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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 916543, 19 pages
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.
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.
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., [1–11] 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, 14–21] 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 : where , the fractional derivative is understood in the Caputo sense (, see Definition 2.3 in Section 2), is an admissible phase space, defined by, , generates a compact and uniformly bounded semigroup on which implies that there exists such that, belongs to with and is a multivalued map to be specified later.
Throughout this paper, is a Banach space with norm , is the Banach space of all linear continuous operators on , , and () is the space of all -valued continuous functions on .
Moreover, we abbreviate as , for any .
We use the notation to denote the family of all nonempty subsets of . Let denote, respectively, the family of all nonempty bounded, closed, compact, convex, and compact-convex subsets of .
Definition 2.1. A linear space consisting of functions from into with norm is called an admissible phase space if has the following properties.(H1)For any and , if is continuous on and , then , is continuous in , and for a positive constant . (H2)There exists a continuous function and a locally bounded function in such that for and as in (H1).(H3)The space is complete.
Remark 2.2. (H1) is equivalent to that for any and , if is continuous on and , then , is continuous in , and for a positive constant .
We set for , and , where is the Gamma function.
Definition 2.3. Let and . Then the expression with is called Riemann-Liouville integral of order of .
Definition 2.4. Let , (). The Caputo fractional derivative of order of is defined by where .
The following concepts are also very basic, which will be used later.
A multivalued map is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if is bounded in for all , that is,
A multivalued map is said to be measurable if for each the function defined by is measurable.
If for each , the set is a nonempty, closed subset of , and for each open set of containing , there exists an open neighborhood of such that , then is called upper semicontinuous (u.s.c.) on .
If for every , is relatively compact, then is said to be completely continuous.
If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, that is, We say that has a fixed point if there is some such that .
For more details on multivalued maps we refer to the book by Deimling .
The following is the multivalued version of the fixed-point theorem due to Martelli .
Lemma 2.5. Let be a Banach space and let be an upper semicontinuous is bounded; then has a fixed point and completely continuous multivalued map. If the set is bounded, then has a fixed point.
Let be the norm of defined by
Remark 2.6. It is not difficult to verify that for , Then, we can see
We define the mild solution to problem (1.1) as follows.
Definition 2.7. A function satisfying the equation is called a mild solution of problem (1.1), where
Remark 2.8. (1) Since we only consider the following case:
we define the mild solution to problem (1.1) in the way as mentioned before.
(2) For general , 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:
3. Results and Proofs
We will require the following assumptions.(A1); is measurable with respect to for each ; for every , the map is u.s.c., and the set
is nonempty.(A2) There exist two functions such that(A3)There exist positive constants and such that(A4)For each , is measurable on and
is bounded on . The map is continuous from to .
The following lemma will be used in the proof of our main result.
Lemma 3.1 (see ). Let be a compact real interval and let be a Banach space. Let be a multivalued map satisfying hypothesis (A1) and let be a linear continuous mapping from . Then, is a closed graph operator in .
To prove the main result, we consider the multivalued map defined by where .
It is clear that the fixed points of are mild solutions to problem (1.1).
For , we define the function then .
Set , .
It is obvious that satisfies (2.19) if and only if satisfies and for , where Let For any , Thus is a Banach space.
Set For , from Definition 2.1, we have where Define the operator by where .
We can see that if has a fixed point in , then has a fixed point in which is a mild solution of problem (1.1).
Assume the following.(A5) The function is completely continuous, and for every bounded set , the set is equicontinuous in .
Then we can deduce that has a fixed point under the assumptions (A1)–(A5). For this purpose, we will show that the multivalued operator 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. is convex for each .
Proof. For , there exist such that for each we have Let . Then for each , we get Since has convex values, is convex, we see that
Proposition 3.3. maps bounded sets into bounded sets in .
Proposition 3.4. maps bounded sets into equicontinuous sets in .
Proof. Let for , and let . Then we have
It follows from the continuity of in the uniform operator topology for that
The equicontinuity of ensures that
For , we obtain
Clearly, the first term and third term on the right-hand side of (3.28) tend to 0 as . The second term on the right-hand side of (3.28) tends to 0 as as a consequence of the continuity of in the uniform operator topology for .
Thus the set is equicontinuous.
Proposition 3.5. is relatively compact for each , where
Proof. Fix . For arbitrary and arbitrary , write where . Since is compact for each and (A5), the set is relatively compact. Moreover, which implies that is relatively compact.
is completely continuous.
Proposition 3.6. has a closed graph.
Proof. Suppose that We claim that In fact, the assumption implies that there exists such that We will show that there exists such that Obviously, as , we have Consider the following linear continuous operator: By virtue of Lemma 3.1, we know that is a closed graph operator. Moreover, we get Since and , it follows from Lemma 3.1 that for some .
Now, we can conclude that 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 provided that .
Then, according to the previous propositions and discussions, we see that we only need to prove that the set is bounded.
Take . Then there exists such that It follows from Definition 2.1 and (A2) that where Denote and let such that . Then, by (3.45), we get Furthermore, It is known from [25, Lemma 7.1.1] that, for any continuous functions , if is nondecreasing and there are constants and such that then there exists a constant such that By virtue of this general fact and (3.49), we see that there exists a constant such that Therefore . This means that the set is bounded.
Thus, it follows from Lemma 2.5 that has a fixed point in . Then has a fixed point which gives rise to a mild solution to problem (1.1).
Example 3.8. Set and define by
Then generates a compact, analytic semigroup of uniformly bounded linear operators, and (see  for more related information).
Consider the following Cauchy problem for a fractional integrodifferential conclusion: where , , is a continuous function and is a u.s.c. multivalued map with compact convex values.
Let , define the space endowed with the norm Clearly, we can see that is an admissible phase space which satisfies (H1)–(H3) with For , , and , let Then problem (3.54) can be written in the abstract form (1.1).
Furthermore, we assume the following.(1)The function is continuous and(2)There exists a continuous function such that(3)The function is continuous in and Then, we can obtain where .
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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