Asymptotic dichotomy for nonoscillatory solutions of a nonlinear difference equation

Abstract

A nonlinear difference equation involving the maximum function is studied. We derive sufficient conditions in order that eventually positive or eventually negative solutions tend to zero or to positive or negative infinity. Nonlinear difference equations involving three or more functional values of the state variable are important as they appear naturally as discrete analogs and as numerical schemes of differential equations which model various natural phenomena. For an introductory exposition, the readers may consult, e.g., Kocic and Ladas [2]. In this paper, we are concerned with the nonlinear difference equation (1) ∆(xn − pnxn−τ ) + qn max n−σ≤s≤n xs = 0, n = 0, 1, . . . , where τ > 0, σ ≥ 0, and {pn}n=0, {qn}n=0 are real sequences. For σ = 0, the above equation reduces to the linear equation (2) ∆(xn − pnxn−τ ) + qnxn = 0, n = 0, 1, 2, . . . , which has been studied by Zhang and Cheng [5]. For σ = 0 and {pn} ≡ 0, (1) reduces further to the second order linear difference equation (3) ∆xn + qnxn = 0, n = 0, 1, 2, . . . , which has been studied by a number of authors. Let μ = max{τ, σ}. If a real sequence x = {xn}n=−μ satisfies the functional relation defined by (1), then it is said to be a solution of (1). Since 1991 Mathematics Subject Classification: Primary 39A10.