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

Sufﬁcient conditions are given for the existence of a unique bounded in the mean solution to a second-order difference equation with jumps of operator coefﬁcients in a Banach space. The question of the proximity of this solution to the stationary solution of the corresponding difference equation with constant operator coefﬁcients is studied


Introduction
Let ( , F, P ) be a complete probability space, X a complex separable Banach space with norm • X and zero element 0 X , L(X) the Banach algebra of bounded linear operators defined on X, and B(X) the σ -algebra of Borel sets in X. Definition 1.A sequence of X-valued random elements {ξ n , n ∈ Z} defined on ( , F, P ) is called -bounded in the mean if sup n∈Z E ξ n X < +∞; -stationary (in the restricted sense) if Consider the difference equation where A, B are fixed operators belonging to L(X), {η n , n ∈ Z} is the given bounded in the mean sequence of X-valued random elements.
Definition 2. A sequence of X-valued random elements {ξ n , n ∈ Z} is called a bounded in the mean solution of equation ( 1) corresponding to a bounded in the mean sequence {η n , n ∈ Z} if the sequence {ξ n , n ∈ Z} is bounded in the mean and equality (1) holds with probability 1 for all n ∈ 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 {ξ n , n ∈ Z} for each bounded in the mean sequence {η n , n ∈ Z} and also to prove that E ξ n − ζ n X → 0, as n → ∞, where {ζ n , n ∈ 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 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.

Consider an operator T
, where I and O are the identity and zero operators in X, respectively.Denote by σ (T A ), ρ(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.Proof.Sufficiency.Since (λ+ 1 λ −2) ∈ ρ(A), the operator λ = λ 2 I −(A+2I )λ+I has a continuous inverse operator −1 λ .Let J be the identity operator in X 2 .It is easy to verify that the operator Necessity.Let us fix λ ∈ ρ(T A ), λ = 0.It suffices to prove that the operator λ has a continuous inverse operator.
Let S = {z ∈ C | |z| = 1} be the unit circle on the complex plane C.

Lemma 2. σ (T A ) ∩ S = ∅ if and only if
Lemma 3. The difference equation ( 1) has a unique bounded in the mean solution {ξ n , n ∈ Z} for each bounded in the mean sequence {η n , n ∈ Z} if and only if the difference equation has a unique bounded in the mean solution {ξ n , n ∈ Z} for each bounded in the mean sequence of X 2 -valued random elements {η n , n ∈ Z} defined on ( , F, P ).
The proof of Lemma 3 is standard and is omitted here.
n ∈ Z is a bounded in the mean solution of equation ( 4) corresponding to the bounded in the mean sequence n−1 with probability 1 for all n ∈ Z and therefore {ξ Denote by Y the Banach space L 1 ( , X) of all equivalence classes of random elements ξ : → X such that ξ Y = E ξ X < +∞.Each operator G belonging to L(X) induces an operator G belonging to L(Y ) and defined by the rule The following lemma is a direct consequence of Definitions 1 and 2.
Lemma 4. The difference equation ( 4) has a unique bounded in the mean solution {ξ n , n ∈ Z} for each bounded in the mean sequence {η n , n ∈ Z} if and only if the deterministic difference equation has a unique bounded solution {ξ n , n ∈ Z} for each sequence {η n , n ∈ Z} bounded in Y 2 .
Let W be a complex Banach space.Suppose that the spectrum σ (U) of the operator U ∈ L(W ) satisfies the condition σ (U) ∩ S = ∅.Let σ − (U ) be the part of the spectrum σ (U) lying inside the circle S and σ + (U ) = σ (U)\σ − (U ).In what follows, we will consider the case when σ − (U ) = ∅, σ + (U ) = ∅.Note that all the results obtained below are also true in the case when one of the sets σ − (U ), σ + (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 ) +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 σ − (U ), σ + (U ), respectively; the spectral radii of the operators U − , U −1 + satisfy the inequalities

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 {ξ n , n ∈ Z} for each bounded in the mean X-valued sequence {η n , n ∈ Z}.
Proof.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.

Remark 2.
In paper [6] it was established that if, in addition, the space X is finitedimensional 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 = C 2 , consider the operators A = . Consequently, the operators T A , T B , respectively.Since these four vectors are linearly dependent, condition (i2) of Theorem 1 is not satisfied.

Proximity of components of the bounded in the mean solutions of the difference equations (1) and (2) for n → ∞
First, consider the deterministic analogs of equations ( 1) and (2).Let U, V be fixed operators belonging to L(W ).In what follows, we need the following statements.
Theorem 2 (See Theorem 1 in [3, p. 9]).The difference equation has a unique bounded solution {u n , n ∈ Z} for each sequence {y n , n ∈ Z} bounded in W if and only if σ (U) ∩ S = ∅.
Remark 3. It follows from the proof of Theorem 2 that if σ (U) ∩ S = ∅ then the unique bounded solution of equation ( 8) corresponding to the bounded sequence {y n , n ∈ 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.
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 ∈ Z} for each sequence {y n , n ∈ Z} bounded in W .
Remark 4. It was also shown in [4] that for equation (10) under conditions (j1), (j2) for each n ≥ 1 the element x n of the unique bounded solution {x n , n ∈ Z} corresponding to a bounded sequence {y n , n ∈ 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 where I W is the identity operator in W . Then Conditions (j1), (j2) ensure the existence of the projectors P U ± , P V ± , P 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 ∈ Z}.
From Theorem 1 with A = B it follows that when σ (A) ∩ [−4; 0] = ∅ holds, the difference equation ( 2) has a unique bounded in the mean solution {ζ n , n ∈ Z} for each bounded in the mean sequence {η n , n ∈ Z}.It also follows from the results established in [4] , 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.Theorem 5. Let the conditions of Theorem 1 be satisfied.Then there exist constants ρ ∈ (0, 1), C > 0, n 0 ∈ N depending only on the operators A and B and such that for each bounded in the mean sequence of X-valued random elements {η n , n ∈ Z} for bounded in the mean solutions {ξ n , n ∈ Z} and {ζ n , n ∈ Z} of equations ( 1) and ( 2) the following estimate holds: (20) Note that when the sequence {η n , n ∈ Z} is, in addition, stationary, then the corresponding solution {ζ n , n ∈ 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 {ζ n , n ∈ Z} when n → ∞, despite the jump in the operator coefficient in (1).
∈ Z} is a bounded in the mean solution of equation (1) corresponding to the sequence {η n , n ∈ Z}.