Oscillation Criteria for First-order Forced Nonlinear Difference Equations

where (i) {p(n)}, {e(n)} are sequences of real numbers; (ii) {qi(n)}, i= 1,2, are sequences of positive real numbers; (iii) λ, μ are ratios of positive odd integers with 0 < μ < 1 and λ > 1. By a solution of equation (1, i), i= 1,2,3, we mean a nontrivial sequence {x(n)}which is defined for n ≥ n0 ∈ N = {0,1,2, . . .} and satisfies equation (1, i), i = 1,2,3, and n = 1,2, . . . . A solution {x(n)} of any of the equations (1, i), i= 1,2,3, is said to be oscillatory if for every n1 ∈N, n1 > 0, there exists an n ≥ n1 such that x(n)x(n+ 1) ≤ 0, otherwise, it is nonoscillatory. Any of the equations (1, i), i= 1,2,3, is said to be oscillatory if all its solutions are oscillatory. In recent years, there has been an increasing interest in studying the oscillation and nonoscillation of solutions of difference equations. For example, see [1, 3–5] and the