Linearized oscillation theorems for certain nonlinear difference equations with continuous arguments

This paper is concerned with the nonlinear difference equation y(t) −y(t−τ) + ∑i-1m pifi (y(t−σi)) = 0, where 0 < τ ≤ σ1 < … < σm, pi > 0, fi ϵ C(R, R), ufi(u) > 0 for u ≠ 0, limu → 0 ((fi(u))u) = 1, i = 1,2,…, m. Some linearized oscillation theorems for this equation are obtained, i.e., we will prove that under some conditions on f, this equation has the same oscillatory character as an associated linear equation.Linearized oscillation theorems for a nonautonomous nonlinear difference equation y(t) −y(t−τ) + ∑i-1m hi (y(t−σi)) = 0, are obtained also.