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Existence theorems for nonlocal multivalued Hadamard fractional integro-differential boundary value problems
Journal of Inequalities and Applications volume 2014, Article number: 454 (2014)
In this paper, we study the existence of solutions for a boundary value problem involving Hadamard type fractional differential inclusions and integral boundary conditions. Some new existence results for convex as well as non-convex multivalued maps are obtained by using standard fixed point theorems for multivalued maps. The paper concludes with an illustrative example.
The intensive development of fractional calculus and its widespread applications in several disciplines clearly indicate the interest of researchers and modelers in the subject. As a matter of fact, the tools of fractional calculus have been effectively used in applied and technical sciences such as physics, mechanics, chemistry, engineering, biomedical sciences, control theory, etc. It has been mainly due to the fact that fractional-order operators can exhibit the hereditary properties of many materials and processes. For a detailed account of applications and recent results on initial and boundary value problems of fractional differential equations and inclusions, we refer the reader to a series of books and papers [1–13]. However, it has been noticed that most of the work on the topic involves Riemann-Liouville and Caputo type fractional differential operators. Another class of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard, introduced in 1892 . This derivative contains logarithmic function of arbitrary exponent in the kernel of the integral in its definition. Preliminary concepts and properties of Hadamard fractional derivative and integral can be found in [2, 15–22].
In this paper, we study the following boundary value problem with an integral nonlocal boundary condition:
where is the Hadamard fractional derivative of order α, is the Hadamard fractional integral of order γ, is a multivalued map, is the family of all subsets of ℝ and A, B, c are real constants. Further, it is assumed that .
The present paper is motivated by a recent paper of the authors , where problem (1.1) was considered for a single-valued case.
We establish some existence results for the problem (1.1), when the right-hand side is convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semi-continuous multivalued maps with nonempty closed and decomposable values, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. The methods used are well known, however, their exposition in the framework of problem (1.1) is new.
Definition 2.1 ()
The Hadamard derivative of fractional order q for a function is defined as
where denotes the integer part of the real number q and .
Definition 2.2 ()
The Hadamard fractional integral of order q for a function g is defined as
provided the integral exists.
Definition 2.3 A function is called a solution of problem (1.1) if there exists a function with , a.e. such that , , a.e. and , , .
Lemma 2.4 ()
Given , the unique solution of the problem
is given by
Remark 2.5 Observe that solution (2.2) for corresponds to the one for a boundary value problem of a Cauchy-Euler type differential equation:
3 Existence results
For a normed space , let , , , and . A multivalued map is convex (closed) valued if is convex (closed) for all . The map G is bounded on bounded sets if is bounded in X for all (i.e. ). G is called upper semi-continuous (u.s.c.) on X if for each , the set is a nonempty closed subset of X, and if for each open set N of X containing , there exists an open neighborhood of such that . G is said to be completely continuous if is relatively compact for every . If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., , , imply . G has a fixed point if there is such that . The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map is said to be measurable if for every , the function
Let denote a Banach space of continuous functions from into ℝ with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
3.1 The Carathéodory case
Definition 3.1 A multivalued map is said to be Carathéodory if
is measurable for each ;
is upper semi-continuous for almost all ;
Further a Carathéodory function F is called -Carathéodory if
for each , there exists such that
for all and for a.e. .
For each , define the set of selections of F by
For the forthcoming analysis, we need the following lemmas.
Lemma 3.2 (Nonlinear alternative for Kakutani maps) 
Let E be a Banach space, C a closed convex subset of E, U an open subset of C and . Suppose that is a upper semi-continuous compact map. Then either
F has a fixed point in , or
there is a and with .
Lemma 3.3 ()
Let X be a Banach space. Let be an -Carathéodory multivalued map and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
Now we are in a position to prove the existence of the solutions for the boundary value problem (1.1) when the right-hand side is convex valued.
Theorem 3.4 Assume that:
(H1) is Carathéodory and has nonempty compact and convex values;
(H2) there exists a continuous nondecreasing function and a function such that
(H3) there exists a constant such that
Then the boundary value problem (1.1) has at least one solution on .
Proof Define the operator by
for . We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that is convex for each . This step is obvious since is convex (F has convex values), and therefore we omit the proof.
In the second step, we show that maps bounded sets (balls) into bounded sets in . For a positive number r, let be a bounded ball in . Then, for each , , there exists such that
Then, for , we have
Now we show that maps bounded sets into equicontinuous sets of . Let with and . For each , we obtain
Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that for each ,
Thus it suffices to show that there exists such that for each ,
Let us consider the linear operator given by
Thus, it follows by Lemma 3.3 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
Finally, we show there exists an open set with for any and all . Let and . Then there exists with such that, for , we have
Using the computations of the second step above we have
which implies that
In view of (H3), there exists M such that . Let us set
Note that the operator is upper semi-continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that has a fixed point which is a solution of the problem (1.1). This completes the proof. □
3.2 The lower semi-continuous case
As a next result, we study the case when F is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo for lower semi-continuous maps with decomposable values.
Let X be a nonempty closed subset of a Banach space E and be a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set is open for any open set B in E. Let A be a subset of . A is measurable if A belongs to the σ-algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in ℝ. A subset of is decomposable if for all and measurable , the function , where stands for the characteristic function of .
Definition 3.5 Let Y be a separable metric space and let be a multivalued operator. We say N has the property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with F as
which is called the Nemytskii operator associated with F.
Definition 3.6 Let be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator ℱ is lower semi-continuous and has nonempty closed and decomposable values.
Lemma 3.7 ()
Let Y be a separable metric space and let be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) such that for every .
Theorem 3.8 Assume that (H2), (H3), and the following condition holds:
(H4) is a nonempty compact-valued multivalued map such that
is lower semi-continuous for each .
Then the boundary value problem (1.1) has at least one solution on .
Proof It follows from (H2) and (H4) that F is of l.s.c. type. Then from Lemma 3.7, there exists a continuous function such that for all .
Consider the problem
Observe that if is a solution of (3.1), then x is a solution to the problem (1.1). In order to transform the problem (3.1) into a fixed point problem, we define the operator as
It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.4. So we omit it. This completes the proof. □
3.3 The Lipschitz case
Now we prove the existence of solutions for the problem (1.1) with a non-convex valued right-hand side by applying a fixed point theorem for a multivalued map due to Covitz and Nadler.
Let be a metric space induced from the normed space . Consider given by
where and . Then is a metric space and is a generalized metric space (see ).
Definition 3.9 A multivalued operator is called:
γ-Lipschitz if and only if there exists such that
a contraction if and only if it is γ-Lipschitz with .
Lemma 3.10 ()
Let be a complete metric space. If is a contraction, then .
Theorem 3.11 Assume that:
(H5) is such that is measurable for each .
(H6) for almost all and with and for almost all .
Then the boundary value problem (1.1) has at least one solution on if
Proof Observe that the set is nonempty for each by the assumption (H5), so F has a measurable selection (see Theorem III.6 ). Now we show that the operator , defined in the beginning of proof of Theorem 3.4, satisfies the assumptions of Lemma 3.10. To show that for each , let be such that () in . Then and there exists such that, for each ,
As F has compact values, we pass onto a subsequence (if necessary) to obtain that converges to v in . Thus, and for each , we have
Next we show that there exists such that
Let and . Then there exists such that, for each ,
By (H6), we have
So, there exists such that
Since the multivalued operator is measurable (Proposition III.4 ), there exists a function which is a measurable selection for U. So and for each , we have .
For each , let us define
Analogously, interchanging the roles of x and , we obtain
Since is a contraction, it follows by Lemma 3.10 that has a fixed point x which is a solution of (1.1). This completes the proof. □
Example 3.12 Consider the problem
Here , , , , , and . With the given values, we find that
Let be a multivalued map given by
For , we have
with , . In this case by the condition
we find that . Hence by Theorem 3.4 the problem (3.2) has a solution on .
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This paper was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.
The authors declare that they have no competing interests.
Each of the authors, BA, SKN, and AA contributed to each part of this work equally and read and approved the final version of the manuscript.