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Some existence results for differential inclusions of fractional order with nonlocal strip conditions
Advances in Difference Equations volume 2013, Article number: 181 (2013)
In this paper, the existence of solutions for differential inclusions of fractional order with nonlocal strip conditions is investigated. Our study includes two cases: (i) the multivalued map involved in the problem is not necessarily convex valued, (ii) the multivalued map consists of non-convex values. We combine the nonlinear alternative of Leray-Schauder type coupled with the selection theorem of Bressan and Colombo to establish the first result, while the second result relies on Wegrzyk’s fixed point theorem for generalized contractions.
MSC:34A08, 34B10, 34B15.
Nonlocal nonlinear boundary value problems of fractional differential equations and inclusions have received considerable attention, and a great deal of work concerning a variety of boundary conditions can be found in the recent literature on the topic. It has been due to the extensive applications of fractional calculus in numerous branches of physics, economics and technical sciences [1–5]. Fractional-order differential operators are found to be effective and realistic mathematical tools for the description of memory and hereditary properties of various materials and processes. For examples and details, we refer the reader to a series of papers [6–26] and the references therein.
In this paper, we discuss the existence of solutions for a boundary value problem of differential inclusions of fractional order with nonlocal strip conditions given by
where denotes the Caputo fractional derivative of order q, is a multivalued map, is a family of all nonempty subsets of ℝ, and satisfy the relation .
The present work is motivated by a recent paper  where the author studied problem (1.1) with as a single-valued mapping. We establish our existence results by means of the nonlinear alternative of Leray-Schauder type, the selection theorem of Bressan and Colombo for lower semi-continuous maps with decomposable values and Wegrzyk’s fixed point theorem for generalized contraction maps.
In this section, we present some basic concepts of multivalued maps and fixed point theorems needed in the sequel.
Let Y denote a normed space with the norm . A multivalued map is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if is bounded in Y for all bounded sets B in Y (i.e., ). is called upper semi-continuous (u.s.c.) on Y if for each , the set is a nonempty closed subset of Y, and if for each open set N of Y containing , there exists an open neighborhood of such that . is said to be completely continuous if is relatively compact for every bounded set B in Y. If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph (i.e., , , imply ). has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by .
Let denote the Banach space of all continuous functions from into ℝ with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by
Let E be a Banach space, X be a nonempty closed subset of E and let 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 D is Borel measurable in ℝ. A subset A of is decomposable if for all and measurable, the function , where stands for the characteristic function of .
Definition 2.1 If is a multivalued map with compact values and , then is of lower semi-continuous type if
is lower semi-continuous with closed and decomposable values.
Let be a metric space associated with the metric d. The Pompeiu-Hausdorff distance of the closed subsets is defined by
Definition 2.2 
A function is said to be a strict comparison function if it is continuous, strictly increasing and for each .
Definition 2.3 A multivalued operator N on X with nonempty values in X is called
γ-Lipschitz if and only if there exists such that
a contraction if and only if it is γ-Lipschitz with ;
a generalized contraction if and only if there is a strict comparison function such that
The following lemmas will be used in the sequel.
Lemma 2.4 
Let Y be a separable metric space and let be a lower semi-continuous multivalued map with closed decomposable values. Then has a continuous selection; i.e., there exists a continuous mapping (single-valued) such that for every .
Lemma 2.5 (Wegrzyk’s fixed point theorem )
Let be a complete metric space. If is a generalized contraction with nonempty closed values, then .
Lemma 2.6 (Covitz and Nadler’s fixed point theorem )
Let be a complete metric space. If is a multivalued contraction with nonempty closed values, then N has a fixed point such that , i.e., .
In order to define the solution of (1.1), we consider the following lemma whose proof is given in .
Lemma 2.2 For , the unique solution of the following problem:
is given by
Definition 2.7 A function is a solution of problem (1.1) if there exists a function such that a.e. on and
3 Existence of solutions
In the sequel, we set
where δ is given by (2.3).
Our first result deals with the case when F is not necessarily convex valued. We establish this result by means of the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo  for lower semi-continuous maps with decomposable values.
Theorem 3.1 Assume that
(A1) there exists a continuous nondecreasing function and a positive continuous function p such that
(A2) there exists a number such that
where θ is given by (3.1);
(A3) is a nonempty compact-valued multivalued map such that
is lower semicontinuous for each .
Then boundary value problem (1.1) has at least one solution on .
Proof By the conditions (A1) and (A3), it follows that F is of l.s.c. type. Then from Lemma 2.4, there exists a continuous function such that for all .
Let us consider the problem
Note that if is a solution of (3.2), then x is a solution to problem (1.1). In order to transform problem (3.2) into a fixed point problem, we define the operator
The proof consists of several steps.
ℋ is continuous. Let be a sequence such that in . Then
Thus ℋ is continuous.
ℋ maps bounded sets into bounded sets in . Indeed, it is enough to show that there exists a positive constant such that, for each , we have . From (A1) we have
Taking norm and using (3.1), we get
ℋ maps bounded sets into equicontinuous sets in . Let , and be a bounded set in . Then
As , the right-hand side of the above inequality tends to zero independently of . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Finally, we discuss a priori bounds on solutions. Let x be a solution of (3.2). In view of (A1), for each , we obtain
which, on taking norm and using (3.1), yields
In view of (A2), there exists M such that . Let us set
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 , we deduce that ℋ has a fixed point , which is a solution of problem (3.2). Consequently, it is a solution to problem (1.1). This completes the proof. □
Next, we show the existence of solutions for problem (1.1) with a non-convex valued right-hand side by applying Lemma 2.5 due to Wegrzyk.
Theorem 3.2 Suppose that
(A4) has nonempty compact values and is measurable for each ;
(A5) for almost all and with and for almost all , where is strictly increasing.
Then BVP (1.1) has at least one solution on if is a strict comparison function, where (θ is given by (3.1)).
Proof Suppose that is a strict comparison function. Observe that by the assumptions (A4) and (A5), is measurable and has a measurable selection (see Theorem III.6 ). Also, and
Thus the set is nonempty for each .
Transform problem (1.1) into a fixed point problem. Consider the operator defined by
for . We shall show that the map ℋ satisfies the assumptions of Lemma 2.5. To show that the map is closed for each , let such that in . Then and there exists such that, for each ,
As F has compact values, we pass onto a subsequence to obtain that converges to y in . Thus, and for each ,
So, and hence is closed.
Next, we show that
Let and . Then there exists such that for each ,
From (A5) it follows that
So, there exists such that
Since the multivalued operator is measurable (see Proposition III.4 in ), there exists a function which is a measurable selection for . So, , and for each ,
For each , let us define
By an analogous argument, interchanging the roles of x and , we obtain
for each . So, ℋ is a generalized contraction and thus, by Lemma 2.5, ℋ has a fixed point x which is a solution to (1.1). This completes the proof. □
Remark 3.3 It is important to note that the condition (A5) reduces to
for , where a contraction principle for a multivalued map due to Covitz and Nadler  (Lemma 2.6) is applicable under the condition . Thus, our result dealing with a non-convex valued right-hand side of (1.1) is more general. Furthermore, Theorem 3.2 holds for several values of the function ℓ.
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The author thanks the referees for their useful comments. This paper was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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