International Journal of Nonlinear Analysis and Applications

Non-resonant Nabla fractional boundary value problems

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad, Telangana 500078, India

10.22075/ijnaa.2023.26401.4430

Abstract

We consider two simple non-resonant boundary value problems for a nabla fractional difference equation. First, we construct associated Green's functions and obtain some of their properties. Under suitable constraints on the nonlinear part of the nabla fractional difference equation, we deduce sufficient conditions for the existence of solutions to the considered problems through an appropriate fixed point theorem. We also provide two examples to demonstrate the applicability of the established results.

Keywords

[1] R.P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.
[2] H.M. Ahmed and M. Alessandra Ragusa, Nonlocal controllability of Sobolev-type conformable fractional stochastic evolution inclusions with Clarke subdifferential, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 6, 3239–3253.
[3] K. Ahrendt and C. Kissler, Green’s function for higher-order boundary value problems involving a nabla Caputo fractional operator, J. Difference Equ. Appl. 25 (2019), no. 6, 788–800.
[4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser Boston Inc., Boston, MA, 2001.
[5] C. Chen, M. Bohner, and B. Jia, Existence and uniqueness of solutions for nonlinear Caputo fractional difference equations, Turk. J. Math. 44 (2020), no. 3, 857–869.
[6] Y. Gholami and K. Ghanbari, Coupled systems of fractional ∇-difference boundary value problems, Differ. Equ. Appl. 8 (2016), no. 4, 459–470.
[7] C. Goodrich and A.C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015.
[8] L. Gu, Q. Zhong, and Z. Shao, On multiple positive solutions for singular fractional boundary value problems with Riemann–Stieltjes integrals, J. Funct. Spaces 2023 (2023), 1–7.
[9] A. Ikram, Lyapunov inequalities for nabla Caputo boundary value problems, J. Difference Equ. Appl. 25 (2019), no. 6, 757–775.
[10] J.M. Jonnalagadda, Solutions of fractional nabla difference equations - existence and uniqueness, Opuscula Math. 36 (2016), no. 2, 215–238.
[11] J.M. Jonnalagadda, On two-point Riemann–Liouville type nabla fractional boundary value problems, Adv. Dyn. Syst. Appl. 13 (2018), no. 2, 141–166.
[12] J.M. Jonnalagadda and N.S. Gopal, Green’s function for a discrete fractional boundary value problem, Differ. Equ. Appl. 14 (2022), no. 2, 163–178.
[13] Z. Laadjal, T. Abdeljawad, and F. Jarad, Some results for two classes of two-point local fractional proportional boundary value problems, Filomat 37 (2023), no. 21, 7199–7216.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 17 November 2024
  • Receive Date: 05 May 2023
  • Revise Date: 07 October 2023
  • Accept Date: 17 October 2023