Oscillation theorems for non-linear difference equation of the second order


E. M. ElabbasySh. R. Elzeiny


Abstract

carpathian_2009_25_1_061_072_abstract

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carpathian_2009_25_1_061_072

We obtain some oscillation criteria for the solutions of the non-linear difference equation of the form ∆ (rnψ (xn) f (∆xn)) + qnϕ (g (xn+1) , rn+1ψ (xn+1) f (∆xn+1)) = 0, n = 0, 1, 2, …, where u ϕ (u, v) > 0 for all u 6= 0, x g (x) > 0 and xf (x) > 0 for all x 6= 0, ψ (x) > 0 for all x ∈ R, {rn}∞n=0 is sequence of positive real numbers and {qn}∞n=0 is sequence of real values. The relevance of our theorems becomes clear due to a carefully selected examples.