Title: Fractional Differential Equations and Inclusions with Nonlocal Generalized Riemann-Liouville Integral Boundary Conditions
Author(s): Bashir Ahmad, Sotiris K. Ntouyas, Ahmed Alsaedi
Pages: 231-247
Cite as:
Bashir Ahmad, Sotiris K. Ntouyas, Ahmed Alsaedi, Fractional Differential Equations and Inclusions with Nonlocal Generalized Riemann-Liouville Integral Boundary Conditions, Int. J. Anal. Appl., 13 (2) (2017), 231-247.

Abstract


In this paper, we study a new kind of nonlocal boundary value problems of nonlinear fractional differential equations and inclusions supplemented with nonlocal and generalized Riemann-Liouville fractional integral boundary conditions. In case of single valued maps (equations), we make use of contraction mapping principle, fixed point theorem due to Sadovski, Krasnoselskii-Schaefer fixed point theorem due to Burton and Kirk, and fixed point theorem due to O’Regan to obtain the desired existence results. On the other hand, the existence results for inclusion case are based on Krasnoselskii’s fixed point theorem for multivalued maps and nonlinear alternative for contractive maps. Examples illustrating the main results are also constructed.

Full Text: PDF

 

References


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

  2. 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.

  3. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston, 2012.

  4. R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59 (2010), 1095-1100.

  5. D. Baleanu, O.G. Mustafa, R.P. Agarwal, On L p -solutions for a class of sequential fractional differential equations, Appl. Math. Comput. 218 (2011), 2074-2081.

  6. B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011 (2011), Art. ID 36.

  7. B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ. 2011 (2011), Art. ID 107384.

  8. D. O’Regan, S. Stanek, Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 71 (2013), 641-652.

  9. B. Ahmad, S.K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. 2013 (2013) Art. ID 320415.

  10. B. Ahmad, J.J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl. 2013 (2013), Art. ID 149659.

  11. L. Zhang, B. Ahmad, G. Wang, R.P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math. 249 (2013), 51-56.

  12. X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Differ. Equ. 2013 (2013), Art. ID 126.

  13. S.K. Ntouyas, S. Etemad, On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Appl. Math. Comp. 266 (2015), 235–243.

  14. S.K. Ntouyas, S. Etemad, J. Tariboon, Existence results for multi-term fractional differential inclusions, Adv. Differ. Equ. 2015 (2015), Art. ID 140.

  15. R.P. Agarwal, S.K. Ntouyas, B. Ahmad, A.K. Alzahrani, Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments, Adv. Difference Equ. 2016 (2016), Art. ID 92.

  16. B. Ahmad, R.P. Agarwal, A. Alsaedi, Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions, Bound. Value Probl. 2016 (2016), Art. ID 28.

  17. B. Ahmad, S.K. Ntouyas, Some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 110 (2016), 159-172.

  18. H. Dong, B. Guo, B. Yin, Generalized fractional supertrace identity for Hamiltonian structure of NLS-MKdV hierarchy with self-consistent sources, Anal. Math. Phys. 6 (2) (2016), 199-209.

  19. L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1991), 11-19.

  20. L. Byszewski, Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991) 494-505.

  21. R. Ciegis, A. Bugajev, Numerical approximation of one model of the bacterial self-organization, Nonlinear Anal. Model. Control. 17 (2012), 253-270.

  22. U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2015), 860–865.

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

  24. E. Zeidler, Nonlinear functional analysis and its application: Fixed point-theorems, Springer-Verlag, New York, vol. 1 1986.

  25. B.N. Sadovskii, On a fixed point principle, Funct. Anal. Appl. 1 (1967), 74-76.

  26. T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998), 23-31.

  27. D. O’Regan, Fixed-point theory for the sum of two operators, Appl. Math. Lett. 9 (1996), 1-8.

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

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

  30. G.V. Smirnov, Introduction to the theory of differential inclusions, American Mathematical Society, Providence, RI, 2002.

  31. A. Lasota, 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.

  32. A. Petrusel, Fixed points and selections for multivalued operators, Seminar on Fixed Point Theory Cluj-Napoca 2 (2001), 3-22.

  33. W.V. Petryshyn, P. M. Fitzpatric, A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact maps, Trans. Amer. Math. Soc., 194 (1974), 1-25.