- Open Access
On fractional differential inclusions with anti-periodic type integral boundary conditions
© Ahmad et al.; licensee Springer. 2013
Received: 4 October 2012
Accepted: 12 March 2013
Published: 10 April 2013
This paper investigates the existence of solutions for fractional differential inclusions of order with anti-periodic type integral boundary conditions by means of some standard fixed point theorems for inclusions. Our results include the cases when the multivalued map involved in the problem has convex as well as non-convex values. The paper concludes with an illustrative example.
The topic of fractional differential equations and inclusions has recently emerged as a popular field of research due to its extensive development and applications in several disciplines such as physics, mechanics, chemistry, engineering, etc. [1–5]. An important characteristic of a fractional-order differential operator, in contrast to its integer-order counterpart, is its nonlocal nature. This feature of fractional-order operators (equations) is regarded as one of the key factors for the popularity of the subject. As a matter of fact, the use of fractional-order operators in the mathematical modeling of several real world processes gives rise to more realistic models as these operators are capable of describing memory and hereditary properties. For some recent results on fractional differential equations, see [6–22] and the references cited therein, whereas some recent work dealing with fractional differential inclusions can be found in [23–28].
where denotes the Caputo derivative of fractional order q, denotes j th derivative of with , is a multivalued map, is the family of all subsets of ℝ, are given continuous functions and ().
The present work is motivated by a recent paper , where the authors considered (1.1) with F as a single-valued map. The existence of solutions for problem (1.1) has been discussed for the cases when the right-hand side is convex as well as non-convex valued. The first result is based on the nonlinear alternative of Leray-Schauder type, whereas the second result is established by combining the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values. In the third result, we use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. Though the methods used are well known, their exposition in the framework of problem (1.1) is new. We recall some preliminary facts about fractional calculus and multivalued maps in Section 2, while the main results are presented in Section 3.
2.1 Fractional calculus
where denotes the integer part of the real number ν.
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 , .
In the sequel, the following lemma plays a pivotal role.
Lemma 2.4 ()
2.2 Basic concepts of multivalued analysis
Let denote a normed space equipped with the norm . A multivalued map is
convex (closed) valued if is convex (closed) for all ;
bounded on bounded sets if is bounded in for all bounded sets B in , that is, ;
upper semi-continuous (u.s.c.) on if for each , the set is a nonempty closed subset of , and if for each open set of containing , there exists an open neighborhood of such that ;
completely continuous if is relatively compact for every bounded set B in .
Remark 2.5 If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, that is, , , imply that .
Definition 2.6 The multivalued map has a fixed point if there is such that . The fixed point set of the map is denoted by FixG.
Definition 2.8 A multivalued map is called Carathéodory if is measurable for each and is upper semicontinuous for almost all . A Carathéodory function is said to be -Carathéodory if, for each , there exists such that for all and for a.e. .
Let denote a nonempty closed subset of a Banach space E, and let be a multivalued operator with nonempty closed values. The map 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 2.9 Let Y be a separable metric space. A multivalued operator 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 2.10 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.
where and . Then is a metric space and is a generalized metric space (see ), where , and .
Definition 2.11 A multivalued operator is called γ-Lipschitz if and only if there exists such that for each and is a contraction if and only if it is γ-Lipschitz with .
3 Existence results
3.1 The Carathéodory case
We recall the following lemmas to prove the existence of solutions for problem (1.1) when the multivalued map F in (1.1) is of Carathéodory type.
Lemma 3.1 (Nonlinear alternative for Kakutani maps) 
F has a fixed point in , or
there is an and with .
Lemma 3.2 ()
is a closed graph operator in .
Theorem 3.3 Suppose 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 the 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, it follows by the Ascoli-Arzelá theorem that is completely continuous.
for some .
Note that the operator is upper semicontinuous 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 3.1), we deduce that has a fixed point which is a solution of problem (1.1). This completes the proof. □
3.2 The lower semicontinuous case
This section deals with 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.
Lemma 3.4 (Bressan and Colombo )
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 3.5 Assume that (H2), (H3), (H4) and the following condition hold:
is lower semicontinuous 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 3.4, 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 3.3. So, we omit it. This completes the proof. □
3.3 The Lipschitz case
Here we show the existence of solutions for problem (1.1) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler .
Lemma 3.6 ()
Let be a complete metric space. If is a contraction, then .
Theorem 3.7 Assume that the following conditions hold:
(A1) is such that is measurable for each ;
(A2) 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 3.6 that has a fixed point x which is a solution of (1.1). This completes the proof. □
and , , , .
we find that . Thus, all the conditions of Theorem 3.3 are satisfied. So, there exists at least one solution of problem (3.2) on .
The authors are grateful to the anonymous referees for their useful comments. The research of B. Ahmad and A. Alsaedi was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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