By Saber Elaydi; et al
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Additional info for Proceedings of the Eighth International Conference on Difference Equations and Applications
The lemma is proved since all remaining affirmations are elementary consequences of the theory of number sequences. Let us formulate an elementary property of solutions. The corresponding proof is omitted. 4. Solutions u(k), U (k), k ∈ N (a) of two initial conditions for equation (1): u(a) = α, and U (a) = β with 0 < α < β satisfy the inequalities u(k) < U (k) for every k ∈ N (a). 5. Let the inequalities (6), (7) be valid for every k ∈ N (a). Then a) The solution u = u∗cs (k), k ∈ N (a) of the initial condition u∗cs (a) = ucs , s ∈ N∗ (14) for the equation (1) satisfies the relations u∗cs (k) ∈ ω(k) (15) for every k = a, a + 1, .
We use the spectral transformation y = ck−k0 x (k0 ∈ Z) to transform (2) into y = ck+1−k0 f (k, ck0 −k y). (6) Then the following connection between the global pullback attractor A 1 of system (6) and the pseudo-unstable manifold R0 of system (2) holds: A1 (k) = ck−k0 R0 (k) for all k ∈ Z. If the system (2) is invertible, then we have the following connection between the global pullback attractor A2 of the inverted version of (6), y = c−k−k0 −1 f −1 (−k − 1, ck0 +k y), (7) and the pseudo-stable manifold S0 of system (2): A2 (k) = c−k−k0 S0 (−k) for all k ∈ Z.
In this paper we consider a further generalization of the classical theorem by considering nonautonomous difference equations. In this context, the invariant manifolds are no longer subsets of the phase space but of the extended phase space (see Aulbach ). Although their existence is clear, the analytical computation of the invariant manifolds is only possible in rare cases. Therefore it is reasonable to develop numerical tools for the approximation of these sets. For stable and unstable manifolds of autonomous difference equations this topic is well examined.
Proceedings of the Eighth International Conference on Difference Equations and Applications by Saber Elaydi; et al