Solvability of Some Boundary Value Problems for Fractional -Laplacian Equation
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.
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 , charge transport in amorphous semiconductors , and propagations of mechanical waves in viscoelastic media , 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.  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.  considered a two-point boundary value problem at nonresonance, and Bai  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  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  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.  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 .
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 .
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.
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 ). Let . Assume that . Then the following equality holds: where , ; here is the smallest integer greater than or equal to .
Lemma 4 (see ). 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
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
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.
Our first result, based on Schaefer’s fixed point theorem and Lemma 5, is stated as follows.
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.
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.
This work was supported by the Fundamental Research Funds for the Central Universities (2012QNA50) and the National Natural Science Foundation of China (11271364).
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