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

Almalahi, Mohammed A.; Panchal, Satish K.

doi:10.2478/aupcsm-2020-0013

Abstract

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 E_α -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.

References

[1] 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. Cited on 172.
Search in Google Scholar Back to article
[2] Ahmad, Bashir, and Juan J. Nieto. “Riemann-Liouville fractional differential equations with fractional boundary conditions.” Fixed Point Theory 13, no. 2 (2012): 329-336. Cited on 172.
Search in Google Scholar Back to article
[3] 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. Cited on 172.
Search in Google Scholar Back to article
[4] 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. Cited on 172 and 175.
Search in Google Scholar Back to article
[5] 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. Cited on 172.10.1007/s12215-020-00484-8
Search in Google Scholar Back to article
[6] Banaś, Jósef. “On measures of noncompactness in Banach spaces.” Comment. Math. Univ. Carolin. 21, no. 1 (1980): 131-143. Cited on 174.
Search in Google Scholar Back to article
[7] 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. Cited on 172.
Search in Google Scholar Back to article
[8] 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. Cited on 172.
Search in Google Scholar Back to article
[9] 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. Cited on 174.10.1007/978-1-4613-1281-9_2
Search in Google Scholar Back to article
[10] Hilfer, Rudolf (ed). Applications of fractional calculus in physics. River Edge, NJ: World Scientific Publishing Co., Inc., 2000. Cited on 172.
Search in Google Scholar Back to article
[11] 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. Cited on 179.10.1515/9783110870893
Search in Google Scholar Back to article
[12] 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. Cited on 172 and 173.
Search in Google Scholar Back to article
[13] 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. Cited on 182.
Search in Google Scholar Back to article
[14] Mönch, Harald. “Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.” Nonlinear Anal. 4, no. 5 (1980): 985-999. Cited on 175.
Search in Google Scholar Back to article
[15] Otrocol, Diana, and Veronica Ilea. “Ulam stability for a delay differential equation.” Cent. Eur. J. Math. 11, no. 7 (2013): 1296-1303. Cited on 182.
Search in Google Scholar Back to article
[16] 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. Cited on 172.
Search in Google Scholar Back to article
[17] 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. Cited on 172.
Search in Google Scholar Back to article
[18] Shreve, Warren E. “Boundary Value Problems for y″ = f(x, y, λ) on [a, ∞).” SIAM J. Appl. Math. 17, no. 1 (1969): 84-97. Cited on 172.
Search in Google Scholar Back to article
[19] 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. Cited on 172, 175, 179 and 185.10.1016/j.ajmsc.2019.03.001
Search in Google Scholar Back to article
[20] 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. Cited on 173.
Search in Google Scholar Back to article
[21] Vanterler da Costa Sousa, José, and Edmundo Capelas de Oliveira. “A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator.” Differ. Equ. Appl., no. 1 11 (2019): 87-106. Cited on 175 and 182.
Search in Google Scholar Back to article
[22] Wang, JinRong, and Yuruo Zhang. “Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations.” Optimization 63, no. 8 (2014): 1181-1190. Cited on 182.
Search in Google Scholar Back to article
[23] 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. Cited on 172.
Search in Google Scholar Back to article

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

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