- Open Access
Some existence results for differential inclusions of fractional order with nonlocal strip conditions
© Alsulami; licensee Springer 2013
- Received: 8 January 2013
- Accepted: 29 May 2013
- Published: 25 June 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.
- fractional differential inclusions
- integral boundary conditions
- fixed point theorems
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.
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 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 .
is lower semi-continuous with closed and decomposable values.
Definition 2.2 
A function is said to be a strict comparison function if it is continuous, strictly increasing and for each .
- (a)γ-Lipschitz if and only if there exists such that
a contraction if and only if it is γ-Lipschitz with ;
- (c)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 .
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
where θ is given by (3.1);
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 .
- (i)ℋ is continuous. Let be a sequence such that in . Then
- (ii)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
- (iii)ℋ maps bounded sets into equicontinuous sets in . Let , and be a bounded set in . Then
- (iv)Finally, we discuss a priori bounds on solutions. Let x be a solution of (3.2). In view of (A1), for each , we obtain
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)).
Thus the set is nonempty for each .
So, and hence is closed.
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. □
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.
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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