Generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral boundary conditions

Sotiris K. Ntouyas, Bashir Ahmad, Madeaha Alghanmi, Ahmed Alsaedi



In this paper, we establish sufficient criteria for the existence of solutions for generalized fractional differential equations and inclusions supplemented with generalized fractional integral boundary conditions. We make use of the standard fixed point theorems for single-valued and multivalued maps to obtain the desired results, which are well illustrated with the aid of examples.


Differential Equation; Caputo fractional derivative; fractional integral; existence; fixed point

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R.P. Agarwal, Y. Zhou, J.R. Wang and X. Luo, Fractional functional differential equations with causal operators in Banach spaces, Math. Comput. Modelling 54 (2011), 1440–1452.

B. Ahmad, A. Alsaedi, S. Aljoudi and S.K. Ntouyas, On a coupled system of sequential fractional differential equations with variable coefficients and coupled integral boundary conditions, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 60 (2017), no. 108, 3–18.

B. Ahmad, A. Alsaedi, S.K. Ntouyas and J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, 2017.

B. Ahmad and S.K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 266 (2015), 615–622.

B. Ahmad and 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.

B. Ahmad, S.K. Ntouyas and 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.

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

M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008), 1340–1350.

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

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

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

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer–Verlag, Berlin, Heidelberg, 2010.

A. Granas and J. Dugundji, Fixed Point Theory, Springer–Verlag, New York, 2003.

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

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

U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math, Anal. Appl.

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

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

M.A. Krasnosel’skiı̆, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk 10 (1955), 123–127.

V. Lakshmikantham, S. Leela and J.V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.

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.

B. Lupinska and T. Odzijewicz, A Lyapunov-type inequality with the Katugampolafractional derivative, Math. Meth. Appl. Sci. (2018), 1–12.

K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.

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


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