Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Abstract

Abstract This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form Δ x ( n ) + p ( n ) x ( τ ( n ) ) = 0 ,   n ∈ ℕ 0   [ ∇ x ( n ) − q ( n ) x ( σ ( n ) ) = 0 ,   n ∈ ℕ ] , \[\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\ , where ( p ( n ) ) n ≥ 0 and ( q ( n ) ) n ≥ 1 are sequences of nonnegative real numbers and ( τ ( n ) ) n ≥ 0 ,   ( σ ( n ) ) n ≥ 1 \[{(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}\] are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.