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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 398632, 9 pages
A Class of Fractional -Laplacian Integrodifferential Equations in Banach Spaces
1Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
2College of Sciences, Hezhou University, Hezhou, Guangxi 542899, China
Received 17 April 2013; Accepted 26 June 2013
Academic Editor: Paul Eloe
Copyright © 2013 Yiliang Liu and Liang Lu. 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.
We study a class of nonlinear fractional integrodifferential equations with -Laplacian operator in Banach space. Some new existence results are obtained via fixed point theorems for nonlocal boundary value problems of fractional -Laplacian equations. An illustrative example is also discussed.
In this paper, we discuss a class of fractional integrodifferential equations with -Laplacian operator and nonlocal boundary condition in Banach space : where , , , denotes the Caputo fractional derivative of order . is called a -Laplacian operator. is a constant, is the zero element of , and , , , and is a continuous functional.
Recently, fractional differential equations with -Laplacian operator have been widely applied in many fields of physics and natural phenomena, such as non-Newtonian mechanics, fluid mechanics, viscoelasticity mechanics, combustion theory, and material science. There have appeared some results for the existence of solutions of BVPs for fractional differential equations with -Laplacian operator, see [1–11] and the references therein.
In the last few years, the research of antiperiodic BVPs has received considerable attention and become a much important area. The study of antiperiodic solutions for nonlinear evolution equations is closely related to the study of periodic solutions, and it was initiated by Okochi . And antiperiodic boundary conditions appear in physics in a variety of situations (cf. [13–15] and references therein).
As well as we know, it has been shown, first by Tavazoei et al. and later by Kaslik and Sivasundaram (cf. [16–19]), that periodic solution in fractional dynamical systems does not exist. Therefore antiperiodic solutions may not exist for fractional differential equations. However, such kind of boundary condition (the value at endpoint has different signs) still received considerable attention; it may involve some resonance problems. For example, Chen and Liu  studied a kind of BVP for the fractional -Laplacian equation as follows: where , , , is a standard Caputo fractional derivative and is continuous. Under certain nonlinear growth conditions of the nonlinearity, the existence result was obtained by using Schaefer's fixed point theorem.
Alsaedi  proved some existence results for a BVP for fractional differential equations as mentioned later: where denotes the standard Caputo fractional derivative of order , and for , , is a Banach space. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to establish the results in .
Moreover, if , , then (1) reduces to the Langevin equation which has been widely used to describe the evolution of physical phenomena in fluctuating environments. In , the authors studied such type of Langevin equation with two different fractional orders. This new version of fractional Langevin equation gives a fractional Gaussian process parameterized by two indices, which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. In , the fractional oscillator process with two indices was discussed. For more details, see [22–26] and references therein.
In , Ahmad et al. discussed the existence solutions for the three-point BVPs of Langevin equation with two different fractional orders: where denotes the standard Caputo fractional derivative, , , , and is a real number. Ahmad and Nieto  studied a Dirichlet BVP of Langevin equation where , is a real number, . Here, is a Banach space. A. P. Chen and Y. Chen  considered the BVP of Langevin equation with two different fractional orders: where , is continuous and is a real number. By applying contraction mapping principle and Krasnoselskii's fixed point theorem, some existence results are obtained in [24–26].
Motivated by previously mentioned works, we will consider the existence of solutions of fractional -Laplacian BVP (1) with nonlocal boundary condition. To authors' knowledge, there are few results on the existence of solutions of nonlinear fractional -Laplacian differential equations in Banach spaces, and no paper is concerned with the existence results for fractional -Laplacian integrodifferential equation (1). And the main difficulty that, for , it is impossible for us to find a Green's function in the equivalent integral operator since the differential operator is nonlinear. This paper is concerned with BVP (1) by using some known fixed point theorems. Such investigations will provide an important platform for gaining a deeper understanding of nature.
The paper is organized as follows. In Section 2, we present some material. In Section 3, by applying Krasnoselskii's theorem and Schauder's fixed point theorem, the existence of solutions is given for nonlinear fractional BVP (1). Finally, an example is shown in Section 4 to illustrate the usefulness of the main results.
2. Preliminaries and Lemmas
Definition 1. The Riemann-Liouville fractional integral operator of order of a function is given by provided that the right side integral is pointwise defined on .
Definition 2. The Riemann-Liouville derivative of order for a function can be written as where is the smallest integer greater than .
Definition 3. The Caputo fractional derivative of order for a function can be written as
where is the smallest integer greater than .
For , denotes the space of functions which have continuous derivatives up to order on such that : where is the space of absolutely continuous functions on . Then we easily get the following.
Remark 4. If , then it is the standard Caputo fractional derivative where is the smallest integer greater than . Furthermore, the Caputo derivative of a constant is equal to zero.
Lemma 5 (see ). Let . Assume that . Then the following equality holds: where ; here is the smallest integer greater than .
The following famous fixed point theorems will be used to prove the existence results of BVP (1).
Lemma 6 (Krasnoselskii's Theorem ). Let be a closed convex and nonempty subset of a Banach space . Let be the operators such that(i) whenever ; (ii) is compact and continuous; (iii) is a contraction mapping.
Then there exists such that .
Lemma 7 (Schauder's fixed point theorem). If is a nonempty closed bounded convex subset of a Banach space , and is completely continuous, then has a fixed point in .
3. Existence Results
In this section, we deal with the existence of solutions of the BVP (1).
At first, we introduce the Banach space of all continuous functions from to endowed with a topology of uniform convergence with norm defined by , and denotes the norm in Banach space .
In relation to (1), we introduce the following fractional -Laplacian differential equations: where .
Remark 8. Obviously, is invertible and , where such that . and are strictly increasing functions. Furthermore, the nonlinear operator reduces to the linear operator when , and the additive index law holds under reasonable conditions on the function (see [4, 10]).
Lemma 9. Let and ; then the unique solution of the BVP, is given by where is given by
Proof. Assume that satisfies the equation of (14); then by the first equality of (14) and Lemma 5, we have Applying the boundary conditions of (14), thus Through some calculation, we get Thus This completes the proof.
Lemma 10. The function satisfy
Proof. It is easy to see that, for , we obtain This completes the proof.
Proof. Assume that is a solution of (13). Then by (13) and Lemma 5, we have
For , we can get
Combining with , we obtain
Since and , thus (27) yields that
Substituting previous equation into (25), then
which means that
Now, setting , then (13) can be changed into the form of (14). Applying Lemma 9, we can get (23).
Conversely, we can obtain that the solution of (23) is the solution of the BVP (13) by calculation, which completes our proof.
Now, let us consider the existence of solutions of BVP for -Laplacian equations (1). Let be a Nemytskii operator defined by and for , we denote We define an operator as follows: where is given by (16). Clearly, a fixed point of the operator is a solution of the problem (1).
In the sequel, we need the following assumptions. Let be continuous and such that The function is continuous and for all . Moreover, such that Let , and satisfies Let be a continuous functional and satisfies
Proof. Let us define a bounded set and , where
and . Then is a closed convex and nonempty subset of a Banach space . We define two operators and on such that and
Firstly, we show that for any .
From the assumptions () and (), we have and by Lemma 10, for any , we have where , and the definition of yields that .
On the other hand, from (), , where and ; then for , we have for all . By , we can obtain that for .
Next, we will prove that is compact and continuous.
According to (42), it is easy to know that such that for all , which yields that is uniformly bounded in . In view of the Arzelá-Ascoli theorem, we are going to prove that is equicontinuous.
For all , the inequality of (41) guarantees that ; then for , we have Since is uniformly continuous on , we can obtain that is equicontinuous on . Then, the Arzelá-Ascoli theorem yields that is relatively compact in .
Now, we show that is continuous.
Let be a sequence with in ; we will show that . By the continuity of , it is easy to see . Moreover, using the Lebesgue dominated convergence theorem, we have uniformly for . Moreover, by the properties of Green function and , and the uniformly continuity of function , we have uniformly for . Thus we have in . This shows that is continuous.
Finally, we show that is a contraction mapping. For , by (), we obtain Thus , where , which guarantees that is a contraction mapping.
Thus all the assumptions of Lemma 6 are satisfied and the conclusion of Lemma 6 implies that the operator has at least a fixed point in , which is a solution of BVP (1).
The proof is complete.
Obviously, if ()–() hold and with then the following BVP: has at least one solution in .
In what follows, we will use the Schauder's fixed Point theorem to prove the existence of the solutions of BVP (1). We first list the following conditions. Let be continuous, and there exist and such that The function is continuous and for all . Moreover, there exist and such that Let be a continuous functional, and there exist and such that
Theorem 13. Assume that (A1)–(A3) and (H3) hold. Then the BVP (1) has at least one solution in .
Proof. Define a bounded set , where
Now we are going to show that . From the assumptions and , we have
By using of and , we have
Hence, . Similarly to the proof of Theorem 12, it can be shown that the operator is completely continuous. According to the Schauder's fixed point theorem, has at least a fixed point which is a solution of the problem (1).
The proof is completed.
In this section, an example is given to illustrate our results.
Example 1. Consider the nonlocal boundary value problem for nonlinear fractional differential equation as follows where . Then (58) has at least one solution.
Proof. It is easy to see that (58) is a form of (1) in the space . We have and Obviously, , , and is continuous functional. Moreover, satisfy and . Then, we get , , , and such that conditions hold. Moreover, Hence, (58) satisfies all assumptions of Theorem 12. As a result, (58) has at least one solution.
In this paper, we study the existence solutions of nonlinear Caputo fractional integrodifferential equations with -Laplacian operator and nonlocal boundary conditions in Banach spaces. We mainly consider the equivalent integral equations of corresponding fractional differential equations. By using via Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, we obtain some new existence results for this kind of nonlocal boundary value problems (1). An illustrative example is also discussed to show the effectiveness of the results in this paper. In the near future, we will consider Riemann-Liouville Fractional nonlocal boundary problems, which will be more complicated.
This project is supported by NNSF of China Grants nos. 11271087, 61263006, Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116, the open fund of Guangxi Key laboratory of Hybrid Computation and IC Design Analysis no. 2012HCIC07, and the Innovation Project of Guangxi University for Nationalities no. gxun-chx2012096.
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