Oscillation of solutions of some generalized nonlinear α -difference equations

摘要

In this paper, the authors discuss the oscillation of solutions of some generalized nonlinear α-difference equation 1 Δ α ( ? ) ( p ( k ) Δ α ( ? ) u ( k ) ) + q ( k ) f ( u ( k ? τ ( k ) ) ) = 0 , k ∈ [ a , ∞ ) , where the functions p, q, f and τ are defined in their domain of definition and α > 1 , ? is a positive real. Further, u f ( u ) > 0 for u ≠ 0 , p ( k ) > 0 and lim k → ∞ ( k ? τ ( k ) ) = ∞ , where R k = ∑ r = 0 [ k ? ? ? ] 1 α r p ( r ? ) → ∞ as k → ∞ and u ( k ) is defined for k ≥ min i ≥ 0 ( i ? τ ( i ) ) for all k ∈ [ a , ∞ ) for some a ∈ [ 0 , ∞ ) . MSC:39A12.