Existence and Uniqueness of Solution to Two-Point Boundary Value for Two-Sided Fractional Differential Equations ()
1. 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)
(1.1)
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 [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
(1.2)
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 [1].
The left Riemann-Liouville fractional integrals (LFLI) of order
of function
which is defined as follows,
(1.3)
The right Riemann-Liouville fractional integrals (RFLI) of order
of function
which is defined as follows,
(1.4)
The left Riemann-Liouville fractional derivative (LFLD) of order
of function
which is defined as follows,
(1.5)
The right Riemann-Liouville fractional derivative (RFLD) of order
of function
which is defined as follows,
(1.6)
The left Caputo fractional derivative (LCFD) of order
of function
which is defined as follows,
(1.7)
The right Caputo fractional derivative (RCFD) of order
of function
which is defined as follows,
(1.8)
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 [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
or
. 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.
Let
,
, and
. If
, then
(1.9)
We let
,
, and
, if
,
, then, by (1.9) and Remark 1.1, we have that
(1.10)
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
be of class
. Suppose that there exist two closed subspaces X and Y such that
and two continuous non-increasing functions
,
such that


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
.
2. Basic Facts
In [13], authors gave the definition of solution to (1.2) as following Definition 2.1. [13] A function
is called a solution of BVP (1.2) if
(i)
and
are derivative for almost every
, and (ii)
satisfies (1.2).
In [13], in order to establish a variational structure which enables ones to reduce the existence of solutions of BVP (1.2) to the one of critical points of corresponding functional, authors constructed an appropriate function spaces
, which depend on
-integrability of the Riemann-Liouville fractional derivative of a function.
Definition 2.2. [13] Let
,
. The fractional derivative space
is defined by the closure of
with respect to the norm
(2.1)
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. [13] Let
,
. The space
is a reflexive and separable Banach space.
Proposition 2.4. [13] Let
,
. For all
, if
or
, we have

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. [13] Let
,
and
,
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 [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
such that
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.
3. Main Result
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
(3.1)
(3.2)
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 [13], we know that
is a real Hilbert space with the inner product by
(3.3)
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
(3.4)
where
. From the arguments in [13], we know that
(3.5)
for
. By the assumptions and the analogy arguments with, we have that
(3.6)
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).
4. 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.
5. Acknowledgements
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).