INVARIANT MEASURE AND STATISTICAL SOLUTIONS FOR NON-AUTONOMOUS DISCRETE KLEIN-GORDON-SCHRÖDINGER-TYPE EQUATIONS∗

In this article, we first prove the existence of the pullback attractor for no-autonomous discrete Klein-Gordon-Schrödinger-type equations. Then we construct the invariant measure and statistical solutions for this discrete equations via the generalized Banach limit.


Introduction
In this article, we consider the following non-autonomous discrete Klein-Gordon-Schrödinger-type equations: Equations (1.1) can be regarded as a discrete version of the following continuous Klein-Gordon-Schrödinger-type equations on C × R, Equations (1.3) describe the nonlinear interaction between high-frequency electron waves and low-frequency ion plasma waves, where z = z(x, t) ∈ C represents the dimensionless low frequency electron field, and u = u(x, t) ∈ R represents the dimensionless low frequency density, iαz and λu t denote the dissipative mechanism of the system. The term Rez x reflects the contribution of the effect caused by polarization drift to the system of equations. The well-posedness of the continuous version of Klein-Gordon-Schrödinger-type equations (1.3) has been studied extensively, we refer the readers to [1,6,[12][13][14] and the references cited therein. The invariant measures and statistical solutions are very useful concepts to understand the complexity of dynamical systems. For example, in order to analysis the turbulence in the case of Navier-Stokes equations, the measurements of several aspects of turbulent flows are actually measurements of time-average quantities, see [?] for details. Statistical solutions were used to formalize the notion of ensemble average in the conventional statistical theory of turbulence. Foias and Prodi [4] introduced the so-called Foias-Prodi statistical solution, which are associated to some invariant measures defined on the phase space (independent of time t) of the addressed system. Vishik and Furshikov [15] developed the so-called Vishik-Furshikov statistical solution, which are associated to some invariant measures defined on the trajectory space (dependent of time t), see [3] for details. Wang [16] talked about invariant Borel probability measures for discrete long-wave-short-wave resonance equations. Zhao [19] constructed trajectory statistical solutions possess an invariant property and satisfy a Liouville type equation. There are a series of papers concerning the invariant measures for well-posed dissipative systems, see [7-9, 11, 17, 18].
For the autonomous discrete Klein-Gordon-Schrödinger-type equations, Li, Hsu, Lin and Zhao established the existence of global attractor and obtained an upper bound of fractal dimension for the global attractor in [10]. Recently, Zhao, Xue and Łukaszewicz [20] studied the existence of the pullback attractor and invariant measure for a kind of lattice Klein-Gordon-Schrödinger equations, but they do not study the statistical solution for these equations.
In the current article, we first prove the existence of the pullback attractor for noautonomous discrete Klein-Gordon-Schrödinger-type equations. Then we construct the invariant measure and statistical solutions for this discrete equations via the generalized Banach limit. To the best of the authors' knowledge, this is the first reference investigating the statistical solutions for lattice dynamical systems.
The rest of this article is organized as follows. Section 2 are preliminaries. In Section 3, we prove the existence of the pullback attractor. In section 4, we construct the invariant measure and statistical solutions via the generalized Banach limit.

Preliminaries
Set We use X to denote ℓ 2 or L 2 for brevity, and equip X with the inner product and norm by wherev m denote the conjugate of v m . Define a bilinear form (·, ·) 1 and the adjoint operator B * of B respectively as Obviously, the bilinear form (·, ·) 1 is also an inner product in X, and the norm u 1 := (u, u) 1 . Then we have: Thus, the norm · 1 is equivalent to the norm · .
We next denote E = L 2 × ℓ 2 1 × ℓ 2 , and equip it with the inner products and the corresponding norm Then equations (1.2)-(1.4) can be rewritten as following systems For the locally existence of solution to problem (2.1), we have Lemma 2.1 ( [20]). Let f ∈ L 2 and g ∈ l 2 . For any initial data We next verify the boundness of the solutions. Then Proof. Taking the imaginary part of the inner product (L 2 , (·, ·)) of (1.1) with z, we obtain d dt Using the Gronwall inequality, we obtain the estimation (2.4) and the proof is completed.  where c 1 = max{1/α, 4/λ}, σ = min{2β, α}, and β is defined in Lemma 2.2.
By (2.4) and (2.5), we see that for any Thus, Lemma 2.1 implies that the maps of solutions operators

Existence of the pullback attractor
We first prove some estimates of solutions to problem (2.1). To guarantee the existence of pullback absorbing set, we need some assumption on the functions f (t) and g(t).
for some continuous function K(·) on the real line, bounded on intervals of the form hereinafter, the constant σ comes from Lemma 2.4. The class D σ will be called a universe in P(E). Clearly, all fixed bounded subsets of E lie in D σ . We next prove the existence of the pullback-D σ absorbing set for {U (t, τ )} t⩾τ in E.

Lemma 3.1. Let assumption (H) hold. Then the process {U
Using assumption (H) and (2.6), we can establish that for each t Then from (2.5) and (3.3) we conclude that the family The following lemma reveals the pullback-D σ asymptotic nullness of the process

Lemma 3.2. Let assumption (H) hold. Then for any given
Proof. First, for any M ∈ Z + , we set Take inner product with (2.1) by ϕ and take the real part, we have:

Direct calculation implies that
Then, using the definition of χ(x) and Lemma 3.1, we obtain that where Next, we estimate Re(Θψ, ϕ). It is easy to see that By elementary computations, we have Combining these inequality, when t > t 0 , we get Then Therefore, we obtain We derive from (3.7) that (3.8) Finally, we estimate Re(G(ψ, t), ϕ) E . Direct computations show that where Now, we estimate terms I 1 and I 2 . Taking the imaginary part of the inner product of equation (2.7) with w = (w m ) m∈Z and using (3.10), we obtain Then, applying Gronwall inequality ,we have and I 2 is similar to I 1 .

Invariant measures and statistical solutions
In this section, we will first construct the invariant measure for {U (t, τ )} t⩾τ . Then we prove that this invariant measure is a statistical solution of problem (2.1). In the beginning, we recall the definition of generalized Banach limit.

Definition 4.2 ( [5]
). A process {U(t, τ )} t⩾τ is said to be τ -continuous on a metric space X if for every x 0 ∈ X and every t ∈ R, the X-valued function τ → U (t, τ )x 0 is continuous and bounded on (−∞, t]. Theorem 4.1 ( [7]). Let {U(t, τ )} t⩾τ be a τ -continuous evolutionary process in a complete metric space X that has a pullback-D attractor A(·). Fix a generalized Banach limit LIM T →∞ and let γ : R → X be a continuous map such that γ(·) ∈ D. Then there exists a unique family of Borel probability measure {µ t } t∈R in X such that the support of the measure µ t is contained in A t and for any real-valued continuous functional ϕ on X. In addition, µ t is invariant in the sense that Lemma 4.1. Let ψ (1) (t) and ψ (2) (t) be two solutions of problem (2.1) corresponding to the initial conditions ψ (1) τ and ψ (2) 2τ ∈ E, respectively. Then (4.1) Proof. Let ψ 1 (t), ψ 2 (t) ∈ E be the solutions of problem (2.1) with the initial condition ψ 1 (τ ), Thenψ satisfies 2τ .

(4.2)
According to lemma [20, Lemma 3.3], we have For the right side where L(B)) is a positive constant depending on B. Take the real part of the inner product of (4.2) withψ : Coombing (4.3) to (4.5), we have for all s > τ , where σ is defined as Lemma 2.4. Using the Gronwall inequality, we can get Here C is a constant depending on σ and L(B).

Lemma 4.2.
Let f (t) and g(t) satisfy the conditions of (H). Then for every ψ * ∈ E and every t ∈ R, the E − valued function τ → U (t, τ )ψ * is continuous and bounded on (−∞, t]. Proof. Consider any ψ * = (u * , v * , z * ) T ∈ E and t ∈ R. We shall prove that for any ϵ > 0 there exists some δ = δ(ϵ) > 0, such that if r < t, s < t and |r − s| < δ then U (t, r)ψ * − U (t, s)ψ * E < ϵ. We assume r < s without loss of generality. Set Employing Lemma 4.1 and the continuity property of the process, we have Now, we have shown that solutions of problem (4.2) belong to the space C ([τ, +∞), E) , hence for any s ∈ R with s ⩽ t. Therefore, the E-valued function τ −→ U (t, τ )ψ * is continuous with respect to τ ∈ (−∞, t] in the space E . Finally, for above ψ * ∈ E and t ∈ R, we see from lemma 2.4 and theorem 3.1 that for any real-valued continuous functional λ on E. In addition, {m t } t∈R is invariant in the sense that We next prove that the obtained invariant measures {m t } t∈R is a statistical solutions for problem (2.1). We introduce some notations. Definition 4.3 (cf. [5]). We define the class T E of test functions to be the set of real-valued functionals Υ = Υ(ϕ) on E that are bounded on bounded subset of E and satisfy (a) for any ϕ ∈ E, the Fréchet derivative Υ ′ (ϕ) exists: for each ϕ ∈ E there exists an element Υ ′ (ϕ) such that where ·, · is the dual product between E * and E; (b) Υ ′ (ϕ) ∈ E * for all ϕ ∈ E, and the mapping ϕ −→ Υ ′ (ϕ) is continuous and bounded as a function from E to E * .
(ii) For almost every t ∈ [τ, +∞), and the function ψ belongs to L 1 loc ([τ, +∞)), for every v ∈ E, (iii) For any cylindrical test function Ψ in T E , it follows that, for all t ⩾ τ Proof. Let f ∈ L 2 and g ∈ ℓ 2 . There exists a unique local solution ψ(t) = (z(t), ψ(t), v(t)) T ∈ E of equations (2.1) such that for any initial data ψ 0 = (z 0 , ψ 0 , v 0 ) T ∈ E, ψ(·) ∈ C([0, T ), E) ∩ C 1 ((0, T ), E) for some T > 0. So we have Note that the invariant property of m t and the continuity of U(t, τ ), the function t → A Dσ (t) Φ(ϕ)dm t (ϕ) is continuous. It follows from condition (2) that F (t, ψ) = Θψ + G(t, ψ). Using the property of m t in Theorem 4.2, we get (4.10) Combining with the estimates in Lemma 2.4, we derive that (4.11) According to the m t are a unique family of Borel probability measures {m t } t∈R in the space E such that the support of the measure {m t } t∈R is contained in A Dσ (t), the condition (2) satisfies.