Abstract and Applied Analysis

Volume 2013 (2013), Article ID 432509, 6 pages

http://dx.doi.org/10.1155/2013/432509

Research Article

Solvability of Some Boundary Value Problems for Fractional -Laplacian Equation

Taiyong Chen and Wenbin Liu

Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, China

Received 4 July 2013; Accepted 9 September 2013

Academic Editor: Chuanzhi Bai

Copyright © 2013 Taiyong Chen and Wenbin Liu. 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.

This paper considers the existence of solutions for two boundary value problems for fractional -Laplacian equation. Under certain nonlinear growth conditions of the nonlinearity, two new existence results are obtained by using Schaefer's fixed point theorem. As an application, an example to illustrate our results is given.

1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger. This subject, as old as the problem of ordinary differential calculus, can go back to the times when Leibniz and Newton invented differential calculus. As is known to all, the problem for fractional derivative was originally raised by Leibniz in a letter, dated September 30, 1695. A fractional derivative arises from many physical processes, such as a non-Markovian diffusion process with memory [1], charge transport in amorphous semiconductors [2], and propagations of mechanical waves in viscoelastic media [3], and so forth. Moreover, phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are also described by differential equations of fractional order [4–8]. For instance, Pereira et al. [9] considered the following fractional Van der Pol equation:

where is the fractional derivative of order and is a control parameter that reflects the degree of nonlinearity of the system. Equation (1) is obtained by substituting the capacitance by a fractance in the nonlinear RLC circuit model.

Recently, fractional differential equations have been of great interest due to the intensive development of the theory of fractional calculus itself and its applications. For example, for fractional initial value problems, the existence and multiplicity of solutions (or positive solutions) were discussed in [10–13]. On the other hand, for fractional boundary value problems (FBVPs), Agarwal et al. [14] considered a two-point boundary value problem at nonresonance, and Bai [15] considered a -point boundary value problem at resonance. For more papers on FBVPs, see [16–21] and the references therein.

The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson [22] introduced the -Laplacian equation as follows:

where . Obviously, is invertible and its inverse operator is , where is a constant such that .

In the past few decades, many important results relative to (2) with certain boundary value conditions have been obtained. We refer the reader to [23–27] and the references cited therein. For boundary value problems of fractional -Laplacian equations, Chen and Liu [28] considered an antiperiodic boundary value problem with the following form:

where , , , and is Caputo fractional derivative. Under certain nonlinear growth conditions of the nonlinearity, an existence result was obtained by using degree theory. In addition, Yao et al. [29] studied a three-point boundary value problem given by

where is the standard Riemann-Liouville derivative with , , , , and the constant is a positive number satisfying . The monotone iterative technique was applied to establish the existence results on multiple positive solutions in [29].

Motivated by the works mentioned previously, in this paper, we investigate the existence of solutions for fractional -Laplacian equation of the form

subject to either boundary value conditions

or

where , , , is a Caputo fractional derivative, and is continuous.

Note that the nonlinear operator reduces to the linear operator when and the additive index law

holds under some reasonable constraints on the function [30].

The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions, and lemmas. In Section 3, based on Schaefer’s fixed point theorem, we establish two theorems on existence of solutions for FBVP (5) and (6) (Theorem 7) and FBVP (5) and (7) (Theorem 8). Finally, in Section 4, an explicit example is given to illustrate the main results. Our results are different from those of bibliographies listed in the previous texts.

2. Preliminaries

For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory, which can be found, for instance, in [31, 32].

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 Caputo fractional derivative of order of a continuous function is given by

where is the smallest integer greater than or equal to , provided that the right side integral is pointwise defined on .

Lemma 3 (see [33]). Let . Assume that . Then the following equality holds:

where , ; here is the smallest integer greater than or equal to .

Lemma 4 (see [34]). For fixed , let one define

Then the equation has a unique solution .

In this paper, we take with the norm and with the norm . By means of the linear functional analysis theory, we can prove that is a Banach space.

3. Existence Results

In this section, two theorems on existence of solutions for FBVP (5) and (6) and FBVP (5) and (7) will be given under nonlinear growth restriction of .

As a consequence of Lemma 3, we have the following results that are useful in what follows.

Lemma 5. Given , the unique solution of

is

where

Proof. Assume that satisfies the equation of FBVP (13); then Lemma 3 implies that

From the boundary value condition , one has

Thus, we have

By condition , we get . The proof is complete.

Define the operator by

where is the Nemytskii operator defined by

Clearly, the fixed points of the operator are solutions of FBVP (5) and (6).

Lemma 6. Given , the unique solution of

is

where

here is a constant dependent on .

Proof. Assume that satisfies the equation of FBVP (21); then Lemma 3 implies that

From condition , one has

Based on Lemma 4, we know that (25) has a unique solution . Moreover, by the integral mean value theorem, there exists a constant such that , which implies that . Thus, we have

Hence

From condition , we get . The proof is complete.

Define the operator by

where and is the Nemytskii operator defined by (20). Clearly, the fixed points of the operator are solutions of FBVP (5) and (7).

Our first result, based on Schaefer’s fixed point theorem and Lemma 5, is stated as follows.

Theorem 7. Let be continuous. Assume that

there exist nonnegative functions such that

Then FBVP (5) and (6) has at least one solution, provided that

Proof. We will use Schaefer’s fixed point theorem to prove that has a fixed point. The proof will be given in the following two steps.

Step 1. is completely continuous.

Let be an open bounded subset. By the continuity of , we can get that is continuous and is bounded. Moreover, there exists a constant such that , for all , . Thus, in view of the Arzelà-Ascoli theorem, we need only to prove that is equicontinuous.

For and , we have

Since is uniformly continuous on , we can obtain that is equicontinuous. A similar proof can show that is equicontinuous. This, together with the uniform continuity of on , yields that is also equicontinuous.

Step 2 (priori bounds). Set

Now it remains to show that the set is bounded.

From Lemma 3 and boundary value condition , one has

Thus, we get

That is,

For , we have

So, from , we obtain that

which, together with and (35), yields that

In view of (30), from (38), we can see that there exists a constant such that

Thus, from (35), we get

Combining (39) with (40), we have

As a consequence of Schaefer’s fixed point theorem, we deduce that has a fixed point which is the solution of FBVP (5) and (6). The proof is complete.

Our second result, based on Schaefer’s fixed point theorem and Lemma 6, is stated as follows.

Theorem 8. Let be continuous. Suppose that holds; then FBVP (5) and (7) has at least one solution, provided that (30) is satisfied.

Proof. The proof work is similar to the proof of Theorem 7, so we omit the details.

4. An Example

In this section, we will give an example to illustrate our main results.

Example 1. Consider the following fractional -Laplacian equation:

Corresponding to (5), we get that , , , and

Choose , , and . By a simple calculation, we can obtain that , and

Obviously, (42) subject to boundary value conditions (6) (or (7)) satisfies all assumptions of Theorem 7 (or Theorem 8). Hence, FBVP (42) and (6) (or FBVP (42) and (7)) has at least one solution.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (2012QNA50) and the National Natural Science Foundation of China (11271364).

References

R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,” Physica A, vol. 278, no. 1-2, pp. 107–125, 2000.

H. Scher and E. W. Montroll, “Anomalous transit-time dispersion in amorphous solids,” Physical Review B, vol. 12, no. 6, pp. 2455–2477, 1975.

F. Mainardi, “Fractional diffusive waves in viscoelastic solids,” in Nonlinear Waves in Solids, J. L. Wegner and F. R. Norwood, Eds., Fairfield, 1995.

K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity,” in Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., Springer, Heidelberg, Germany, 1999.

L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991.

W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, pp. 46–53, 1995.

F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, New York, NY, USA, 1997.

R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.

E. Pereira, C. A. Monje, B. M. Vinagre, and F. Gordillho, “Matlab toolbox for the analysis of fractional order systems with hard nonlinearities,” in Proceedings of the First IFAC Workshop on Fractional Differentiation and Its Applications (FDA '04), pp. 214–219, Bordeaux, France, July 2004.

A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003.

D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems-I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems-II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002.

R. P. Agarwal, D. O'Regan, and S. Staněk, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 57–68, 2010.

Z. Bai, “On solutions of some fractional -point boundary value problems at resonance,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 37, pp. 1–15, 2010.

M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2391–2396, 2009.

Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3438–3443, 2010.

W. Jiang, “The existence of solutions to boundary value problems of fractional differential equations at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 1987–1994, 2011.

N. Kosmatov, “A boundary value problem of fractional order at resonance,” Electronic Journal of Differential Equations, no. 135, pp. 1–10, 2010.

X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009.

L. S. Leibenson, “General problem of the movement of a compressible fluid in a porous medium,” Izvestiia Akademii Nauk Kirgizskoĭ SSSR, pp. 7–10, 1983 (Russian).

T. Chen, W. Liu, and C. Yang, “Antiperiodic solutions for Liénard-type differential equation with -Laplacian operator,” Boundary Value Problems, vol. 2010, Article ID 194824, 12 pages, 2010.

D. Jiang and W. Gao, “Upper and lower solution method and a singular boundary value problem for the one-dimensional -Laplacian,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 631–648, 2000.

L. F. Lian and W. G. Ge, “The existence of solutions of -point -Laplacian boundary value problems at resonance,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 2, pp. 288–295, 2005.

B. Liu and J. S. Yu, “Existence of solutions for the periodic boundary value problems with -Laplacian operator,” Journal of Systems Science and Mathematical Sciences, vol. 23, no. 1, pp. 76–85, 2003.

J. J. Zhang, W. B. Liu, J. B. Ni, and T. Y. Chen, “Multiple periodic solutions of -Laplacian equation with one-side Nagumo condition,” Journal of the Korean Mathematical Society, vol. 45, no. 6, pp. 1549–1559, 2008.

T. Chen and W. Liu, “An anti-periodic boundary value problem for the fractional differential equation with a -Laplacian operator,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1671–1675, 2012.

S. Yao, G. Wang, Z. Li, and L. Yu, “Positive solutions for three-point boundary value problem of fractional differential equation with -Laplacian operator,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 376938, 7 pages, 2013.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.

R. Manásevich and J. Mawhin, “Periodic solutions for nonlinear systems with -Laplacian-like operators,” Journal of Differential Equations, vol. 145, no. 2, pp. 367–393, 1998.