JAMP  Vol.8 No.7 , July 2020
Existence and Stability Results for Impulsive Fractional q-Difference Equation
Abstract: In this paper, we study the boundary value problem for an impulsive fractional q-difference equation. Based on Banach’s contraction mapping principle, the existence and Hyers-Ulam stability of solutions for the equation which we considered are obtained. At last, an illustrative example is given for the main result.
Cite this paper: Jiang, M. and Huang, R. (2020) Existence and Stability Results for Impulsive Fractional q-Difference Equation. Journal of Applied Mathematics and Physics, 8, 1413-1423. doi: 10.4236/jamp.2020.87107.

[1]   Jackson, F.H. (1910) On q-Definite Integrals. The Quarterly Journal of Pure and Applied Mathematics, 41, 193-203.

[2]   Kac, V. and Cheung, P. (2002) Quantum Calculus. Springer, New York.

[3]   Al-Salam, W.A. (1966) Some Fractional q-Integral and q-Derivatives. Proceedings of the Edinburgh Mathematical Society, 15, 135-140.

[4]   Agarwal, R.P. (1969) Certain Fractional q-Integrals and q-Derivatives. Mathematical Proceedings of the Cambridge Philosophical Society, 66, 365-370.

[5]   Tariboom, T., Ntouvas, S.K. and Agarwal, R. (2015) New Concepts of Fractional Quantum Calculus and Applications to Impulsive Fractional q-Difference Equation. Advances in Difference Equations, 2015, Article No. 18.

[6]   Tariboom, T. and Ntouvas, S.K. (2013) Quantum Calculus on Finite Intervals and Applications to Impulsive Difference Equations. Advances in Difference Equations, 2013, Article No. 282.

[7]   Ahmad, B., Ntouyas, S.K., Tariboon, J., Alsaedi, A. and Alsulami, H.H. (2016) Impulsive Fractional q-Integro-Difference Equations with Separated Boundary Conditions. Applied Mathematics and Computation, 281, 199-213.

[8]   Agarwal, R.P., Wang, G., Ahmad, B., Zhang, L., Hobiny, A. and Monaquel, S. (2015) On Existence of Solutions for Nonlinear q-Difference Equations with Nonlocal q-Integral Boundary Conditions. Mathematical Modelling and Analysis, 20, 604-618.

[9]   Li, X.H., Han, Z.L., Sun, S.R. and Sun, L.Y. (2016) Eigenvalue Problems of Fractional q-Difference Equations with Generalized p-Laplacian. Applied Mathematics Letters, 57, 46-53.

[10]   Ferreira, R. (2011) Positive Solutions for a Class of Boundary Value Problems with Fractional q-Differences. Computers & Mathematics with Applications, 61, 367-373.

[11]   Ge, Q. and Hou, C.M. (2015) Positive Solution for a Class of p-Laplacian Fractional q-Difference Equations Involving the Integral Boundary Condition. Mathematica Aeterna, 5, 927-944.

[12]   Balkani, N., Rezapour, S. and Haghi, R.H. (2019) Approximate Solutions for a Fractional q-Integro-Difference Equation. Journal of Mathematical Extension, 13, 201-214.

[13]   Samei, M.E. and Khalilzadeh Ranjbar, G. (2019) Some Theorems of Existence of Solutions for Fractional Hybrid q-Difference Inclusion. Journal of Advanced Mathematical Studies, 12, 63-76.

[14]   Kalvandi, V. and Samei, M.E. (2019) New Stability Results for a Sum-Type Fractional q-Integro-Differential Equation. Journal of Advanced Mathematical Studies, 12, 201-209.

[15]   Liang, S.H. and Samei, M.E. (2020) New Approach to Solutions of a Class of Singular Fractional q-Differential Problem via Quantum Calculus. Advances in Difference Equations, 2020, Article No. 14.

[16]   Zhai, C.B. and Ren, J. (2018) The Unique Solution for a Fractional q-Difference Equation with Three-Point Boundary Conditions. Indagationes Mathematicae, 29, 948-961.

[17]   Fahd, J., Thabet, A. and Dumitru, B. (2013) Stability of q-Fractional Non-Autonomous Systems. Nonlinear Analysis, 14, 780-784.