Pure and Applicable Analysis

Qualitative Behavior and Solutions of Sixth Rational Difference Equations

E. M. Elasyed, Mai T. Alharthi — Mon, 20 Mar 2023 00:00:00 +0800

In this article, we introduced the solutions of the following difference equations

$$z_{n+1}=\frac{z_{n-4}z_{n-5}}{z_{n}(\pm 1\pm z_{n-4}z_{n-5})}, n=0,1,2,...,$$

where the initial conditions $z_{-5},\ z_{-4},\ z_{-3},\ z_{-2,}\ z_{-1}$ and $z_{0}$ are arbitrary non-zero real numbers. Moreover, we presented the solutions of some special cases of these equations and studied the dynamic behavior of the these equations. Finally, we obtained the estimation of the initial coefficients.

On a Higher-Order Systems of Difference Equations

E. M. Elsayed, Marwa M. Alzubaidi — Thu, 12 Jan 2023 00:00:00 +0800

Our goal in this objective is to study the form of the solutions of a class of rational systems of difference equations:

$$x_{n+1}=\frac{y_{n-5}x_{n-8}}{y_{n-2}\left(1+y_{n-5}x_{n-8}\right)}, y_{n+1}=\frac{x_{n-5}y_{n-8}}{x_{n-2}(\pm1\pm x_{n-5}y_{n-8})}, n=0,1,...,$$

where the initial conditions $x_{-\alpha}$, $y_{-\alpha}$, $\alpha\in\{0,1,...,8\}$ are non-zero real numbers. Keywords: Difference equations, Systems of difference equations, Recursive sequences.

On a Strongly Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

Mohamed El Ouaarabi, Chakir Allalou, Said Melliani — Tue, 10 Jan 2023 00:00:00 +0800

In this paper, we prove the existence and uniqueness of weak solution to a strongly nonlinear degenerate elliptic problem of the type:

−div|ω1a(x, ▽u) + ω2b(x, u, ▽u)| + ω3g(x)u(x) = f (x).

Here, ω1, ω2 and ω3 are Ap-weight functions that will be defined in the preliminaries, where, Ω is a bounded open set of R n (n ≥ 2) and f ∈ L1 (Ω), with b: Ω × R × R n −→ R, a: Ω × R n -> R and g: Ω -> R are functions that satisfy some conditions and f belongs to L p 0 (Ω, ω 1−p 0 1). First, we transformed the problem into an equivalent operator equation; second, we utilized the Browder-Minty Theorem to prove the existence and uniqueness of weak solution to the considered problem.