Discontinuity, Nonlinearity, and Complexity
Existence and Stability of Solutions for Nonlinear Impulsive Nabla Fractional Boundary Value Problems of Order Less Than One
Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 231244  DOI:10.5890/DNC.2023.06.001
$^{1}$ Department of Mathematics, Birla Institute of Technology & Science Pilani, Hyderabad, Telangana, India 
500078
$^{2}$ Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
$^3$ Department of Industrial Engineering, OST.{I}M Technical University, Ankara 06374, T"{u}rkiye
$^{4}$ Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and
Informatics, Comenius University in Bratislava, Mlynsk'a dolina, 842 48 Bratislava, Slovakia
$^{5}$ Mathematical Institute, Slovak Academy of Sciences, v Stef'anikova 49, 814 73 Bratislava, Slovakia
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Abstract
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 nonâ€“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 UlamHyers by the help of generalized Gronwall Inequality. To demonstrate the validity and applicability of the established results, we provide a couple of particular examples.
Acknowledgments
J. Alzabut would like to thank Prince Sultan University for supporting this work. M. Fe\v ckan is thankful to the Slovak Research and Development Agency under the contract No. APVV180308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20.
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