Asymptotically polynomial solutions of difference equations

Abstract

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.

Asymptotic behavior of solutions of discrete Volterra equations

“…As in [15,16], and [17] we use o(n s ) as a measure of approximation. The idea of the proofs of Theorems 3.1, 3.2 and 3.5 is based on the results obtained in [17].…”

“…We need two lemmas. Lemma 3.9 is a consequence of Theorem 1 in [3] and Lemma 3.10 is a consequence of Theorem 2.1 in [15]. …”

“…The results devoted to the study of asymptotically polynomial solutions of ordinary difference equations can be found, for example, in [15,17,18,22] or [23]. In 1986, Popenda [22] gave sufficient conditions under which for any polynomial of degree at most m − 1, there exists a solution x of the form (1.1) with s = 0 for a difference equation…”

“…Recently, J. Migda obtained in [15] and [17] various results concerning solutions of the form (1.1) with arbitrary s ∈ (∞, 0] of the equation…”

“…We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior, especially asymptotically polynomial and asymptotically periodic solutions. As in [15], we use o(n s ) as a measure of approximation. For example, we give sufficient conditions under which, for real constants c 0 , c 1 , .…”