- Open Access
Existence results for fractional integral inclusions via nonlinear alternative for contractive maps
© Alsaedi et al.; licensee Springer. 2014
- Received: 18 November 2013
- Accepted: 10 January 2014
- Published: 30 January 2014
In this paper, a new existence result is obtained for a fractional multivalued problem with fractional integral boundary conditions by applying a (Krasnoselskii type) fixed-point result for multivalued maps due to Petryshyn and Fitzpatric [Trans. Am. Math. Soc. 194:1-25, 1974]. The case for lower semi-continuous multivalued maps is also discussed. An example for the illustration of our main result is presented.
- fractional differential inclusions
- nonlocal boundary conditions
- fixed-point theorems
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 [1–4]. The nonlocal behavior exhibited by a fractional-order differential operator makes it distinct from the integer-order 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 integer-order models to fractional-order 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 [5–18] and the references cited therein.
where , , , denotes the Riemann-Liouville 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 Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semi-continuous 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.
where denotes the integer part of the real number q.
provided the integral exists.
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.
Notice that in view of the given values of the parameters involved in the expression. This completes the proof. □
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 [, Corollary 3.8].
is contraction, and
is upper semi-continuous (u.s.c. for shortly) and compact.
H has a fixed point in or
there is a point and with .
is measurable for each ,
is upper semi-continuous for almost all , and
- (iii)for each real number , there exists a function such that
for all with .
Lemma 2.6 (Lasota and Opial )
is a closed graph operator in .
Before presenting the main results, we define the solutions of the boundary value problem (1.1).
Theorem 3.2 Assume that
(H1) is an Carathéodory multivalued map;
for all and , where is given by (3.2);
(H3) is an Carathéodory multivalued map;
for all ;
where , are given by (3.2) and (3.3), respectively, and .
Then the problem (1.1) has a solution on .
for , , where Q is given by (3.1).
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 .
Step 2. We shall show that the operator is u.s.c. and compact. It is well known [, 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 I maps bounded sets into bounded sets in .
Let be a bounded set in .
Hence is bounded.
Claim II maps bounded sets into equicontinuous sets.
Obviously the right hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Claim III Next we prove that has a closed graph.
for some .
Hence has a closed graph (and therefore it has closed values). In consequence, is compact valued.
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 semi-continuous case
This section is devoted to the study of the case that the maps in (1.1) are not necessarily convex-valued. We establish the existence result for the problem at hand by applying the nonlinear alternative of Leray-Schauder type and a selection theorem due to Bressan and Colombo  for lower semi-continuous 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 semi-continuous (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 semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
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 semi-continuous type (l.s.c. type) if its associated Nemytskii operator ℱ is lower semi-continuous and has nonempty closed and decomposable values.
Lemma 3.5 ()
Let Y be a separable metric space and let be a multivalued operator satisfying the property (BC). Then has a continuous selection, that is, there exists a continuous function (single-valued) such that for every .
Theorem 3.6 Assume that (H2), (H4), (H5), and the following condition hold:
, are measurable,
are lower semicontinuous for each ;
Then the boundary value problem (1.1) has at least one solution on .
Proof It follows from (H2), (H4), and (H6) that F and G are of l.s.c. type. Then from Lemma 3.5, there exist continuous functions such that , for all .
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 single-valued setting  and hence the problem (3.8) has a solution. □
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).
Member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant no. 3-130/1433/HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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