- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 659405, 10 pages
On Higher-Order Sequential Fractional Differential Inclusions with Nonlocal Three-Point Boundary Conditions
1Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Received 18 May 2014; Accepted 27 June 2014; Published 3 August 2014
Academic Editor: Hari M. Srivastava
Copyright © 2014 Bashir Ahmad and Sotiris K. Ntouyas. 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.
We study a nonlinear three-point boundary value problem of sequential fractional differential inclusions of order with , . Some new existence results for convex as well as nonconvex multivalued maps are obtained by using standard fixed point theorems. The paper concludes with an example.
The topic of fractional differential equations has attracted a great attention in the recent years. It is mainly due to the intensive development of the theory and applications of fractional calculus. In fact, the tools of fractional calculus have considerably improved the modeling of several real world phenomena in physics, chemistry, bioengineering, etc. The systematic development of theory, methods, and applications of fractional differential equations can be found in [1–6]. For some recent results on fractional differential equations and inclusions, see [7–23] and the references cited therein.
In this paper, we study the following boundary value problem: where is the Caputo fractional derivative, is the ordinary derivative, is a multivalued map, is the family of all subsets of , , is a positive real number, and is a real number.
The present work is motivated by a recent paper of the authors , where the problem (1) was considered for a single-valued case. The existence of solutions for the given multivalued problem is discussed for three cases: (a) convex-valued maps; (b) not necessarily convex-valued maps; (c) nonconvex-valued maps. To establish the existence results, we make use of nonlinear alternative for Kakutani maps, nonlinear alternative of Leray-Schauder type for single-valued maps, selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, and a fixed point theorem for contractive multivalued maps due to Covitz and Nadler. The tools employed in this paper are standard; however, their exposition in the framework of the problem at hand is new.
The paper is organized as follows: in Section 2 we recall some preliminary facts that we used in the sequel. Section 3 contains the main results and an example. In Section 4, we summarize the work obtained in this paper and discuss some special cases.
Definition 1. For -times absolutely continuous function , the Caputo derivative of fractional order is defined as where denotes the integer part of the real number .
Definition 2. The Riemann-Liouville fractional integral of order is defined as provided the integral exists.
Definition 3. A function is called a solution of problem (1) if there exists a function with , a.e. , such that , a.e. , and , , , and .
For the forthcoming analysis, we define Furthermore, we assume the nonresonance condition, that is, for and , we choose such that
3. Existence Results
Let denote a Banach space of continuous functions from into with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
Let denote a normed space. Then we define
Definition 5. A multivalued map is convex- (closed-) valued if is convex (closed) for all .
Definition 6. The map is bounded on bounded sets if is bounded in for all (i.e., ).
Definition 7. is called upper semicontinuous (u.s.c.) on if for each , the set is a nonempty closed subset of , and if, for each open set of containing , there exists an open neighborhood of such that .
Definition 8. is said to be completely continuous if is relatively compact for every .
If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph; that is, , , and imply that . has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by .
Definition 9. A multivalued map is said to be measurable if for every , the function is measurable.
3.1. The Carathéodory Case
Definition 10. A multivalued map is said to be Carathéodory if (i) is measurable for each ;(ii) is upper semicontinuous for almost all .Further a Carathéodory function is called -Carathéodory if (iii)for each , there exists such that for all and for .
For each , define the set of selections of by
For the forthcoming analysis, we need the following lemmas.
Lemma 11 (nonlinear alternative for Kakutanimaps ). Let be a Banach space, a closed convex subset of , an open subset of , and . Suppose that is an upper semicontinuous compact map. Then either(i) has a fixed point in , or(ii)there is a and with .
Lemma 12 (see ). Let be a Banach space. Let be an -Carathéodory multivalued map and let be a linear continuous mapping from to . Then the operator is a closed graph operator in .
Now we are in a position to prove the existence of the solutions for the boundary value problem (1) when the right-hand side is convex-valued.
Theorem 13. Assume that the nonresonance condition (6) holds. In addition, we suppose that(H1) is Carathéodory and has nonempty compact and convex values;(H2)there exist a continuous nondecreasing function and a function such that (H3)there exists a constant such that where ( is defined in (4) and (5)). Then the boundary value problem (1) has at least one solution on .
Proof. Define the operator by
for . We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that is convex for each . This step is obvious since is convex ( has convex values), and therefore we omit the proof.
In the second step, we show that maps bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then, for each , there exists such that Then for , we have Consequently,
Now we show that maps bounded sets into equicontinuous sets of . Let and . For each , we obtain
Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, therefore it follows from the Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that has 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 from Lemma 12 that is a closed graph operator. Further, we have . Since , therefore, we have for some .
Finally, we show that there exists an open set with for any and all . Let and . Then there exists with such that, for , we have Using the computations of the second step above we have Consequently, we have In view of , there exists such that . Let us set 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 Leray-Schauder type (Lemma 11), we deduce that has a fixed point which is a solution of the problem (1). This completes the proof.
Remark 14. The condition in the statement of Theorem 13 may be replaced with the following one.
There exists a constant such that where is the same as defined in .
3.2. The Lower Semicontinuous Case
As a next result, we study the case when is not necessarily convex-valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo  for lower semicontinuous maps with decomposable values.
Let be a nonempty closed subset of a Banach space and let 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 -algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in . A subset of is decomposable if, for all and measurable , the function , where stands for the characteristic function of .
Definition 15. Let 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 as which is called the Nemytskii operator associated with .
Definition 16. Let be a multivalued function with nonempty compact values. We say is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator is lower semicontinuous and has nonempty closed and decomposable values.
Lemma 17 (see ). Let 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 18. Assume that , , and the following condition hold:(H4) is a nonempty compact-valued multivalued map such that(a) is measurable,(b) is lower semicontinuous for each .Further the nonresonance condition (6) holds. Then the boundary value problem (1) has at least one solution on .
Proof. It follows from and that is of l.s.c. type. Then from Lemma 17, there exists a continuous function such that for all .
Consider the problem
Observe that if is a solution of (32), then is a solution to the problem (1). In order to transform the problem (32) into a fixed point problem, we define the operator as It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 13. So we omit it. This completes the proof.
3.3. The Lipschitz Case
Let be a metric space induced from the normed space . Consider given by where and . Then is a metric space and is a generalized metric space (see ).
Definition 19. A multivalued operator is called(a)-Lipschitz if and only if there exists such that (b)a contraction if and only if it is -Lipschitz with .
Lemma 20 (see ). Let be a complete metric space. If is a contraction, then .
Theorem 21. Assume that the nonresonance condition (6) holds. In addition, suppose that the following conditions hold: (H5) is such that is measurable for each ;(H6) for almost all and with and for almost all .Then the boundary value problem (1) has at least one solution on if
Proof. Observe that the set is nonempty for each by the assumption , so has a measurable selection (see Theorem ). Now we show that the operator , defined in the beginning of proof of Theorem 13, satisfies the assumptions of Lemma 20. To show that for each , let be such that in . Then and there exists such that, for each ,
As has compact values, we pass onto a subsequence (if necessary) to obtain that converges to in . Thus, and, for each , we have
Next we show that there exists such that Let and . Then there exists such that, for each ,
By , we have So, there exists such that
Define by Since the multivalued operator is measurable (Proposition ), there exists a function which is a measurable selection for . So and for each , we have .
For each , let us define
Analogously, interchanging the roles of and , we obtain
Since is a contraction, it follows from Lemma 20 that has a fixed point which is a solution of (1). This completes the proof.
Example 23. Consider the problem
Here, , , , , , and is a multivalued map given by For , we have Thus, with . In this case By the condition , that is, we find that with . Therefore, it follows from Theorem 13 that problem (49) has at least one solution.
In this paper, we have solved a three-point boundary value problem of Caputo-type sequential fractional differential inclusions of an arbitrary order . The existence of solutions for the given problem with the convex-valued map is obtained by means of nonlinear alternative for Kakutani maps, while the existence result for not necessarily convex-valued map is established by combining 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 decomposable values. The nonconvex-valued case relies on a fixed point theorem for contractive multivalued maps due to Covitz and Nadler. Some new existence results follow by fixing the parameters involved in the given problem. For instance, by taking , our results correspond to a two-point Caputo-type multivalued problem of an arbitrary order , while the results for sequential differential inclusions of order can be obtained by fixing in the results of this paper.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos), World Scientific, Boston, Mass, USA, 2012.
- 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 B.V., Amsterdam, The Netherland, 2006.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
- J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
- A. Aghajani, Y. Jalilian, and J. J. Trujillo, “On the existence of solutions of fractional integro-differential equations,” Fractional Calculus and Applied Analysis, vol. 15, no. 1, pp. 44–69, 2012.
- R. P. Agarwal, B. Ahmad, A. Alsaedi, and N. Shahzad, “Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions,” Advances in Difference Equations, vol. 2012, article 74, 2012.
- B. Ahmad and S. K. Ntouyas, “Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 71, pp. 1–17, 2010.
- B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 480–487, 2010.
- B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.
- B. Ahmad, J. J. Nieto, and A. Alsaedi, “A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders,” Advances in Difference Equations, vol. 2012, article 54, 16 pages, 2012.
- B. Ahmad and S. K. Ntouyas, “Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions,” Electronic Journal of Differential Equations, vol. 2012, no. 98, pp. 1–22, 2012.
- B. Ahmad and S. K. Ntouyas, “A higher-order nonlocal three-point boundary value problem of sequential fractional differential equations,” Miskolc Mathematical Notes. To appear.
- Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010.
- K. Balachandran and J. J. Trujillo, “The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4587–4593, 2010.
- D. Baleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010.
- S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011.
- Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.
- J. Henderson and A. Ouahab, “Fractional functional differential inclusions with finite delay,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 5, pp. 2091–2105, 2009.
- V. Keyantuo and C. Lizama, “A characterization of periodic solutions for time-fractional differential equations in spaces and applications,” Mathematische Nachrichten, vol. 284, no. 4, pp. 494–506, 2011.
- M. Klimek, “Sequential fractional differential equations with Hadamard derivative,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4689–4697, 2011.
- Z. Wei, Q. Li, and J. Che, “Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 260–272, 2010.
- K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, Germany, 1992.
- S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, The Netherlands, 1997.
- A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2005.
- A. Lasota and Z. Opial, “An application of the Kakutani—KY Fan theorem in the theory of ordinary differential equations,” Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques, et Physiques, vol. 13, pp. 781–786, 1965.
- A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, no. 1, pp. 69–86, 1988.
- M. Frigon, “Théorèmes d'existence de solutions d'inclusions différentielles,” in Topological Methods in Differential Equations and Inclusions, A. Granas and M. Frigon, Eds., vol. 472 of NATO ASI Series C, pp. 51–87, Kluwer Academic Publisher, Dordrecht, Germany, 1995.
- H. Covitz and S. B. Nadler, “Multi-valued contraction mappings in generalized metric spaces,” Israel Journal of Mathematics, vol. 8, pp. 5–11, 1970.
- M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
- C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer, Berlin, Germany, 1977.