Three-dimensional systems of difference equations – the asymptotic behavior of their solutions Josef Diblík Irena Růžičková

(1)

The solution of system (1) is defined as an infinite sequence of number vectors

The existence and uniqueness of the solution of initial problem (1),(2) with a prescribed initial value is obvious. (2)

If for every fixed the right hand sides are continuous on then the solution of initial problem (1),(2) depends continuously on the initial data.

Problem Find sufficient conditions with respect to the right-hand side of system (1) in order to guarantee the existence of at least one solution satisfying for every.

The boundary of the set is where

A point is called the first consequent point of a point (and we write ) if

A point is a point of the type of strict egress for the set with respect to system (1) if and only if for some or

The set  is of Liapunov type with respect to the j-th variable with respect to system (1) if for every

Theorem 1

Proof The proof of Theorem 1 will be performed by contradiction. We suppose that there exists no solution satisfying inequalities (3) for every In such situation we prove that there exists a continuous mapping of a closed interval onto its both endpoints.

Retract and retraction retraction retract

The general scheme of the proof