𝑎𝐶1𝜃𝑑𝑠11𝑎𝐶1̃1+𝑘𝜃2𝑇𝑞𝑞1𝑎𝐶1=𝜁.(3.52) Therefore 𝑢𝑇𝜁. This means that the set Ω is bounded.
Thus, it follows from Lemma 2.5 that 𝒩 has a fixed point in 𝒫0[0,𝑇]. Then 𝒩 has a fixed point which gives rise to a mild solution to problem (1.1).

Example 3.8. Set 𝑋=𝐿2([0,𝜋],𝐑) and define 𝐴 by 𝐷(𝐴)=𝐻2(0,𝜋)𝐻10(0,𝜋),𝐴𝑢=𝑢.(3.53) Then 𝐴 generates a compact, analytic semigroup 𝑆() of uniformly bounded linear operators, and 𝑆(𝑡)1 (see [26] for more related information).
Consider the following Cauchy problem for a fractional integrodifferential conclusion: 𝜕𝑞𝜕𝑡𝑞𝑣(𝑡,𝜉)𝑡𝜕𝛾(𝑠𝑡)𝑣(𝑠,𝜉)𝑑𝑠2𝜕𝜉2𝑣(𝑡,𝜉)𝑡+𝛾(𝑠𝑡)𝑣(𝑠,𝜉)𝑑𝑠𝑡0(𝑡𝑠)𝑠[],𝜂(𝑠,𝜏𝑠,𝜉)𝐻(𝑠,𝑣(𝜏,𝜉))𝑑𝜏𝑑𝑠,𝑡0,1𝑣(𝑡,0)𝑡𝛾(𝑠𝑡)𝑣(𝑠,0)𝑑𝑠=0,𝑣(𝑡,𝜋)𝑡𝛾(𝑠𝑡)𝑣(𝑠,𝜋)𝑑𝑠=0,𝑣(𝜃,𝜉)=𝑣0(𝜃,𝜉),<𝜃0,(3.54) where 0<𝑞<1, 𝜉[0,𝜋], 𝑣0(,0]×[0,𝜋]𝐑 is a continuous function and 𝐻[0,1]×𝐑𝔅(𝐑) is a u.s.c. multivalued map with compact convex values.
Let 𝜛<0, define the space ]𝒫=𝜑𝐶((,0,𝑋)lim𝜃𝑒𝜛𝜃𝜑(𝜃)existsin𝑋(3.55) endowed with the norm 𝜑𝒫=sup<𝜃0𝑒𝜛𝜃𝜑(𝜃)𝐿2.(3.56) Clearly, we can see that 𝒫 is an admissible phase space which satisfies (H1)–(H3) with 𝐶1(𝑡)=max1,𝑒𝜛𝑡,𝐶2(𝑡)=𝑒𝜛𝑡.(3.57) For 𝑡[0,1], 𝜉[0,𝜋], and 𝜑𝒫, let 𝜙𝑥(𝑡)(𝜉)=𝑣(𝑡,𝜉),(𝜃)(𝜉)=𝑣0],(𝜃,𝜉),𝜃(,0𝑔(𝑡,𝜑)(𝜉)=0𝛾(𝜃)𝜑(𝜃)(𝜉)𝑑𝜃,𝐾(𝑡,𝑠)=𝑡𝑠,𝐹(𝑡,𝑥(𝑡),𝜑)(𝜉)=0𝜂(𝑡,𝜃,𝜉)𝐻(𝑡,𝜑(𝜃)(𝜉))𝑑𝜃.(3.58) Then problem (3.54) can be written in the abstract form (1.1).

Furthermore, we assume the following.(1)The function 𝛾(,0]𝐑 is continuous and𝑀21=2𝜛0𝛾2(𝜃)𝑑𝜃1/2<.(3.59)(2)There exists a continuous function 𝑣1(𝑡) such that||||𝐻(𝑡,𝜑)𝑣1(𝑡)𝜑(𝜃)𝐿2.(3.60)(3)The function 𝜂(𝑡,𝜃,𝜉)0 is continuous in [0,1]×(,0]×[0,𝜋] and0𝜂(𝑡,𝜃,𝜉)𝑒𝜛𝜃𝑑𝜃=𝑝(𝑡,𝜉)<.(3.61) Then, we can obtain𝐹(𝑡,𝑥(𝑡),𝜑)𝐿2=𝜋0||||0||||𝜂(𝑡,𝜃,𝜉)𝐻(𝑡,𝜑(𝜃)(𝜉))𝑑𝜃2𝑑𝜉1/2𝜋00𝜂(𝑡,𝜃,𝜉)𝑣2(𝑡)𝜑(𝜃)𝐿2𝑑𝜃2𝑑𝜉1/2=𝜋00𝜂(𝑡,𝜃,𝜉)𝑣2(𝑡)𝑒𝜛𝜃𝑒𝜛𝜃𝜑(𝜃)𝐿2𝑑𝜃2𝑑𝜉1/2𝜋00𝜂(𝑡,𝜃,𝜉)𝑒𝜛𝜃𝑑𝜃2𝑑𝜉1/2𝑣2(𝑡)𝜑𝒫𝜋0𝑝2(𝑡,𝜉)𝑑𝜉1/2𝑣2(𝑡)𝜑𝒫=𝑝(𝑡)𝑣2(𝑡)𝜑𝒫,,(3.62) where 𝑝(𝑡)=𝑝(𝑡,)𝐿2.

Moreover,𝑔(𝑡,𝜑)𝐿2=𝜋00𝛾(𝜃)𝜑(𝜃)(𝜉)𝑑𝜃2𝑑𝜉1/2𝜋00𝛾2(𝜃)𝑑𝜃0𝜑2(𝜃)(𝜉)𝑑𝜃𝑑𝜉1/2=0𝛾2(𝜃)𝑑𝜃1/2𝜋00𝜑2(𝜃)(𝜉)𝑑𝜃𝑑𝜉1/2=0𝛾2(𝜃)𝑑𝜃1/20𝜑(𝜃)2𝐿2𝑑𝜃1/2=0𝛾2(𝜃)𝑑𝜃1/20𝑒2𝜛𝜃𝑒2𝜛𝜃𝜑(𝜃)2𝐿2𝑑𝜃1/20𝛾2(𝜃)𝑑𝜃1/20𝑒2𝜛𝜃sup<𝜃0𝑒𝜛𝜃𝜑(𝜃)𝐿22𝑑𝜃1/2=0𝛾2(𝜃)𝑑𝜃1/20𝑒2𝜛𝜃𝑑𝜃1/2𝜑𝒫=𝑀2𝜑𝒫.(3.63) Therefore, by virtue of Theorem 3.7, problem (3.54) has a mild solution when 𝑒𝜛𝑀2<1.

Acknowledgments

F. Li acknowledges support from the NSF of Yunnan Province (2009ZC054M). T.- J. Xiao acknowledges support from the NSF of China (11071042), the Chinese Academy of Sciences and the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900). H.- K. Xu acknowledges support from NSC 100-2115-M-110-003-MY2 (Taiwan).

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