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In this paper, existence and uniqueness of solution to two-point boundary value for two-sided fractional differential equations involving Caputo fractional derivative is discussed, by means of the Min-Max Theorem.

In this paper, using the Min-Max Theorem, we will devote to considering the existence and uniqueness result of solution to the following two-sided fractional differential equations boundary value problems (BVP for short)

where denote the right-side and left-side Caputo fractional derivative of order, respectively, is a continuous differential function with respect to all variables, and.

In particular, if, BVP (1.1) reduces to the standard second order boundary value problem of the following form

Recently, fractional differential equations have been verified to be valuable tools in the modeling of many phenomena in various fields of science and engineering. There have many papers which are concerning with the existence of solutions for fractional differential equations boundary value problems, by means of some classic fixed point theorems and monotone iterative methods, such as [1-11], etc. But, as far as we known, there are few papers which considered the existence of solutions for fractional differential equations boundary value problems using the variable method, such as the direct method, the critical point theory. Recently, there appeared some interesting works [12,13] considering existence of solution to fractional differential problems, by means of the variable way, In [

where and are the right and left RiemannLiouville fractional derivatives of order respectively, is a given function satisfying some assumptions and is the gradient of at. This is a very interesting and meaning works, this is the first time that the existence of solutions for fractional differential equation two-point boundary value problem via the critical point theory.

The following are definitions and some properties of Riemann-Liouville fractional integral and derivative, the Caputo fractional derivative, for the details, please see [

The left Riemann-Liouville fractional integrals (LFLI) of order of function which is defined as follows,

The right Riemann-Liouville fractional integrals (RFLI) of order of function which is defined as follows,

The left Riemann-Liouville fractional derivative (LFLD) of order of function which is defined as follows,

The right Riemann-Liouville fractional derivative (RFLD) of order of function which is defined as follows,

The left Caputo fractional derivative (LCFD) of order of function which is defined as follows,

The right Caputo fractional derivative (RCFD) of order of function which is defined as follows,

Remark 1.1. Obviously, if, then

; it, then

.

It is well known that there are several kinds of fractional derivatives, such as, Riemann-liouville fractional derivative, Marchaud fractional derivative, Caputo derivative, Griinwald-Letnikov fractional derivative, etc. Since as cited in [

The following is the rule of fractional integration by parts for LFLI and RFLI.

Let, , and. If

, then

We let, , and, if

,

, then, by (1.9) and Remark 1.1, we have that

Inspired by [12,13], in this paper, we will consider the unique existence of solution to problem (1.1), by means of the following Min-Max Theorem.

Min-Max Theorem (Manasevich). [

for all and, and

for all and. Then 1) there exists a unique such that

;

2)

Here, and denote the gradient and the Hessian of at, respectively. In this case,

is a mapping and is a bounded self-adjoins linear operator on.

In [

(i) and are derivative for almost every, and (ii) satisfies (1.2).

In [

Definition 2.2. [

It is obvious that space is the space of functions having an -order fractional derivative and

. Furthermore, it is easy to verify that is a reflexive and separable Banach space.

Theorem 2.3. [

Proposition 2.4. [

with this property, one can consider with respect to the norm

If, the following theorem is useful for us to establish the variational structure on the space for BVP (1.2).

Theorem 2.5. [

, be measurable in t for each and continuously differentiable in for almost every. If there exists; and

;, such that, for a.e.

and every, one has

where, then the functional defined by

is continuously differentiable on, and

, we have

From, we known that, for a solution

of BVP (1.2) such that, multiplying (1.2) by yields

According these facts, authors [

Definition 2.6. [

and satisfies the above equality for all.

Using the direct method and the Mountain pass theorem, authors obtain two existence results of weak solution to (1.2), please see [

Basing on some deductions, authors verified that a weak of (1.2) is also its solution.

From the Remark 1.1 and Definition 2.2, we will use function space in the following arguments.

Theorem 3.1. Assume that is continuous differentiable with respect to its two variablesthere is a constant such that

for all. Then problem (1.1) exists unique solution.

Proof. We can decompose (1.1) into the following two problems

according to the linearity of and, we can easily know that if are solution of (3.1), (3.2), respectively, then is a solution of (1.1). Obviously, is unique solution of (3.2). Next, we will verify that (3.1) exists unique solution, by means of the Min-Max Theorem (Manasevich).

From [

It follows from assumptions on function that we can easily know that g satisfies assumption of Theorem 2.5.

We let, clearly, we have . From the Algebra knowledge, it is well know that and are closed subsets of From the previous arguments, we can complete this proof through two steps.

The first step, we will consider the existence of critical point of functional defined as following

where. From the arguments in [

for. By the assumptions and the analogy arguments with, we have that

for.

For all and, by Proposition 2.4, we have that

For all and, we have that

which implies that

holds for all and. Obviously, functions satisfy assumption conditions of the Min-Max Theorem. Hence, the MinMax Theorem assures that there exists unique such that, which means that is a unique weak solution of (3.1).

It follows from that, hence the left Caputo fractional equal to the left Riemann-Liouville fractional derivative. Hence, by the similar proofs of lemma theorem, we know that this weak solution is also a solution of (3.1). Thus, we obtain that is unique solution of (1.1).

In this paper, using a Min-Max Theorem (Manasevich), we considered the existence and uniqueness of solution to some class of two-sided fractional differential equations with two-point boundary value problems.

The authors would like to thank those listed references for their helpful suggestions, which helped to improve the quality of the paper. This research supported by 2013 Science and Technology Research Project of Beijing Municipal Education Commission (KM201310016001) and 2011 Science and Technology Research Project of Beijing Municipal Education Commission (KM2011100160 12).