# Positive solutions of nabla fractional boundary value problem

• N. S. Gopal

## Abstract

In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions.

\begin{align*}\begin{cases} -\big{(}\nabla^{\nu}_{\rho(a)}u\big{)}(t) + \lambda u(t) = f(t, u(t)), \quad t \in \mathbb{N}^{b}_{a + 2}, \\u(a) = u(b) = 0, \end{cases} \end{align*}

where $$1 < \nu < 2$$, $$a,b \in \mathbb{R}$$ with $$b-a\in\mathbb{N}_{3}$$, $$\mathbb{N}^b_{a+2} = \{a+2,a+3, . . . ,b\}$$, $$|\lambda| < 1$$, $$\nabla^{\nu}_{\rho(a)}u$$ denotes the $$\nu^{\text{th}}$$-order Riemann–Liouville nabla difference of $$u$$ based at $$\rho(a)=a-1$$, and $$f : \mathbb{N}^{b}_{a + 2} \times \mathbb{R} \rightarrow \mathbb{R}^{+}$$.

We make use of Guo–Krasnosels'kiÄ­ and Leggett–Williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. We establish sufficient requirements for at least one, at least two, and at least three positive solutions of the considered boundary value problem. We also provide an example to demonstrate the applicability of established results.

## Keywords

• Pages: 467–484
• Date Published: 2022-12-21
• Vol. 24 No. 3 (2022)

D. Anderson, R. Avery and A. Peterson, “Three positive solutions to a discrete focal boundary value problem”, J. Comput. Appl. Math., vol. 88, no. 1, pp. 103–118, 1998.

F. M. AtÄ±cÄ± and P. W. Eloe, “Discrete fractional calculus with the nabla operator”, Electron. J. Qual. Theory Differ. Equ., Special Edition I, Paper No. 3, 12 pages, 2009.

F. M. Atici and P. W. Eloe, “Linear systems of fractional nabla difference equations”, Rocky Mountain J. Math., vol. 41, no. 2, pp. 353–370, 2011.

F. M. AtÄ±cÄ± and P. W. Eloe, “Two-point boundary value problems for finite fractional difference equations”, J. Difference Equ. Appl., vol. 17, no. 4, pp. 445–456, 2011.

F. M. AtÄ±cÄ± and P. W. Eloe, “Gronwall‘s inequality on discrete fractional calculus”, Comput. Math. Appl., vol. 64, no. 10, pp. 3193–3200, 2012.

M. Bohner and A. Peterson, Dynamic equations on time scales. An introduction with applications, Boston: Birkhäuser Boston, 2001.

P. Eloe and J. Jonnalagadda, “Mittag-Leffler stability of systems of fractional nabla difference equations”, Bull. Korean Math. Soc. vol. 56, no. 4, pp. 977–992, 2019.

P. Eloe and Z. Ouyang, “Multi-term linear fractional nabla difference equations with constant coefficients”, Int. J. Difference Equ., vol. 10, no. 1, pp. 91–106, 2015.

R. A. C. Ferreira, Discrete fractional calculus and fractional difference equations, Springer Briefs in Mathematics. Cham: Springer, 2022.

Y. Gholami and K. Ghanbari, “Coupled systems of fractional âˆ‡-difference boundary value problems”, Differ. Equ. Appl., vol. 8, no. 4, pp. 459–470, 2016.

J. St. Goar, “A Caputo boundary value problem in nabla fractional calculus”, Ph. D. dissertation, Univ. Nebraska–Lincoln, Nebraska, 2016.

C. Goodrich and A. C. Peterson, Discrete fractional calculus, Cham: Springer, 2015.

R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, 2nd. ed., Berlin: Springer, 2020.

N. S. Gopal and J. M. Jonnalagadda, “Existence and uniqueness of solutions to a nabla fractional difference equation with dual nonlocal boundary conditions”, Foundations, vol. 2, pp. 151–166, 2022.

H. L. Gray and N. F. Zhang, “On a new definition of the fractional difference”, Math. Comp., vol. 50, no. 182, pp. 513–529, 1988.

J. Henderson, “Existence of local solutions for fractional difference equations with Dirichlet boundary conditions”, J. Difference Equ. Appl., vol. 25, no. 6, pp. 751–756, 2019.

J. Henderson and J. T. Neugebauer, “Existence of local solutions for fractional difference equations with left focal boundary conditions”, Fract. Calc. Appl. Anal., vol. 24, no. 1, pp. 324–331, 2021.

A. Ikram, “Lyapunov inequalities for nabla Caputo boundary value problems”, J. Difference Equ. Appl., vol. 25, no. 6, pp. 757–775, 2019.

J. M. Jonnalagadda, “On two-point Riemann-Liouville type nabla fractional boundary value problems”, Adv. Dyn. Syst. Appl., vol. 13, no. 2, pp. 141–166, 2018.

J. M. Jonnalagadda, “Existence results for solutions of nabla fractional boundary value problems with general boundary conditions”, Adv. Theory Non-linear Anal. Appl., vol. 4, no. 1, pp. 29–42, 2020.

J. M. Jonnalagadda and N. S. Gopal, “On hilfer-type nabla fractional differences”, Int. J. Differ. Equ., 2020, vol. 15, no. 1, pp. 91–107, 2020.

J. M. Jonnalagadda and N. S. Gopal. “Linear Hilfer nabla fractional difference equations”, Int. J. Dyn. Syst. Differ. Equ., vol. 11, no. 3–4, pp. 322–340, 2021.

J. M. Jonnalagadda and N. S. Gopal, “Green‘s function for a discrete fractional boundary value problem”, Differ. Equ. Appl., vol. 14, no. 2, pp. 163–178, 2022.

M. A. Krasnosel‘skiÄ­, Positive solutions of operator equations, The Netherlands: P. Noordhoff Ltd., 1964.

M. K. Kwong, “On Krasnoselskii‘s cone fixed point theorem”, Fixed Point Theory Appl., Art. ID 164537, 18 pages, 2008.

R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces”. Indiana Univ. Math. J., vol. 28, no. 4, pp. 673–688, 1979.

K. Mehrez and S. M. Sitnik, “Functional inequalities for the Mittag-Leffler functions”. Results Math., vol. 72, no. 1–2, pp. 703–714, 2017.

K. S. Miller and B. Ross, “Fractional difference calculus” in Univalent functions, fractional calculus, and their applications, Ellis Horwood Series in Mathematics and its Applications, H. M. Srivastava and S. Owa, Chichester: Ellis Horwood Limited, 1989, pp. 139–152.

P. Ostalczyk, Discrete fractional calculus: Applications in control and image processing, Singapore: World Scientific Publishing Co. Pte. Ltd, 2016.

J. D. Paneva-Konovska, From Bessel to multi-index Mittag-Leffler functions. Enumerable families, series in them and convergence, London: World Scientific Publishing, 2017.

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering 198, San Diego: Academic Press, Inc., 1999.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Switzerland: Gordon & Breach Science Publishers, 1993.

J. Spanier and K. B. Oldham, “The Pochhammer Polynomials (x)n”, in An Atlas of functions, Washington, DC: Hemisphere Publishing Corporation, 1987, pp. 149–165.

H. M. Srivastava and S. Owa (Eds.), Univalent functions, fractional calculus, and their applications, Ellis Horwood Series in Mathematics and its Applications, Chichester: Ellis Horwood Limited, 1989.

2022-12-21

## How to Cite

[1]
N. S. Gopal and J. M. Jonnalagadda, “Positive solutions of nabla fractional boundary value problem”, CUBO, vol. 24, no. 3, pp. 467–484, Dec. 2022.

Articles