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Fractional differential inclusions with fractional separated boundary conditions

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Abstract

This paper studies a new class of boundary value problems of nonlinear fractional differential inclusions of order q ∈ (1, 2] with fractional separated boundary conditions. New existence results are obtained for this class of problems by using some standard fixed point theorems. A possible generalization for the inclusion problem with fractional separated integral boundary conditions is also discussed. Some illustrative examples are presented.

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References

  1. R.P. Agarwal and B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, No 3 (2011), 1200–1214.

    Article  MathSciNet  MATH  Google Scholar 

  2. R.P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Difference Equ. 2009, Art. ID 981728, 47 pp.

  3. A. Aghajani, Y. Jalilian and J.J. Trujillo, On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 15, No 1 (2012), 44–69; DOI: 10.2478/s13540-012-0005-4; http://www.springerlink.com/content/1311-0454/15/1/

    MathSciNet  Google Scholar 

  4. B. Ahmad, Existence results for fractional differential inclusions with separated boundary conditions. Bull. Korean Math. Soc. 47, No 4 (2010), 805–813.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations. Appl. Math. Lett. 23, No 4 (2010), 390–394.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Ahmad and R.P. Agarwal, On nonlocal fractional boundary value problems. Dynam. Contin. Discrete Impuls. Systems 18, No 4 (2011), 535–544.

    MathSciNet  MATH  Google Scholar 

  7. B. Ahmad and J.J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, No 9 (2009), 1838–1843.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Ahmad and J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, No 2 (2010), 295–304.

    MathSciNet  MATH  Google Scholar 

  9. B. Ahmad and J.J. Nieto, Anti-periodic fractional boundary value problem with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 451–462; DOI: 10.2478/s13540-012-0032-1; same issue; http://www.springerlink.com/content/1311-0454/15/3/

    MathSciNet  Google Scholar 

  10. B. Ahmad and J.J. Nieto, J. Pimentel, Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, No 3 (2011), 1238–1250.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Ahmad and S.K. Ntouyas, Some existence results for boundary value problems fractional differential inclusions with non-separated boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2010, No 71 (2010), 1–17.

    MathSciNet  Google Scholar 

  12. B. Ahmad and S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Difference Equ. 2011, Art. ID 107384, 11 pp.

  13. B. Ahmad and S.K. Ntouyas, A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order. Electron. J. Qual. Theory Differ. Equ. 2011, No 22, 15 pp.

  14. B. Ahmad and S.K. Ntouyas, Boundary value problems for q-difference inclusions. Abstr. Appl. Anal. 2011, Article ID 292860, 15 pp.

  15. B. Ahmad and S.K. Ntouyas, A note on fractional differential equations with fractional separated boundary conditions. Abstr. Appl. Anal. 2012, Article ID 818703, 11 pp.

  16. B. Ahmad and S. Sivasundaram, Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions. Commun. Appl. Anal. 13, No 1 (2009), 121–228.

    MathSciNet  MATH  Google Scholar 

  17. B. Ahmad and S. Sivasundaram, On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput. 217, No 2 (2010), 480–487.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Baleanu, O.G. Mustafa and R.P. Agarwal, An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 23, No 9 (2010), 1129–1132.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values. Studia Math. 90, No 1 (1988), 69–86.

    MathSciNet  MATH  Google Scholar 

  20. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

    MATH  Google Scholar 

  21. A. Cernea, On the existence of solutions for nonconvex fractional hyperbolic differential inclusions. Commun. Math. Anal. 9, No 1 (2010), 109–120.

    MathSciNet  MATH  Google Scholar 

  22. A. Cernea, A note on the existence of solutions for some boundary value problems of fractional differential inclusions. Fract. Calc. Appl. Anal. 15, No 2 (2012), 183–194; DOI: 10.2478/s13540-012-0013-4; http://www.springerlink.com/content/1311-0454/15/2/

    Google Scholar 

  23. A. Cernea, On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38, No 1–2 (2012), 133–143.

    Article  Google Scholar 

  24. Y.-K. Chang and J.J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Modelling 49, No 9–10 (2009), 605–609.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Covitz and S.B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5–11.

    Article  MathSciNet  MATH  Google Scholar 

  26. K. Deimling, Multivalued Differential Equations. Walter De Gruyter, Berlin-New York, 1992.

    Book  MATH  Google Scholar 

  27. N.J. Ford and M. Luisa Morgado, Fractional boundary value problems: Analysis and numerical methods. Fract. Calc. Appl. Anal. 14, No 4 (2011), 554–567; DOI: 10.2478/s13540-011-0034-4; http://www.springerlink.com/content/1311-0454/14/4/

    MathSciNet  Google Scholar 

  28. A. Granas and J. Dugundji, Fixed Point Theory. Springer-Verlag, New York, 2005.

    Google Scholar 

  29. E. Hernandez and D. O’Regan, K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, No 10 (2010), 3462–3471.

    Article  MathSciNet  MATH  Google Scholar 

  30. Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, 1997.

    MATH  Google Scholar 

  31. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

    Book  MATH  Google Scholar 

  32. M. Kisielewicz, Differential Inclusions and Optimal Control. Kluwer, Dordrecht, 1991.

    Google Scholar 

  33. A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781–786.

    MathSciNet  MATH  Google Scholar 

  34. R.L. Magin, Fractional Calculus in Bioengineering. Begell House Publisher, Inc., Connecticut, 2006.

    Google Scholar 

  35. A. Ouahab, Fractional semilinear differential inclusions. Comput. Math. with Appl., Available online 11 April 2012; http://dx.doi.org/10.1016/j.camwa.2012.03.039

  36. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.

    MATH  Google Scholar 

  37. J. Sabatier, O.P. Agrawal and J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, 2007.

    MATH  Google Scholar 

  38. J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl. 12, No 6 (2011), 3642–3653.

    Article  MathSciNet  MATH  Google Scholar 

  39. G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford, 2005.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Bashir Ahmad

  2. Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece

    Sotiris K. Ntouyas

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  1. Bashir Ahmad
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  2. Sotiris K. Ntouyas
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Correspondence to Bashir Ahmad.

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Ahmad, B., Ntouyas, S.K. Fractional differential inclusions with fractional separated boundary conditions. fcaa 15, 362–382 (2012). https://doi.org/10.2478/s13540-012-0027-y

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