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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 901942, 18 pages
A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations
1School of Mathematics, Yunnan Normal University, Kunming 650092, China
2Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
4Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
Received 12 February 2012; Accepted 3 March 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.
This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.
Nonlocal Cauchy problem for equations is an initial problem for the corresponding equations with nonlocal initial data. Such a Cauchy problem has better effects than the normal Cauchy problem with the classical initial data when we deal with many concrete problem coming from engineering and physics (cf., e.g., [1–10] and references therein). Therefore, the study of this type of Cauchy problem is important and significant. Actually, as we have seen from the just mentioned literature, there have been many significant developments in this field.
On the other hand, fractional differential and integrodifferential equations arise from various real processes and phenomena appeared in physics, chemical technology, materials, earthquake analysis, robots, electric fractal network, statistical mechanics biotechnology, medicine, and economics. They have in recent years been an object of investigations with much increasing interest. For more information on this subject see for instance [9, 11–18] and references therein.
Throughout this paper, is a separable Banach space; is the Banach space of all linear bounded operators on ; is the generator of an analytic and uniformly bounded semigroup on with for a constant , and is the space of all -valued continuous functions on with the supremum norm as follows:
Let , , , , and . The nonlocal Cauchy problem for abstract fractional integrodifferential equations, with which we are concerned, is in the following form: where and are given functions to be specified later and the fractional derivative is understood in the Caputo sense, this means that, the fractional derivative is understood in the following sense: and where is the Riemann-Liouville derivative of order of , where is the Gamma function.
Our main purpose is to establish an existence theorem for the mild solutions to the nonlocal Cauchy problem based on a special measure of noncompactness under weak assumptions on the nonlinearity and the semigroup generated by .
2. Existence Result and Proof
As usual, we abbreviate with , for any .
Definition 2.1. 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 in .
Definition 2.2. Let be a Banach space, and is continuous. Let be a measure of noncompactness in such that(i)for any with , (ii) for every , , If for every bounded set which is not relatively compact, then we say that is condensing with respect to the measure of noncompactness (or -condensing).
Definition 2.3. Let
be a one-sided stable probability density, and
For any , we define operators and by
If a continuous function satisfies
then the function is called a mild solution of (1.2).
Our main result is as follows.
Theorem 2.4. Assume that (1) and are measurable for each ; is measurable for each ;(2) is continuous for a.e. ; is completely continuous; is continuous for a.e. ; the map is continuous from to ;(3)there exist two positive functions and two positive functions and on with
for all , and
for any bounded set , where is the Hausdorff measure of noncompactness:
for a positive constant , and
is bounded on .
Then the mild solutions set of problem (1.2) is a nonempty compact subset of the space , in the case of
Proof. First of all, let us prove our definition of the mild solution to problem (1.2) is well defined and reasonable. Actually, the proof is basic. We present it here for the completeness of the proof as well as the convenience of reading.
Write Clearly, the nonlocal Cauchy problem (1.2) can be written as the following equivalent integral equation: provided that the integral in (2.18) exists. Formally taking the Laplace transform to (2.18), we have Therefore, if the related integrals exist, then we obtain Now using the uniqueness of the Laplace transform (cf., e.g., [21, Theorem 1.1.6]), we deduce that Consequently, we see that the mild solution to problem (1.2) given by Definition 2.3 is well defined.
Next, we define the operator as follows: It is clear that the operator is well defined.
The operator can be written in the form , where the operators are defined as follows: The following facts will be used in the proof. (1) which implies that (2) which implies that Let such that for an . Then by the assumptions, we know that for almost every and : Therefore, for sufficiently large , we have where Hence, Thus, since (2.27) implies that By (2.33) and our assumptions, we see that is continuous.
Since is the Hausdorff measure of noncompactness in , 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 every , , (iii)for every relatively compact set , , (iv)for each , , (v) is equivalent to the relative compactness of .Noting that for any , we have So, there exists a positive constant such that For every bounded subset , we define Then is the module of equicontinuity of , and is a measure of noncompactness in the space with values in the cone .
Let be a nonempty, bounded set such that By the assumptions and the continuity of in the uniform operator topology for , we get Clearly, Let , such that and . Then It is not hard to see that the right-hand side of (2.45) tend to 0 as . Thus, the set is equicontinuous, then . Combining with (2.43), we have , which implies from (2.42). Next, we show that .
It is easy to see that For any , we define We consider the multifunction : Obviously, is integrable, that is, admits a Bochner integrable selection , and From (2.27) and our assumptions, it follows that is integrably bounded, that is, there exists a function such that Moreover, we have the following estimate for a.e. : Therefore, since is a separable Banach space, we know by [20, Theorem 4.2.3] that So Similarly, if we set then we see that the multifunction , is integrable and integrably bounded. Thus, we obtain the following estimate for a.e. : Now, from (2.53) and (2.56), it follows that where . Then by (2.42), we get . Hence . Thus, is relatively compact due to the regularity property of . This means that is -condensing.
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 .
From the assumptions, we have Moreover, Therefore, Dividing both sides of (2.62) by , and taking , we have This is a contradiction. Hence for some positive number , . According to the following known fact.
Let be a bounded convex closed subset of and a -condensing map. Then is nonempty. we see that problem (1.2) 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. Indeed, let , for , we have
Noting that the Hölder inequality, we have Therefore, from (2.66), we obtain We denote Let such that . Then, by (2.68), we can see By a generalization of Gronwall’s lemma for singular kernels ([22, Lemma 7.1.1]), we deduce that there exists a constant such that Hence, .
Now we consider a closed ball: We take the radius large enough to contain the set inside itself. Moreover, from the proof above, is -condensing. Consequently, the following known fact implies our conclusion: Let be a bounded open neighborhood of zero and a -condensing map satisfying the boundary condition: for all and . Then, is nonempty compact.
In this section, let , we consider the following nonlocal Cauchy problem for an integrodifferential problem: where is the Caputo fractional partial derivative of order ; ; is a constant to be specified later; are continuous functions and there exists a positive constant such that For , , we set On the other hand, it is known that the operator ( with ) generates an analytic semigroup and uniformly bounded semigroup on with . Therefore, (3.1) is a special case of (1.2).
Moreover, we havefor all , for any , that is, for any bounded set , for a.e. ;for almost all , where , and that is, for any bounded set , where , and Therefore, Theorem 2.4 implies that the problem (3.1) has at least a mild solution when
The work was supported partly by the Chinese Academy of Sciences, the NSF of Yunnan Province (2009ZC054M) and the NSF of China (11171210).
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