On Bounded Solutions of a Difference Equation with Jumps of the Operator Coefficient

Bounded in the mean solutions of a second-order difference equation

“…, where the operators T A , T B are defined according to (5). Therefore, applying Theorem 4 to the difference equation ( 6) and then using Lemmas 3, 4, Theorem 1 and Remark 1, we conclude that the following theorem holds.…”

“…Condition (i1) and Lemma 2 imply that σ (T A ) ∩ S = ∅, σ (T B ) ∩ S = ∅. Also, using condition (i2) and Theorem 2 from [5], we conclude that the difference equation ( 4) has a unique bounded in the mean solution {ξ n , n ∈ Z} for every bounded in the mean sequence {η n , n ∈ Z}. Therefore the assertion of the theorem holds by Lemma 3.…”

“…Bounded solutions of second-order deterministic difference equations with constant operator coefficients are studied in [3,8], stationary solutions of the secondorder equation (2) in [3,2], bounded in the mean solutions of a first-order difference equation with a jump of the operator coefficient in [5], and bounded solutions of a deterministic analogue of equation ( 1) in [6]. Some applications of difference equations with operator coefficients in the deterministic case are given in [3,7,10,1], and in the stochastic case in [3,2,9] and in references therein.…”