OSCILLATIONS IN CERTAIN DIFFERENCE-EQUATIONS

Consider the difference equation [GRAPHICS] where J is a nonempty (finite or infinite) subset of N = {0,1,2,...}, pj is-an-element-of (0, infinity for j is-an-element-of J, and k(j) is-an-element-of J so that k(j1) not-equal k(j2) if j1, j2 is-an-element-of J with j1 not-equal j2. We prove that: A necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation [GRAPHICS] has no roots in (1, infinity).