In recent years, great progress had been made in the study of non-autonomous infinite dimensional dynamical systems. See, e.g.,      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.,  -  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.,  -  and etc.
As to the dissipative autonomous Klein-Gordon-Schrödinger (Hereafter KGS) lattice systems, many authors have studied them. For example, Abdallah in , Abounouh, Goubet and Hakim in , Yin and Zhou et al. in  investigated the existence, regularity, upper semicontinuity, Kolmogorov entropy of global attractor and so forth. Meanwhile, the following dissipative non-autonomous KGS lattice system
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 , 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  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  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 , 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  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 is a compact set in a metric space such that the fractal dimension of is less or equal to for some , then there exists an injective Lipschitz mapping 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.
To begin, we introduce
where , and denote the integral, real and complex numbers, respectively.
Write or , and endow H with the inner product and norm as below
where is the conjugate of . Clearly, H is a Hilbert space.
Define linear operators A and B as follows
For any , define a bilinear form by means of
where 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 , and as
and equip it with the following norm and inner product
and denote by and respectively the set of continuous and bounded functions from into and .
Definition 2.1. A two-parameter family of mappings is called to be a process in a Hilbert space , if
3) (identity operator of ), .
Definition 2.2. A function , is said to be a complete trajectory of the process , if . The kernel of the process consists of all bounded complete trajectories of , i.e.,
and the kernel sections of the kernel at time is
Definition 2.3. The fractal dimension of a compact set in a metric space is defined as follows, namely
where is the minimal number of closed sets of radius which cover the set .
The criterion below is directly cited from Zhou et al. .
Lemma 2.1. Let be a continuous process on a Hilbert space and be a family of compact, negatively invariant (i.e., for all ) subsets of . Assume that
1) there exists a uniform finite covering of closed subsets with diameter 2 of for all , that is, there exists closed balls of with diameter 2 covering for all , where is independent of t;
2) for any , there exists and , which are all independent of such that for ,
a) there exists yields
i.e., is Lipschitz on ;
b) there exists finite-dimensional orthoprojector P of satisfies
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
where , ; , ; , ; , ; is the imaginary numbers’ unit; , , and are positive constants; , , denotes or .
Thus, (3.1) can be written as below
where , , , and
From Zhao and Zhou , we can see, for given with , with , the solution mappings of (3.2), that is, ,
, , generate a family of continuous processes in . Moreover, the family of processes , possess a family of compact kernel sections , where is included in a uniformly bounded set and satisfies , here
In the sequel, we get an upper bound of the fractal dimension of the compact kernel sections , which is generated by the process of the dissipative non-autonomous KGS lattice system (3.1).
Suppose , , then for , . Set , then by (3.2), we have
where , and , , .
Lemma 3.1. For any , is Lipschitz on , i.e.,
and as in (3.3) and (3.4), respectively.
For brief, we denote by and respectively the real part and imaginary part of inner product .
Proof. Taking the real part of the inner product of (3.5) with , we have
By simple computation, we get
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
Set , , 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 of and such that
Proof. For this purpose, we choose an increasingly smooth function , yielding
and at the same time, there exists a constant such that , .
Let M be a fixed positive integer, set , . Taking the real part of the inner product in (3.5) with , we get
Similar to Zhou , we have
and analogous to (3.11), we obtain
Combining (3.15)-(3.18), we get
From Zhao and Zhou , we know that for , there exist and an integer , which satisfies
Thus, for any and , we have
By (3.13), it can easily obtain
From (3.19) and (3.20), we get
Furthermore, by Gronwall’s inequality, we have
and define , it is clear that . Let be the finite dimensional orthoprojector from to , then for , (3.14) holds with
where and 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 has a finite fractal dimension , which satisfies
where and as in (3.3), (3.4), (3.7), (3.21) and (3.22), respectively.
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. , 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 with , . In this case, operator A possesses the following decomposition
where means the norm in space H, K is a positive constant. Here, linear operator and its adjoint operator are defined by
where and .
The author would like to thank the anonymous referees for their helpful comments and thank the editors for their help.
 Cheban, D.N., Kloeden, P.E. and Schmalfuß, B. (2002) The Relationship between Pullback, Forward and Global Attractors of Non-Autonomous Dynamical Systems. Nonlinear Dynamical Systems Theory, 2, 9-28.
 Li, Y., Zha, T., Wu, H. and Wei, J. (2016) Pullback Exponential Attractors for Non-Autonomous Dynamical System in Space of Higher Regularity. Journal of Applied Analysis and Computation, 6, 242-253.
 Zhou, S. and Wang, L.S. (2003) Kernel Sections for Non-Autonomous Damped Wave Equations with Critical Exponent. Discrete and Continous Dynamical Systems, 9, 399-412.
 Gu, A., Zhou, S. and Jin, Q. (2015) Random Attractors for Partly Dissipative Stochastic Lattice Dynamical Systems with Multiplicaive White Noises. Acta Mathematicae Applicatae Sinica (English Series), 31, 567-576.
 Jia, Q., Zhou, S. and Yin, F. (2003) Kolmogorov Entropy of Global Attrator for Dissipative Lattice Dynamical Systems. Journal of Mathematical Physics, 44, 5804-5810.
 Li, X.J. and Zhong, C.K. (2005) Attractors for Partly Diassipative Lattice Dynamical Systems in . Journal of Computational and Applied Mathematics, 177, 159-174.
 Li, X.J. and Wang, D.B. (2007) Attractors for Partly Diassipative Lattice Dynamical Systems in Weighted Spaces. Journal of Mathematical Analysis and Application, 325, 141-156.
 Zhou, S. (2008) Fractal Dimension of Global Attractors for Some Dissipative Lattice Systems. International Journal of Bifurcation and Chaos, 18, 3447-3454.
 Zhao, X. and Zhou, S. (2008) Kernel Sections for Process and Non-Autonomous Lattice Systems. Discrete and Continous Dynamical Systems (Series B), 9, 763-785.
 Zhou, S., Zhao, C. and Liao, X. (2007) Compact Uniform Attractors for Dissipative Non-Autonomous Lattice Dynamical Systems. Communications on Pure and Applied Analysis, 6, 1087-1111.
 Zhou, S., Zhou, C. and Wang, Y. (2008) Finite Dimensionality and Upper Semocontinuity of Compact Kernel Section of Non-Autonomous Lattice Systems. Discrete and Continous Dynamical Systems, 21, 1259-1277.
 Zhou, S., Huang, J. and Han, X. (2010) Compact Kernel Sections for Dissipative Non-Autonomous Zakharov Equation on Infinite Lattices. Communications on Pure and Applied Analysis, 9, 193-210.
 Zhou, S. and Han, X. (2012) Pullback Exponential Attractors for Non-Autonomous Lattice Systems. Journal of Dynamics and Differential Equations, 24, 601-631.
 Abdallah, A.Y. (2006) Asymptotic Bahaviour of the Klein-Gordon-Schrödinger Lattice Dynamical Systems. Communications on Pure and Applied Analysis, 5, 55-69.
 Yin, F., Zhou, S., Yin, C. and Xiao, C. (2007) Global Attractor for Klein-Gordon-Schrödinger Lattice Systems. Applied Mathematics and Mechanics (English Edition), 28, 695-706.
 Abdallah, A.Y. (2015) Uniform Exponential Attractors for Non-Autonomous Klein-Gordon-Schrödinger Lattice Systems in Weighted Spaces. Nonlinear Analysis: Theory, Methods & Applications, 127, 279-297.
 Huang, J., Han, X. and Zhou, S. (2009) Uniform Attractors for Non-Autonomous Klein-Gordon-Schrödinger Lattice Systems. Applied Mathematics and Mechanics (English Edition), 30, 1597-1607.
 Zhao, C. and Zhou, S. (2007) Compact Kernel Sections for Non-Autonomous Klein-Gordon-Schrödinger Equation on Infinite Lattices. Journal of Mathematical Analysis and Application, 332, 32-56.
 Zhou, S. and Han, X. (2013) Uniform Exponential Attractors for Non-Autonomous KGS and Zakharov Lattice Systems with Quasiperiodic External Forces. Nonlinear Analysis: Theory, Methods & Applications, 78, 141-155.