Oscillation of a Certain Class of Third Order Nonlinear Difference Equations

In this paper, we are concerned with oscillation of the nonlinear di¤erence equation (cn [ (dn xn)] ) + qnf(xg(n)) = 0; n n0; where > 0 is the quotient of odd positive integers, cn, dn and qn are positive sequences of real numbers, g(n) is a sequence of nonnegative integers and f 2 C(R;R) such that uf(u) > 0 for u 6= 0:We establish some new su¢ cient conditions for oscillation by employing the Riccati substitution and the analysis of the associated Riccati di¤erence inequality. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results. 1. Introduction In recent years, the asymptotic properties and oscillation of di¤erence equations and their applications have been and still are receiving intensive attention. In fact, in the last few years several monographs and hundreds of research papers have been written, see for example the monographs [1, 3, 6, 11]. Determination of oscillatory behavior for solutions of …rst and second order di¤erence equations has occupied a great part of researchers’ interest. Compared to the …rst and second order di¤erence equations, the study of third order di¤erence equations has received considerably less attention in the literature, even though such equations arise in the study of economics, mathematical biology, and other areas of mathematics which discrete models are used (see for example [4]). For contributions, we refer the reader to the papers [2, 5, 7, 8, 13, 14, 15, 16, 17, 18, 19] and the references cited therein. For completeness and comparison, we present below some of these results. In this paper, we are concerned with oscillation of the nonlinear di¤erence equation (1.1) (cn [ (dn xn)] ) + qnf(xg(n)) = 0; n n0; where > 0 is quotient of odd positive integers. Throughout this paper, we will assume the following hypotheses: (h1): cn; dn; qn are positive sequences of real numbers, g(n) : N! Z; limn!1 g(n) = 1, (h2): f : R! R is continuous, f( u) = f(u), for u 6= 0;and f(u)=u > K > 0: Equation (1.1) is called a delay equation if g(n) < n and is called an advanced equation if g(n) > n: Since, we are interested in oscillation and asymptotic behavior of solutions near in…nity, we make a standing hypothesis that the equation under consideration does possess such solutions and the solutions vanishing in some neighborhood of in…nity will be excluded from our consideration. Our attention is restricted to those solutions of (1.1) 1991 Mathematics Subject Classi…cation. 34K11, 39A10.