Advances in Mathematical Physics

Volume 2013 (2013), Article ID 823961, 9 pages

http://dx.doi.org/10.1155/2013/823961

Research Article

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

Nemat Nyamoradi,¹ Dumitru Baleanu,2,3,4 and Ravi P. Agarwal5,6

¹Department 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

³Institute 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

⁵Department 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

Received 12 January 2013; Accepted 22 January 2013

Academic Editor: José Tenreiro Machado

Copyright © 2013 Nemat Nyamoradi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion , where is the standard Riemann-Liouville fractional derivative, , satisfies ,  and   is a set-valued map. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or nonconvex values.

1. Introduction

In this paper, we will consider the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion

where is the standard Riemann-Liouville fractional derivative, , , satisfies , and is a set-valued map.

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 [2–4] and the references therein.

The existence of solutions of initial value problems for fractional order differential equations has been studied in the literature [5–17] and the references therein. The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim [18]. Also, recently, several qualitative results for fractional differential inclusions were obtained in [19–23] 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.

In Section 2 we recall some preliminary facts that we need in the sequel, and in Section 3 we prove our main results. Finally, in Section 4, an example is given to demonstrate the application of one of our main results.

2. 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 . We denote by the -algebra of all Lebesgue measurable subsets of , by the family of all nonempty subsets of , and by the family of all Borel subsets of . If then denotes the characteristic function of . For any subset , we denote by the closure of .

Recall that the Pompeiu-Hausdorff distance of the closed subsets is defined by the following:

where . Define

Also, we denote by the Banach space of all continuous functions endowed with the norm and by the Banach space of all (Bochner) integrable functions endowed with the norm .

Let and be two metric spaces. If 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.

We define the graph of to be the set and recall a useful result regarding connection between closed graphs and upper semicontinuity.

Lemma 1 (see [24, Proposition 1.2]). If is upper semicontinuous, then is a closed subset of , that is, for every sequence and , if when , , , and , then . Conversely, if is completely continuous and has a closed graph, then it is upper semicontinuous.

For the convenience of the reader, we present here the following nonlinear alternative of the Leray-Schauder type and its consequences.

Theorem 2 (nonlinear alternative for Kakutani maps [25]). Let be a Banach space, a closed convex subset of , an open subset of , and . Suppose that is an upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets of . Then, either

(i)has a fixed point in or(ii)there is a and with .

Definition 3. The multifunction is said to be lower semicontinuous if for any closed subset , the subset is closed.

If is a set-valued map with compact values and , we define

Then, is of a lower semicontinuous type if is a lower semicontinuous with closed and decomposable values.

Theorem 4 (see [26]). Let be a separable metric space and be a lower semicontinuous set-valued map with closed decomposable values. Then has a continuous selection (i.e., there exists a continuous mapping such that for all ).

Definition 5. Consider the following.

(i)A set-valued map with nonempty compact convex values is said to be measurable if for any the function is measurable.(ii)A set-valued map is said to be Carathéodory if is measurable for all and is upper semicontinuous for almost all .(iii) is said to be -Carathéodory if for any there exists such that .

Finally, the following results are easily deduced from the theoretical limit set properties.

Lemma 6 (see [27, Lemma ]). Let be a sequence of subsets where is a compact subset of a separable Banach space . Then,

where refers to the closure of the convex hull of .

Lemma 7 (see [27, Lemma ]). Let and be two metric spaces. If is an upper semicontinuous, then, for each ,

Definition 8. Let be a Banach space. A sequence is said to be semicompact if

(a)it is integrably bounded; that is, there exists such that

(b)the image sequence is relatively compact in for a.e. .

The following important result follows from the Dunford-Pettis theorem (see [28, Proposition ]).

Lemma 9. Every semicompact sequence is weakly compact in .

When the nonlinearity takes convex values, Mazur’s Lemma, 1933, may be useful.

Lemma 10 (see [29, Theorem 21.4]). Let be a normed space and a sequence weakly converging to a limit . Then, there exists a sequence of convex combinations with for and which converges strongly to .

Lemma 11 (see [30]). Let be defined as before and . Then is relatively compact in if the following conditions hold:

(a) is uniformly bounded in ;(b)the functions from are equicontinuous on any compact interval of ;(c)the functions from are equiconvergent; that is, for any given , there exists a such that , for any , .

Definition 12 (see [6]). The Riemann-Liouville fractional integral operator of order , of function , is defined as

where is the Euler gamma function.

Definition 13 (see [31]). The Riemann-Liouville fractional derivative of order , , is defined as

where the function has absolutely continuous derivatives up to order .

Lemma 14 (see [31]). The equality , holds for .

Lemma 15 (see [31]). Let and . Then, the differential equation

has a unique solution , , , where .

Lemma 16 (see [31]). Let . Then, the following equality holds for , :

, , where .

By we denote the space of continuous real-valued functions whose first derivative exists and it is absolutely continuous on . 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, , , and . Next, we need the following technical result proved in [1].

Lemma 17 (see [1]). For any , the problem

has a unique solution that

where

Note that , (e.g., Lemma  3.2 in [1]) and from the definition of , we have the following (e.g., Remark  3.1 in [1]):

where

Also, one can get

where

Lemma 18. The function defined by (16) satisfies

By calculation, it is easy to prove that Lemma 18 holds. So, we omit its proof here.

3. Main Results

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

3.1. The Upper Semicontinuous Case

To obtain the complete continuity of existence solutions operator, the following lemma is still needed.

Lemma 19 (see [32]). Let . If   is equicontinuous on any compact interval of and equiconvergent at infinity, then is relatively compact on .

Definition 20. is called equiconvergent at infinity if and only if for all , there exists such that for all , it holds

Theorem 21. The Carathéodory multivalued map has nonempty, compact, convex values and satisfies the following.

   There exists a continuous nondecreasing function and , such that  and  each  . There exists a constant such that

Then, problem (1) has at least one solution.

Proof. Let and consider 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 set-valued map, is defined by

We show that satisfies the hypotheses of Theorem 2.

Claim 1. We show that is convex for any . If , then, there exist such that for any one has

Let . Then, for any , we have

The values of are convex; thus, is a convex set and hence .

Claim 2. We show that is bounded on bounded sets of . Let be any bounded subset of . Then, there exists such that for all . If , then there exists such that . One may write the following for any :

On the other hand,

and therefore

for all ; that is, is bounded.

Claim 3. We show that maps bounded the sets into equi-continuous sets. Let be any bounded subset of as before and for some . Then, there exists such that . So, for any and , without loss of generality, we may assume that and one can get the following:

On the other hand, we get

Similar to (34), we have

From (34) and (35), we have

Similar to (36), one can get

Therefore, is an equicontinuous set in .

Claim 4. We show that is equiconvergent at . Let for some . Then, there exists such that . So, we have the following:

and, similarly, one has

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 such that and , as . Then, there exists such that . We will prove that means that there exists such that, for a.e. , we have . Then, we need to show that .

Condition (H1) implies that . Then, is integrably bounded in . Since has compact values, we deduce that is semicompact. By Lemma 9, there exists a subsequence, still denoted as , which converges weakly to some limit . Moreover, the mapping defined by

is a continuous linear operator. Then, it remains continuous if these spaces are endowed with their weak topologies [29, 33]. Moreover, for a.e. , converges to . Then, we have

It remains to prove that , a.e. . Mazur’s Lemma (see Lemma 10) yields the existence of , such that and the sequence of convex combinations converges strongly to in . Using Lemma 6, we obtain that

However, the fact that the multivalued is upper semicontinuous and has compact values, together with Lemma 7, implies that

This along with (42) yields that . Finally, has closed, convex values; hence, , a.e. . Thus, , proving that has a closed graph. Finally, with Lemma 1 and the compactness of , we conclude that is upper semicontinuous.

Claim 6. A priori bounds on solutions. Let be a solution of (1). Then, there exists with such that . In view of (H1), and using the computations in Claim 2 above, for each , we obtain

Consequently,

In view of (H2), there exists such that . Let us consider the following:

Note that the operator is upper semicontinuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of the Leray-Schauder type (Theorem 2), we deduce that has a fixed point which is a solution of the problem (1). This completes the proof.

3.2. 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].

Definition 22. A multivalued operator is called the following:

(a)-Lipschitz if and only if there exists such that for each ;(b)a contraction if and only if it is -Lipschitz with .

Lemma 23 (Covitz-Nadler, [34]). Let be a complete metric space. If is a contraction, then .

Definition 24. A measurable multivalued function is said to be integrably bounded if there exists a function such that for all , for a.e. .

Theorem 25. Assume that the following condition holds:

   such that is measurable for each ; There exist which are not identical zero on any closed subinterval of , and

such that for almost all ,

for all , and with for almost all .

Then, the boundary value problem (1) has at least one solution on if

Proof. We transform problem (1) into a fixed point problem. Consider the set-valued map defined at the beginning of the proof of Theorem 21. It is clear that the fixed point of are solutions of the problem (1).

Note that since the set-valued map is measurable with the measurable selection theorem (e.g., Theorem in [35]) it admits a measurable selection . Moreover, since is integrably bounded, . Therefore, .

We will prove that fulfills the assumptions of Covitz-Nadler contraction principle (Lemma 23).

First, we note that since , for any .

Second, we prove that is closed for any . Let such that in . Then and there exists such that

Since has compact values, we may pass onto a subsequence (if necessary) to obtain that converges to in . In particular, , and for any we have

that is, and is closed.

Next we show that is a contraction on . Let and . Then there exist such that

Consider the set-valued map

By (H5), we have

hence has nonempty closed values. Moreover, since is measurable (e.g., Proposition in [35]), there exists which is a measurable selection of . It follows that and for any ,

Define

and one can get

Similarly, we have

Therefore,

From an analogous reasoning by interchanging the roles of and , it follows that

Since is a contraction, it follows by the Lemma 23 that admits a fixed point which is a solution to problem (1).

4. Application

Consider the fractional boundary value problem,

Here , , and , and is a multivalued map given by

For , we have

Thus,

with , .

Also, by direct calculation, we can obtain that and . Further, by using the following condition:

we find that . Clearly, all the conditions of Theorem 21 are satisfied. So, there exists at least one solution of problem (1) on .

Acknowledgments

The authors thank the referees for their careful reading of this paper and useful suggestions. This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.

References

S. Liang and J. Zhang, “Existence of multiple positive solutions for -point fractional boundary value problems on an infinite interval,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1334–1346, 2011.

A. M. A. El-Sayed, “Nonlinear functional-differential equations of arbitrary orders,” Nonlinear Analysis. Theory, Methods & Applications, vol. 33, no. 2, pp. 181–186, 1998.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002.

D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

I. Podlubny, The Laplace Transform Method for Linear Differential Equations of Fractional Order, Slovac Academy of Science, Slovak Republic, 1994.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, The Netherlands, 1993.

R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.

V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.

N. Nyamoradi, “Existence of solutions for multi point boundary value problems for fractional differential equations,” Arab Journal of Mathematical Sciences, vol. 18, no. 2, pp. 165–175, 2012.

N. Nyamoradi, “Positive solutions for multi-point boundary value problem for nonlinear fractional differential equations,” Journal of Contemporary Mathematical Analysis. Accepted for publishing

N. Nyamoradi, “A Six-point nonlocal integral boundary value problem for fractional differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 43, no. 5, pp. 429–454, 2012.

N. Nyamoradi and T. Bashiri, “Multiple positive solutions for nonlinear fractional differential systems,” Fractional Differential Calculus, vol. 2, no. 2, pp. 119–128, 2012.

N. Nyamoradi and T. Bashiri, “Existence of positive solutions for fractional differential systems with multi point boundary conditions,” Annali Dell'Universita di Ferrara, 2012.

R. P. Agarwal and B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1200–1214, 2011.

B. Ahmad and S. K. Ntouyas, “Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions,” Boundary Value Problems, vol. 2012, article 55, 21 pages, 2012.

A. M. A. El-Sayed and A.-G. Ibrahim, “Multivalued fractional differential equations,” Applied Mathematics and Computation, vol. 68, no. 1, pp. 15–25, 1995.

B. Ahmad and S. K. Ntouyas, “Existence of solutions for fractional differential inclusions with nonlocal strip conditions,” Arab Journal of Mathematical Sciences, vol. 18, no. 2, pp. 121–134, 2012.

M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional functional differential inclusions with infinite delay and applications to control theory,” Fractional Calculus & Applied Analysis, vol. 11, no. 1, pp. 35–56, 2008.

A. Cernea, “Continuous version of Filippov's theorem for fractional differential inclusions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 1, pp. 204–208, 2010.

N. Nyamoradi and M. Javidi, “Exictence of multiple positive soulutions for fractional differential inclusion with m-point boundary conditions and two fractional orders,” Electronic Journal of Differential Equations, vol. 2012, p. 126, 2012.

A. Ouahab, “Some results for fractional boundary value problem of differential inclusions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 11, pp. 3877–3896, 2008.

K. Deimling, Multivalued Differential Equations, vol. 1 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 1992.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003.

A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, no. 1, pp. 70–85, 1988.

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Birkhäuser, Boston, Mass, USA, 1990.

M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, Germany, 2001.

J. Musielak, Introduction to Functional Analysis, PWN, Warsaw, Poland, 1976.

R. P. Agarwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.

Y. Liu, “Existence and unboundedness of positive solutions for singular boundary value problems on half-line,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 543–556, 2003.

L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, The Macmillan, New York, NY, USA, 1964.

H. Covitz and S. B. Nadler, Jr., “Multi-valued contraction mappings in generalized metric spaces,” Israel Journal of Mathematics, vol. 8, pp. 5–11, 1970.

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, vol. 580 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.