ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

On the Solutions of Some Boundary Value Problems for Integro-differential Inclusions of Fractional Order

Journal of Applied Nonlinear Dynamics 6(2) (2017) 173--179 | DOI:10.5890/JAND.2017.06.004

Aurelian Cernea$^{1}$,$^{2}$

$^{1}$ Faculty of Mathematics and Informatics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania

$^{2}$ Academy of Romanian Scientists, Splaiul Independen¸tei 54, 050094 Bucharest, Romania

Abstract

We study the existence of solutions for fractional integro-differential inclusions with nonlocal boundary conditions and with multi-order fractional integral conditions. We establish Filippov type existence results in the case of nonconvex set-valued maps.

References

1.  [1] Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier: Amsterdam.
2.  [2] Miller, K. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, John Wiley: New York, 1993.
3.  [3] Podlubny, I. (1999), Fractional Differential Equations, Academic Press: San Diego.
4.  [4] Almeida, R., Pooseh, S., and Torres, D.F.M. (2015), Computational Methods in the Fractional Calculus of Variations, Imperial College Press: London.
5.  [5] Malinowska, A.B., Odzijewicz, T., and Torres, D.F.M. (2015), Advanced Methods in the Fractional Calculus of Variations, Springer: Cham.
6.  [6] Băleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012), Fractional Calculus Models and Numerical Methods, World Scientific: Singapore.
7.  [7] Caputo, M. (1969), Elasticità e Dissipazione, Zanichelli: Bologna.
8.  [8] Cernea, A. (2014), On a fractional integrodifferential inclusion, Electronic J. Qual. Theory Differ. Equ., 2014(25), 1-11.
9.  [9] Cernea, A. (2015), Continuous selections of solutions sets of fractional integrodifferential inclusions, Acta Math. Scientia, 35B, 399-406.
10.  [10] Cernea, A. (2016), A note on the solution set of a fractional integrodifferential inclusion, Progress Fractional Differentiation Appl., 2, 1-7.
11.  [11] Chalishajar, D.N. and Karthikeyan, K. (2013), Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces, Acta Math. Scientia, 33B, 758-772.
12.  [12] Karthikeyan, K. and Trujillo, J.J. (2012), Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Commun Nonlinear Sci. Numer. Simulat. 17, 4037-4043.
13.  [13] Wang, J.R., Wei, W., and Yang, Y. (2010) Fractional nonlocal integro-differential equations of mixed type with time varying generating operators and optimal control, Opuscula Math. 30, 361-381.
14.  [14] Filippov, A.F. (1967), Classical solutions of differential equations with multivalued right hand side, SIAM J. Control, 5, 609-621.
15.  [15] Cernea, A. (2010), Continuous version of Filippov’s theorem for fractional differential inclusions, Nonlinear Anal., 72, 204-208.
16.  [16] Cernea, A. (2015), Filippov lemma for a class of Hadamard-type fractional differential inclusions, Fract. Calc. Appl. Anal., 18, 163-171.
17.  [17] Ahmad, B. and Ntouyas, S.K. (2014), On higher-order sequential differential inclusions with nonlocal three- point boundary conditions, Abstract Appl. Anal., 2014, ID 659405, 1-10.
18.  [18] Aphithana, A., Ntouyas, S.K., and Tariboon, S. (2015), Existence and uniqueness of symmetric solutions for fractional differential equations with multi-order fractional integral conditions, Boundary Value Problems, 2015(68), 1-14.
19.  [19] Aubin, J.P. and Frankowska, H. (1990), Set-valued Analysis, Birkhauser: Basel.