- Open Access
Existence theorems for nonlocal multivalued Hadamard fractional integro-differential boundary value problems
© Ahmad et al.; licensee Springer. 2014
- Received: 30 May 2014
- Accepted: 27 October 2014
- Published: 11 November 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.
- fractional differential inclusions
- nonlocal boundary conditions
- fixed point theorems
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].
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 ()
where denotes the integer part of the real number q and .
Definition 2.2 ()
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 ()
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
is measurable for each ;
is upper semi-continuous for almost all ;
- (iii)for each , there exists such that
for all and for a.e. .
For the forthcoming analysis, we need the following lemmas.
Lemma 3.2 (Nonlinear alternative for Kakutani maps) 
F has a fixed point in , or
there is a and with .
Lemma 3.3 ()
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;
Then the boundary value problem (1.1) has at least one solution on .
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.
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.
for some .
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.
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:
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 .
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.
where and . Then is a metric space and is a generalized metric space (see ).
- (a)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 .
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 .
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. □
we find that . Hence by Theorem 3.4 the problem (3.2) has a solution on .
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.
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