A General Version of the Retract Method for Discrete Equations

Ważewski theorem on time scales with a set of egress points that is not the whole boundary

“…Note that the results obtained in the paper are new even in the case of difference equations. It is worth mentioning the paper [11], where convex discrete tubes generated by so-called connecting functions are considered as a set of constraints, and a recent paper [12], where discrete tubes with rectangular sections are studied.…”

How to find initial data generating bounded solutions of discrete equations

“…It is easy to see that if we take s as the initial value of n instead of a, Theorem 2.1 remains valid, i.e., if f is continuous and all the points (n, u) ∈ ∂ω, n ∈ Z ∞ s , are points of strict egress, then there exists a initial condition u(s) = u * * ∈ ω(s) (9) such that the corresponding solution u = u * * (n) satisfies the relation…”

Discrete and Differential Equations in Applied Mathematics

“…A lot of effort was devoted to investigation of existence of bounded solutions of different solutions via different methods that are used in the previous part. With so called retract type technique and Lyapunov technique, new methods and results were published, e.g., in [14,20,21,35]. Part of results obtained was extended to some special and arbitrary time-scales.…”

“…A general qualitative result about the structure of solutions of linear differential equations with delay was proved in [15]. Let us consider a linear equation (14) x t c t x t r t…”