A problem of Hartman and Wintner: Approximation for discrete perturbations
Consider the second-order self-adjoint difference equation Delta (c(n)Deltax(n)) + (a(n) + f(n)) x(n+1) = 0 as a perturbation of the eventually disconjugate difference equation Delta (c(n)Deltaz(n)) + a(n) z(n+1) = 0, where c(n) not equal 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 c(n) is allowed to be oscillatory, and we do not assume absolute summability conditions on f(n),. Results obtained are applied to a second-order Poincare difference equation whose unperturbed equation has a double characteristic root. Nonlinear and nonhomogeneous perturbations are also considered. An example is given for illustration. (C) 2001 Elsevier Science Ltd. All rights reserved.
asymptotic expression self adjoint difference equation