---

Journal of Applied Nonlinear Dynamics 2(2) (2013) 151--160 | DOI:10.5890/JAND.2013.04.004

Aurelian Cernea

Faculty of Mathematics and Informatics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania

Download Full Text PDF

 

Abstract

We study a multi point boundary value problem for a fractional order differential inclusion involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.

References

1. [1]	Bǎleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2012), Fractional Calculus Models and Numerical Methods, World Scientific, Singapore.

2. [2]	Bǎleanu, D., Mustafa, O.G., and Agarwal, R.P. (2010), An existence result for a superlinear frsctional differential equation, Applied Mathematics Letters, 23, 1129-1132.

3. [3]	D. Bǎleanu, Mustafa, O.G. and Agarwal, R.P. (2010), On the solution set for a class of sequential fractional differential equations, Journal of Physics A: Mathematical and Theoretical, 43, 385-409.

4. [4]	Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

5. [5]	Miller, K. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.

6. [6]	Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.

7. [7]	Benchohra, M., Henderson, J., Ntouyas, S.K., and Ouahab, A. (2008), Existence results for fractional order functional differential inclusions with infinite delay and applications to control theory, Fractional Calculus and Applied Analysis, 11, 35-56.

8. [8]	Cernea, A. (2010), Continuous version of Filippov's theorem for fractional differential inclusions, Nonlinear Analysis, 72, 204-208.

9. [9]	Cernea, A. (2011), Some remarks on a fractional differential inclusion with non-separated boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 45, 1-14.

10. [10]	Cernea, A. (2012), A note on the existence of solutions for some boundary value problems of fractional differential inclusions, Fractional Calculus and Applied Analysis, 15, 183-194.

11. [11]	Cernea, A. (2013), On a multi point boundary value problem for a fractional order differential inclusion, Arab Journal of Mathematical Sciences, 19, 73-83.

12. [12]	Ouahab, A. (2009), Some results for fractional boundary value problem of differential inclusions, Nonlinear Analysis, 69, 3871-3896.

13. [13]	Nyamoradi, N. (2012), Existence of solutions for multi point boundary value problems for fractional differential equations, Arab Journal of Mathematical Sciences, 18, 165-175.

14. [14]	Filippov, A.F. (1967), Classical solutions of differential equations with multivalued right hand side, SIAM Journal on Control and Optimization, 5, 609-621.

15. [15]	Marano, S. and Staicu, V. (1997), On the set of solutions to a class of nonconvex nonclosed differential inclusions, Acta Mathematica Hungarica, 76, 287-301.

16. [16]	Marano, S. (1994), Fixed points of multivalued contractions with nonclosed, nonconvex values, Atti. Accad. Naz. Lincei Rend. Cl. Sci.Fis. Mat. Natur., 5, 203-212.

17. [17]	Aubin, J.P. and Frankowska, H. (1990), Set-valued Analysis, Birkhauser: Basel.

18. [18]	Kuratowski, K. and Ryll-Nardzewski, C. (1965), A general theorem on selectors, Bulletin of the Polish Academy of Sciences, 13, 397-403.