Existence of solutions on a nonlinear boundary value problem including Hilfer fractional q-derivative

Document Type : Original Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Bu-Ali Sina, Hamedan, IRAN

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

Abstract

In this paper, we introduce an extension of the Hilfer fractional derivative, the "Hilfer fractional quantum derivative", and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach's contraction mapping principle. Examples illustrating the obtained results are also presented.

Keywords

Main Subjects


1] Agarwal, R. P., 1969. Certain fractional q-integrals and q-derivatives. Mathematical Proceedings of the Cambridge Philosophical Society, 66, pp. 365–370. doi: 10.1017/S0305004100045060
[2] Ahmad, B., Tariboon, J., Ntouyas, S. K., Alsulami, H. H. and Monaquel, S., 2016. Existence results
for impulsive fractional q-difference equations with anti-periodic boundary conditions, Boundary
Value Problems, 2016, pp. 15. doi: 10.1186/s13661-016-0521-y
[3] Ahmadi, A. and Samei, M. E., 2020. On existence and uniqueness of solutions for a class of coupled system of three term fractional q-differential equations, Journal of Advanced Mathematical Studies, 14(1), pp. 69–80.
[4] Al-Salam, W. A., 1966. Some fractional q-integrals and q-derivatives, Proceedings of the Edinburgh Mathematical Society, 15, pp. 135–140. doi: 10.1017/S0013091500011469
[5] Annaby, M. H. and Mansour, Z. S., 2012. q-Fractional Calculus and Equations. Heidelberg, New York, Springer.
[6] Berhail, A., Tabouche, N., Alzabut, J. and Samei, M. E., 2022. Using Hilfer-Katugampola fractional derivative in initial value Mathieu fractional differential equations with application on particle in the plane, Advances in Continuous and Discrete Models: Theory and Applications, 2022, pp. 44. doi:10.1186/s13662-022-03716-6
[7] Boutiara, A., Kaabar, M. K. A., Siri, Z., Samei, M. E. and Yue, X. G., 2022. Investigation of a generalized proportional Langevin and Sturm-Liouville fractional differential equations via variable coefficients and anti-periodic boundary conditions with a control theory application arising from complex networks, Mathematical Problems in Engineering, 2022, pp. 21 pages. doi: 10.1155/2022/7018170
[8] Eswari, R., Alzabut, J. and Samei, M. E., Tunç, C. and Jonnalagad, J. M., 2022. New results on the
existence of periodic solutions for Rayleigh equations with state-dependent delay, Nonautonomous Dynamical Systems, 9, pp. 103–115. doi: 10.1515/msds-2022-0149
[9] Hedayati, V. and Samei, M. E., 2019. Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous Dirichlet boundary conditions, Boundary Value Problems, 2019, pp. 141. doi: 10.1186/s13661-019-1251-8
[10] Hilfer, R., 2000. Applications of Fractional Calculus in Physics, Singapore, World Scientific.
[11] Houas, M. and Samei, M. E., 2022. Existence and Mittag-Leffler-Ulam-stability results for duffing type problem involving sequential fractional derivatives, International Journal of Applied and Computational Mathematics, 8, pp. 185. doi: 10.1007/s40819-022-01398-y
[12] Houas, M. and Samei, M. E., 2023. Existence and stability of solutions for linear and nonlinear damping of q−fractional Duffing-Rayleigh problem, Mediterranean Journal of Mathematics, 20, pp. 148. doi: 10.1007/s00009-023-02355-9
[13] Izadi, M. and Samei, M. E., 2022. Time accurate solution to Benjamin-Bona-Mahony Burgers equation via Taylor-Boubaker series scheme, Boundary Value Problems, 2022, pp. 17.doi:10.1186/s13661-022-01598-x
[14] Jackson, F. H., 1910. q-difference equations, American Journal of Mathematic, 32(4), pp. 305–314.doi: 10.2307/2370183
[15] Mishra, S. K., Rajković, P., Samei, M. E., Chakraborty, S. K., Ram, B. and Kaabar, M. K. A., 2021. A
q-gradient descent algorithm with quasi-Fejér convergence for unconstrained optimization problems, Fractal and Fractional, 5(3), pp. 110. doi: 10.3390/fractalfract5030110
[16] Ntouyas, S. K., and M. E. Samei, 2019. Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus, Advances in Difference Equations, 2019, pp. 475. doi: 10.1186/s13662-019-2414-8
[17] Samei, M. E., Hedayati, V. and Rezapour, S., 2019. Existence results for a fraction hybrid differential inclusion with Caputo-Hadamard type fractional derivative, Advances in Difference Equations, 2019,
pp. 163. doi: 10.1186/s13662-019-2090-8
[18] Samei, M. E., Karimi, L. and Kaabar, M. K. A., 2022. To investigate a class of multi-singular pointwise defined fractional q-integro-differential equation with applications, AIMS Mathematics, 7(5),pp. 7781–7816. doi: 10.3934/math.2022437
[19] Suantai, S., Ntouyas, S. K., Asawasamrit, S. and Tariboon, J., 2015. A coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions, Advances in Difference Equations, 2015, pp. 124. doi: 10.1186/s13662-015-0462-2
[20] Subramanian, M., Alzabut, J., Dumitru, D., Samei, M. E. and Zadaf, A., 2021. Existence, uniqueness and stability analysis of a coupled fractional-order differential systems involving Hadamard derivatives and associated with multi-point boundary conditions, Advances in Difference Equations, 2021, pp. 267. doi: 10.1186/s13662-021-03414-9
[21] Tariboon, J. and Ntouyas, S. K., 2013. Quantum calculus on finite intervals and applications to impulsive difference equations, Advances in Difference Equations, 2013, 282. doi: 10.1186/1687- 1847-2013-282
[22] Tariboon, J., and Ntouyas, S. K. and Agarwal, P., 2015. New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations, Advances in Difference Equations, 2015, pp. 18. doi: 10.1186/s13662-014-0348-8