Coupled systems of fractional ∇-difference boundary value problems

Gholami, Yousef; Ghanbari, Kazem

doi:10.7153/dea-08-26

Cited by 14 publications

(9 citation statements)

References 2 publications

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“…For our convenience, in this section, we present a few useful definitions and fundamental facts of nabla fractional calculus, which can be found in [1,2,3,8,9,10,16,18,20].…”

Section: Preliminariesmentioning

confidence: 99%

A coupled system of fractional difference equations with anti-periodic boundary conditions

Jonnalagadda¹

2023

Stud. Univ. Babes-Bolyai Math.

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“…For our convenience, in this section, we present a few useful definitions and fundamental facts of nabla fractional calculus, which can be found in [1,2,3,8,9,10,16,18,20].…”

Section: Preliminariesmentioning

confidence: 99%

A coupled system of fractional difference equations with anti-periodic boundary conditions

Jonnalagadda¹

2023

Stud. Univ. Babes-Bolyai Math.

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“…Brackins [15] studied a particular class of self-adjoint Riemann-Liouville nabla BVPs and derived the Green's function associated with it along with a few of its properties. Gholami et al [16] obtained the Green's function for a non-homogeneous Riemann-Liouville nabla BVP with Dirichlet boundary conditions. Jonnalagadda [17][18][19][20][21][22] analyzed some qualitative properties of twopoint non-linear Riemann-Liouville nabla BVPs associated with a variety of boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions

Gopal

Jonnalagadda

2022

Foundations

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In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient requirements for at least one and at least two positive solutions of the boundary value problem. Next, we discuss the existence and uniqueness of solutions to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem, respectively. Finally, we provide a few examples to illustrate the applicability of established results.

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“…To name a few, Goar [11] and Ikram [18] worked with self-adjoint Caputo nabla BVPs. Gholami et al [10] obtained the Green's function for a non-homogeneous Riemann-Liouville nabla BVP with Dirichlet boundary conditions. Jonnalagadda [19,20,23] analysed some qualitative properties of two-point non-linear Riemann-Liouville nabla fractional BVPs associated with a variety of boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of nabla fractional boundary value problem

Gopal

Jonnalagadda

2022

Cubo (Temuco, Online)

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In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions. \begin{equation*} \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{equation*} where $1 < \nu < 2$, $a,b \in \mathbb{R}$ with $b-a\in\mathbb{N}_{3}$, $\mathbb{N}^b_{a+2} = \{a+2,a+3,\hdots,b\}$, $|\lambda| < 1$, $\nabla^{\nu}_{\rho(a)}u$ 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'ki\v{\i} 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.

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Coupled systems of fractional ∇-difference boundary value problems

Abstract: Abstract. In this paper, we study the existence of solutions for a coupled system of two-point fractional ∇ -difference boundary value problems of the form Mathematics subject classification (2010): 34A08, 39A12, 34A12, 34B15, 39A10.

Cited by 14 publications

References 2 publications

A coupled system of fractional difference equations with anti-periodic boundary conditions

A coupled system of fractional difference equations with anti-periodic boundary conditions

Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions

Positive solutions of nabla fractional boundary value problem

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