LINEARIZED OSCILLATION OF NONLINEAR DIFFERENCE EQUATIONS WITH ADVANCED ARGUMENTS

Abstract

This paper is concerned with the nonlinear advanced difference equation with constant coefficients x(n+1) - x(n) + Sigma(m)(i=1) p(in)f(i) (x(n-ki)) = 0, n = 0, 1, . . . where p(i) subset of ( infinity, 0) and k(i) subset of {. . . , 2, 1} for i = 1, 2, . . . , m. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients x(n+1) - x(n) + Sigma(m)(i=1) p(in)f(i) (x(n-ki)) = 0, n = 0, 1, . . . where p(in) <= 0 and k(i) is an element of{. . . , -2, -1} for i = 1, 2, . . . , m.