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

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(1)

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

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

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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.

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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.

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The boundary of the set is where

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

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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

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

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Theorem 1

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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.

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Retract and retraction retraction retract

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The general scheme of the proof

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