Existence results for fractional differential inclusions arising from real estate asset securitization and HIV models
© Ahmad and Ntouyas; licensee Springer 2013
Received: 3 May 2013
Accepted: 5 July 2013
Published: 22 July 2013
This paper studies a new class of boundary value problems of nonlinear differential inclusions with Riemann-Liouville integral boundary conditions arising from real estate asset securitization and HIV models. Some new existence results are obtained by using standard fixed point theorems when the right-hand side of the inclusion has convex as well as non-convex values. Some illustrative examples are also discussed.
where is the standard Riemann-Liouville fractional derivative of order , , , is a multivalued map, is the family of all subsets of ℝ, denotes the Riemann-Stieltjes integral, and A is a function of bounded variation.
The subject of fractional differential equations has evolved as an interesting and important field of research in view of its numerous applications in physics, mechanics, chemistry, engineering (like traffic, transportation, logistics etc.), and so forth [1–3]. The tools of fractional calculus have played a key role in improving the mathematical modeling of many real world processes based on classical calculus. The nonlocal characteristic of a fractional order differential operator distinguishes it from a classical integer-order differential operator. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of some important and useful materials and processes. For some recent development on the topic, see [4–19] and the references cited therein.
On the other hand, the nonlocal condition given by a Riemann-Stieltjes integral is due to Webb and Infante in  and gives a unified approach to many BVPs. Since covers a multipoint BVP and an integral BVP as special cases, the fractional differential equations with the Riemann-Stieltjes integral condition were extensively studied by many authors, see [21, 22].
where λ is a parameter, , , and A, B are functions of bounded variation.
In the present paper, we are motivated by some recent papers [18, 24, 29], which considered problem (1.1) with F being single-valued and provided results on the existence and nonexistence of positive solutions. Here we extend the results to cover the multivalued case.
We establish existence results for problem (1.1), when the right-hand side is convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. 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 semicontinuous multivalued maps with nonempty closed and decomposable values; while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.
The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel, and in Section 3 we prove our main results. Examples illustrating the obtained results are presented in Section 4.
In this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper.
Let denote a Banach space of continuous functions from into ℝ with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
where and .
is convex (closed) valued if is convex (closed) for all ;
is bounded on bounded sets if is bounded in X for all (i.e., );
is called upper semicontinuous (u.s.c.) on X if, for each , the set is a nonempty closed subset of X, and if, for each open set N of X containing , there exists an open neighborhood of such that ;
G is lower semicontinuous (l.s.c.) if the set is open for any open set B in E;
is said to be completely continuous if is relatively compact for every ;
- (vi)is said to be measurable if, for every , the function
has a fixed point if there is such that . The fixed point set of the multivalued operator G will be denoted by FixG.
- (a)γ-Lipschitz if and only if there exists such that
a contraction if and only if it is γ-Lipschitz with .
is measurable for each ;
is upper semicontinuous for almost all .
- (iii)for each , there exists such that
for all and for a.e. .
We define the graph of G to be the set and recall two useful results regarding closed graphs and upper-semicontinuity.
Lemma 2.3 [, Proposition 1.2]
If is u.s.c., then is a closed subset of ; i.e., for every sequence and , if when , , and , then . Conversely, if G is completely continuous and has a closed graph, then it is upper semicontinuous.
Lemma 2.4 
is a closed graph operator in .
provided the integral exists, where denotes the integer part of the real number q.
provided the integral exists.
Lemma 2.7 
where , ().
- (1)If , , then(2.1)
- (2)If , , then(2.2)
for some , ().
and BVP (2.4) is equivalent to BVP (1.1).
Lemma 2.9 
Throughout the paper we always assume that the following holds.
Lemma 2.11 
We end this section by recalling two well-known fixed point theorems which will be used in the sequel, the nonlinear alternative of Leray-Schauder for multivalued maps and Covitz and Nadler fixed point theorem.
Lemma 2.12 (Nonlinear alternative for Kakutani maps) 
F has a fixed point in , or
there is a and with .
Lemma 2.13 
Let be a complete metric space. If is a contraction, then .
3 Existence results
Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problem (1.1) is new.
3.1 Convex case
Theorem 3.1 Assume that () holds. In addition we assume that:
() is Carathéodory and has nonempty compact and convex values;
for each ;
Then boundary value problem (1.1) has at least one solution on .
We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that is convex for each . This step is obvious since is convex (F has convex values), and therefore we omit the proof.
Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, it follows by the Arzelá-Ascoli theorem that is completely continuous.
for some .
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 (Lemma 2.12), we deduce that has a fixed point , which is a solution of problem (1.1). This completes the proof. □
3.2 Non-convex case
In this subsection, we study the case when F is not necessarily convex-valued.
Theorem 3.4 Assume that (), (), () and the following condition holds:
is lower semicontinuous for each .
Then boundary value problem (1.1) has at least one solution on .
Proof It follows from (), () and Lemma 4.1 of  that F is of l.s.c. type. Then, from the selection theorem due to Bressan and Colombo  for lower semicontinuous maps with decomposable values, there exists a continuous function such that for all .
It can easily be shown that is continuous and completely continuous and satisfies all the conditions of the nonlinear alternative of Leray-Schauder type for single-valued maps . The remaining part of the proof is similar to that of Theorem 3.1. So we omit it. This completes the proof. □
Now we prove our second existence result for problem (1.1) with a non-convex-valued right-hand side by applying a fixed point theorem for a multivalued map due to Covitz and Nadler .
Theorem 3.5 Assume that the following conditions hold:
() is such that is measurable for each ;
() for almost all and with and for almost all .
We show that the operator satisfies the assumptions of Lemma 2.13. The proof will be given in two steps.
i.e., and hence F is integrably bounded. Therefore, .
Since the multivalued operator is measurable (Proposition III.4 in ), there exists a function which is a measurable selection for U. So, and for each , we have .
Since is a contraction, it follows by Lemma 2.13 that has a fixed point x which is a solution of (1.1). This completes the proof. □
In this section, we give two examples to show the applicability of our results.
Thus , and is increasing, so () holds.
and hence () holds.
Clearly, all the conditions of Theorem 3.1 are satisfied. Hence the conclusion of Theorem 3.1 applies to problem (4.1).
Hence all the assumptions of Theorem 3.5 are satisfied and by the conclusion of it, BVP (4.3) has at least one solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
The authors thank the reviewers 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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