1 Introduction
Let $(\Omega ,\mathcal{F},P)$ be a complete probability space, X a complex separable Banach space with norm $\| \cdot {\| _{X}}$ and zero element ${0_{X}}$, $\mathcal{L}(X)$ the Banach algebra of bounded linear operators defined on X, and $\mathcal{B}(X)$ the σ-algebra of Borel sets in X.
Definition 1.
A sequence of X-valued random elements $\{{\xi _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ defined on $(\Omega ,\mathcal{F},P)$ is called
-
– bounded in the mean if $\underset{n\in \mathbb{Z}}{\sup }E\| {\xi _{n}}{\| _{X}}<+\infty $;
-
– stationary (in the restricted sense) if$\forall \hspace{0.2778em}m\in \mathbb{N}$ $\forall \hspace{0.2778em}{n_{1}},{n_{2}},\dots ,{n_{m}}\in \mathbb{Z}$ $\forall \hspace{0.2778em}{Q_{1}},{Q_{2}},\dots ,{Q_{m}}\in \mathcal{B}(X)$:$P\{{\xi _{{n_{k}}+1}}\in {Q_{k}},1\leqslant k\leqslant m\}=P\{{\xi _{{n_{k}}}}\in {Q_{k}},1\leqslant k\leqslant m\}$.
Consider the difference equation
where $A,B$ are fixed operators belonging to $\mathcal{L}(X)$, $\{{\eta _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ is the given bounded in the mean sequence of X-valued random elements.
(1)
\[ \left\{\begin{array}{l}{\xi _{n+1}}-2{\xi _{n}}+{\xi _{n-1}}=A{\xi _{n}}+{\eta _{n}},\hspace{0.1667em}n\ge 1,\hspace{1em}\\ {} {\xi _{n+1}}-2{\xi _{n}}+{\xi _{n-1}}=B{\xi _{n}}+{\eta _{n}},\hspace{0.1667em}n\le 0,\hspace{1em}\end{array}\right.\]Definition 2.
A sequence of X-valued random elements $\{{\xi _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ is called a bounded in the mean solution of equation (1) corresponding to a bounded in the mean sequence $\{{\eta _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ if the sequence $\{{\xi _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ is bounded in the mean and equality (1) holds with probability 1 for all $n\in \mathbb{Z}$.
The purpose of this article is to obtain sufficient conditions for the operators $A,B$ under which the difference equation (1) has a unique bounded in the mean solution $\{{\xi _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ for each bounded in the mean sequence $\{{\eta _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ and also to prove that $E\| {\xi _{n}}-{\zeta _{n}}{\| _{X}}\to 0$, as $n\to \infty $, where $\{{\zeta _{n}},\hspace{0.1667em}n\in \mathbb{Z}\}$ is the unique bounded in the mean solution of the difference equation with a constant operator coefficient A
Bounded solutions of second-order deterministic difference equations with constant operator coefficients are studied in [3, 8], stationary solutions of the second-order 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.
2 Auxiliary statements
Put ${X^{2}}=\Big\{\left(\begin{array}{c}{x^{(1)}}\\ {} {x^{(2)}}\end{array}\right)|\hspace{0.2778em}{x^{(1)}},{x^{(2)}}\in X\Big\}$. Then ${X^{2}}$ will be a complex separable Banach space with coordinatewise addition and multiplication by a scalar and with norm $||\overline{x}|{|_{{X^{2}}}}=||{x^{(1)}}|{|_{X}}+||{x^{(2)}}|{|_{X}}$, $\overline{x}=\left(\begin{array}{c}{x^{(1)}}\\ {} {x^{(2)}}\end{array}\right)\in {X^{2}}$. If operators $E,F,G,H$ belong to $\mathcal{L}(X)$, then, as in the case of numerical matrices $T=\left(\begin{array}{c@{\hskip10.0pt}c}E& F\\ {} G& H\end{array}\right)$ defines an operator belonging to $\mathcal{L}({X^{2}})$ by the rule $T\hspace{0.1667em}\overline{x}=\left(\begin{array}{c}E{x^{(1)}}+F{x^{(2)}}\\ {} G{x^{(1)}}+H{x^{(2)}}\end{array}\right)$, $\overline{x}=\left(\begin{array}{c}{x^{(1)}}\\ {} {x^{(2)}}\end{array}\right)\in {X^{2}}$.
Consider an operator ${T_{A}}=\left(\begin{array}{c@{\hskip10.0pt}c}A+2I& -I\\ {} I& O\end{array}\right)$, where I and O are the identity and zero operators in X, respectively. Denote by $\sigma ({T_{A}})$, $\rho ({T_{A}})$, $r({T_{A}})$ the spectrum, resolvent set and spectral radius of the operator ${T_{A}}$, respectively. In what follows, we will use the following statements.
Lemma 1.
The number $\lambda \ne 0$ belongs to $\rho ({T_{A}})$ if and only if $\lambda +\frac{1}{\lambda }-2$ belongs to $\rho (A)$.
Proof.
Sufficiency. Since $(\lambda +\frac{1}{\lambda }-2)\in \rho (A)$, the operator ${\Delta _{\lambda }}={\lambda ^{2}}I-(A+2I)\lambda +I$ has a continuous inverse operator ${\Delta _{\lambda }^{-1}}$. Let J be the identity operator in ${X^{2}}$. It is easy to verify that the operator
\[ {({T_{A}}-\lambda J)^{-1}}=\left(\begin{array}{c@{\hskip10.0pt}c}-\lambda {\Delta _{\lambda }^{-1}}& {\Delta _{\lambda }^{-1}}\\ {} -{\Delta _{\lambda }^{-1}}& (A+2I-\lambda I){\Delta _{\lambda }^{-1}}\end{array}\right)\]
is a continuous inverse operator to ${T_{A}}-\lambda J$. Therefore, $\lambda \in \rho ({T_{A}})$.Necessity. Let us fix $\lambda \in \rho ({T_{A}})$, $\lambda \ne 0$. It suffices to prove that the operator ${\Delta _{\lambda }}$ has a continuous inverse operator.
From the Banach theorem on the inverse operator, it follows that if ${\Delta _{\lambda }^{-1}}$ does not exist, then one of the following conditions is satisfied:
If condition $(a1)$ is satisfied then $({T_{A}}-\lambda J)\left(\begin{array}{c}\lambda u\\ {} u\end{array}\right)=\left(\begin{array}{c}{0_{X}}\\ {} {0_{X}}\end{array}\right)$. This contradicts inclusion $\lambda \in \rho ({T_{A}})$.
Since $\lambda \in \rho ({T_{A}})$, the equation
has a solution. Writing the equation 3 coordinatewise, we successively obtain the equalities ${x^{(1)}}=\lambda {x^{(2)}}$, $(A+2I-\lambda I)\lambda {x^{(2)}}-{x^{(2)}}=v$. Hence, the equation ${\Delta _{\lambda }}x=v$ has a solution $x=-{x^{(2)}}$. Thus, condition $(a2)$ is also not satisfied. □
(3)
\[ \left(\begin{array}{c@{\hskip10.0pt}c}A+2I-\lambda I& -I\\ {} I& -\lambda I\end{array}\right)\left(\begin{array}{c}{x^{(1)}}\\ {} {x^{(2)}}\end{array}\right)=\left(\begin{array}{c}v\\ {} {0_{X}}\end{array}\right)\]Let $S=\{z\in \mathbb{C}\hspace{0.1667em}\mid |z|=1\}$ be the unit circle on the complex plane $\mathbb{C}$.
Since $\{\lambda +\frac{1}{\lambda }-2\mid \lambda \in S\}=[-4;0]$, Lemma 2 is a direct consequence of Lemma 1.
Lemma 3.
The difference equation (1) has a unique bounded in the mean solution $\{{\xi _{n}},n\in \mathbb{Z}\}$ for each bounded in the mean sequence $\{{\eta _{n}},n\in \mathbb{Z}\}$ if and only if the difference equation
has a unique bounded in the mean solution $\{{\overline{\xi }_{n}},n\in \mathbb{Z}\}$ for each bounded in the mean sequence of ${X^{2}}$-valued random elements $\{{\overline{\eta }_{n}},n\in \mathbb{Z}\}$ defined on $(\Omega ,\mathcal{F},P)$.
(4)
\[ \left\{\begin{array}{l}{\overline{\xi }_{n+1}}={T_{A}}{\overline{\xi }_{n}}+{\overline{\eta }_{n}},n\ge 1,\hspace{1em}\\ {} {\overline{\xi }_{n+1}}={T_{B}}{\overline{\xi }_{n}}+{\overline{\eta }_{n}},n\le 0,\hspace{1em}\end{array}\right.\]The proof of Lemma 3 is standard and is omitted here.
Remark 1.
If $\Big\{\left(\begin{array}{c}{\xi _{n}^{(1)}}\\ {} {\xi _{n}^{(2)}}\end{array}\right)n\in \mathbb{Z}\Big\}$ is a bounded in the mean solution of equation (4) corresponding to the bounded in the mean sequence $\Big\{\left(\begin{array}{c}{\eta _{n}}\\ {} {0_{X}}\end{array}\right),n\in \mathbb{Z}\Big\}$, then ${\xi _{n}^{(2)}}={\xi _{n-1}^{(1)}}$ with probability 1 for all $n\in \mathbb{Z}$ and therefore $\{{\xi _{n}^{(1)}},n\in \mathbb{Z}\}$ is a bounded in the mean solution of equation (1) corresponding to the sequence $\{{\eta _{n}},n\in \mathbb{Z}\}$.
Denote by Y the Banach space ${\mathcal{L}_{1}}(\Omega ,X)$ of all equivalence classes of random elements $\xi :\Omega \to X$ such that $\| \xi {\| _{Y}}=E\| \xi {\| _{X}}<+\infty $. Each operator G belonging to $\mathcal{L}(X)$ induces an operator $\widetilde{G}$ belonging to $\mathcal{L}(Y)$ and defined by the rule
Lemma 4.
The difference equation (4) has a unique bounded in the mean solution $\{{\overline{\xi }_{n}},n\in \mathbb{Z}\}$ for each bounded in the mean sequence $\{{\overline{\eta }_{n}},n\in \mathbb{Z}\}$ if and only if the deterministic difference equation
has a unique bounded solution $\{{\overline{\xi }_{n}},n\in \mathbb{Z}\}$ for each sequence $\{{\overline{\eta }_{n}},n\in \mathbb{Z}\}$ bounded in ${Y^{2}}$.
(6)
\[ \left\{\begin{array}{l}{\overline{\xi }_{n+1}}={\widetilde{T}_{A}}{\overline{\xi }_{n}}+{\overline{\eta }_{n}},n\ge 1,\hspace{1em}\\ {} {\overline{\xi }_{n+1}}={\widetilde{T}_{B}}{\overline{\xi }_{n}}+{\overline{\eta }_{n}},n\le 0,\hspace{1em}\end{array}\right.\]Let W be a complex Banach space. Suppose that the spectrum $\sigma (U)$ of the operator $U\in \mathcal{L}(W)$ satisfies the condition $\sigma (U)\cap S=\varnothing $. Let ${\sigma _{-}}(U)$ be the part of the spectrum $\sigma (U)$ lying inside the circle S and ${\sigma _{+}}(U)=\sigma (U)\setminus {\sigma _{-}}(U)$. In what follows, we will consider the case when ${\sigma _{-}}(U)\ne \varnothing $, ${\sigma _{+}}(U)\ne \varnothing $. Note that all the results obtained below are also true in the case when one of the sets ${\sigma _{-}}(U),{\sigma _{+}}(U)$ is empty, with obvious changes in the formulas obtained.
From the theorem on the spectral decomposition of an operator in a Banach space (see, for example, [3, p. 8]) it follows that the space W is represented as a direct sum $W={W_{-}}(U)\dot{+}{W_{+}}(U)$ of subspaces ${W_{-}}(U),{W_{+}}(U)$, for which the following conditions are satisfied:
the subspaces ${W_{-}}(U),{W_{+}}(U)$ are invariant under the operator U;
the restrictions ${U_{-}},{U_{+}}$ of the operator U to the subspaces ${W_{-}}(U),{W_{+}}(U)$ have the spectra ${\sigma _{-}}(U),{\sigma _{+}}(U)$, respectively;
3 The bounded in the mean solutions of the difference equation (1)
The following theorem is one of the main results of this article.
Theorem 1.
Let the operators $A,B$ satisfy the following conditions:
Then the difference equation (1) has a unique bounded in the mean solution $\{{\xi _{n}},n\in \mathbb{Z}\}$ for each bounded in the mean X-valued sequence $\{{\eta _{n}},n\in \mathbb{Z}\}$.
Proof.
Condition $(i1)$ and Lemma 2 imply that $\sigma ({T_{A}})\cap S=\varnothing ,\sigma ({T_{B}})\cap S=\varnothing $. Also, using condition $(i2)$ and Theorem 2 from [5], we conclude that the difference equation (4) has a unique bounded in the mean solution $\{{\overline{\xi }_{n}},n\in \mathbb{Z}\}$ for every bounded in the mean sequence $\{{\overline{\eta }_{n}},n\in \mathbb{Z}\}$. Therefore the assertion of the theorem holds by Lemma 3. □
Remark 2.
In paper [6] it was established that if, in addition, the space X is finite-dimensional and the matrices of the operators $A,B$ have the Jordan normal form in the same basis, then condition $(i1)$ implies condition $(i2)$.
Example 1.
In the complex Euclidean space $X={\mathbb{C}^{2}}$, consider the operators $A=\left(\begin{array}{c@{\hskip10.0pt}c}1/2& 0\\ {} 0& 4/3\end{array}\right),\hspace{0.2778em}B=\left(\begin{array}{c@{\hskip10.0pt}c}1/2& 0\\ {} -5/6& 4/3\end{array}\right)$. It is easy to verify that $\sigma (A)=\sigma (B)=\{1/2,4/3\}$, $\sigma ({T_{A}})=\sigma ({T_{B}})=\{1/2,2,1/3,3\}$. It follows from the proof of Lemma 1 that if $\lambda \ne 0$, then $Au=(\lambda +\frac{1}{\lambda }-2)u$ if and only if ${T_{A}}\left(\begin{array}{c}\lambda u\\ {} u\end{array}\right)=\lambda \left(\begin{array}{c}\lambda u\\ {} u\end{array}\right)$. Consequently, ${X_{-}^{2}}({T_{A}}),{X_{+}^{2}}({T_{B}})$ are, respectively, the linear spans of the eigenvectors $\left(\begin{array}{c}1\\ {} 0\\ {} 2\\ {} 0\end{array}\right),\left(\begin{array}{c}0\\ {} 1\\ {} 0\\ {} 3\end{array}\right)$ and $\left(\begin{array}{c}2\\ {} 2\\ {} 1\\ {} 1\end{array}\right),\left(\begin{array}{c}0\\ {} 3\\ {} 0\\ {} 1\end{array}\right)$ of the operators ${T_{A}},{T_{B}}$. These four vectors are linearly independent. Therefore, for the operators $A,B$, conditions $(i1)$ and $(i2)$ of Theorem 1 are satisfied.
Example 2.
Let A be the operator from Example 1 and
\[ B=\frac{1}{21}\left(\begin{array}{c@{\hskip10.0pt}c}14\cdot 17+15\cdot 50& -64\cdot 15\\ {} 64\cdot 17& \hspace{1em}-(14\cdot 15+17\cdot 50)\end{array}\right).\]
Then $\sigma (B)=\{4/3,-100/21\},\sigma ({T_{B}})=\{1/3,3,-3/7,-7/3\}$ and also ${X_{-}^{2}}({T_{A}})$, ${X_{+}^{2}}({T_{B}})$ are the linear spans of the eigenvectors $\left(\begin{array}{c}1\\ {} 0\\ {} 2\\ {} 0\end{array}\right),\left(\begin{array}{c}0\\ {} 1\\ {} 0\\ {} 3\end{array}\right)$ and $\left(\begin{array}{c}3\\ {} 3\\ {} 1\\ {} 1\end{array}\right),\left(\begin{array}{c}-7\cdot 15\\ {} -7\cdot 17\\ {} 3\cdot 15\\ {} 3\cdot 17\end{array}\right)$ of the operators ${T_{A}},{T_{B}}$, respectively. Since these four vectors are linearly dependent, condition $(i2)$ of Theorem 1 is not satisfied.4 Proximity of components of the bounded in the mean solutions of the difference equations (1) and (2) for $n\to \infty $
First, consider the deterministic analogs of equations (1) and (2). Let $U,V$ be fixed operators belonging to $\mathcal{L}(W)$. In what follows, we need the following statements.
Theorem 2 (See Theorem 1 in [3, p. 9]).
Remark 3.
It follows from the proof of Theorem 2 that if $\sigma (U)\cap S=\varnothing $ then the unique bounded solution of equation (8) corresponding to the bounded sequence $\{{y_{n}},n\in \mathbb{Z}\}$ has the form
where ${P_{-}^{U}},{P_{+}^{U}}$ are the projectors in W onto the subspaces ${W_{-}}(U)$ and ${W_{+}}(U)$, respectively. Due to inequalities (7), the series in (9) converge.
(9)
\[ \Bigg\{{u_{n}}={\sum \limits_{j=0}^{\infty }}{U_{-}^{j}}{P_{-}^{U}}{y_{n-1-j}}-{\sum \limits_{j=-\infty }^{-1}}{U_{+}^{j}}{P_{+}^{U}}{y_{n-1-j}},n\in \mathbb{Z}\Bigg\},\]Theorem 3 (See Theorem 1 in [4]).
Assume that the following conditions are fulfilled:
Then the difference equation
has a unique bounded solution $\{{x_{n}},n\in \mathbb{Z}\}$ for each sequence $\{{y_{n}},n\in \mathbb{Z}\}$ bounded in W.
(10)
\[ \left\{\begin{array}{l}{x_{n+1}}=U{x_{n}}+{y_{n}},n\ge 1,\hspace{1em}\\ {} {x_{n+1}}=V{x_{n}}+{y_{n}},n\le 0,\hspace{1em}\end{array}\right.\]Remark 4.
It was also shown in [4] that for equation (10) under conditions (j1), (j2) for each $n\ge 1$ the element ${x_{n}}$ of the unique bounded solution $\{{x_{n}},n\in \mathbb{Z}\}$ corresponding to a bounded sequence $\{{y_{n}},n\in \mathbb{Z}\}$ can be obtained as follows. Let ${P_{-}^{0}},{P_{+}^{0}}$ be projectors in W onto the subspaces ${W_{-}}(U),{W_{+}}(V)$, respectively, corresponding to the representation $W={W_{-}}(U)\dot{+}{W_{+}}(V)$. Put
where ${I_{W}}$ is the identity operator in W. Then
Conditions (j1), (j2) ensure the existence of the projectors ${P_{\pm }^{U}}$, ${P_{\pm }^{V}}$, ${P_{\pm }^{0}}$, and also, taking into account inequalities (7), the convergence in the norm in W of the series from (12) and the boundedness of the sequence $\{{x_{n}},n\in \mathbb{Z}\}$.
(11)
\[ \forall \hspace{0.1667em}n\ge 1:{P_{+}^{n}}={U^{n}}\hspace{0.1667em}{P_{+}^{0}}\hspace{0.1667em}{U_{+}^{-n}}\hspace{0.1667em}{P_{+}^{U}},\hspace{0.1667em}{P_{-}^{n}}={I_{W}}-{P_{+}^{n}},\](12)
\[\begin{aligned}{}& \forall \hspace{0.1667em}n\ge 1:{x_{n}}={P_{-}^{n-1}}{y_{n-1}}+{U_{-}}{P_{-}^{n-2}}{y_{n-2}}+\cdots +{U_{-}^{n-2}}{P_{-}^{1}}{y_{1}}\\ {} & +{\sum \limits_{j=-\infty }^{0}}{U_{-}^{n-1}}\hspace{0.1667em}{P_{-}^{0}}\hspace{0.1667em}{V_{-}^{|j|}}\hspace{0.1667em}{P_{-}^{V}}\hspace{0.1667em}{y_{j}}-{\sum \limits_{j=n}^{\infty }}{P_{+}^{n-1}}\hspace{0.1667em}{U_{+}^{n-1-j}}\hspace{0.1667em}{P_{+}^{U}}\hspace{0.1667em}{y_{j}}.\end{aligned}\]Theorem 4.
Let conditions (j1), (j2) of Theorem 3 be satisfied. Then there exist constants $\rho \in (0;1)$, $C>0$, ${n_{0}}\in \mathbb{N}$ depending only on the operators $U,V$ and such that for each sequence $\{{y_{n}},n\in \mathbb{Z}\}$ bounded in W, for bounded solutions $\{{u_{n}},n\in \mathbb{Z}\}$ and $\{{x_{n}},n\in \mathbb{Z}\}$ of equations (8) and (10) corresponding to the sequence $\{{y_{n}},n\in \mathbb{Z}\}$, the following estimate holds:
Proof.
From (7) it follows that the spectral radii of the operators ${U_{-}},{U_{+}^{-1}},{V_{-}}$ are less than one. Therefore, there exist constants $\rho \in (0,1)$, ${m_{0}}\in \mathbb{N}$ such that
Fix a bounded sequence $\{{y_{n}},n\in \mathbb{Z}\}$ and, for $n\ge {m_{0}}+2$, estimate $\| {u_{n}}-{x_{n}}{\| _{W}}$ using (9), (12). Since ${P_{-}^{0}}$ is a projector onto ${W_{-}}(U)$, then if we also use (11), we get
Here $\| y{\| _{\infty }}=\underset{n\in \mathbb{Z}}{\sup }\| {y_{n}}{\| _{W}}$.
(14)
\[ \forall \hspace{0.1667em}m\ge {m_{0}}:\max \big(\| {U_{-}^{m}}\| ,\| {U_{+}^{-m}}\| ,\| {V_{-}^{m}}\| \big)\le {\rho ^{m}}.\]
\[\begin{aligned}{}& \forall \hspace{0.1667em}0\le k\le n-2:\| {U_{-}^{k}}{P_{-}^{U}}{y_{n-1-k}}-{U_{-}^{k}}{P_{-}^{n-1-k}}{y_{n-1-k}}{\| _{W}}\\ {} & =\| {U_{-}^{k}}\big({P_{-}^{U}}-{I_{W}}+{P_{+}^{n-1-k}}\big){y_{n-1-k}}{\| _{W}}=\| {U_{-}^{k}}\big({P_{+}^{n-1-k}}-{P_{+}^{U}}\big){y_{n-1-k}}{\| _{W}}\\ {} & =\| {U_{-}^{k}}\big({U^{n-1-k}}{P_{+}^{0}}{U_{+}^{-(n-1-k)}}-{U^{n-1-k}}{U_{+}^{-(n-1-k)}}\big){P_{+}^{U}}{y_{n-1-k}}{\| _{W}}\\ {} & =\| -{U_{-}^{k}}{U^{n-1-k}}{P_{-}^{0}}{U_{+}^{-(n-1-k)}}{P_{+}^{U}}{y_{n-1-k}}{\| _{W}}\\ {} & =\| {U_{-}^{n-1}}{P_{-}^{0}}{U_{+}^{-(n-1-k)}}{P_{+}^{U}}{y_{n-1-k}}{\| _{W}}.\end{aligned}\]
Therefore denoting by ${C_{1}}$ the maximum of the squared norms of the operators ${P_{\pm }^{0}}$, ${P_{\pm }^{U}}$, ${P_{\pm }^{V}}$ we obtain From (11) and the properties of the projectors it follows that
Also
Note that the constants in (15)–(19) depend only on the operators U and V.
(17)
\[\begin{aligned}{}& \forall \hspace{0.1667em}k\ge 0:\| {U_{+}^{-1-k}}{P_{+}^{U}}{y_{n+k}}-{P_{+}^{n-1}}{U_{+}^{-1-k}}{P_{+}^{U}}{y_{n+k}}{\| _{W}}\\ {} & =\| \big({U^{n-1}}{U_{+}^{-n+1}}{P_{+}^{U}}-{U^{n-1}}{P_{+}^{0}}{U^{-n+1}}{P_{+}^{U}}\big){U_{+}^{-1-k}}{P_{+}^{U}}{y_{n+k}}{\| _{W}}\\ {} & =\| {U_{-}^{n-1}}{P_{-}^{0}}{U_{+}^{-n-k}}{P_{+}^{U}}{y_{n+k}}{\| _{W}}\le {\rho ^{n-1}}{\rho ^{n+k}}{C_{1}}\| y{\| _{\infty }}.\end{aligned}\](18)
\[\begin{aligned}{}& {\left\| {\sum \limits_{j=n-1}^{\infty }}{U_{-}^{j}}{P_{-}^{U}}{y_{n-1-j}}{\| _{W}}\le {C_{1}}\| y\right\| _{\infty }}\frac{{\rho ^{n-1}}}{1-\rho },\end{aligned}\](19)
\[\begin{aligned}{}& {\left\| {\sum \limits_{j=-\infty }^{0}}{U_{-}^{n-1}}{P_{-}^{0}}{V^{|j|}}{P_{-}^{V}}{y_{j}}\right\| _{W}}\\ {} & \le {\rho ^{n-1}}{C_{1}}\| y{\| _{\infty }}\bigg({m_{0}}\underset{0\le k\le {m_{0}}-1}{\max }\| {V_{-}^{k}}\| +\frac{{\rho ^{{m_{0}}}}}{1-\rho }\bigg).\end{aligned}\]From Theorem 1 with $A=B$ it follows that when $\sigma (A)\cap [-4;0]=\varnothing $ holds, the difference equation (2) has a unique bounded in the mean solution $\{{\zeta _{n}},n\in \mathbb{Z}\}$ for each bounded in the mean sequence $\{{\eta _{n}},n\in \mathbb{Z}\}$. It also follows from the results established in [4] that if $\sigma ({T_{A}})\cap S=\varnothing $, $\sigma ({T_{B}})\cap S=\varnothing $, ${X^{2}}={X_{-}^{2}}({T_{A}})\dot{+}{X_{+}^{2}}({T_{B}})$, then $\sigma ({\widetilde{T}_{A}})=\sigma ({T_{A}})$, $\sigma ({\widetilde{T}_{B}})=\sigma ({T_{B}})$, ${Y^{2}}={Y_{-}^{2}}({\widetilde{T}_{A}})\dot{+}{Y_{+}^{2}}({\widetilde{T}_{B}})$, where the operators ${\widetilde{T}_{A}},{\widetilde{T}_{B}}$ 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.
Theorem 5.
Let the conditions of Theorem 1 be satisfied. Then there exist constants $\rho \in (0,1)$, $C>0$, ${n_{0}}\in \mathbb{N}$ depending only on the operators A and B and such that for each bounded in the mean sequence of X-valued random elements $\{{\eta _{n}},n\in \mathbb{Z}\}$ for bounded in the mean solutions $\{{\xi _{n}},n\in \mathbb{Z}\}$ and $\{{\zeta _{n}},n\in \mathbb{Z}\}$ of equations (1) and (2) the following estimate holds:
Note that when the sequence $\{{\eta _{n}},n\in \mathbb{Z}\}$ is, in addition, stationary, then the corresponding solution $\{{\zeta _{n}},n\in \mathbb{Z}\}$ of equation (2) is also stationary. According to (20), in this case, the elements of the solution to equation (1) are close to the stationary sequence $\{{\zeta _{n}},n\in \mathbb{Z}\}$ when $n\to \infty $, despite the jump in the operator coefficient in (1).
