- Open Access
Existence results for fractional differential inclusions with three-point fractional integral boundary conditions
© Fu; licensee Springer. 2013
- Received: 15 May 2013
- Accepted: 16 October 2013
- Published: 8 November 2013
This paper is concerned with fractional differential inclusions with three-point fractional integral boundary conditions. We consider the fractional differential inclusions under both convexity and nonconvexity conditions on the multivalued term. Some new existence results are obtained by using standard fixed point theorems. Two examples are given to illustrate the main results.
MSC:34A60, 26A33, 34B15.
- fractional differential inclusions
- boundary value problems
- existence results
- multivalued maps
Fractional differential equations have recently gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many physical phenomena [1–3]. For some recent developments on the existence results of fractional differential equations, we can refer, for instance, to [4–17] and the references therein.
Differential inclusions arise in the mathematical modeling of certain problems in economics, optimal control, etc. and are widely studied by many authors, see [18, 19] and the references therein. For some recent works on differential inclusions of fractional order, we refer the reader to the references [4, 5, 20–29].
where denotes the Caputo fractional derivative of order p, the Riemann-Liouville fractional integral of order q, is a multifunction and a, b, c are real constants with .
We remark that when , and third variable of the function F in (1) vanishes, problem (1) reduces to a three-point fractional integral boundary value problem (see  with a given continuous function).
The rest of this paper is organized as follows. In Section 2 we present the notations, definitions and give some preliminary results that we need in the sequel, Section 3 is dedicated to the existence results of problem (1), in the final Section 4, two examples are given to illustrate the main results.
In this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper.
Let be a normed space. We use the notations: , , , , and so on.
A multivalued map is convex (closed) valued if is convex (closed) for all . F is said to be completely continuous if is relatively compact for every . F is called upper semicontinuous on X if, for every , the set is a nonempty closed subset of X, and for every open set O of X containing , there exists an open neighborhood U of x such that . Equivalently, F is upper semicontinuous if the set is open for any open set O of X. F is called lower semicontinuous if the set is open for each open set O in X. If a multivalued map F is completely continuous with nonempty compact values, then F is upper semicontinuous if and only if F has a closed graph, i.e., if and , then implies .
A multivalued map is said to be measurable if, for every , the function is a measurable function.
- (1)γ-Lipschitz if there exists such that
a contraction if it is γ-Lipschitz with .
is measurable for each ;
is upper semicontinuous for a.e. .
- (3)for each , there exists such that
for all , and a.e. .
The following lemmas will be used in the sequel.
Lemma 2.1 (see )
is a closed graph operator in .
Definition 2.3 ()
provided the integral exists.
Definition 2.4 ()
where denotes the integer part of the real number q.
Lemma 2.2 ()
here , , .
Substituting the values of , , we obtain the result. This completes the proof. □
Let us define what we mean by a solution of problem (1).
Let be the space of all continuous functions defined on . Define the space endowed with the norm . Obviously, is a Banach space.
Theorem 2.1 (Nonlinear alternative of Leray-Schauder type)
Let X be a Banach space, C be a closed convex subset of X, U be an open subset of C with . Suppose that is an upper semicontinuous compact map. Then either (1) F has a fixed point in , or (2) there are and such that .
Theorem 2.2 (Covitz and Nadler)
Let be a complete metric space. If is a contraction, then F has a fixed point.
In this section, three existence results of problem (1) are presented. The first one concerns the convex valued case, and the others are related to the nonconvex valued case.
Now let us begin with the convex valued case.
Theorem 3.1 Suppose that the following (H1), (H2) and (H3) are satisfied.
(H1) is a Carathéodory multivalued map.
for , .
Then boundary value problem (1) has at least one solution on .
Clearly, by Lemma 2.3, we know that the fixed points of N are solutions of problem (1). From (H1) and (H2), we have, for each , that the set is nonempty . Next we will show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof is given in the following five steps.
Step 1: is convex valued. Since F is convex valued, we know that is convex and therefore it is obvious that for each , is convex.
independently of and .
Step 4: N has a closed graph. Let , and , we need to show that . Since , there exists such that for . We must prove that there exists such that for .
for some . This implies that .
As a consequence of Steps 1-4, together with the Arzela-Ascoli theorem, we can obtain that is an upper semicontinuous and completely continuous map. From the choice of U, there is no such that for some . Hence, by Theorem 2.1, we deduce that N has a fixed point which is a solution of problem (1). This is the end of the proof. □
Next we study the case when F is not necessarily convex valued.
Let A be a subset of . A is measurable if A belongs to the σ-algebra generated by all sets of the form , where J is Lebesgue measurable in and D is a Borel set of ℝ. A subset A of is decomposable if for all and Lebesgue measurable, then , where χ stands for the characteristic function.
Theorem 3.2 Let (H2) and (H3) hold and assume:
(H4) is such that: (1) is measurable; (2) the map is lower semicontinuous for a.e. .
Then problem (1) has at least one solution on .
with the boundary conditions in (2). Note that if is a solution of problem (7), then x is a solution to problem (1).
It can easily be shown that is continuous and completely continuous and satisfies all conditions of the Leray-Schauder nonlinear alternative for single-valued maps . By a discussion similar to the one in Theorem 3.1, Theorem 3.2 follows. □
Theorem 3.3 We assume that:
then problem (1) has at least one solution on .
Proof From (H5), for each , the multivalued map is measurable and closed valued. Hence it has measurable selection (Theorem 2.2.1 ) and the set is nonempty. Let N be defined in (5). We will show that N satisfies the requirements of Theorem 2.2.
This implies that and is closed.
Therefore from condition (8), Theorem 2.2 implies that N has a fixed point which is a solution of problem (1). This completes the proof. □
In this section, we give two examples to illustrate the results.
Thus, by the conclusion of Theorem 3.1, boundary value problem (10) has at least one solution on .
Hence it follows from Theorem 3.3 that problem (11) has at least one solution on .
The author carried out the proofs of the theorems and approved the final manuscript.
The author would like to express his thanks to the referees for their helpful suggestions.
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