On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

On Some Fractional Systems of Difference Equations

This paper deal with the solutions of the systems of difference equations $$x_{n+1}=frac{y_{n-3}y_nx_{n-2}}{y_{n-3}x_{n-2}pm y_{n-3}y_n pm y_nx_{n-2}}, ,y_{n+1}=frac{y_{n-2}x_{n-1}}{ 2y_{n-2}pm x_{n-1}},,nin mathbb{N}_{0},$$ where $mathbb{N}_{0}=mathbb{N}cup left{0right}$, and initial values $x_{-2},, x_{-1},,x_{0},,y_{-3},,y_{-2},,y_{-1},,y_{0}$ are non-zero real numbers.

Stability analysis of fractional-order nonlinear Systems via Lyapunov method

‎In this paper‎, ‎we study stability of fractional-order nonlinear dynamic systems by means of Lyapunov‎ ‎method‎. ‎To examine the obtained results‎, ‎we employe the developed techniques on test examples‎.

Stability of nonlinear h-difference systems with n fractional orders

On the Definitions of Nabla Fractional Operators

and Applied Analysis 3 Definition 2.2. Let ρ t t − 1 be the backward jump operator. Then i the nabla left fractional sum of order α > 0 starting from a is defined by ∇−α a f t 1 Γ α t ∑ s a 1 ( t − ρ s α−1f s , t ∈ Na 1 2.4 ii the nabla right fractional sum of order α > 0 ending at b is defined by b∇−αf t 1 Γ α b−1 ∑ s t ( s − ρ t α−1f s 1 Γ α b−1 ∑ s t σ s − t α−1f s , t ∈b−1 N. 2.5 We want to...

Nabla discrete fractional calculus and nabla inequalities

Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincaré and Sobolev type inequalities .