Oscillatory behavior of advanced difference equations with general arguments


Şeyda Öcalan, Özkan Öcalan, Mustafa Kemal Yildiz




In this paper, we introduce some oscillation criteria for the first-order advanced difference equations with general arguments ∇x(n) − m∑ i=1 pi(n)x (τi(n)) = 0, n ≥ 1, n ∈N, where {pi(n)}(i = 1, 2, . . . ,m) are sequences of positive real numbers, {τi(n)}(i = 1, 2, . . . ,m) are sequences of integers and are not necessarily monotone such that τi(n) ≥ n (i = 1, 2, . . . ,m). An example illustrating the results is also given. 1. Introduction In this paper, we study the oscillatory behavior of all solutions of the first-order advanced difference equations ∇x(n) − m∑ i=1 pi(n)x (τi(n)) = 0, n ∈N, n ≥ 1, (1) where { pi(n) } (i = 1, 2, · · · ,m) are sequences of positive real numbers, {τi(n)} (i = 1, 2, · · · ,m) are sequences of integers and are not necessarily monotone such that τi(n) ≥ n for n ≥ 1. (2).