Asymptotic Behavior of the Solutions of the Fuzzy Difference Equation

Tollu

2019

We have studied recently solvability and semi‐cycles of eight systems of difference equations of the following form: xn=a+pn−1qn−2pn−1+qn−2,yn=a+rn−1sn−2rn−1+sn−2,n∈double-struckN0, where a ∈ [0, + ∞), the sequences pn, qn, rn, sn are some of the sequences xn and yn, with positive initial values x−j,y−j, j = 1,2, in detail. This paper is devoted to the study of the other eight systems of the form. We show that these systems are also solvable in closed form and describe semi‐cycles of their solutions complementing our previous results on such systems of difference equations.

“…(a) If a = 0, then the general solution to the system is given by formulas (29) and (30). (b) If a > 0, then the general solution to the system is given by formulas (37) and (38).…”

Section: Case Min{xmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solvability of eight classes of nonlinear systems of difference equations

Tollu

2019

“…Let us only mention, that Stević's method from 2004 for solving the following nonlinear difference equation

x_{n + 1} = \frac{x_{n - 1}}{a + b x_{n - 1} x_{n}}, n \in {double-struckN}_{0},

has attracted some attention and was later used and suitably modified in several papers. () When for a solution to the equation holds x n ≠ 0 for every n ≥ − 1, we transformed the equation to a solvable linear one by using a suitable change of variables. The same idea was later used for several other classes of difference equations and systems.…”

Section: Introductionmentioning

confidence: 99%

More on a hyperbolic‐cotangent class of difference equations

Iričanin

Kosmala

2019

We present a natural method for solving the difference equation xn=xn−kxn−l+axn−k+xn−l,n∈double-struckN0, where k,l∈double-struckN, parameter a, and initial values x−j, j=true1,t‾, t=maxfalse{k,lfalse}, are real or complex numbers. As a concrete example, the case k = 1, l = 2 is studied in detail, and several interesting formulas for solutions and objects used in the analysis of the equation are given. A useful remark on solvability of difference equations on complex domains is presented and used here.

“…A 2004 note of ours on solvability of a nonlinear difference equation, whose results were later extended and developed in previous studies, 13,14 a bit unexpectedly, attracted considerable attention. Namely, it turned out that many nontrivial new classes of nonlinear difference equations can be transformed to known solvable ones, which motivated many authors to try to solve some equations in this way (see, eg, previous studies 15,16,[33][34][35][36] and the related references therein). Studying concrete systems of difference equations is an area popularized by Papaschinopoulos and Schinas and their collaborators (see, eg, previous studies [37][38][39][40][41] ).…”

Section: Introductionmentioning

confidence: 99%

“…Case b = 1. Since b = 1, from Equation33, it follows that a must be equal to zero. Hence, matrix Equation 32 becomes…”

mentioning

confidence: 99%

Solving a class of nonautonomous difference equations by generalized invariants

2019

By using generalized invariants, we describe a method for solving a class of higher-order nonautonomous difference equations. Solvability of the equations of order two, three, and four are studied in detail. Our results explain and extend some problems in the literature. As far as we know, the case when the order is four is considered for the first time in this paper. For the equations of second order, we also give an explanation how they can be obtained in a natural way.