1. Introduction

In recent years, great progress had been made in the study of non-autonomous infinite dimensional dynamical systems. See, e.g., [1] [2] [3] [4] [5] and the references therein. Lattice dynamical systems (Hereafter LDSs) are infinite dimensional ordinary differential equations, which were widely and deeply investigated in the past decades due to its wide application in many fields such as laser systems, material science, electrical engineering, biology, chemical reaction theory and etc. See, e.g., [6] - [17] and so on. Nowadays, the study of non-autonomous LDSs appealed to more and more researchers, but there are few papers for non-autonomous LDSs until now. See e.g., [18] - [23] and etc.

As to the dissipative autonomous Klein-Gordon-Schrödinger (Hereafter KGS) lattice systems, many authors have studied them. For example, Abdallah in [24], Abounouh, Goubet and Hakim in [25], Yin and Zhou et al. in [26] investigated the existence, regularity, upper semicontinuity, Kolmogorov entropy of global attractor and so forth. Meanwhile, the following dissipative non-autonomous KGS lattice system

$\{\begin{array}{l}i{\stackrel{\dot{}}{w}}_m-\left(w_{m-1}+w_{m+1}-2w_m\right)+i\alpha w_m+u_m{w}_m=f_m\left(t\right)\mathrm{,}\hfill \\ {\ddot{u}}_m+\beta {\stackrel{\dot{}}{u}}_m+\left(u_{m-1}+u_{m+1}-2u_m\right)+\gamma u_m-\mu {\left|w_m\right|}^2=g_m\left(t\right)\mathrm{,}\hfill \end{array}m\in Z\mathrm{,}t>\tau \in R\mathrm{,}$ (1.1)

was investigated by many researchers either. Specifically, the existence of uniform exponential attractors for the dissipative non-autonomous KGS lattice system (1.1) with quasi-periodic symbols is studied in weighted spaces of infinite sequences by Abdallah in [27], simultaneously, some main results that the solution semigroup associated with such a system is Lipschitz continuous, α-contraction and satisfies the squeezing property, are obtained under some premise. Huang et al. in [28] proved the existence of a compact uniform attractor and obtained an upper bound of the Kolmogorov entropy of the compact uniform attractor. In addition, an upper semicontinuity of the compact uniform attractor is established as well. Zhao and Zhou in [29] proved the existence of compact kernel sections and obtained an upper bound of the Kolmogorov entropy of the compact kernel sections, but they didn’t study the fractal dimension of the compact kernel sections. In Zhou and Han [30], some sufficient conditions for the existence of a uniform exponential attractor for a family of continuous processes on separable Hilbert spaces and the space of infinite sequences are presented at first, and then the existence of uniform exponential attractors for the dissipative non-autonomous KGS lattice system (1.1) and for the dissipative non-autonomous Zakharov lattice system driven by quasi-periodic external forces in the spaces of infinite sequences is studied. However, what’s more important, so far to our knowledge, this problem that the fractal dimension of the compact kernel sections was not studied in Zhao and Zhou [29] is still an open topic till today. In view of this point, this paper is to estimate the fractal dimension of the compact kernel sections for the dissipative non-autonomous KGS lattice system (1.1). For our purpose, we first mention that as we all know, if $\mathcal{A}$ is a compact set in a metric space such that the fractal dimension of $\mathcal{A}$ is less or equal to $n/2$ for some $n\in N$, then there exists an injective Lipschitz mapping $F\mathrm{<mo>\; :\; </mo>}\mathcal{A}\to R^n$ such that its inverse is Hölder continuous. In the sequel of this paper, we will present a criterion for estimating the fractal dimension of a family of compact subsets of a separable Hilbert space and then apply this criterion to obtain an upper bound of the fractal dimension of the compact kernel sections associated with the dissipative non-autonomous KGS lattice system (1.1).

The remaining of this paper is organized as below. We give the preliminaries in Section 2. In Sections 3, a criterion is used to estimate the fractal dimension of the compact kernel sections for the dissipative non-autonomous KGS lattice system, and an upper bound is obtained. Lastly, Section 4 presents the conclusions.

2. Preliminaries

To begin, we introduce

${\mathcal{l}}^2=\left\{x={\left(x_m\right)}_{m\in Z}\mathrm{<mo>\; |\; </mo>}{x}_m\in R\mathrm{,}{\displaystyle \underset{m\in Z}{\sum }}{x}_m^2<+\infty \right\}\mathrm{,}$

$l^2=\left\{x={\left(x_m\right)}_{m\in Z}\mathrm{<mo>\; |\; </mo>}{x}_m\in C\mathrm{,}{\displaystyle \underset{m\in Z}{\sum }}{\left|x_m\right|}^2<+\infty \right\}\mathrm{,}$

where $Z$, $R$ and $C$ denote the integral, real and complex numbers, respectively.

Write $H={\mathcal{l}}^2$ or $l^2$, and endow H with the inner product and norm as below

$\left(x\mathrm{,}y\right)={\displaystyle \underset{m\in Z}{\sum }}{x}_m{\bar{y}}_m\mathrm{,}{\Vert x\Vert }^2=\left(x\mathrm{,}x\right)\mathrm{,}\forall x={\left(x_m\right)}_{m\in Z}\mathrm{,}y={\left(y_m\right)}_{m\in Z}\in H\mathrm{,}$

where ${\bar{y}}_m$ is the conjugate of $y_m$. Clearly, H is a Hilbert space.

Define linear operators A and B as follows

${\left(Ax\right)}_m=2x_m-x_{m-1}-x_{m+1}\mathrm{,}{\left(Bx\right)}_m=x_{m+1}-x_m\mathrm{,}\forall x={\left(x_m\right)}_{m\in Z}\in H\mathrm{.}$

For any $x\mathrm{,}y\in H$, define a bilinear form by means of

${\left(x\mathrm{,}y\right)}_{\gamma }=\left(Bx\mathrm{,}By\right)+\gamma \left(x\mathrm{,}y\right)\mathrm{,}$

where $\gamma$ as in the dissipative non-autonomous KGS lattice system (1.1) presented above. This bilinear form is obviously an inner product in Hilbert space H.

In the end, we express Hilbert spaces ${\mathcal{l}}_{\gamma }^2$, ${\mathcal{l}}^2$ and $l^2$ as

${\mathcal{l}}_{\gamma }^2=\left({\mathcal{l}}^2\mathrm{,}{\left(\cdot \mathrm{,}\cdot \right)}_{\gamma }\mathrm{,}{\Vert \cdot \Vert }_{\gamma }\right)\mathrm{,}{\mathcal{l}}^2=\left({\mathcal{l}}^2\mathrm{,}\left(\cdot \mathrm{,}\cdot \right)\mathrm{,}\Vert \cdot \Vert \right)\mathrm{,}{l}^2=\left(l^2\mathrm{,}\left(\cdot \mathrm{,}\cdot \right)\mathrm{,}\Vert \cdot \Vert \right)\mathrm{.}$

Set

$H_{\gamma }={\mathcal{l}}_{\gamma }^2\times {\mathcal{l}}^2\times l^2\mathrm{,}$

and equip it with the following norm and inner product

${\Vert \phi \Vert }_{H_{\gamma }}^2={\left(\phi \mathrm{,}\phi \right)}_{H_{\gamma }}\mathrm{,}\forall \phi \in H_{\gamma }\mathrm{<mo>\; ;\; </mo>}$

${\left({\phi }^{\left(1\right)}\mathrm{,}{\phi }^{\left(2\right)}\right)}_{H_{\gamma }}={\displaystyle \underset{m\in Z}{\sum }}\left({\left(B{u}^{\left(1\right)}\right)}_m{\left(B{u}^{\left(2\right)}\right)}_m+\gamma u_m^{\left(1\right)}{u}_m^{\left(2\right)}+v_m^{\left(1\right)}{v}_m^{\left(2\right)}+w_m^{\left(1\right)}\overline{w_m^{\left(2\right)}}\right)\mathrm{,}$

where ${\phi }^{\left(\iota \right)}={\left(u^{\left(\iota \right)}\mathrm{,}{v}^{\left(\iota \right)}\mathrm{,}{w}^{\left(\iota \right)}\right)}_{m\in Z}\in H_{\gamma }\mathrm{,}\iota =\mathrm{1,2}$.

Define

$\begin{array}{l}\mathcal{M}=\{h\left(t\right)={\left(h_m\left(t\right)\right)}_{m\in Z}\in {\mathcal{C}}_b\left(R\mathrm{,}H\right)\mathrm{<mo>\; :\; </mo>}\text{foreach}\tau \in R\text{and}\forall ϵ>\mathrm{0,}\begin{array}{c}\\ \end{array}\\ \exists \mathcal{N}\left(ϵ\right)\in N\text{suchthat}{\displaystyle \underset{\left|m\right|\ge \mathcal{N}\left(ϵ\right)}{\sum }}{\left|h_m\left(ı\right)\right|}^2\le ϵ\text{forany}ı\le \tau \}\mathrm{,}\end{array}$

and denote by ${\mathcal{C}}_b\left(R\mathrm{,}{\mathcal{l}}^2\right)$ and ${\mathcal{C}}_b\left(R\mathrm{,}{l}^2\right)$ respectively the set of continuous and bounded functions from $R$ into ${\mathcal{l}}^2$ and $l^2$.

Definition 2.1. A two-parameter family of mappings ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }$ is called to be a process in a Hilbert space $H$, if

1) $U\left(t\mathrm{,}\tau \right)\mathrm{<mo>\; :\; </mo>}H\to H\mathrm{,}t\ge \tau$ ;

2) $U\left(t\mathrm{,}s\right)U\left(s\mathrm{,}\tau \right)=U\left(t\mathrm{,}\tau \right)\mathrm{,}\forall t\ge s\ge \tau \mathrm{,}\tau \in R$ ;

3) $U\left(\tau \mathrm{,}\tau \right)=I$ (identity operator of $H$ ), $\tau \in R$.

Definition 2.2. A function $\phi \left(s\right)\mathrm{,}s\in R$, is said to be a complete trajectory of the process ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }$, if $U\left(t\mathrm{,}\tau \right)\phi \left(\tau \right)=\phi \left(t\right)\mathrm{,}\forall t\ge \tau \mathrm{,}\tau \in R$. The kernel $\mathcal{K}$ of the process ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }$ consists of all bounded complete trajectories of ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }$, i.e.,

$\mathcal{K}=\left\{\phi (\cdot )\mathrm{<mo>\; :\; </mo>}U\left(t\mathrm{,}\tau \right)\phi \left(\tau \right)=\phi \left(t\right)\mathrm{,}t\ge \tau \mathrm{,}\tau \in R\mathrm{,}{\Vert \phi \left(s\right)\Vert }_H\le {\mathcal{M}}_{\phi }\mathrm{,}s\in R\right\}\mathrm{,}$

and the kernel sections $\mathcal{K}\left(s\right)\subset H$ of the kernel $\mathcal{K}$ at time $s\in R$ is

$\mathcal{K}\left(s\right)=\left\{\phi \left(s\right)\mathrm{<mo>\; :\; </mo>}\phi (\cdot )\in \mathcal{K}\right\}\mathrm{.}$

Definition 2.3. The fractal dimension ${\dim }_f\mathcal{A}$ of a compact set $\mathcal{A}$ in a metric space $\mathcal{H}$ is defined as follows, namely

${\dim }_f\mathcal{A}=\underset{\epsilon \to 0}{\lim }\sup \frac{\ln N\left(\mathcal{A}\mathrm{,}\epsilon \right)}{\ln \left(1/\epsilon \right)}\mathrm{,}$

where $N\left(\mathcal{A}\mathrm{,}\epsilon \right)$ is the minimal number of closed sets of radius $\epsilon$ which cover the set $\mathcal{A}$.

The criterion below is directly cited from Zhou et al. [21].

Lemma 2.1. Let ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }$ be a continuous process on a Hilbert space $H$ and ${\left\{A\left(t\right)\right\}}_{t\in R}$ be a family of compact, negatively invariant (i.e., $A\left(t\right)\subset U\left(t\mathrm{,}\tau \right)A\left(\tau \right)$ for all $t\ge \tau \mathrm{,}\tau \in R$ ) subsets of $H$. Assume that

1) there exists a uniform finite covering of closed subsets with diameter 2 of $A\left(t\right)$ for all $t\in R$, that is, there exists $N^{\star }$ closed balls of $\mathcal{M}$ with diameter 2 covering $A\left(t\right)$ for all $t\in R$, where $N^{\star }$ is independent of t;

2) for any $\tau \in R$, there exists $\Gamma >0$ and $0<\eta <1$, which are all independent of $\tau \in R$ such that for ${\omega }_{\iota }\in A\left(\tau \right)\mathrm{,}\iota =\mathrm{1,2}$,

a) there exists $L>0$ yields

${\Vert U\left(\Gamma +\tau \mathrm{,}\tau \right){\omega }_1-U\left(\Gamma +\tau \mathrm{,}\tau \right){\omega }_2\Vert }_H\le L{\Vert {\omega }_1-{\omega }_2\Vert }_H\mathrm{,}$

i.e., $U\left(\Gamma +\tau \mathrm{,}\tau \right)$ is Lipschitz on $A\left(\tau \right)$ ;

b) there exists finite-dimensional orthoprojector P of $H$ satisfies

${\Vert \left(I-P\right)\left(U\left(\Gamma +\tau \mathrm{,}\tau \right){\omega }_1-U\left(\Gamma +\tau \mathrm{,}\tau \right){\omega }_2\right)\Vert }_H\le \eta {\Vert {\omega }_1-{\omega }_2\Vert }_H\mathrm{<mo>\; ;\; </mo>}$

then

${\dim }_{f}A\le \dim P\cdot \ln \left(1+\frac{8\left(1+L\right)}{1-\eta }\right)\cdot {\left(\ln \frac{2}{1+\eta }\right)}^{-1}\mathrm{.}$

3. Fractal Dimension of Compact Kernel Sections for Dissipative Non-Autonomous KGS Lattice System

Consider the dissipative non-autonomous KGS lattice system with the initial conditions as vector form

$\{\begin{array}{l}i\stackrel{\dot{}}{w}-Aw+i\alpha w+uw=f\left(t\right)\mathrm{,}\hfill \\ \ddot{u}+\beta \stackrel{\dot{}}{u}+Au+\gamma u-\mu {\left|w\right|}^2=g\left(t\right)\mathrm{,}t>\tau \mathrm{,}\tau \in R\hfill \\ w\left(\tau \right)={\left(w_m\left(\tau \right)\right)}_{m\in Z}\mathrm{,}u\left(\tau \right)={\left(u_m\left(\tau \right)\right)}_{m\in Z}\mathrm{,}\stackrel{\dot{}}{u}\left(\tau \right)={\left({\stackrel{\dot{}}{u}}_m\left(\tau \right)\right)}_{m\in Z}\mathrm{,}\hfill \end{array}$ (3.1)

where $w={\left(w_m\right)}_{m\in Z}$, $w_m\in C$ ; $u={\left(u_m\right)}_{m\in Z}$, $u_m\in R$ ; $uw={\left(u_m{w}_m\right)}_{m\in Z}$, ${\left|w\right|}^2={\left({\left|w_m\right|}^2\right)}_{m\in Z}$ ; $f\left(t\right)={\left(f_m\left(t\right)\right)}_{m\in Z}$, $g\left(t\right)={\left(g_m\left(t\right)\right)}_{m\in Z}$ ; $i$ is the imaginary numbers’ unit; $\alpha$, $\beta$, $\gamma$ and $\mu$ are positive constants; ${\left(Az\right)}_m=2z_m-z_{m-1}-z_{m+1}$, $m\in Z$, $z$ denotes $u$ or $w$.

We set

$v=\stackrel{\dot{}}{u}+\lambda u,\lambda =\frac{\beta \gamma }{{\beta }^2+4\gamma }>0.$

Thus, (3.1) can be written as below

$\{\begin{array}{l}\stackrel{\dot{}}{\phi }+G\phi =F\left(\phi \mathrm{,}t\right)\mathrm{,}t>\tau \mathrm{<mo>\; ;\; </mo>}\hfill \\ \phi \left(\tau \right)={\left(u\left(\tau \right)\mathrm{,}v\left(\tau \right)\mathrm{,}w\left(\tau \right)\right)}^{\text{T}}={\left(u\left(\tau \right)\mathrm{,}\stackrel{\dot{}}{u}\left(\tau \right)+\lambda u\left(\tau \right)\mathrm{,}w\left(\tau \right)\right)}^{\text{T}}\mathrm{,}\tau \in R;\hfill \end{array}$ (3.2)

where $\phi ={\left(u\mathrm{,}v\mathrm{,}w\right)}^{\text{T}}$, $v=\stackrel{\dot{}}{u}+\lambda u$, $F\left(\phi \mathrm{,}t\right)={\left(\mathrm{0,}\mu {\left|w\right|}^2+g\left(t\right)\mathrm{,}iuw-if\left(t\right)\right)}^{\text{T}}$, and

$G=\left(\begin{array}{ccc}\lambda I& -I& 0\\ A+\gamma I+\lambda \left(\lambda -\beta \right)I& \beta -\lambda & 0\\ 0& 0& iA+\lambda I\end{array}\right)\mathrm{.}$

From Zhao and Zhou [29], we can see, for given $f\left(t\right)={\left(f_m\left(t\right)\right)}_{m\in Z}\in \mathcal{M}$ with $H=l^2$, $g\left(t\right)={\left(g_m\left(t\right)\right)}_{m\in Z}\in \mathcal{M}$ with $H={\mathcal{l}}^2$, the solution mappings of (3.2), that is, $U\left(t\mathrm{,}\tau \right)\mathrm{<mo>\; :\; </mo>}\phi \left(\tau \right)={\left(u\left(\tau \right)\mathrm{,}v\left(\tau \right)\mathrm{,}w\left(\tau \right)\right)}^{\text{T}}\in H_{\gamma }\to \phi \left(t\right)={\left(u\left(t\right)\mathrm{,}v\left(t\right)\mathrm{,}w\left(t\right)\right)}^{\text{T}}\in H_{\gamma }$,

$\forall t\ge \tau$, $\tau \in R$, generate a family of continuous processes ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }$ in $H_{\gamma }$. Moreover, the family of processes ${\left\{U\left(t\mathrm{,}\tau \right)\right\}}_{t\ge \tau }\mathrm{,}\tau \in R$, possess a family of compact kernel sections ${\left\{\mathcal{K}\left(\tau \right)\right\}}_{\tau \in R}$, where $\mathcal{K}\left(\tau \right)$ is included in a uniformly bounded set $B=\left\{\phi \in H_{\gamma }\mathrm{<mo>\; :\; </mo>}{\Vert \phi \Vert }_{H_{\gamma }}^2\le R_0^2\right\}$ and satisfies $U\left(t\mathrm{,}\tau \right)\mathcal{K}\left(\tau \right)=\mathcal{K}\left(t\right)\mathrm{,}\forall t\ge \tau \mathrm{,}\tau \in R$, here

$R_0={\left(\frac{1}{{\delta }_0}\left(\frac{\underset{t\in R}{{\displaystyle \sup }}{\Vert f\left(t\right)\Vert }^2}{\alpha }+\frac{\underset{t\in R}{{\displaystyle \sup }}{\Vert g\left(t\right)\Vert }^2}{\beta }+\frac{2{\mu }^2\underset{t\in R}{{\displaystyle \sup }}{\Vert f\left(t\right)\Vert }^4}{{\alpha }^4\beta }\right)\right)}^{1/2}\mathrm{,}$ (3.3)

${\delta }_0=\min \left\{\delta ,\frac{\alpha }{4}\right\},\delta =\frac{\beta \gamma }{\sqrt{{\beta }^2+4\gamma }\left(\sqrt{{\beta }^2+4\gamma }+\beta \right)}>0.$ (3.4)

In the sequel, we get an upper bound of the fractal dimension of the compact kernel sections $\mathcal{K}\left(\tau \right)$, which is generated by the process of the dissipative non-autonomous KGS lattice system (3.1).

Suppose ${\phi }^{\left(\iota \right)}\left(\tau \right)\in \mathcal{K}\left(\tau \right)$, $\tau \in R$, then ${\phi }^{\left(\iota \right)}\left(t\right)=U\left(t\mathrm{,}\tau \right){\phi }^{\left(\iota \right)}\left(\tau \right)={\left(u^{\left(\iota \right)}\left(t\right)\mathrm{,}{v}^{\left(\iota \right)}\left(t\right)\mathrm{,}{w}^{\left(\iota \right)}\left(t\right)\right)}^{\text{T}}\in \mathcal{K}\left(\tau \right)\subseteq B$ for $t-\tau \ge T\left(\tau \mathrm{,}B\right)$, $\iota =1,2$. Set $\varphi \left(t\right)={\phi }^{\left(1\right)}\left(t\right)-{\phi }^{\left(2\right)}\left(t\right)$, then by (3.2), we have

$\{\begin{array}{l}\stackrel{\dot{}}{\varphi }+G\varphi =F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\hfill \\ \varphi \left(\tau \right)={\left(\xi \left(\tau \right)\mathrm{,}\zeta \left(\tau \right)\mathrm{,}\varsigma \left(\tau \right)\right)}^{\text{T}}\mathrm{,}\hfill \end{array}t>\tau \mathrm{,}\tau \in R\mathrm{,}$ (3.5)

where $\varphi ={\left(\xi \mathrm{,}\zeta \mathrm{,}\varsigma \right)}^{\text{T}}={\left({\xi }_m\mathrm{,}{\zeta }_m\mathrm{,}{\varsigma }_m\right)}_{m\in Z}^{\text{T}}={\left({\varphi }_m\right)}_{m\in Z}$, and $\xi =u^{\left(1\right)}-u^{\left(2\right)}$, $\zeta =\stackrel{\dot{}}{\xi }+\lambda \xi =v^{\left(1\right)}-v^{\left(2\right)}$, $\varsigma =w^{\left(1\right)}-w^{\left(2\right)}$.

Lemma 3.1. For any $\Gamma >0$, $U\left(\Gamma +\tau ,\tau \right)$ is Lipschitz on $\mathcal{K}\left(\tau \right)$, i.e.,

${\Vert {\phi }^{\left(1\right)}\left(\Gamma +\tau \right)-{\phi }^{\left(2\right)}\left(\Gamma +\tau \right)\Vert }_{H_{\gamma }}\le {\text{e}}^{\left(C_0{R}_0^2-{\delta }_0\right)\Gamma }{\Vert {\phi }^{\left(1\right)}\left(\tau \right)-{\phi }^{\left(2\right)}\left(\tau \right)\Vert }_{H_{\gamma }}\mathrm{,}$ (3.6)

where

$C_0=\frac{2{\mu }^2}{\beta }+\frac{1}{3\alpha }$ (3.7)

$R_0$ and ${\delta }_0$ as in (3.3) and (3.4), respectively.

For brief, we denote by $Re\left(\cdot \mathrm{,}\cdot \right)$ and $Im\left(\cdot \mathrm{,}\cdot \right)$ respectively the real part and imaginary part of inner product $\left(\cdot \mathrm{,}\cdot \right)$.

Proof. Taking the real part of the inner product ${\left(\cdot \mathrm{,}\cdot \right)}_{H_{\gamma }}$ of (3.5) with $\varphi$, we have

$Re{\left(\stackrel{\dot{}}{\varphi }\mathrm{,}\varphi \right)}_{H_{\gamma }}+Re{\left(G\varphi \mathrm{,}\varphi \right)}_{H_{\gamma }}=Re{\left(F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\varphi \right)}_{H_{\gamma }}\mathrm{.}$ (3.8)

By simple computation, we get

$Re{\left(\stackrel{\dot{}}{\varphi }\mathrm{,}\varphi \right)}_{H_{\gamma }}=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2\mathrm{,}$ (3.9)

$\begin{array}{c}Re{\left(G\varphi \mathrm{,}\varphi \right)}_{H_{\gamma }}\ge \delta \left({\Vert \xi \Vert }_{\gamma }^2+{\Vert \zeta \Vert }^2\right)+\frac{\beta }{2}{\Vert \zeta \Vert }^2+\alpha {\Vert \varsigma \Vert }^2\\ \ge {\delta }_0{\Vert \varphi \Vert }_{H_{\gamma }}^2+\frac{\beta }{2}{\Vert \zeta \Vert }^2+\frac{3\alpha }{4}{\Vert \varsigma \Vert }^2\mathrm{,}\end{array}$ (3.10)

$\begin{array}{l}Re{\left(F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\varphi \right)}_{H_{\gamma }}\\ \le \frac{\beta }{2}{\Vert \zeta \Vert }^2+\frac{3\alpha }{4}{\Vert \varsigma \Vert }^2+\left(\frac{{\mu }^2}{2\beta }+\frac{1}{12\alpha }\right){\Vert \varphi \Vert }_{H_{\gamma }}^2{\left(\left|w^{\left(1\right)}\right|+\left|w^{\left(2\right)}\right|\right)}^2\mathrm{.}\end{array}$ (3.11)

Actually

$\begin{array}{l}Re{\left(F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\varphi \right)}_{H_{\gamma }}\\ =Re\mu \left({\left|w^{\left(1\right)}\right|}^2-{\left|w^{\left(2\right)}\right|}^2\mathrm{,}\zeta \right)+Re\left(i\left(u^{\left(1\right)}{w}^{\left(1\right)}-u^{\left(2\right)}{w}^{\left(2\right)}\right)\mathrm{,}\varsigma \right)\\ \le \mu \Vert \zeta \Vert \Vert \varsigma \Vert \left(\left|w^{\left(1\right)}\right|+\left|w^{\left(2\right)}\right|\right)+Re\left(i\left(u^{\left(1\right)}{w}^{\left(1\right)}-u^{\left(2\right)}{w}^{\left(2\right)}\right)\mathrm{,}\varsigma \right)\mathrm{,}\end{array}$

and

$\begin{array}{l}Re\left(i\left(u^{\left(1\right)}{w}^{\left(1\right)}-u^{\left(2\right)}{w}^{\left(2\right)}\right)\mathrm{,}\varsigma \right)\\ =Re\left(i\frac{\left(u^{\left(1\right)}-u^{\left(2\right)}\right)\left(w^{\left(1\right)}+w^{\left(2\right)}\right)+\left(u^{\left(1\right)}+u^{\left(2\right)}\right)\left(w^{\left(1\right)}-w^{\left(2\right)}\right)}{2}\mathrm{,}\varsigma \right)\\ =-Im\left(\frac{\left(u^{\left(1\right)}-u^{\left(2\right)}\right)\left(w^{\left(1\right)}+w^{\left(2\right)}\right)}{2}\mathrm{,}\varsigma \right)\\ \le \frac{1}{2}{\Vert \xi \Vert }_{\gamma }\Vert \varsigma \Vert \left(\left|w^{\left(1\right)}\right|+\left|w^{\left(2\right)}\right|\right)\mathrm{,}\end{array}$

thus

$\begin{array}{l}Re{\left(F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\varphi \right)}_{H_{\gamma }}\\ \le \mu \Vert \zeta \Vert \Vert \varsigma \Vert \left(\left|w^{\left(1\right)}\right|+\left|w^{\left(2\right)}\right|\right)+\frac{1}{2}{\Vert \xi \Vert }_{\gamma }\Vert \varsigma \Vert \left(\left|w^{\left(1\right)}\right|+\left|w^{\left(2\right)}\right|\right)\mathrm{.}\end{array}$ (3.12)

Applying Young’s inequality to (3.12), it is obvious to know that (3.11) holds.

Taking (3.8)-(3.11) into account, we see

$\begin{array}{c}\frac{\text{d}}{\text{d}t}{\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2\le -2{\delta }_0{\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2+\left(\frac{{\mu }^2}{\beta }+\frac{1}{6\alpha }\right){\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2{\left(\left|w^{\left(1\right)}\right|+\left|w^{\left(2\right)}\right|\right)}^2\\ \le 2\left(C_0{R}_0^2-{\delta }_0\right){\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2\mathrm{.}\end{array}$ (3.13)

Set $t=\Gamma +\tau$, $\Gamma >0$, and then apply Gronwall’s inequality to (3.13), it is easy to see that (3.6) holds. The proof is completed.

Lemma 3.2. There exists a finite dimensional orthoprojector $P$ of $H_{\gamma }$ and $\tilde{\eta }\in \left(\mathrm{0,1}\right)$ such that

$\begin{array}{l}{\Vert \left(I-P\right)\left(U\left({\Gamma }_0+\tau \mathrm{,}\tau \right){\phi }^{\left(1\right)}\left(\tau \right)-U\left({\Gamma }_0+\tau \mathrm{,}\tau \right){\phi }^{\left(2\right)}\left(\tau \right)\right)\Vert }_{H_{\gamma }}\\ \le \tilde{\eta }{\Vert {\phi }^{\left(1\right)}\left(\tau \right)-{\phi }^{\left(2\right)}\left(\tau \right)\Vert }_{H_{\gamma }}\mathrm{.}\end{array}$ (3.14)

Proof. For this purpose, we choose an increasingly smooth function $\chi \left(\theta \right)\in {\mathcal{C}}^1\left(R^{+}\mathrm{,}\left[\mathrm{0,1}\right]\right)$, yielding

$\{\begin{array}{ll}\chi \left(\theta \right)=0,\hfill & 0\le \theta \le 1;\hfill \\ 0\le \chi \left(\theta \right)\le 1,\hfill & 1\le \theta \le 2;\hfill \\ \chi \left(\theta \right)=1,\hfill & \theta \ge 2;\hfill \end{array}$

and at the same time, there exists a constant ${\chi }_0$ such that $\left|{\chi }^{\prime }\left(\theta \right)\right|\le {\chi }_0$, $\forall \theta \in R^{+}$.

Let M be a fixed positive integer, set $\psi ={\left({\psi }_m\right)}_{m\in Z}={\left(\chi \left(\frac{\left|m\right|}{M}\right){\varphi }_m\right)}_{m\in Z}$, ${\varphi }_m={\left({\xi }_m\mathrm{,}{\zeta }_m\mathrm{,}{\varsigma }_m\right)}_{m\in Z}^{\text{T}}$. Taking the real part of the inner product ${\left(\cdot \mathrm{,}\cdot \right)}_{H_{\gamma }}$ in (3.5) with $\psi$, we get

$Re{\left(\stackrel{\dot{}}{\varphi }\mathrm{,}\psi \right)}_{H_{\gamma }}+Re{\left(G\varphi \mathrm{,}\psi \right)}_{H_{\gamma }}=Re{\left(F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\psi \right)}_{H_{\gamma }}\mathrm{.}$ (3.15)

Similar to Zhou [14], we have

$Re{\left(\stackrel{\dot{}}{\varphi }\mathrm{,}\psi \right)}_{H_{\gamma }}\ge \frac{1}{2}\frac{\text{d}}{\text{d}t}{\displaystyle \underset{m\in Z}{\sum }}\chi \left(\frac{\left|m\right|}{M}\right){\Vert {\varphi }_m\Vert }_{H_{\gamma }}^2+\frac{2{\chi }_0\left(\gamma +\lambda +1\right)}{M\gamma }{\Vert \varphi \Vert }_{H_{\gamma }}^2\mathrm{,}$ (3.16)

$\begin{array}{l}Re{\left(G\varphi \mathrm{,}\psi \right)}_{H_{\gamma }}\\ \ge {\displaystyle \underset{m\in Z}{\sum }}\chi \left(\frac{\left|m\right|}{M}\right)\left(\delta \left({\Vert {\xi }_m\Vert }_{\gamma }^2+{\left|{\zeta }_m\right|}^2\right)+\frac{\beta }{2}{\zeta }_m^2+\alpha {\left|{\varsigma }_m\right|}^2\right)-\frac{{\chi }_0\left(3\gamma +2\lambda +2\right)}{M\gamma }{\Vert \varphi \Vert }_{H_{\gamma }}^2\\ \ge {\displaystyle \underset{m\in Z}{\sum }}\chi \left(\frac{\left|m\right|}{M}\right)\left({\delta }_0{\Vert {\varphi }_m\Vert }_{H_{\gamma }}^2+\frac{\beta }{2}{\zeta }_m^2+\frac{3\alpha }{4}{\left|{\varsigma }_m\right|}^2\right)-\frac{{\chi }_0\left(3\gamma +2\lambda +2\right)}{M\gamma }{\Vert \varphi \Vert }_{H_{\gamma }}^2\mathrm{,}\end{array}$ (3.17)

and analogous to (3.11), we obtain

$\begin{array}{l}Re{\left(F\left({\phi }^{\left(1\right)}\mathrm{,}t\right)-F\left({\phi }^{\left(2\right)}\mathrm{,}t\right)\mathrm{,}\psi \right)}_{H_{\gamma }}\\ \le {\displaystyle \underset{m\in Z}{\sum }}\chi \left(\frac{\left|m\right|}{M}\right)\left(\frac{\beta }{2}{\zeta }_m^2+\frac{3\alpha }{4}{\left|{\varsigma }_m\right|}^2+\frac{C_0}{4}{\Vert {\varphi }_m\Vert }_{H_{\gamma }}^2{\left(\left|w_m^{\left(1\right)}\right|+\left|w_m^{\left(2\right)}\right|\right)}^2\right)\mathrm{.}\end{array}$ (3.18)

Combining (3.15)-(3.18), we get

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\displaystyle \underset{m\in Z}{\sum }}\chi \left(\frac{\left|m\right|}{M}\right){\Vert {\varphi }_m\Vert }_{H_{\gamma }}^2\\ \le {\displaystyle \underset{m\in Z}{\sum }}\chi \left(\frac{\left|m\right|}{M}\right)\left(-2{\delta }_0{\Vert {\varphi }_m\Vert }_{H_{\gamma }}^2+\frac{C_0}{2}{\Vert {\varphi }_m\Vert }_{H_{\gamma }}^2{\left(\left|w_m^{\left(1\right)}\right|+\left|w_m^{\left(2\right)}\right|\right)}^2\right)+\frac{2{\chi }_0}{M}{\Vert \varphi \Vert }_{H_{\gamma }}^2\mathrm{.}\end{array}$

From Zhao and Zhou [29], we know that for ${\eta }_0=\frac{{\delta }_0}{2C_0}>0$, there exist ${\Gamma }_0={\Gamma }_0\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)\ge T\left(\tau \mathrm{,}B\right)$ and an integer $M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)\in N$, which satisfies

$\frac{1}{2\alpha }{\displaystyle \underset{\left|m\right|\ge M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)}{\sum }}{\left|f_m\left(\tau \right)\right|}^2+\frac{1}{\beta }{\displaystyle \underset{\left|m\right|\ge M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)}{\sum }}{g}_m{\left(\tau \right)}^2+\frac{\widetilde{C_0}}{M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)}\le \frac{3{\delta }_0{\eta }_0}{10}\mathrm{,}$

where

$\widetilde{C_0}=\frac{2{\mu }^2{\chi }_0{R}_0^4}{\alpha \beta }+\frac{{\chi }_0{R}_0^2\left(5\gamma +4\lambda +4\right)}{\gamma }\mathrm{,}$

such that

${\displaystyle \underset{\left|m\right|\ge 2M}{\sum }}{\Vert {\phi }^{\left(\iota \right)}\left(t\right)\Vert }_{H_{\gamma }}^2\le {\eta }_0=\frac{{\delta }_0}{2C_0}\mathrm{,}\iota =\mathrm{1,2,}\forall M\ge M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)\mathrm{,}t-\tau \ge {\Gamma }_0\mathrm{.}$

Thus, for any $M\ge M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)$ and $t-\tau \ge {\Gamma }_0$, we have

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\left|m\right|\ge 2M}{\sum }}{\Vert {\varphi }_m\left(t\right)\Vert }_{H_{\gamma }}^2\le -{\delta }_0{\displaystyle \underset{\left|m\right|\ge 2M}{\sum }}{\Vert {\varphi }_m\left(t\right)\Vert }_{H_{\gamma }}^2+\frac{2{\chi }_0}{M}{\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2\mathrm{.}$ (3.19)

By (3.13), it can easily obtain

${\Vert \varphi \left(t\right)\Vert }_{H_{\gamma }}^2\le {\text{e}}^{2\left(C_0{R}_0^2-{\delta }_0\right)\left(t-\tau \right)}{\Vert \varphi \left(\tau \right)\Vert }_{H_{\gamma }}^2\mathrm{.}$ (3.20)

From (3.19) and (3.20), we get

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\left|m\right|\ge 2M}{\sum }}{\Vert {\varphi }_m\left(t\right)\Vert }_{H_{\gamma }}^2\le -{\delta }_0{\displaystyle \underset{\left|m\right|\ge 2M}{\sum }}{\Vert {\varphi }_m\left(t\right)\Vert }_{H_{\gamma }}^2+\frac{2{\chi }_0}{M}{\text{e}}^{2\left(C_0{R}_0^2-{\delta }_0\right)\left(t-\tau \right)}{\Vert \varphi \left(\tau \right)\Vert }_{H_{\gamma }}^2\mathrm{.}$

Furthermore, by Gronwall’s inequality, we have

${\displaystyle \underset{\left|m\right|\ge 2M}{\sum }}{\Vert {\varphi }_m\left(t\right)\Vert }_{H_{\gamma }}^2\le \left({\text{e}}^{-{\delta }_0\left(t-\tau \right)}+\frac{2{\chi }_0}{M}\cdot \frac{{\text{e}}^{\left(2C_0{R}_0^2-{\delta }_0\right)\left(t-\tau \right)}-1}{2C_0{R}_0^2-{\delta }_0}\right){\Vert \varphi \left(\tau \right)\Vert }_{H_{\gamma }}^2\mathrm{.}$

Set

$\begin{array}{l}\tilde{M}=\max \left\{M\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)\mathrm{,}\frac{2{\chi }_0\left({\text{e}}^{\left(2C_0{R}_0^2-{\delta }_0\right){\Gamma }_0}-1\right)}{\left(1-{\text{e}}^{-{\delta }_0{\Gamma }_0}\right)\left(2C_0{R}_0^2-{\delta }_0\right)}+1\right\}\mathrm{,}\\ {\Gamma }_0={\Gamma }_0\left({\eta }_0\mathrm{,}\tau \mathrm{,}B\right)>T\left(\tau \mathrm{,}B\right)\mathrm{,}\end{array}$ (3.21)

and define $H_{2\tilde{M}}=\left\{\varphi ={\left({\varphi }_m\right)}_{m\in Z}\in H_{\gamma }\mathrm{<mo>\; |\; </mo>}{\varphi }_m=\mathrm{0,}\left|m\right|>2\tilde{M}\right\}$, it is clear that $\dim H_{2\tilde{M}}={\left(4\tilde{M}+1\right)}^4<+\infty$. Let $P\mathrm{<mo>\; :\; </mo>}{H}_{\gamma }\to H_{2\tilde{M}}$ be the finite dimensional orthoprojector from $H_{\gamma }$ to $H_{2\tilde{M}}$, then for $t={\Gamma }_0+\tau >\tau$, (3.14) holds with

$0<\tilde{\eta }={\left({\text{e}}^{-{\delta }_0{\Gamma }_0}+\frac{2{\chi }_0}{\tilde{M}}\cdot \frac{{\text{e}}^{\left(2C_0{R}_0^2-{\delta }_0\right){\Gamma }_0}-1}{2C_0{R}_0^2-{\delta }_0}\right)}^{1/2}<1,$ (3.22)

where $R_0,{\delta }_0,C_0$ and $\tilde{M}\mathrm{,}{\Gamma }_0$ as in (3.3), (3.4), (3.7) and (3.21), respectively. The proof is completed.

As a straightforward consequence of Lemma 2.1, Lemma 3.1 and Lemma 3.2, we get the following Theorem 3.1.

Theorem 3.1. The compact kernel sections $\mathcal{K}\left(\tau \right)$ has a finite fractal dimension ${\dim }_f\mathcal{K}\left(\tau \right)$, which satisfies

${\dim }_f\mathcal{K}\left(\tau \right)\le {\left(4\tilde{M}+1\right)}^4\cdot \ln \left(1+\frac{8\left(1+{\text{e}}^{\left(C_0{R}_0^2-{\delta }_0\right){\Gamma }_0}\right)}{1-\tilde{\eta }}\right)\cdot {\left(\ln \frac{2}{1+\tilde{\eta }}\right)}^{-1}\mathrm{,}$ (3.23)

where $R_0\mathrm{,}{\delta }_0\mathrm{,}{C}_0,\tilde{M}\mathrm{,}{\Gamma }_0$ and $\tilde{\eta }$ as in (3.3), (3.4), (3.7), (3.21) and (3.22), respectively.

4. Conclusions

This paper studied the fractal dimension of the compact kernel sections which is generated by the process of the dissipative non-autonomous KGS lattice system described in (3.1) by applying a criterion given in Lemma 2.1 cited directly from Zhou et al. [21], and then an upper bound of the fractal dimension is obtained in (3.23) presented in Theorem 3.1.

Remark. We can use the argument in this paper to study the dissipative non-autonomous Klein-Gordon-Schrödinger lattice system defined on $Z^n$ with $n\ge 2$ , $n\in N$ . In this case, operator A possesses the following decomposition

$A=A_1+A_2+\cdot \cdot \cdot +A_n\mathrm{,}$

meanwhile

$A_j=B_j{\bar{B}}_j={\bar{B}}_j{B}_j,{\Vert B_j\Vert }_H\le K,j=1,2,\cdot \cdot \cdot ,n,$

where ${\Vert \cdot \Vert }_H$ means the norm in space H, K is a positive constant. Here, linear operator $B_j\mathrm{<mo>\; :\; </mo>}H\to H\left(H={\mathcal{l}}^2\mathrm{<mi>\; </mi>}\text{or}\mathrm{<mi>\; </mi>}{l}^2\right)$ and its adjoint operator ${\bar{B}}_j$ are defined by

${\left(B_{j}z\right)}_m={\displaystyle \underset{k=-m_0}{\overset{m_0}{\sum }}}{b}_{j\mathrm{,}k}{z}_{m_{jk}}\mathrm{,}{\left({\bar{B}}_{j}z\right)}_m={\displaystyle \underset{k=-m_0}{\overset{m_0}{\sum }}}{b}_{j\mathrm{,}-k}{z}_{m_{jk}}\mathrm{,}z={\left(z_m\right)}_{m\in Z^n}\in H\mathrm{,}$

where $j=1,2,\cdot \cdot \cdot ,n$ and $m_{jk}=\left(m_1\mathrm{,}{m}_2\mathrm{,}\cdot \cdot \cdot \mathrm{,}{m}_{j-1}\mathrm{,}{m}_j+k\mathrm{,}{m}_{j+1}\mathrm{,}\cdot \cdot \cdot \mathrm{,}{m}_n\right)\in Z^n$ .

Acknowledgements

The author would like to thank the anonymous referees for their helpful comments and thank the editors for their help.