Existence and Uniqueness of Solution to Two-Point Boundary Value for Two-Sided Fractional Differential Equations

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.


Introduction
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 0 T D D      denote the right-side and left-side Caputo fractional derivative of order 0  1 is a continuous differential function with respect to all variables, and .a b R   In particular, if 1   , 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][2][3][4][5][6][7][8][9][10][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 [13], by the critical point theory, author considered the existence of solutions of the following a twopoint boundary value problem for some class of fractional differential equation containing the left and right Riemann-Liouville fractional derivative operators where t T D  and 0 t D  are the right and left Riemann-Liouville fractional derivatives of order 0  1 is a given function satisfying some assumptions and is the gradient of F at x .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 [1].
The left Riemann-Liouville fractional integrals (LFLI) of order 0 3) The right Riemann-Liouville fractional integrals (RFLI) of order 0 The left Riemann-Liouville fractional derivative (LFLD) The right Riemann-Liouville fractional derivative The left Caputo fractional derivative (LCFD) of order 0 1 The right Caputo fractional derivative (RCFD) of or- 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 [2] there have appeared a number of works, especially in the theory of viscous elasticity and in hereditary solid mechanics, where fractional derivatives are used for a better description of material properties.Mathematical modeling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations.Applied problems require definitions of fractional derivatives allowing the utilization of physically in interpretable initial conditions, which contain .
, etc".In fact, the same requirements apply for boundary conditions.Therefore, we cannot impose initial and boundary conditions, such as on problems involving the Riemann-Liouville fractional derivative 0 t   u d e D  or t t D  .We find that Caputo fractional derivative exactly satisfies these demands.Therefore in this article, we deal with boundary value problem for fractional differential equation involving Caputo derivative.
The following is the rule of fractional integration by parts for LFLI and RFLI.
We let 0 1    , , and 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).[14] Let H be a real Hilbert space and let f H R   be of class .Suppose that there exist two closed subspaces X and Y such that for all x X y Y    and , and Here, f  and denote the gradient and the Hessian of

Basic Facts
In [13], authors gave the definition of solution to (1.2) as following Definition 2.1.[13] A function having an  -order fractional derivative and with this property, one can consider 0 p E   with respect to the norm , the following theorem is useful for us to establish the variational structure on the space 0 p E   for BVP (1.2).
Theorem 2.5.[13] Let1 where 1 1 1 p q   , then the functional defined by  is continuously differentiable on , and From, we known that, for a solution According these facts, authors [13] gave the definition of weak solution for BVP (1.2) as follows.
Definition 2.6.[13] By the weak solution of BVP (1.2), we mean that the function 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 [13].
Basing on some deductions, authors verified that a weak of (1.2) is also its solution.

Main Result
From the Remark 1.1 and Definition 2.2, we will use function space E  in the following arguments.
Theorem 3.1.Assume that is continuous differentiable with respect to its two variables, there is a constant Proof.We can decompose (1.1) into the following two problems according to the linearity of T D   and 0 D   , we can easily know that if 1 2 u u  are solution of (3.1), (3.2), respectively, then (3.2).Next, we will verify that (3.1) exists unique , by means of the Min-Max Theorem (Manasevich).
From [13], we know that E  is a real Hilbert space with the inner product by It follows from assumptions on function g that we can easily know that g satisfies assumption of Theorem 2.5.
We let are closed subsets of .E  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 argu- for u v E    .By the assumptions and the analogy arguments with, we have that For all x X y Y    and , by Proposition 2.4, we have that  u E   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).

Conclusion
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.

1 u
holds for all x X y Y    and .Obviously, func-Min-Max Theorem.Hence, the Min-Max Theorem assures that there exists unique 1It follows from 1 it is easy to verify that p E   is a reflexive and separable Banach space.