Fractional Nabla Difference Research Articles

Mathematical Study of Nabla Fractional Difference Tech Layoff Model

Mathematical Study of Nabla Fractional Difference Tech Layoff Model

Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results.

Data Dependence and Existence and Uniqueness for Hilfer Nabla Fractional Difference Equations

In the present article, we establish sufficient conditions for the existence of a unique bounded solution using a prominent fixed-point theorem for the non-linear initial value problem involving the recently introduced Hilfer nabla fractional difference operator. where 0 <ℵ<1, 0 ≤ ß ≤1, ι =ℵ+ ß −ℵß and j : Nx × Rn →Rn. We also analyze the Ulam-Hyers stability of the considered problem and make some interesting observations on the dependence of its solutions on initial conditions and parameters. Finally, we conclude this article by constructing suitable examples to illustrate the applicability of established results.

Asymptotic Stability of a Linear Nabla Fractional Difference Equation

Asymptotic Stability of a Linear Nabla Fractional Difference Equation

Quasi-synchronization of discrete-time fractional-order quaternion-valued memristive neural networks with time delays and uncertain parameters

In this paper, quasi-synchronization of discrete-time fractional-order quaternion-valued memristive neural networks (DTFQMNNs) with time delays and uncertain parameters is investigated. Firstly, some related preliminaries are introduced and two new inequalities are obtained based on nabla fractional difference, quaternion theory and nabla h-Laplace transform. Then, a quaternion-valued controller with time delay is designed to realize quasi-synchronization goal, and some sufficient criteria are derived to guarantee quasi-synchronization of DTFQMNNs by using our created inequalities and some analysis techniques. Finally, a numerical example is provided to verify the correctness of our theoretical results.

A Comparison Result for the Nabla Fractional Difference Operator

This article establishes a comparison principle for the nabla fractional difference operator ∇ρ(a)ν, 1<ν<2. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green’s function. I prove that this Green’s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem.

Quasi-projective synchronization of discrete-time fractional-order quaternion-valued neural networks

In this article, without decomposing the quaternion-valued neural networks (QVNNs) into two complex-valued subsystems or four real-valued subsystems, quasi-projective synchronization of discrete-time fractional-order QVNNs is investigated. To this end, the sign function for quaternion number is introduced and some related properties are given. Then, two inequalities are built according to the nabla fractional difference and quaternion theory. Subsequently, a simple linear quaternion-valued controller is designed, and some synchronization conditions are given by means of our created inequalities. Finally, numerical simulations are given to prove the feasibility and correctness of the theoretical results.

Existence and Stability of Solutions for Nabla Fractional Difference Systems with Anti-periodic Boundary Conditions

In this paper, we propose sufficient conditions on existence, uniqueness and Ulam-Hyers stability of solutions for coupled systems of fractional nabla difference equations with anti-periodic boundary conditions, by using fixed point theorems. We also support these results through a couple of examples.

Global Mittag–Leffler stability and synchronization of discrete-time fractional-order delayed quaternion-valued neural networks

This paper is devoted to investigating discrete-time fractional-order delayed quaternion-valued neural networks (DFDQNNs) by utilizing direct quaternion approach. Firstly, a novel lemma and its two corresponding corollaries have been proposed for estimating nabla fractional difference of the quaternion-valued Lyapunov function. Then, the existence and uniqueness of equilibrium point for DFDQNNs is proved by constructing a new quaternion-valued contraction mapping. In addition, by means of our designed Lyapunov functions and the effective feedback controller as well as neoteric nabla difference inequalities, some sufficient criteria have been obtained to ensure the global Mittag–Leffler stability and Mittag–Leffler synchronization of DFDQNNs, respectively. Finally, some numerical examples are provided to verify the yielded results.

On Hilfer Generalized Proportional Nabla Fractional Difference Operators

In this paper, the Hilfer type generalized proportional nabla fractional differences are defined. A few important properties in the left case are derived and the properties in the right case are proved by Q-operator. The discrete Laplace transform in the sense of the left Hilfer generalized proportional fractional difference is explored. Furthermore, An initial value problem with the new operator and its generalized solution are considered.

Existence and asymptotic behaviors of nonlinear neutral Caputo nabla fractional difference equations

Existence and asymptotic behaviors of nonlinear neutral Caputo nabla fractional difference equations

Monotonicity Results for Nabla Riemann–Liouville Fractional Differences

Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the monotonicity of the discrete nabla fractional differences of the Riemann–Liouville type by considering the positivity of ∇b0RLθg(z) combined with a condition on g(b0+2), g(b0+3) and g(b0+4), successively. The article ends with a relationship between the discrete nabla fractional and integer differences of the Riemann–Liouville type, which serves to show the monotonicity of the discrete fractional difference ∇b0RLθg(z).

Iterative system of nabla fractional difference equations with two-point boundary conditions

Iterative system of nabla fractional difference equations with two-point boundary conditions

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

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.

Stability criteria for volterra type linear nabla fractional difference equations

Stability criteria for volterra type linear nabla fractional difference equations

Relationships between the discrete Riemann-Liouville and Liouville-Caputo fractional differences and their associated convexity results

<abstract><p>In this study, we have presented two new alternative definitions corresponding to the basic definitions of the discrete delta and nabla fractional difference operators. These definitions and concepts help us in establishing a relationship between Riemann-Liouville and Liouville-Caputo fractional differences of higher orders for both delta and nabla operators. We then propose and analyse some convexity results for the delta and nabla fractional differences of the Riemann-Liouville type. We also derive similar results for the delta and nabla fractional differences of Liouville-Caputo type by using the proposed relationships. Finally, we have presented two examples to confirm the main theorems.</p></abstract>

Impulsive nabla fractional difference equations

Impulsive nabla fractional difference equations

Monotonicity results for CFC nabla fractional differences with negative lower bound

Abstract We consider the sequential CFC-type nabla fractional difference ( CFC ∇ a + 1 ν ∇ a μ CFC u ) ( t ) {(^{\mathrm{CFC}}\nabla^{\nu}_{a+1}{}^{\mathrm{CFC}}\nabla^{\mu}_{a}u)(t)} and show that one can derive monotonicity-type results even in the case where this difference satisfies a strictly negative lower bound. This illustrates some dissimilarities between the integer-order and fractional-order cases.

Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems

In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green’s function as a series of functions with the help of spectral theory, and obtain some of its properties. Under suitable conditions on the nonlinear part of the nabla fractional difference equation, we deduce two existence results of the considered nonlinear problem by means of two Leray–Schauder fixed point theorems. We provide a couple of examples to illustrate the applicability of the established results.

Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels ( {}^{ABR}_{a-1}{nabla }^{delta ,gamma }y )(eta ) of order 0<delta <0.5, beta =1, 0<gamma leq 1 starting at a-1. If ({}^{ABR}_{a-1}{nabla }^{delta ,gamma }y ) ( eta )geq 0, then we deduce that y(eta ) is delta ^{2}gamma -increasing. That is, y(eta +1)geq delta ^{2} gamma y(eta ) for each eta in mathcal{N}_{a}:={a,a+1,ldots}. Conversely, if y(eta ) is increasing with y(a)geq 0, then we deduce that ({}^{ABR}_{a-1}{nabla }^{delta ,gamma }y )(eta ) geq 0. Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.