Abstract
In this paper, we study the existence of solutions for a new class of boundary value problems of nonlinear fractional integro-differential equations and inclusions of arbitrary order with initial and non-separated boundary conditions. In the case of inclusion, the existence results are obtained for convex as well as non-convex multifunctions. Our results rely on the standard tools of fixed point theory and are well illustrated with the aid of examples.
Similar content being viewed by others
1 Introduction
The subject of fractional calculus has recently been investigated in an extensive manner. The publication of several books, special issues, and a huge number of articles in journals of international repute, exploring numerous aspects of this branch of mathematics, clearly indicates the popularity of the topic. One of the key factors accounting for the utility of the subject is that fractional-order operators are nonlocal in nature in contrast to the integer-order operators and can describe the hereditary properties of many underlying phenomena and processes. Owing to this characteristic, the principles of fractional calculus have played a significant role in improving the modeling techniques for several real world problems [1]–[4].
Many researchers have focused their attention on fractional differential equations and inclusions, and a variety of interesting and important results concerning existence and uniqueness of solutions, stability properties of solutions, analytic and numerical methods of solutions of these equations have been obtained and the surge for investigating more and more results is still under way. For details and examples, we refer the reader to a series of papers [5]–[29] and the references therein. Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes and some works have been published in this area, for instance; see [30]–[34] and the references therein.
In this paper, for , , , , , we investigate the fractional integro-differential equation
and related fractional integro-differential inclusion
supplemented with initial boundary conditions
where cD denotes the Caputo fractional derivative, is a continuous function, is a multifunction, is the family of all non-empty subsets of and the maps ϕ and ψ are defined by
and
where and are continuous maps, (), (), (), , , and for .
The paper is organized as follows. In Section 2, we recall some preliminary facts that we used in the sequel. Section 3 deals with the existence result for single-valued initial boundary value problem, while the results for multivalued problem are presented in Section 4. We present some examples illustrating the main results in Section 5.
2 Preliminaries
Let be a normed space, the set of all non-empty subsets of X, the set of all non-empty closed subsets of X, the set of all non-empty bounded subsets of X, the set of all non-empty compact subsets of X and the set of all non-empty compact and convex subsets of X[35]. A multivalued map is said to be convex (closed) valued whenever is convex (closed) for all [35]. The multifunction G is called bounded on bounded sets whenever is bounded subset of X for all , that is, for all [35]. Also, the multifunction is called upper semi-continuous whenever for each the set is a non-empty closed subset of X, and for every open set N of X containing , there exists an open neighborhood of such that [36], [37]. The multifunction is called compact whenever is relatively compact for all and also is called completely continuous whenever G is upper semi-continuous and compact [38], [39]. It is well known that a compact multifunction G with non-empty compact valued is upper semi-continuous if and only if G has a closed graph, that is, , for all n, and imply [37]. We say that is a fixed point of a multifunction G whenever [40]. Let and a multifunction. We say that G is measurable whenever the function is measurable for all [38], [39].
One can find basic notions of fractional calculus in [1] and [2]. We recall two necessary ones here.
The Riemann-Liouville fractional integral of order with the lower limit zero for a function is defined by for provided the integral exists.
The Caputo fractional derivative of order for a function can be written as
where and .
To define the solution for problems (1.1)-(1.3) and (1.2)-(1.3), we establish the following lemma.
Lemma 2.1
Let. Then the integral solution of the linear equation
subject to the initial boundary conditions (1.3) is given by
where
and
Proof
It is well known that the solution of (2.1) can be written as
where are arbitrary constants. Using the initial conditions , we find that . Since for all constant c, , , , , , , , , , , , and , therefore
Now using the conditions , , , and , we obtain
Substituting the values of in (2.3), we get the solution (2.2). □
3 Existence results for problem (1.1)-(1.3)
Consider the space endowed with the norm
Obviously is a Banach space.
We need the following result [40] in the sequel.
Theorem 3.1
Let E be a Banach space, a completely continuous operator and
a bounded set. Then S has a fixed point in E.
Let us define the operator by
where
For the sake of convenience, we set
Theorem 3.2
The operatoris completely continuous.
Proof
First, we show that the operator is continuous. Let be a sequence in X with and . Then we have
Since , uniformly on I. Similarly, uniformly on I for , uniformly on I for . Also, we get , , and uniformly on I for . Since
using the continuity of f, , , we get . Thus, T is continuous on X. Now, let be a bounded subset. Then there exists a positive constant such that for all and . We show that T Ω is a bounded set. We have
and
for all . Hence, we get
This implies that the operator T maps bounded sets of X into bounded sets. Now, we prove that the sets , , , are equicontinuous on I. For , we have
In a similar manner, one can find that
and
Clearly the right-hand sides of the above inequalities tend to zero as . So T is completely continuous. This completes the proof. □
Theorem 3.3
Assume that there exist positive constants, (), (), (), (), (), such that
for all and and
for, all, and all. In addition, assume that
whereand. Then problem (1.1)-(1.3) has at least one solution.
Proof
In view of Theorem 3.2, the operator is completely continuous. Next we show that the set is bounded. Let and . Then we have
and . Thus, we get
and
This implies that
and so . Thus, the set V is bounded. Hence it follows by Theorem 3.1 that the operator T has at least one fixed point, which in turn implies that problem (1.1)-(1.3) has a solution. □
Theorem 3.4
Assume thatandare continuous functions and there exist constants (), (), (), (), () such that
for all, and, and also
for, , and. In addition, suppose that
Then problem (1.1)-(1.3) has a unique solution.
Proof
Set , for , and choose . We show that , where . Let . Then we have
In a similar way, we can obtain
and
Hence,
and so . Also, we have
for all and . Similarly, one can obtain
and
for all and . This implies that
Since , the operator T is a contraction. In consequence, by the Banach contraction principle, T has a unique fixed point which corresponds to the unique solution of problem (1.1)-(1.3). □
4 Existence results for problem (1.2)-(1.3)
This section is concerned with the existence of solutions for problem (1.2)-(1.3). As before, let the space be endowed with the norm
A multivalued map is said to be Carathéodory whenever the map
is measurable for all for and the map
is upper semi-continuous for almost all . Also, a Carathéodory function F is said to be -Carathéodory whenever for each there exists such that
for all () and almost all [35], [37].
Lemma 4.1
([41])
Let E be a Banach space, an-Carathéodory multifunction, and θ a linear continuous mapping fromto. Then the operator
defined byis a closed graph operator, where
Let E be a Banach space. The multivalued map is said to be lower semi-continuous (l.s.c) type whenever is lower semi-continuous and has non-empty closed and decomposable values [42].
Now we state some well-known results which are needed in the sequel.
Lemma 4.2
([42])
Let Y be a separable metric space anda lower semi-continuous multivalued map with closed decomposable values. Then N has a continuous selection, that is, there exists a continuous mappingsuch thatfor all.
Theorem 4.3
(Nonlinear alternative of Leray-Schauder type [43])
Let E be a Banach space, C a closed and convex subset of E, and U an open subset of C such that. Ifis an upper semi-continuous compact map, then either F has a fixed point inor there is aandsuch that.
Theorem 4.4
(Covitz and Nadler [44])
Letbe a complete metric space. Ifis a contraction, then F has a fixed point.
For and , let the multifunction be defined by
and the set of selections of F by . For the sake of brevity, we set
Now, we are in a position to give our first existence result for problem (1.2)-(1.3).
Theorem 4.5
Suppose thatis a Carathéodory multivalued map and there exist continuous nondecreasing functionsfor, for, , andand nonnegative functionsfor, for, andsuch that
for all, for, , , and. Assume that there exist positive constantsandforsuch that
for, all, and allfor. If there exists a constantsuch that, then problem (1.2)-(1.3) has at least one solution, where
Proof
Let . Observe that the first property of the multifunction F and Theorem 1.3.5 in [45] imply that is non-empty. Define an operator by
where
We show that the operator Ω satisfies the hypothesis of the nonlinear alternative of the Leray-Schauder type result (Theorem 4.3). First, we show that is convex for all . Let and . Choose such that
for all . Then we have
for all . Since F has convex values, it is easy to check that is convex and so . Now, we show that Ω maps bounded sets into bounded sets in X. Let , , and . Choose such that
for all . In a similar manner, we obtain
and
for all . Thus, we get
This implies that Ω maps bounded sets into bounded sets in X. Now, we show that Ω maps bounded sets of X into equicontinuous sets. Let with , , and . Then we have
Proceeding as before, one can obtain
and
It is easy to see that the right-hand side of the above inequalities tends to zero as (independent on ). Thus, by using the Arzela-Ascoli theorem, we see that is a compact multivalued map. Next, we show that Ω has a closed graph. Let , for all n and . We show that . Since for all n, there exists such that
for all . Thus, we have to show that there exists such that
for all . Consider the linear continuous operator defined by , where
for all . Since θ is a linear continuous map, therefore, by Lemma 4.1, it follows that is a closed graph operator. Note that for all n. Since and , there exists such that
for all . Let and , then there exists such that
for all . Hence
and so . Letting , the operator is upper semi-continuous and compact. In view of the choice of U, there is no such that for some and so Ω has a fixed point by virtue of Theorem 4.3. Obviously, each fixed point of Ω is a solution of problem (1.2)-(1.3). This completes the proof. □
Our next result deals with the case that F is not necessary convex valued.
Theorem 4.6
Suppose that is a multifunction such that the map
is measurable and the map
is lower semi-continuous for almost all. Assume that there exist continuous nondecreasing functionsfor, for, , andand nonnegative functionsfor, for, , andsuch that
for all, for, , , and. Furthermore, there exist positive constants, andforsuch that
for, and allfor. If there exists a constantsuch that, then problem (1.2)-(1.3) has at least one solution, where
Proof
In view of the given assumptions and Lemma 4.1 in [46], it follows that F is of lower semi-continuous type. Thus, by Lemma 4.2, there exists a continuous function such that for all . Now, consider the equation
supplemented with boundary conditions (1.3). Note that each solution of problem (4.1)-(1.3) with the given conditions is a solution of problem (1.2)-(1.3). Define the operator by
for all . Following the procedure employed in the last result, one can show that is continuous and completely continuous and satisfies all conditions of the nonlinear alternative of Leary-Schauder type for single-valued maps. Consequently, there exists a solution for problem (4.1)-(1.3). This completes the proof. □
Finally, we establish the existence of a solution for the case that the right-hand side of (1.2) is non-convex valued.
Theorem 4.7
Assume that is a multifunction such that the map
is measurable for all, the mapis integrably bounded for almost alland there are nonnegative functions, such that
for almost alland all, . Also, suppose that there existsuch that
for all, , and. If, then problem (1.2)-(1.3) has at least one solution, where
Proof
With the given assumptions and Theorem III-6 (the measurable selection theorem) in [47], one can infer that F admits a measurable selection . Since F is integrably bounded, , so for all . Now, we show that the operator Ω satisfies the assumptions of Theorem 4.4. First, we show that for all . Let for all and for some . For each n, choose such that
for all . Since F has compact values, there is a subsequence of that converges to v in . Thus, and
for all . This implies that . Next, we show that there exists such that
for all . Let and . Choose such that
for all . On the other hand we have
for almost all . Hence, there exists such that
for almost all . Define by for all . By Theorem III-41 in [47], it follows that V is measurable. Since the multivalued operator is measurable (Proposition III-4 in [47]), there exists a function such that
for almost all . Define
for all . Then we have
Further, interchanging the roles of z and , we get
Since , Ω is a contraction and so by Theorem 4.4, Ω has a fixed point which corresponds to a solution of problem (1.2)-(1.3). □
5 Examples
This section is devoted to the illustration of Theorems 4.5 and 4.7.
Example 5.1
Consider the fractional differential inclusion
supplemented with the boundary conditions , , , , and where
Here , , , , , , , , , , , , , , , , , , , , , . Define the multifunction by
and note that
for all and . It is clear that F has convex and compact values and is of Carathéodory type. Let , , , , , , , , , , , , , , , , , and for all and . Hence,
for all and . With the given values, it is found that . Letting , all the conditions of Theorem 4.5 hold and consequently problem (5.1) has at least one solution.
Example 5.2
Consider the fractional differential inclusion
with the boundary value problems , , , , and , where
Here , , , , , , , , , , , , , , , , , . We define the multifunction as
for all and . It is clear that F has compact values and
for all and . Fix , , , . As in the previous example, it is found that and
As all the conditions of Theorem 4.7 are satisfied, the inclusion problem (5.2) has at least one solution.
References
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.
Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 2010, 109: 973-1033. 10.1007/s10440-008-9356-6
Bhalekar S, Daftardar-Gejji V, Baleanu D, Magin R: Fractional Bloch equation with delay. Comput. Math. Appl. 2011, 61: 1355-1365. 10.1016/j.camwa.2010.12.079
Ahmad B, Nieto JJ: Sequential fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 2012, 64: 3046-3052. 10.1016/j.camwa.2012.02.036
Bai ZB, Sun W: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 2012, 63: 1369-1381. 10.1016/j.camwa.2011.12.078
Sakthivel R, Mahmudov NI, Nieto JJ: Controllability for a class of fractional-order neutral evolution control systems. Appl. Math. Comput. 2012, 218: 10334-10340. 10.1016/j.amc.2012.03.093
Agarwal RP, O’Regan D, Stanek S: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 2012, 285: 27-41. 10.1002/mana.201000043
Ahmad B, Ntouyas SK: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 2012, 15(3):362-382. 10.2478/s13540-012-0027-y
Liu X, Liu Z: Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative. Abstr. Appl. Anal. 2012., 2012:
Wang JR, Zhou Y, Medved M: Qualitative analysis for nonlinear fractional differential equations via topological degree method. Topol. Methods Nonlinear Anal. 2012, 40: 245-271.
Cabada A, Wang G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 2012, 389: 403-411. 10.1016/j.jmaa.2011.11.065
Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013., 2013:
Cernea A: On a multi point boundary value problem for fractional order differential inclusion. Arab J. Math. Sci. 2013, 19(1):73-83. 10.1016/j.ajmsc.2012.07.001
Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56. 10.1016/j.cam.2013.02.010
Baleanu D, Rezapour S, Mohammadi H: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2013., 371(1990): 10.1098/rsta.2012.0144
Baleanu D, Agarwal RP, Mohammadi H, Rezapour S: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-112
Zhou WX, Chu YD, Baleanu D: Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations. Adv. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-114
Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-20
Baleanu D, Nazemi SZ, Rezapour S: Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations. Adv. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-368
Nieto JJ, Ouahab A, Prakash P: Extremal solutions and relaxation problems for fractional differential inclusions. Abstr. Appl. Anal. 2013., 2013: 10.1155/2013/292643
Zhai C, Hao M: Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-85
Graef JR, Kong L, Wang M: Existence and uniqueness of solutions for a fractional boundary value problem on a graph. Fract. Calc. Appl. Anal. 2014, 17: 499-510. 10.2478/s13540-014-0182-4
Baleanu D, Nazemi SZ, Rezapour S: Attractivity for a k -dimensional system of fractional functional differential equations and global attractivity for a k -dimensional system of nonlinear fractional differential equations. J. Inequal. Appl. 2014., 2014: 10.1186/1029-242X-2014-31
Baleanu D, Nazemi SZ, Rezapour S: The existence of solution for a k -dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary value problems. Abstr. Appl. Anal. 2014., 2014:
Punzo F, Terrone G: On the Cauchy problem for a general fractional porous medium equation with variable density. Nonlinear Anal. 2014, 98: 27-47. 10.1016/j.na.2013.12.007
Liu X, Liu Z, Fu X: Relaxation in nonconvex optimal control problems described by fractional differential equations. J. Math. Anal. Appl. 2014, 409: 446-458. 10.1016/j.jmaa.2013.07.032
Ahmad B:Existence of solutions for fractional differential equations of order with anti-periodic boundary conditions. J. Appl. Math. Comput. 2010, 34: 385-391. 10.1007/s12190-009-0328-4
Wang F, Liu Z: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv. Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-116
Alsaedi A, Ahmad B, Assolami A: On antiperiodic boundary value problems for higher-order fractional differential equations. Abstr. Appl. Anal. 2012., 2012:
Wang X, Guo X, Tang G: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. J. Appl. Math. Comput. 2013, 41: 367-375. 10.1007/s12190-012-0613-5
Ahmad B, Nieto JJ, Alsaedi A, Mohamad N: On a new class of antiperiodic fractional boundary value problems. Abstr. Appl. Anal. 2013., 2013:
Deimling K: Multivalued Differential Equations. de Gruyter, Berlin; 1992.
Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Kluwer Academic, Dordrecht; 1997.
Kisielewicz M: Differential Inclusions and Optimal Control. Kluwer Academic, Dordrecht; 1991.
Aubin JP, Cellina A: Differential Inclusions. Springer, Berlin; 1984.
Aubin JP, Frankowska H: Set-Valued Analysis. Birkhäuser, Boston; 1990.
Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1980.
Lasota A, Opial Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 781-786.
Bressan A, Colombo G: Extensions and selections of maps with decomposable values. Stud. Math. 1988, 90: 69-86.
Granas A, Dugundji J: Fixed Point Theory. Springer, Berlin; 2005.
Covitz H, Nadler SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5-11. 10.1007/BF02771543
Kamenskii M, Obukhovskii V, Zecca P: Condensing Multi-Valued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter, Berlin; 2001.
Tolstonogov AA: Bogolyubov’s theorem under constraints generated by a controlled second-order evolution system. Izv. Ross. Akad. Nauk, Ser. Mat. 2003, 67(5):177-206. 10.4213/im456
Castaing C, Valadier M: Convex Analysis and Measurable Multifunctions. Springer, Berlin; 1977.
Acknowledgements
This paper was funded by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ahmad, B., Alsaedi, A., Nazemi, S.Z. et al. Some existence theorems for fractional integro-differential equations and inclusions with initial and non-separated boundary conditions. Bound Value Probl 2014, 249 (2014). https://doi.org/10.1186/s13661-014-0249-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-014-0249-5