On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval

1 Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran 2Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey 3 Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele, Bucharest, Romania 4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 5 Department of Mathematics, Texas A & M University-Kingsville, 700 University Boulevard, Kingsville, USA 6Department of Mathematics, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

The present paper is motivated by a recent paper of Liang and Zhang [1], where it is considered problem (1) with  single valued, and several existence results are provided.
Fractional differential equations have been of great interest recently.This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, and engineering.For details, see [2][3][4] and the references therein.
The aim here is to establish existence results for problem (1) when the right hand side is convex as well as nonconvex valued.In the first result (Theorem 21), we consider the case when the right hand side has convex values and prove an existence result via nonlinear alternative for Kakutani maps.In the second result (Theorem 25), we will use the fixed point theorem for contraction multivalued maps according to Covitz and Nadler.The paper is organized as follows.

Preliminaries
In this section, we present some notations and preliminary lemmas that will be used in the proof of the main result.
Let (, ) be a metric space with the corresponding norm ‖ ⋅ ‖ and let  = [0, +∞).We denote by L() the -algebra of all Lebesgue measurable subsets of , by B() the family of all nonempty subsets of , and by P() the family of all Borel subsets of .If  ⊂  then   :  → {0, 1} denotes the characteristic function of .For any subset  ⊂ , we denote by  the closure of .
Finally, the following results are easily deduced from the theoretical limit set properties.
Lemma 6 (see [27,Lemma 1.1.9]).Let {  } ∈N ⊂  ⊂  be a sequence of subsets where  is a compact subset of a separable Banach space .Then, where co() refers to the closure of the convex hull of .
By  1 ([0, +∞), R) we denote the space of continuous real-valued functions whose first derivative exists and it is absolutely continuous on [0, +∞).In this paper, we will use the following space  to the study (1) which is denoted by From [32], we know that  is a Banach space equipped with the norm In what follows,  = [0,+∞),  ∈ (2, 3), and . Next, we need the following technical result proved in [1].

Main Results
Now we are able to present the existence results for problem (1).

The Upper Semicontinuous Case.
To obtain the complete continuity of existence solutions operator, the following lemma is still needed.
Proof.Let  =  and consider  > 0 as in (25).It is obvious that the existence of solutions to problem (1) is reduced to the existence of the solutions of the integral inclusion where (, ) is defined by ( 16) and (17).Consider the setvalued map,  :  → P() is defined by We show that  satisfies the hypotheses of Theorem 2.
Claim 2. We show that  is bounded on bounded sets of .
Let  be any bounded subset of .
and therefore for all V ∈ (); that is, () is bounded.
Claim 3. We show that  maps bounded the sets into equicontinuous sets.Let  be any bounded subset of  as before and V ∈ () for some  ∈ .Then, there exists  ∈   () such that V() = ∫ +∞ 0 (, )().So, for any  0 ∈ (0, +∞) and  1 ,  2 ∈ [0,  0 ], without loss of generality, we may assume that  2 >  1 and one can get the following: On the other hand, we get Similar to (34), we have From ( 34) and ( 35), we have Advances in Mathematical Physics and, similarly, one has lim Therefore, () is equiconvergent at infinity.Therefore, with Lemma 11, Lemma 19 and Claims 2-4, we conclude that  is completely continuous.Claim 5.  is upper semicontinuous.To this end, it is sufficient to show that  has a closed graph.Let V  ∈ (  ) such that V  → V and   → , as  → +∞.Then, there exists  > 0 such that ‖  ‖ ≤ .We will prove that V ∈ () means that there exists   ∈   (  ) such that, for a.e. ∈ , we have V  () = ∫ +∞ 0 (, )  ().Then, we need to show that V ∈ ().

The Lipschitz
Case.Now we prove the existence of solutions for the problem (1) with a nonconvex-valued right hand side by applying a fixed point theorem for multivalued maps according to Covitz and Nadler [34] Proof.We transform problem (1) into a fixed point problem.Consider the set-valued map  :  1 [0, +∞) → P( 1 [0, +∞)) defined at the beginning of the proof of Theorem 21.It is clear that the fixed point of  are solutions of the problem (1).

( 4 )
Then,  is of a lower semicontinuous type if   (⋅) is a lower semicontinuous with closed and decomposable values.
P() is a set-valued map, then a point  ∈  is called a fixed point for  if  ∈ (). is said to be bounded on bounded sets if () := ∪ ∈ () is a bounded subset of  for all bounded sets  in . is said to be compact if () is relatively compact for any bounded sets  in . is said to be totally compact if () is a compact subset of . is said to be upper semicontinuous if for any open set  ⊂ , the set { ∈  : () ⊂ } is open in . is called completely continuous if it is upper semicontinuous and, for every bounded subset  ⊂ , () is relatively compact.It is well known that a compact set-valued map  with nonempty compact values is upper semicontinuous if and only if  has a closed graph.