Existence and Stability of Solutions for Nonlinear Impulsive Nabla Fractional Boundary Value Problems of Order Less Than One
DOI:
https://doi.org/10.5890/DNC.2023.06.001Abstract
In this paper, we establish sufficient conditions on existence and uniqueness of solutions for a class of nonlinear impulsive nabla fractional difference equations of order $\alpha$, $0 < \alpha \leq 1$, associated with non–periodic boundary conditions. The right hand side of the proposed equation may grow linearly, or sublinearly in its second argument. We employ the classical fixed point theorem of Schaefer, and the Nonlinear Alternative to prove the existence and uniqueness of solutions. Further, we study stability of solutions in sense of Ulam--Hyers by the help of generalized Gronwall Inequality. To demonstrate the validity and applicability of the established results, we provide a couple of particular examples.References
[1] Bainov, D. and Simeonov, P. (1993), Impulsive differential equations: periodic solutions and applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, 66, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York.
[2] Benchohra, M., Henderson, J., and Ntouyas, S. (2006), Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2, Hindawi Publishing Corporation, New York.
[3] Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6, World Scientific Publishing Co., Inc., Teaneck, NJ.
[4] Ahmad, B. and Sivasundaram, S. (2009), Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(3), 251-258.
[5] Feckan, M., Zhou, Y., and Wang, J. (2012), On the concept and existence of solution for impulsive fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17(7), 3050-3060.
[6] Wang, J., Zhou, Y., and Feckan, M. (2012), Nonlinear impulsive problems for fractional differential equations and Ulam stability, Computers and Mathematics with Applications, 64(10), 3389-3405.
[7] Wang, J., Feckan, M., and Zhou, Y. (2016), A survey on impulsive fractional differential equations, Fractional Calculus and Applied Analysis, 19(4), 806-831.
[8] Wu, G.C., Zeng, D.Q., and Baleanu, D. (2019), Fractional impulsive differential equations: exact solutions, integral equations and short memory case, Fractional Calculus and Applied Analysis, 22(1), 180-192.
[9] Wu, G.C. and Baleanu, D. (2018), Stability analysis of impulsive fractional difference equations, Fractional Calculus and Applied Analysis, 21(2), 354-375.
[10] Jonnalagadda, J., Impulsive nabla fractional difference equations, Fractional Differential Calculus, To appear.
[11] Bohner, M. and Peterson, A. (2001), Dynamic equations on time scales. An introduction with applications, Birkhäuser Boston, Inc., Boston, MA.
[12] Goodrich, C. and Peterson, A.C. (2015), Discrete fractional calculus, Springer, Cham.
[13] Kelley, W.G. and Peterson, A.C. (2001), Difference equations. An introduction with applications, Second edition, Harcourt / Academic Press, San Diego, CA.
[14] Alzabut, J. and Abdeljawad, T. (2018), A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system, Applicable Analysis and Discrete Mathematics, 12(1), 36-48.
[15] Agarwal, R.P., Meehan, M., and O'Regan, D. (2001), Fixed point theory and applications, Cambridge Tracts in Mathematics, 141, Cambridge University Press, Cambridge.
[16] Smart, D.R. (1974), Fixed point theorems, Cambridge Tracts in Mathematics, No. 66, Cambridge University Press, London - New York.
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