Vol. 235, No. 1, 2008
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Optimal oscillation criteria for first order difference equations with delay argument

George E. Chatzarakis & Roman Koplatadze & Ioannis P. Stavroulakis

Vol. 235 (2008), No. 1, 15-33
Abstract

Consider the first order linear difference equation

Δu (k) + p(k)u(τ(k)) = 0, k in N,

where Δu(k) = u(k + 1) − u(k), p : N → R+, τ : N → N, τ(k) ≤ k − 2 and limk→+∞τ(k) = +∞. Optimal conditions for the oscillation of all proper solutions of this equation are established. The results lead to a sharp oscillation condition, when k − τ(k) → +∞ as k → +∞. Examples illustrating the results are given.
Keywords

difference equation, proper solution, positive solution, oscillatory

Mathematical Subject Classification

Primary: 39A11

Secondary: 39A12
Authors
George E. Chatzarakis
Department of Mathematics
University of Ioannina
451 10 Ioannina
Greece
Roman Koplatadze
Department of Mathematics
University of Tbilisi
University Street 2
Tbilisi 0143
Georgia
Ioannis P. Stavroulakis
Department of Mathematics
University of Ioannina
451 10 Ioannina
Greece