A problem of hartman and wintner: approximation for discrete perturbations

Consider the second-order self-adjoint difference equation Δ(cnΔxn) + (an + fn) xn+1 = 0 as a perturbation of the eventually disconjugate difference equation Δ(cnΔzn) + an zn+1 = 0, where cn ≠ 0. Asymptotic approximation for the fundamental system of solutions of the perturbed equation are expressed explicitly in terms of the coefficients and the principal (or recessive) solution of the unperturbed equation. In particular, the coefficient cn is allowed to be oscillatory, and we do not assume absolute summability conditions on fn. Results obtained are applied to a second-order Poincaré difference equation whose unperturbed equation has a double characteristic root. Nonlinear and nonhomogeneous perturbations are also considered. An example is given for illustration