Nonlinear Functional Analysis of Boundary Value Problems: Novel Theory, Methods, and ApplicationsView this Special Issue
Research Article | Open Access
Multiple Solutions for a Class of Fractional Boundary Value Problems
We study the multiplicity of solutions for the following fractional boundary value problem: where and are the left and right Riemann-Liouville fractional integrals of order , respectively, is a real number, is a given function, and is the gradient of at . The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.
In this paper, we consider the fractional boundary value problem of the following form: where and are the left and right Riemann-Liouville fractional integrals of order , respectively, is a real number, is a given function, and is the gradient of at .
Fractional calculus and fractional differential equations can find many applications in various fields of physical science such as viscoelasticity, diffusion, control, relaxation processes, and modeling phenomena in engineering, see [1–12]. Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by use of techniques of nonlinear analysis, such as fixed-point theory (including Leray-Schauder nonlinear alternative) (see [13, 14]), topological degree theory (including coincidence degree theory) (see [15, 16]), and comparison method (including upper and lower solutions methods and monotone iterative method) (see [17, 18]). However, it seems that the popular methods mentioned above are not appropriate for discussing and , as the equivalent integral equation is not easy to be obtained.
In the past, there were investigations of the eigenvalue problems for fractional differential equations. For more detailed information on this topic, we refer to Zhang et al. [19–21], Wang et al. [22, 23], and Jiang et al. .
Recently, there are many papers dealing with the existence of solutions for problem . In , Jiao and Zhou obtained the existence of solutions for by mountain pass theorem under the Ambrosetti-Rabinowitz condition. Chen and Tang  studied the existence and multiplicity of solutions for the system when the nonlinearity is superquadratic, asymptotically quadratic, and subquadratic, respectively.
But so far, few papers discuss the two solutions of the system via critical point theory. The aim of the present paper is to study the existence of at least two solutions for the system as the parameter for some constant .
The paper is organized as follows. We first introduce some basic preliminary results and a well-known lemma in Section 2, including the fractional derivative space , where . In Section 3, we give the main result and its proof. In Section 4, we give the summary of this paper.
In this section, we recall some related preliminaries and display the variational setting which has been established for our problem.
Definition 2.1 (see ). Let be a function defined on and . The left and right Riemann-Liouville fractional integrals of order for function denoted by and , respectively, are defined by provided the right-hand sides are pointwise defined on , where is the gamma function.
Definition 2.2 (see ). Let be a function defined on . The left and right Riemann-Liouville fractional derivatives of order for function denoted by and , respectively, are defined by where , , and .
The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belong-ing to the space of absolutely continuous functions, which we denote by . is the space of functions such that . In particular, .
Definition 2.3 (see ). Let and . If and , then the left and right Caputo fractional derivatives of order for function denoted by and , respectively, which exist a.e. on . and are represented by respectively, where .
Definition 2.4 (see ). Define and . The fractional derivative space is defined by the closure of with respect to the norm where denotes the set of all functions with . It is obvious that the fractional derivative space is the space of functions having an -order Caputo fractional derivative and .
Proposition 2.5 (see ). Let and . The fractional derivative space is a reflexive and separable space.
Proposition 2.7 (see ). Define and . Assume that , and the sequence converges weakly to , that is, . Then in , that is, , as .
Making use of Definition 2.3, for any , problem is equivalent to the following problem: where .
In the following, we will treat problem in the Hilbert space with the corresponding norm . It follows from [25, Theorem 4.1] that the functional given by is continuously differentiable on . Moreover, for , we have
Definition 2.8 (see ). A function is called a solution of if
(i) is derivative for a.e. ,
(ii) satisfies , where .
Proposition 2.9 (see ). If , then for any , one has
Proposition 2.10 (see ). Let be satisfied. If , then the functional denoted by is convex and continuous on .
In order to prove the existence of two solutions for problem , firstly, we recall some well-known results. Their proofs can be found in many books. Please refer to the references and its references therein.
Lemma 2.11 (see ). If is a Banach space, , , and , such that and and if satisfies the PS condition, with then is a critical value of .
3. The Main Result and Proof of the Theorem
Our hypotheses on nonsmooth potential are as follows.
: is a function such that a.e. on and satisfies the following facts:(i)for all , is measurable,(ii)for a.a. , is continuously differentiable,(iii) there exist and , such that (iv)there exist and , such that (v)there exist , , and , such that where .
Remark 3.1. It is easy to verify that satisfies the whole assumption in [25, Theorem 4.1]. So, is a solution of the corresponding Euler equation , then is a solution of problem which, of course, corresponds to the solution of problem . For great details, please see [25, Theorem 4.2].
Proof. The proof is divided into four steps as follows.
Step 1. We will show that is coercive in this step.
Firstly, by (iii), (2.6), and (2.10), we have where .
Step 2. We will show that the is weakly lower semicontinuous.
Let weakly in , and by Proposition 2.7, we obtain the following results:
By Fatou's lemma, On the other hand, by Proposition 2.10, we have , that is,
Hence, by the Weierstrass theorem, we deduce that there exists a global minimizer such that
Step 3. We will show that there exists such that for each , .
By the condition by (v), there exists such that , a.e. . It is clear that
Now we denote where , and is given in the condition (v). A simple calculation shows that the function is positive whenever and . Thus, is well defined and .
We will show that for each , the problem has two nontrivial solutions. In order to do this, for , let us define
Then and Hence, By (iii) and (v), we have
For , by Proposition 2.9, we have so that whenever .
Step 4. We will check the PS condition in the following.
Suppose that such that and .
Since is coercive and is bounded in and passed to a subsequence, which still denote , we may assume that there exists , such that weakly in ; thus, we have as . Moreover, according to Proposition 2.7, we have , as . Observing that combining this with (3.17), it is easy to verify that , as , and hence that in . Thus, satisfies the PS condition.
From the mean value theorem and (v), we have for all , a.e. .
It follows from the conditions (iii) and all and a.e. that and this together with (3.19) yields that for all and a.e. , for some positive constant .
For all , , and , we have
So, for small enough, there exists a such that and . So by the mountain pass theorem (cf. Lemma 2.11), we can get which satisfies
Therefore, is another nontrivial critical point of .
So far, the results involved potential functions exhibiting sublinear. The next theorem concerns problems where the potential function is superlinear.
Our hypotheses on nonsmooth potential are as follows.
: is a function such that a.e. on and satisfies the following facts:(i)for all , is measurable,(ii)for a.a. , is continuously differentiable,(iii)there exist and , such that (iv)there exist and , such that (v)there exist , , and , such that , a.e. ,where ,(vi)for a.a. and all , we have
Proof. The steps are similar to those of Theorem 3.2. In fact, we only need to modify Step 1 and Step 4 as follows: 1′ shows that is coercive under the condition (vi); 4′ shows that there exists a second nontrivial solution under the conditions (iii) and (iv). Then from Steps 1′, 2, 3, and 4′ above, problem has at least two nontrivial solutions.
Step 1'. By (vi), for all , , we have where .
Step 4'. Because of hypothesis (iii), we have for a.e. and all with .
Combining (3.19) and (3.29), it follows that for a.e. and all .
Thus, for all , and , we have where and are positive constants.
So, for small enough, there exists a such that and .
Arguing as in proof of Step 4 of Theorem 3.2, we conclude that satisfies the PS condition. So by the mountain pass theorem (cf. Lemma 2.11), we can get that satisfies
Therefore, is another nontrivial critical point of .
(A) If , then . Therefore, by Theorems 3.2 and 3.4, we actually obtain the existence of two weak solutions of the following eigenvalue problem: where is a real number, is a given function, and is the gradient of at . Although many excellent results have been worked out on the existence of solutions for second-order system (e.g., [28, 29]), it seems that no similar results were obtained in the literature for fractional system .
(B) We give an example in the following to illustrate our viewpoint in Remark 3.1. We consider where , , for all .
Obviously, hypotheses (i), (ii), and (v) are satisfied. Moreover,
Therefore, So, condition (iii) holds.
On the other hand, uniformly for a.a. , so condition (iv) holds.
(C) We can find the following potential functions satisfying the conditions stated in Theorem 3.4: where , for all .
It is clear that () for a.e. , and hypotheses (i) and (ii) are satisfied. A direct verification shows that conditions (v) and (vi) are satisfied. Note that So, which shows that assumptions (iii) and (iv) are fulfilled.
This work was supported by the National Natural Science Foundation of China (nos. 11126286, 11001063, and 11201095), the Fundamental Research Funds for the Central Universities (no. 2012), China Postdoctoral Science Foundation funded project (no. 20110491032), and China Postdoctoral Science (Special) Foundation (no. 2012T50325).
- R. Goreno and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.
- B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008.
- W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.
- D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000.
- 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., pp. 291–348, Springer, Wien, Austria, 1997.
- J. W. Kirchner, X. Feng, and C. Neal, “Frail chemistry and its implications for contaminant transport in catchments,” Nature, vol. 403, no. 6769, pp. 524–527, 2000.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993.
- S. Aizicovici, N. S. Papageorgiou, and V. Staicu, “Multiple nontrivial solutions for nonlinear periodic problems with the -Laplacian,” Journal of Differential Equations, vol. 243, no. 2, pp. 504–535, 2007.