Existence of Solutions for Fractional Boundary Value Problem with Nonlinear Derivative Dependence

Abstract

We investigate the existence of solutions for fractional boundary value problem including both left and right fractional derivatives by using variational methods and iterative technique.

1. Introduction

Fractional differential equations appear naturally in a number of fields such as physics, chemistry, biology, economics, control theory, signal and image processing, and blood flow phenomena. During last decades, the theory of fractional differential equations is an area intensively developed, due mainly to the fact that fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes (see [1–4] and the references therein). Therein, the composition of fractional differential operators has got much attention from many scientists, mainly due to its wide applications in modeling physical phenomena exhibiting anomalous diffusion. Specifically, the models involving a fractional differential oscillator equation, which contains a composition of left and right fractional derivatives, are proposed for the description of the processes of emptying the silo [5] and the heat flow through a bulkhead filled with granular material [6], respectively. Their studies show that the proposed models based on fractional calculus are efficient and describe well the processes.

In the aspect of theory, the study of fractional boundary value problem including both left and right fractional derivatives has attracted much attention by using variational methods [7–12]. It is not easy to use the critical point theory to study the fractional differential equations including both left and right fractional derivatives, since it is often very difficult to establish a suitable space and a variational functional for the fractional boundary value problem.

For the first time, Jiao and Zhou in [7] showed that the critical point theory is an effective approach to tackle the existence of solutions for the following fractional boundary value problem: where and are the left and right Riemann-Liouville fractional derivatives of order , respectively, is a given function satisfying some assumptions, and is the gradient of at .

In [8], by performing variational methods combined with iterative technique, Sun and Zhang investigated the solvability of the following fractional boundary value problem: where , , , and denote left and right Riemann-Liouville fractional integrals of order , respectively, and is continuous.

Motivated by the above works and [13, 14], in this paper, we attempt to use Mountain Pass theorem and iterative technique to study the existence of solutions for the following nonlinear fractional boundary value problem with dependence on fractional derivative: where and are the left and right Riemann-Liouville fractional derivatives of order , respectively, and and with , for .

In particular, if , problem (3) reduces to the standard second order boundary value problem of the following form: where is a given function.

In fact, for problem (3), due to the appearance of left and right Riemann-Liouville fractional integral, we cannot deal with problem (3) by using fixed point theorems, because it is difficult to find the equivalent integral equation corresponding to problem (3). And since problem (3) is not variational, we cannot find some functional such that its critical point is the solution corresponding to problem (3). However, when there is no presence of the fractional derivative of the solution in the nonlinearity term and is a constant function, then problem (3) is studied by establishing corresponding variational structure in some suitable fractional space and applying the critical point theorems [7, 10, 12].

In order to use variational methods, we consider a family of the following fractional boundary value problem with no dependence on the fractional derivative of the solution; that is, for each (which is defined in Section 2), we consider the following problem: From [7, 10, 12], we know that problem (5) is variational and we can treat it by variational methods. Thus, for each , we can find a solution with some bounds. Next, by iterative methods we can show that there exists a solution for problem (3).

2. Preliminaries and Several Lemmas

In this section, we introduce some basic definitions of fractional calculus and several lemmas which are used further in this paper.

Definition 1 (see [3]). Let be a function defined on . The left and right Riemann-Liouville fractional integrals of order for function denoted by and , respectively, are defined by provided in both cases that the right-hand side is pointwise defined on .

Definition 2 (see [3]). Let be a function defined on . The left and right Riemann-Liouville fractional derivatives of order for function denoted by and , respectively, exist almost everywhere on , , and and are represented by

Proposition 3 (see [7, 10]). If , and , , or , , , then

Proposition 4 (see [7, 10]). If , , and , or , , and , then

In order to establish a variational structure which enables us to reduce the existence of solutions of problem (5) to one of finding critical points of corresponding functional, it is necessary to construct appropriate function spaces.

Let us recall that for any fixed and ,

Definition 5. Let . The fractional derivative space is defined by the closure of with respect to the weighted norm where denote by the set of all functions with .

Lemma 6 (see [7]). Let ; for all , one has(i) (ii)

Remark 7. Let , by (12) and (13); then

At this point, by (14), we can consider with respect to the norm which is equivalent to (11).

Similarly to some results in [7], the properties of the fractional derivative space are given, as follows.

Lemma 8. Let . The fractional derivative space is a reflexive and a separable Banach space.

Lemma 9. Assume that and the sequence converges weakly to in ; that is, . Then in ; that is, , as .

One is now in a position to give the definition for the solution of problem (3).

Definition 10. A function is called a solution of problem (3), if(i) and are derivatives for all ,(ii) satisfies (3).

Definition 11. A function is called a weak solution of problem (3), if for all .

Associated to the boundary value problem (5), for given , we have the functional defined by where and . Clearly, by the continuity hypothesis on and , we have and for all , Moreover, similarly to the proof of [7, Theorem 5.1], we know that the critical point of is a solution of problem (5).

3. Main Results

First, we make the following assumptions:, where as uniformly for , and ; denote .There exist constants and such that There exist constants and such that There exist positive constants such that  where and .For , , , the function satisfies the following Lipschitz conditions:  where and is a positive constant, which is given in (49).

We note that it obviously follows from and that for all , there exists a positive constant , independent of , such that

The main result of this paper is the following.

Theorem 12. Assume that hold and that . If there exist and such that then problem (3) has one nontrivial solution.

Proof. In order to prove Theorem 12, we proceed by three steps.
Step 1. Let ; with , which is given in (48), we show that has a nontrivial critical point in by the Mountain Pass theorem.
Firstly, by and (24) we have By Remarks (15), (18), (28), and Hölder inequality By (25) we can choose such that Hence, let ; with , we know that there exist , such that for , uniformly for .
Secondly, for given , with , it follows from and that Then we have that for Since , taking large enough and letting , then with .
Thirdly, we show that satisfies the Palais-Smale condition.
Let , such that We have Then by (33) and , where . Combining with , as , we know that is bounded in .
Since is a reflexive space, we can assume that . In , according to Remark 7 and Lemma 9 we have that is bounded in and . By the assumption , one gets Notice that Moreover, so , as . That is, converges strongly to in .
Obviously, ; therefore, by Mountain Pass theorem, has a nontrivial critical point in , with where .
Step 2. We construct iterative sequence and estimate its norm in .
We consider the solutions of the following problem: starting with . By iterative technique, we can get a sequence of , the nontrivial critical points obtained by Step 1, provided that there exists some constant , such that .
In the following, we estimate the norm of by the induction method. In fact, we need to prove that if we assume that for some , , then , the nontrivial critical point of , also satisfies .
By the Mountain Pass characterization of the critical level, and and Cauchy’s inequality with positive constant , which is given in the sequel, it follows that Let and since , can achieve its maximum at some . Hence By (35), , and , we have where is independent of . For convenience, we denote , , and ; then (44) can be written as which implies Let together with the assumption ; by simple calculation, for enough small we obtain Hence we can choose , such that .
Step 3. We show that the iterative sequence constructed in Step 2 is convergent to a nontrivial solution of problem (3).
By Step 2, we know , and by Remark 7, there exists positive constant , such that By (19), , and , we obtain hence By it follows that hence Since , we know that is a Cauchy sequence in , so there exists a such that converges strongly to in .
In order to show that is a solution of problem (3), we need to prove that It suffices to show that Indeed, it follows from the assumption that Then, is a solution of (3), and since for , we obtain that is a nontrivial solution of problem (3).

4. Example

In this section, we provide an explicit example to illustrate our main results.

Example 1. Consider the following fractional boundary value problem: Obviously, , , , and . Let , , and , for all , ; it is not difficult to verify that function satisfies the assumptions .
For , for all , , , it follows that Then we choose , , where . Since , it obviously follows that and that (25) is valid. It suffices to verify that From (48), now we estimate the value of , where is dependent on and Since , we can choose , such that with .
By direct calculation, we have that ; moreover, Then ; by the arbitrariness of and , we can choose enough small , , such that , , , and .
Let ; we get Then all conditions of Theorem 12 are satisfied. Consequently we obtain that problem (57) has a nontrivial solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

Wenzhe Xie and Jing Xiao contributed equally to this work.

Acknowledgments

This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (13K029), the Innovation Fund Project For Graduate Student of Hunan Province (CX2013B219), partially supported by the National Natural Science Foundation of China (61170320), and Nature Science Foundation of Guangdong Medical College (B2012053).