Comparison Theorems for the Asymptotic Behavior of Solutions of Nonlinear Difference Equations

Applied Mathematics and Computation

2016

Section: Asymptotically Polynomial Solutionsmentioning

confidence: 91%

“…Convergence of the sequence ∆ m z n is comparatively easy to verify and condition (18) appears in many papers, see for example [6], [14], [18], [23] or the proof of Theorem 3.1 in [22].…”

Section: Asymptotically Polynomial Solutionsmentioning

confidence: 99%

Asymptotically polynomial solutions to difference equations of neutral type

Applied Mathematics and Computation

2016

“…In recent years, the equation (E) and similar equations have been studied in many papers, see for example, [1], [4], [5], [9], [12], [15], [17], [18], [21]. Some classical results on asymptotic behavior of solutions can be found in [2], [3], [6]- [8], [10]- [14], [22]- [24].…”

Section: Introductionmentioning

confidence: 99%

Approximation of Solutions to Nonautonomous Difference Equations

Tatra Mountains Mathematical Publications

2018

We study the asymptotic properties of solutions to nonautonomous difference equations of the form $${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$ Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.

Asymptotically polynomial solutions of difference equations

2013

Adv Differ Equ

Asymptotic properties of solutions of a difference equation of the form m x n = a n f (n, x σ (n)) + b n are studied. We present sufficient conditions under which, for any polynomial ϕ(n) of degree at most m-1 and for any real s ≤ 0, there exists a solution x of the above equation such that x n = ϕ(n) + o(n s). We give also sufficient conditions under which, for given real s ≤ m-1, all solutions x of the equation satisfy the condition x n = ϕ(n) + o(n s) for some polynomial ϕ(n) of degree at most m-1.