On the set of solutions of fractional order Riemann-Liouville integral inclusions

Demonstratio Mathematica

Volumn 46, Issue 2, 2013, Pages 271-281

  Abbas, Saïd ^a   Benchohra, Mouffak ^b  

^a UNIVERSITY OF SAIDA   (Algeria)

^b UNIVERSITY OF SIDI BEL ABBES   (Algeria)

Author keywords

Fixed point; Integral inclusion; Left sided mixed Riemann Liouville integral of fractional order; Solution set

Indexed keywords

EID: 84882955840     PISSN: 04201213     EISSN: 23914661     Source Type: Journal    
DOI: 10.1515/dema-2013-0458     Document Type: Article

References (34)

1. S. Abbas, R. P. Agarwal, M. Benchohra, Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay, Nonlinear Anal. Hybrid Syst. 4 (2010), 818-829.
2. S. Abbas, M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62-72.
3. S. Abbas, M. Benchohra, The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses, Discuss. Math. Differ. Incl. 30(1) (2010), 141-161.
4. S. Abbas, M. Benchohra, Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay, Fract. Calc. Appl. Anal. 13(3) (2010), 225-244.
5. S. Abbas, M. Benchohra, L. Górniewicz, Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative, Sci. Math. Jpn. online e- 2010, 271-282.
6. S. Abbas, M. Benchohra, J. J. Nieto, Global uniqueness results for fractional order partial hyperbolic functional differential equations, Adv. Differential Equ. 2011, Art. ID 379876, 25 pp.
7. A. Belarbi, M. Benchohra, A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal. 85 (2006), 1459-1470.
8. M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions, Appl. Anal. 87(7) (2008), 851-863.
9. M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12.
10. M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340-1350.
11. F. S. De Blasi, J. Myjak, On the set of solutions of a differential inclusion, Bull. Inst. Math. Acad. Sinica 14 (1986), 271-275.
12. F. S. De Blasi, G. Pianigiani, V. Staicu, On the solution sets of some nonconvex hyperbolic differential inclusions, Czechoslovak Math. J. 45 (1995), 107-116. (Pubitemid 126175742)
13. A. Cernea, On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions, Czechoslovak Math. J. 52(127)(1) (2002), 215-224.
14. A. Cernea, On the solution set of a nonconvex nonclosed second order differential inclusion, Fixed Point Theory 8(1) (2007), 29-37.
15. A. Cernea, Arcwise connectedness of the solution set of a nonclosed nonconvex integral inclusion, Miskolc Math. Notes 9(1) (2008), 33-39.
16. K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248.
17. A. Fryszkowski, Fixed Point Theory for Decomposable Sets. Topological Fixed Point Theory and Its Applications, 2. Kluwer Academic Publishers, Dordrecht, 2004.
18. W. G. Glockle, T. F. Nonnenmacher, A fractional calculus approach of selfsimilar protein dynamics, Biophys. J. 68 (1995), 46-53.
19. L. Górniewicz, T. Pruszko, On the set of solutions of the Darboux problem for some hyperbolic equations, Bull. Acad. Polon. Sci. Math. Astronom. Phys. 38 (1980), 279-285.
20. A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
21. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
22. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
23. A. A. Kilbas, S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84-89. (Pubitemid 40551100)
24. M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
25. F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in "Fractals and Fractional Calculus in Continuum Mechanics" (A. Carpinteri and F. Mainardi, Eds), pp. 291-348, Springer-Verlag, Wien, 1997.
26. S. Marano, V. Staicu, On the set of solutions to a class of nonconvex nonclosed differential inclusions, Acta Math. Hungar. 76 (1997), 287-301. (Pubitemid 127337416)
27. F. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), 7180-7186.
28. K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
29. K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974.
30. I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
31. I. Podlubny, I. Petraš, B. M. Vinagre, P. O'Leary, L. Dorcak, Analogue realizations of fractional-order controllers, fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281-296. (Pubitemid 34945402)
32. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
33. A. N. Vityuk, A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7(3) (2004), 318-325.
34. S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Differential Equations 36 (2006), 1-12. (Pubitemid 43448954)