Existence results of ψ-Hilfer integro-differential equations with fractional order in Banach space

Mohammed A. Almalahi, Satish K. Panchal


In this article we present the existence and uniqueness results for fractional integro-differential equations with ψ-Hilfer fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Mönch fixed point theorem and the Banach fixed point theorem. Furthermore, we discuss Eα-Ulam-Hyers stability of the presented problem. Also, we use the generalized Gronwall inequality with singularity to establish continuous dependence and uniqueness of the δ-approximate solution.


ψ-Hilfer fractional derivative; Mönch fixed point theorem; Eα-Ulam-Hyers stability; δ-approximate solution

Mathematics Subject Classification (2010)

34A08; 34B15; 34A12; 47H10


Agarwal, Ravi P., and Mouffak Benchohra, and Samira Hamani. "A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions." Acta Appl. Math. 109, no. 3 (2010): 973-1033.

Ahmad, Bashir, and Juan J. Nieto. "Riemann-Liouville fractional differential equations with fractional boundary conditions." Fixed Point Theory 13, no. 2 (2012): 329-336.

Almalahi, Mohammed A., and Satish K. Panchal. "Eα-Ulam-Hyers stability result for ψ-Hilfer Nonlocal Fractional Differential Equation." Discontinuity, Nonlinearity, and Complexity. In press.

Almalahi, Mohammed A., and Mohammed S. Abdo, and Satish K. Panchal. ψ-Hilfer fractional functional differential equation by Picard operator method, Journal of Applied Nonlinear Dynamics. In press.

Almalahi, Mohammed A., and Mohammed S. Abdo, and Satish K. Panchal. "Existence and Ulam-Hyers-Mittag-Leffler stability results of ψ-Hilfer nonlocal Cauchy problem." Rend. Circ. Mat. Palermo, II. Ser (2020). DOI: 10.1007/s12215-020-00484-8.

Banas, Jósef. "On measures of noncompactness in Banach spaces." Comment. Math. Univ. Carolin. 21, no. 1 (1980): 131-143.

Benchohra, Mouffak, and Soufyane Bouriah. "Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order." Moroccan Journal of Pure and Applied Analysis 1 (2015): 22-37.

Benchohra, Mouffak, and John R. Graef, and Samira Hamani. "Existence results for boundary value problems with non-linear fractional differential equations." Appl. Anal. 87, no. 7 (2008): 851-863.

Guo, Dajun, and Vangipuram Lakshmikantham, and Xinzhi Liu. Nonlinear integral equations in abstract spaces. Vol 373 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 1996.

Hilfer, Rudolf (ed). Applications of fractional calculus in physics. River Edge, NJ: World Scientific Publishing Co., Inc., 2000.

Kamenskii, Mikhail, and Valeri Obukhovskii, and Pietro Zecca. Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Vol. 7 of De Gruyter Series in Nonlinear Analysis and Applications. Berlin: Walter de Gruyter & Co., 2001.

Kilbas, Anatoly A., and Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations. Vol. 204 of North-Holland Mathematics Studies. Amsterdam: Elsevier Science B.V., 2006.

Liu, Kui, and JinRong Wang, and Donal O’Regan. "Ulam-Hyers-Mittag-Leffler stability for ψ-Hilfer fractional-order delay differential equations." Adv. Differ. Equ. 2019 (2019): Art id. 50.

Mönch, Harald. "Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces." Nonlinear Anal. 4, no. 5 (1980): 985-999.

Otrocol, Diana, and Veronica Ilea. "Ulam stability for a delay differential equation." Cent. Eur. J. Math. 11, no. 7 (2013): 1296-1303.

Podlubny, Igor. Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198 of Mathematics in Science and Engineering. San Diego, CA: Academic Press, Inc., 1999.

Samko, Stefan G., and Anatoly A. Kilbas, and Oleg I. Marichev. Fractional integrals and derivatives. Theory and applications. Translated from the 1987 Russian original. Yverdon: Gordon and Breach Science Publishers, 1993.

Shreve,Warren E. "Boundary Value Problems for y00 = f(x, y, lambda) on [a,infty)." SIAM J. Appl. Math. 17, no. 1 (1969): 84-97.

Thabet, Sabri T.M., and Bashir Ahmad, and Ravi P. Agarwal. "On abstract Hilfer fractional integrodifferential equations with boundary conditions." Arab Journal of Mathematical Sciences (2019) DOI: 10.1016/j.ajmsc.2019.03.001.

Vanterler da C. Sousa, José, and Edmundo Capelas de Oliveira. "On the ψ-Hilfer fractional derivative." Commun. Nonlinear Sci. Numer. Simul. 60 (2018): 72-91.

Vanterler da Costa Sousa, José, and Edmundo Capelas de Oliveira. "A Gronwall inequality and the Cauchy-type problem by means of -Hilfer operator." Differ. Equ. Appl., no. 1 11 (2019): 87-106.

Wang, JinRong, and Yuruo Zhang. "Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations." Optimization 63, no. 8 (2014): 1181-1190.

Zhang, Shuqin. "Existence of solution for a boundary value problem of fractional order." Acta Math. Sci. Ser. B (Engl. Ed.) 26, no. 2 (2006): 220-228.

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