Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data

We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give necessary and sufficient conditions for the scattering data in the case of perturbations with finite second (or higher) moment.Comment: 23 page

Positive periodic solutions for nonlinear difference equations with periodic coefficients

WOS: 000079460000010We prove the existence of positive solutions of second-order nonlinear difference equations on a finite discrete segment with periodic boundary conditions. Obtained results yield positive periodic solutions of the equation on the whole discrete axis, provided that the coefficients are periodic. (C) 1999 Academic Press

On Green's functions and positive solutions for boundary value problems on time scales

WOS: 000175021700007In this paper we offer a form of self-adjoint differential equations on time scales so that the associated Green's function is found symmetric in the usual sense. For this purpose together with the delta derivative we employ the nabla derivative as well, We introduce the concepts of Lebesgue delta and nabla integrals on time scales. Next, sign properties of the Green's function are investigated and existence results for positive solutions of nonlinear boundary value problems are established. Upper and lower bounds for these positive solutions also are given, (C) 2002 Elsevier Science B.V. All rights reserved