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A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions

1 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2 Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Special Issues: New trends of numerical and analytical methods with application to real world models for instance RLC with new nonlocal operators

In this paper, we introduce a new kind of generalized fractional integral boundary conditions and develop the existence theory for a fractional di erential equation involving generalized Caputotype fractional derivative equipped with these conditions. We also study the inclusion case of the given problem. Examples are constructed to demonstrate the application of the obtained results.
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1. N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000), 298-305.

2. N. Laskin, Fractals and quantum mechanics, Chaos, 10 (2000), 780-790.

3. B. N. N. Achar, B. T. Yale, J. W. Hanneken, Time fractional Schr\"odinger equation revisited, Adv. Math. Phys., 2018 (2013), 1-11.

4.D. Baleanu, G. C. Wu, S. D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Soliton. Fract., 102 (2017), 99-105.    

5.G. C. Wu, D. Baleanu, Z. G. Deng, et al. Lattice fractional diffusion equation in terms of a Riesz-Caputo difference, Physica A-statistical Mechanics and Its Applications, 438 (2015), 335-339.    

6. U. N. Katugampola, New Approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.

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

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

9. B. Ahmad, A. Alsaedi, S. K. Ntouyas, et al. Hadamard-type fractional differential equations, inclusions and inequalities, Springer, Cham, 2017.

10. M. Benchohra, J. Henderson, S. K. Ntouyas, et al. Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340-1350.    

11. J. R. Wang, Y. Zhang, Analysis of fractional order differential coupled systems, Math. Methods Appl. Sci., 38 (2015), 3322-3338.    

12. B. Ahmad, S. K. Ntouyas, J. Tariboon, A study of mixed Hadamard and Riemann-Liouville fractional integro-differential inclusions via endpoint theory, Appl. Math. Lett., 52 (2016), 9-14.    

13. B. Ahmad, R. Luca, Existence of solutions for a sequential fractional integro-differential system with coupled integral boundary conditions, Chaos Soliton. Fract., 104 (2017), 378-388.    

14. Z. Y. Gao, J. R. Wang, Y. Zhou, Analysis of a class of fractional nonlinear multidelay differential systems, Discrete Dyn. Nat. Soc., 2017 (2017), 1-15.

15. B. Ahmad, S. K. Ntouyas, Existence results for fractional differential inclusions with Erdelyi-Kober fractional integral conditions, An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat., 25 (2017), 5-24.

16. M. Shabibi, Sh. Rezapour, S. M. Vaezpour, A singular fractional integro-differential equation, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 109-118.

17. M. Xu, Z. Han, Positive solutions for integral boundary value problem of two-term fractional differential equations, Bound. Value Probl., 2018 (2018), 100.

18. J. Henderson, R. Luca, Positive solutions for a system of coupled fractional boundary value problems, Lith. Math. J., 58 (2018), 15-32.    

19. R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dynam., 11 (2016), 61017.

20. B. Ahmad, M. Alghanmi, S. K. Ntouyas, et al. Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions, Appl. Math. Lett., 84 (2018), 111-117.    

21. U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.

22. F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619.    

23. A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

24.D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Am. Math. Soc., 20 (1969), 485-464.

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

26. 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.

27. M. Kisielewicz, Stochastic Differential Inclusions and Applications. Springer Optimization and Its Applications, Vol. 80, Springer, New York, 2013.

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

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

Copyright Info: © 2019, Bashir Ahmad, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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