On the spectral and scattering properties of eigenparameter dependent discrete impulsive Sturm-Liouville equations

This work develops scattering and spectral analysis of a discrete impulsive Sturm?Liouville equation with spectral parameter in boundary condition. Giving the Jost solution and scattering solutions of this problem, we find scattering function of the problem. Discussing the properties of scattering function, scattering solutions, and asymptotic behavior of the Jost solution, we find the Green function, resolvent operator, continuous and point spectrum of the problem. Finally, we give an example in which the main results are made explicit. Discrete impulsive equations, that is, difference equations involving impulsive effect, appear as a natural description of observed evolution phenomena of several real world problems. It is well-known that the theory of impulsive difference equations takes form under favor of the theory of the differential equations with impulses. In that way, for the mathematical theory of such impulsive equations, we refer to the monographs [2, 3, 7, 22, 27]. Impulsive difference equations are a basic tool to study dynamics that are subjected to sudden changes in their states. The theory of these equations has been motivated by a number of applied problems arising, in particular, in control theory, mechanical systems with impact, biological systems such as heart beats, blood flows, population dynamics, theoretical physics, chemistry, pharmacokinetics, mathematical economy, electric technology, metallurgy, ecology, infectious diseases, medicine, industrial robotics, biotechnology processes, engineering, navigational control of ships, and aircraft (see [4, 10, 11, 15, 17, 18, 20, 21, 23, 26]). The theory of difference equations with impulses is a new and important branch of difference equations. This work develops scattering and spectral analysis of a discrete impulsive Sturm?Liouville equation with spectral parameter in boundary condition. Giving the Jost solution and scattering solutions of this problem, we find scattering function of the problem. Discussing the properties of scattering function, scattering solutions, and asymptotic behavior of the Jost solution, we find the Green function, resolvent operator, continuous and point spectrum of the problem. Finally, we give an example in which the main results are made explicit.