Abstract
In this paper, we investigate the existence of solution for two systems of fractional differential inclusions via some integral boundary value conditions. For this purpose, we use an endpoint result for multifunctions which has been proved in 2010 by Amini-Harandi (Nonlinear Anal. 72:132-134, 2010). Finally, we give an example for illustrating one of our results.
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1 Introduction
As we know, diverse classes of fractional differential equations have been studied by researchers (see for example, [1–15] and the references therein). Much attention has been devoted to the fractional differential inclusions (see for example, [16–32] and the references therein). Also, there have been provided many applications of this field (see for example, [33, 34] and [35]).
It is the aim of this paper to investigate the existence of solutions for two systems of fractional differential inclusions, subject to some integral boundary value conditions. In this respect, we use an endpoint result for multifunctions due to Amini-Harandi, [36]. We provide an example for illustrating one of our results.
2 Preliminaries
As is well known, the Riemann-Liouville fractional integral of order of a function is given by , provided the right side is pointwise defined on (see [10, 13] and [14]). The Caputo fractional derivative of order α for a continuous function f is defined by , where (see [10, 13] and [14]).
Recall that a multifunction is said to be measurable whenever the function is measurable for all , where [37].
Let be a metric space. We have the well-known Pompeiu-Hausdorff metric (see [38])
where . Then is a metric space and is a generalized metric space, where is the set of closed and bounded subsets of X and is the set of closed subsets of X (see [27]).
Let be a multifunction. An element is called an endpoint of T whenever [36]. Also, we say that T has the approximate endpoint property whenever [36]. A function is called upper semi-continuous whenever for all sequences with [36].
In 2010, Amini-Harandi proved the next result [36].
Lemma 2.1 Let be an upper semi-continuous function such that and , for all , a complete metric space and a multifunction such that for all . Then T has a unique endpoint if and only if T has approximate end point property.
In 2011, Ahmad et al. investigated the fractional inclusion problem , via the integral boundary conditions for , where F is a multifunction (see for more details [20]).
In this paper, we are going to extend the problem in a sense. In this respect, we first investigate the existence of solution for the fractional differential inclusion problem
via integral boundary value conditions
where , , , , and is a multifunction, are continuous functions and is the standard Caputo differentiation. Here, is the set of all compact subsets of ℝ.
Also, we investigate the existence of solution for the fractional differential inclusion problem
via integral boundary value conditions
where , , , for all , , , , and is a multifunction.
3 Main results
Now, we are ready to state and prove our main results. First, we give the following one.
Lemma 3.1 Let , , , and be continuous functions. The unique solution of the fractional differential problem
via the boundary value conditions (2) is given by
Proof It is known that the general solution of (5) is
that is
where , , are real arbitrary constants (see [10, 13] and [14]). Thus,
and . Hence,
and
By using the boundary conditions, we obtain
and
This is a linear system of equations of triangular form, having , , and as unknowns. We solve by back substitution and find
and
Now, we replace , , and in (6) and find the solution as we stated. This completes the proof. □
Let endowed with the norm . Then is a Banach space. For , define
For the study of problem (1) and (2), we shall consider the following conditions.
(H1) is an integrable bounded multifunction such that is measurable for all ;
(H2) be continuous functions, a nondecreasing upper semi-continuous map such that and for all ;
(H3) There exist such that
and for all , and , where
and , and finally
(H4) is given by
where
Theorem 3.1 Assume that (H1)-(H4) are satisfied. If the multifunction N has the approximate endpoint property, then the boundary value inclusion problem (1) and (2) has a solution.
Proof We show that the multifunction has a endpoint which is a solution of the problem (1) and (2).
Note that the multivalued map is measurable and has closed values for all . Hence, it has measurable selection and so is nonempty for all . First, we show that is closed subset of X for all .
Let and be a sequence in with . For each , choose such that
for all .
Since F has compact values, has a subsequence which converges to some . We denote this subsequence again by .
It is easy to check that and
for all . This implies that and so N has closed values.
Since F is a compact multivalued map, it is easy to check that is a bounded set for all .
Now, we show that .
Let and . Choose such that
for almost all .
Since
for all , there exists such that
for all .
Consider the multivalued map defined by
Since and are measurable, the multifunction is measurable.
Choose such that
for all .
Now, consider the element , which is defined by
for all . Thus,
and
Hence,
Thus, it is easy to get for all .
Since the multifunction N has approximate endpoint property, by using Lemma 2.1 there exists such that . Hence by using Lemma 3.1, is a solution of the problem (1) and (2). □
Now, we investigate the existence of solution for the fractional differential inclusion problem
via integral boundary value conditions
where , , , , for all , , and is a multifunction.
Lemma 3.2 Let , , , and for . The unique solution of the fractional differential problem via the boundary value conditions
with , is
where is the Green function given by
whenever ,
whenever ,
whenever ,
whenever ,
whenever and
whenever , where and .
Proof It is known that the general solution of the equation is
where are arbitrary constants (see [10, 13] and [14]). Thus,
and . Hence,
and
By using the boundary conditions, we obtain
and
Thus,
Hence,
This completes the proof. □
Suppose that endowed with the norm . Then is a Banach space [15]. For , define
Now, put
and for all .
Theorem 3.2 Let a nondecreasing upper semi-continuous map such that and for all , a multifunction such that is measurable and integrable bounded for all . Assume that there exists such that
Define by
If the multifunction Ω has the approximate endpoint property, then the boundary value inclusion problem (3) and (4) has a solution.
Proof We show that the multifunction has a endpoint which is a solution of the problem (3) and (4).
First, we show that is closed subset of X for all .
Let and be a sequence in with . For each , choose such that for all . Since F has compact values, has a subsequence which converges to some . We denote this subsequence again by .
It is easy to check that and for all . This implies that and so Ω has closed values. Since F is a compact multivalued map, it is easy to check that is a bounded set for all .
Now, we show that for all , .
Let and . Choose such that for almost all . Since
for all , there exists such that
for all . Consider the multivalued map defined by the rule
Since and
are measurable, the multifunction
is measurable.
Choose such that
for all . Now, consider the element which is defined by for all .
Thus,
and
for all . Hence,
Analogously, interchanging the roles of x, y, we obtain . Since the multifunction Ω has the approximate endpoint property, by using Lemma 3.2 there exists such that . □
4 Example
Here, we give an example to illustrate our first main result.
Example 4.1 Consider the fractional differential inclusion problem via the integral boundary conditions
where . Define the maps
and by the rule
where
Put , , , , , , , , . Then we have , , and .
It is easy to check that
and for all , and . Note that and so . Hence, N has the approximate endpoint property. Now by using Theorem 3.1, the system of fractional differential inclusions has at least one solution.
5 Concluding remarks
This work contains our dedicated study to develop and improve methods for studying two fractional differential inclusions via integral boundary value conditions. We introduced our result by using an endpoint result for multifunctions, due to Amini-Harandi [36]. This study is motivated by relevant applications for solving many real-world problems which give rise to mathematical models in the sphere of boundary value problems.
References
Ahmad B, Nieto JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 2010, 35: 295–304.
Bai Z, 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
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: Article ID 112
Baleanu D, Mohammadi H, Rezapour S: The existence of solutions for a nonlinear mixed problem of singular fractional differential equations. Adv. Differ. Equ. 2013., 2013: Article ID 359
Baleanu D, Mohammadi H, Rezapour S: Positive solutions of a boundary value problem for nonlinear fractional differential equations. Abstr. Appl. Anal. 2012., 2012: Article ID 837437
Baleanu D, Mohammadi H, Rezapour S: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2013., 371: Article ID 20120144
Baleanu D, Mohammadi H, Rezapour S: On a nonlinear fractional differential equation on partially ordered metric spaces. Adv. Differ. Equ. 2013., 2013: Article ID 83
Baleanu D, Nazemi Z, Rezapour S: The existence of positive solutions for a new coupled system of multi-term singular fractional integro-differential boundary value problems. Abstr. Appl. Anal. 2013., 2013: Article ID 368659
Baleanu D, Nazemi Z, Rezapour S: Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations. Adv. Differ. Equ. 2013., 2013: Article ID 368
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Miller KS, Ross B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York; 1993.
Mohammadi H, Rezapour S: Two existence results for nonlinear fractional differential equations by using fixed point theory on ordered Gauge spaces. J. Adv. Math. Stud. 2013,6(2):154–158.
Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.
Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 2009, 22: 64–69. 10.1016/j.aml.2008.03.001
Agarwal RP, Ahmad B: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. J. Appl. Math. Comput. 2011, 62: 1200–1214. 10.1016/j.camwa.2011.03.001
Agarwal RP, Belmekki M, Benchohra M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009., 2009: Article ID 981728
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
Ahmad B, Ntouyas SK: Boundary value problem for fractional differential inclusions with four-point integral boundary conditions. Surv. Math. Appl. 2011, 6: 175–193.
Ahmad B, Ntouyas SK, Alsedi A: On fractional differential inclusions with anti-periodic type integral boundary conditions. Bound. Value Probl. 2013., 2013: Article ID 82
Alsaedi A, Ntouyas SK, Ahmad B: Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach space. Abstr. Appl. Anal. 2013., 2013: Article ID 869837
Aubin J, Ceuina A: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin; 1984.
Bragdi M, Debbouche A, Baleanu D: Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach space. Adv. Math. Phys. 2013., 2013: Article ID 426061
Benchohra M, Hamidi N: Fractional order differential inclusions on the Half-Lin. Surv. Math. Appl. 2010, 5: 99–111.
Benchohra M, Ntouyas SK: On second order differential inclusions with periodic boundary conditions. Acta Math. Univ. Comen. 2000,LXIX(2):173–181.
El-Sayed AMA, Ibrahim AG: Multivalued fractional differential equations. Appl. Math. Comput. 1995, 68: 15–25. 10.1016/0096-3003(94)00080-N
Kisielewicz M: Differential Inclusions and Optimal Control. Kluwer Academic, Dordrecht; 1991.
Liu X, Liu Z: Existence result for fractional differential inclusions with multivalued term depending on lower-order derivative. Abstr. Appl. Anal. 2012., 2012: Article ID 423796
Nieto JJ, Ouahab A, Prakash P: Extremal solutions and relaxation problems for fractional differential inclusions. Abstr. Appl. Anal. 2013., 2013: Article ID 292643
Phung PD, Truong LX: On a fractional differential inclusion with integral boundary conditions in Banach space. Fract. Calc. Appl. Anal. 2013,16(3):538–558.
Ouahab A: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 2008, 69: 3877–3896. 10.1016/j.na.2007.10.021
Wang J, Ibrahim AG: Existence and controllability results for nonlocal fractional impulsive differential inclusions in Banach spaces. J. Funct. Spaces Appl. 2013., 2013: Article ID 518306
Heikkila S, Lakshmikantam V: Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations. Dekker, New York; 1994.
Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 2006,45(5):765–772. 10.1007/s00397-005-0043-5
Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
Amini-Harandi A: Endpoints of set-valued contractions in metric spaces. Nonlinear Anal. 2010, 72: 132–134. 10.1016/j.na.2009.06.074
Deimling K: Multi-Valued Differential Equations. de Gruyter, Berlin; 1992.
Berinde V, Pacurar M: The role of the Pompeiu-Hausdorff metric in fixed point theory. Creative Math. Inform. 2013,22(2):35–42.
Acknowledgements
Research of the second and fourth authors was supported by Azarbaijan Shahid Madani University.
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Ghorbanian, R., Hedayati, V., Postolache, M. et al. On a fractional differential inclusion via a new integral boundary condition. J Inequal Appl 2014, 319 (2014). https://doi.org/10.1186/1029-242X-2014-319
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DOI: https://doi.org/10.1186/1029-242X-2014-319