Abstract
Abstract The paper studies the existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equation of the form: X ( t + n ) + ∑ r = 1 n - 1 A r ( t ) X ( t + r ) + A 0 ( t ) X ( t ) = f ( t , X ( t ) ) , X\left( {t + n} \right) + \sum\limits_{r = 1}^{n - 1} {{A_r}\left( t \right)X\left( {t + r} \right) + {A_0}\left( t \right)X\left( t \right) = f\left( {t,X\left( t \right)} \right),} n ∈ , by means of discrete dichotomy techniques.
Highlights
The study of almost periodicity which generalizes the notion of periodicity is an area of interest in its own right and has sundry applications in elds like Physics
We study the existence of almost periodic solutions to the class of higher-order nonautonomous stochastic di erence equations of the form: n−
Following the same lines as in the proof of Theorem 3.5 it follows that Eq(1.2) has a unique almost periodic solution given by the mapping t → Z(t) := X(t), X(t + ), X(t + ), . . . , X(t + n − ) T
Summary
An L (Ω; B)-valued random sequence X = {X(t)}t∈Z is said to be Bohr almost periodic in mean if for each ε > there exists N (ε) > such that among any N consecutive integers there exists at least an integer p > for which. The collection of all B-valued random sequences X = {X(t)}t∈Z which are Bohr almost periodic in mean is denoted by AP(Z; L (Ω; B)). A B-valued random sequence X = {X(t)}t∈Z is said to be almost periodic in probability if for each ε > , and η > there exists N (ε, η) > such that among any N consecutive integers there exists at least an integer p > for which. In view of the above, the space AP(Z; L (Ω; B)) of almost periodic random sequences equipped with the sup norm · ∞ is a Banach space.
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