Abstract
In this paper, a new existence result is obtained for a fractional multivalued problem with fractional integral boundary conditions by applying a (Krasnoselskii type) fixedpoint result for multivalued maps due to Petryshyn and Fitzpatric [Trans. Am. Math. Soc. 194:125, 1974]. The case for lower semicontinuous multivalued maps is also discussed. An example for the illustration of our main result is presented.
MSC: 34A60, 34A08.
Keywords:
fractional differential inclusions; nonlocal boundary conditions; fixedpoint theorems1 Introduction
The theory of fractional differential equations and inclusions has developed into an important field of investigation due to its extensive applications in numerous branches of physics, economics, and engineering sciences [14]. The nonlocal behavior exhibited by a fractionalorder differential operator makes it distinct from the integerorder differential operator. It means that the future state of a dynamical system or process involving fractional derivatives depends on its current state as well its past states. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of several materials and processes. This characteristic of fractional calculus has contributed to its popularity and has convinced many researchers of the need to shift their focus from classical integerorder models to fractionalorder models. There has been a great surge in developing new theoretical aspects such as periodicity, asymptotic behavior, and numerical methods for fractional equations. For some recent work on the topic, see [518] and the references cited therein.
In this paper, we consider the following boundary value problem of fractional differential inclusions with fractional integral boundary conditions:
where , , , denotes the RiemannLiouville fractional derivative of order , are multivalued maps, is the family of all nonempty subsets of ℝ and A, B are real constants.
We establish two new existence results for the problem (1.1). The first result relies on a nonlinear alternative for contractive maps, while in the second result, we shall combine the nonlinear alternative of LeraySchauder type for singlevalued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values.
The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel and Section 3 deals with the main results.
2 Preliminaries
Let us recall some basic definitions of the fractional calculus [13].
Definition 2.1 The RiemannLiouville derivative of fractional order q for a continuous function is defined as
where denotes the integer part of the real number q.
Definition 2.2 The RiemannLiouville fractional integral of order q for a function is defined as
provided the integral exists.
Observe that the substitution transforms the problem (1.1) to the following form:
To define the solutions of the problem (1.1), we need the following lemma. Though the proof of this lemma involves standard arguments, we trace its proof for the convenience of the reader.
Lemma 2.3For any, the unique solution of the linear fractional boundary value problem
is
Proof It is well known [3] that the solution of fractional differential equation in (2.2) can be written as
where are arbitrary constants. Using the boundary conditions in (2.2), we find that , , and
Substituting these values in (2.3) yields
Notice that in view of the given values of the parameters involved in the expression. This completes the proof. □
Thus, the solution of the equation subject to the boundary conditions given by (1.1) can be written as
where we have used the substitution in the integral of the last term. Using the relation for the Beta function ,
we find that
Let denote the Banach space of all continuous functions from endowed with the norm defined by .
To establish the main results of this paper, we use the following form of the nonlinear alternative for contractive maps [[19], Corollary 3.8].
Theorem 2.4LetXbe a Banach space, andDa bounded neighborhood of. Let (heredenotes the family of all nonempty, compact and convex subsets ofX) andtwo multivalued operators satisfying
(b) is upper semicontinuous (u.s.c. for shortly) and compact.
Definition 2.5 A multivalued map is said to be Carathéodory if
(ii) is upper semicontinuous for almost all , and
(iii) for each real number , there exists a function such that
Denote
Lemma 2.6 (Lasota and Opial [20])
LetXbe a Banach space. Letbe anCarathéodory multivalued map and let Θ be a linear continuous mapping fromto. Then the operator
3 Existence results
Before presenting the main results, we define the solutions of the boundary value problem (1.1).
Definition 3.1 A function is said to be a solution of the problem (1.1) if , and there exist functions , such that
where
In the sequel, we set
Theorem 3.2Assume that
(H_{1}) is anCarathéodory multivalued map;
(H_{2}) there exists a functionsuch that
for alland, whereis given by (3.2);
(H_{3}) is anCarathéodory multivalued map;
(H_{4}) there exists a functionwithfor a.e. and a nondecreasing functionsuch that
(H_{5}) there exists a numbersuch that
where, are given by (3.2) and (3.3), respectively, and.
Then the problem (1.1) has a solution on.
Proof To transform the problem (1.1) to a fixedpoint problem, let us define an operator by
for , , where Q is given by (3.1).
We study the integral inclusion in the space of all continuous real valued functions on with supremum norm . Define two multivalued maps by
Observe that . We shall show that the operators and satisfy all the conditions of Theorem 2.4 on . For the sake of clarity, we split the proof into a sequence of steps and claims.
Step 1. We show that is a multivalued contraction on.
for some . Since , there exists such that . Thus the multivalued operator U is defined by , where
has nonempty values and is measurable. Let be a measurable selection for U (which exists by KuratowskiRyllNardzewski’s selection theorem [21,22]). Then and a.e. on .
Define
Taking the supremum over the interval , we obtain
Combining the inequality (3.5) with the corresponding one obtained by interchanging the roles of x and y, we get
for all . This shows that is a multivalued contraction as
Step 2. We shall show that the operator is u.s.c. and compact. It is well known [[23], Proposition 1.2] that if an operator is completely continuous and has a closed graph, then it is u.s.c. Therefore we will prove that is completely continuous and has a closed graph. This step involves several claims.
Claim Imaps bounded sets into bounded sets in.
Now for each , there exists a such that
which implies that
Claim IImaps bounded sets into equicontinuous sets.
As in the proof of Claim I, let be a bounded set and for some . Then there exists such that
Obviously the right hand side of the above inequality tends to zero independently of as . Therefore it follows by the ArzeláAscoli theorem that is completely continuous.
Claim IIINext we prove thathas a closed graph.
Let , and . Then we need to show that . Associated with , there exists such that for each ,
Thus it suffices to show that there exists such that for each ,
Let us consider the linear operator given by
Observe that
as . Thus, it follows by Lemma 2.6 that is a closed graph operator. Further, we have . Since , we have
Hence has a closed graph (and therefore it has closed values). In consequence, is compact valued.
Therefore the operators and satisfy all the conditions of Theorem 2.4. So the conclusion of Theorem 2.4 applies and either condition (i) or condition (ii) holds. We show that the conclusion (ii) is not possible. If for , then there exist and such that
By hypothesis (H_{2}), for all , we have
Thus,
Now, if condition (ii) of Theorem 2.4 holds, then there exist and such that . Then x is a solution of (3.6) with and consequently, the inequality (3.7) yields
which contradicts (3.4). Hence, has a fixed point in by Theorem 2.4, which in fact is a solution of the problem (1.1). This completes the proof. □
3.1 The lower semicontinuous case
This section is devoted to the study of the case that the maps in (1.1) are not necessarily convexvalued. We establish the existence result for the problem at hand by applying the nonlinear alternative of LeraySchauder type and a selection theorem due to Bressan and Colombo [24] for lower semicontinuous maps with decomposable values. Before presenting this result, we revisit some basic concepts.
Let be a nonempty closed subset of a Banach space and be a multivalued operator with nonempty closed values. is lower semicontinuous (l.s.c.) if the set is open for any open set in . Let be a subset of . is measurable if 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 3.3 Let Y be a separable metric space and let be a multivalued operator. We say has a property (BC) if is lower semicontinuous (l.s.c.) and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with F as
which is called the Nemytskii operator associated with F.
Definition 3.4 Let be a multivalued function with nonempty compact values. We say F is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator ℱ is lower semicontinuous and has nonempty closed and decomposable values.
Lemma 3.5 ([25])
LetYbe a separable metric space and letbe a multivalued operator satisfying the property (BC). Thenhas a continuous selection, that is, there exists a continuous function (singlevalued) such thatfor every.
Theorem 3.6Assume that (H_{2}), (H_{4}), (H_{5}), and the following condition hold:
(H_{6}) are nonempty compactvalued multivalued maps such that
(b) are lower semicontinuous for each;
Then the boundary value problem (1.1) has at least one solution on.
Proof It follows from (H_{2}), (H_{4}), and (H_{6}) that F and G are of l.s.c. type. Then from Lemma 3.5, there exist continuous functions such that , for all .
Consider the problem
Observe that if is a solution of (3.8), then x is a solution to the problem (1.1). Now, we define two multivalued operators by
and
Clearly are continuous. Also the argument in Theorem 3.2 guarantees that and satisfy all the conditions of the nonlinear alternative for contractive maps in the singlevalued setting [26] and hence the problem (3.8) has a solution. □
Example 3.7 Consider the following fractional boundary value problem:
where
We have
with . Using the given data, we find that
Clearly , and by the condition:
it is found that , where . Thus, all the assumptions of Theorem 3.2 are satisfied. Hence, the conclusion of Theorem 3.2 applies to the problem (3.9).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, AA, SKN, BA and HHA, contributed to each part of this work equally and read and approved the final version of the manuscript.
Authors’ information
Member of Nonlinear Analysis and Applied Mathematics (NAAM)Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant no. 3130/1433/HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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