Existence and stability results for a class of nonlinear fractional $q$-integro-differential equation

Esmael Samei, Mohammad; Fathipour, Azam

doi:10.22075/ijnaa.2022.7128

Existence and stability results for a class of nonlinear fractional $q$-integro-differential equation

Document Type : Research Paper

Authors

Mohammad esmael Samei

Azam Fathipour

Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran

10.22075/ijnaa.2022.7128

Abstract

This paper deals with the stability results for the solution of a fractional $q$--integro-differential problem with integral conditions. Using Krasnoselskii's, and Banach's fixed point theorems, we prove the existence and uniqueness of results. Based on the results obtained, conditions are provided that ensure the generalized Ulam stability of the original system on a time scale. The results are illustrated by the examples under the numerical technique.

Keywords

$q$-integro-differential problem

Krasnoselskii's theorem

stability

[1] N. Abdellouahab, B. Tellab, and K. Zennir, Existence and uniqueness of solutions to fractional Langevin equations involving two fractional orders, Kragujevac J. Math. 46 (2022), no. 5, 685–699.

[2] C.R. Adams, The general theory of a class of linear partial q−difference equations, Trans. Amer. Math. Soc. 26 (1924), 283–312.

[3] M.H. Annaby and Z.S. Mansour, q−Fractional Calculus and Equations, Springer Heidelberg, Cambridge, 2012.

[4] F. Atici and P.W. Eloe, Fractional q−calculus on a time scale, J. Nonlinear Math. Phys. 14 (2007), no. 3, 341–352.

[5] H. Baghani, Existence and uniqueness of solutions to fractional Langevin equations involving two fractional orders, J. Fixed Point Theory Appl. 20 (2018), no. 2, 7 pages.

[6] R.A.C. Ferreira, Nontrivials solutions for fractional q−difference boundary value problems, Electronic J. Qual. Theory Differ. Equ. 70 (2010), 1–101.

[7] S.N. Hajiseyedazizi, M.E. Samei, J. Alzabut, and Y. Chu, On multi-step methods for singular fractional q–integrodifferential equations, Open Math. 19 (2021), 1378––1405.

[8] F.H. Jackson, q−difference equations, Amer. J. Math. 32 (1910), 305–314.

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

[10] P.M. Rajkovic, S.D. Marinkovic, and M.S. Stankovic, Fractional integrals and derivatives in q−calculus, Appl. Anal. Discrete Math. 1 (2007), 311–323.

[11] M. E. Samei, V. Hedayati, and Sh. Rezapour, Existence results for a fraction hybrid differential inclusion with Caputo-Hadamard type fractional derivative, Adv. Differ. Equ. 2019 (2019), 163.

[12] M.E. Samei, R. Ghaffari, S.W. Yao, M.K.A. Kaabar, F. Mart´ınez, and M. Inc, Existence of solutions for a singular fractional q−differential equations under Riemann–Liouville integral boundary condition, Symmetry 13 (2021), 135.

[13] M.E. Samei, H. Zanganeh, and S.M. Aydogan, Investigation of a class of the singular fractional integro-differential quantum equations with multi-step methods, J. Math. Ext. 15 (2021), 1–54.