Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

and Applied Analysis 3 For convenience, we will express (1)–(3) as an abstract Cauchy problem of first-order ODE with respect to time t in E μ . To this end, we put u = (u m ) m∈Z, z = (zm)m∈Z, zu = (z m u m ) m∈Z, |z| 2 = (|z m | 2 ) m∈Z, f(t) = (fm(t))m∈Z, g(t) = (g m (t)) m∈Z, zτ = (zm,τ)m∈Z, and uτ = (um,τ)m∈Z, u 1τ = (u 1m,τ ) m∈Z. Then, we rewrite (1)–(3) in a vector form as i?̇? − Az + iαz + zu = f (t) , t > τ, (16) ?̈? + ]?̇? + Au + μu − β|z| 2 = g (t) , t > τ, (17) z (τ) = z τ , u (τ) = u τ , ?̇? (τ) = u 1τ , τ ∈ R. (18) Set V = ?̇? + δu, where δ = μ] 2 (μ + ]) > 0, (19) ψ = (u, V, z)T, F(ψ, t) = (0, β|z|2 + g(t), izu − if(t))T,


Introduction
Lattice dynamical systems (LDSs for short), including coupled ordinary differential equations (ODEs), coupled map lattices, and cellular automata [1], are spatiotemporal systems with discretization in some variables.In some cases, LDSs arise as the spatial discretization of partial differential equations (PDEs) on unbounded or bounded domains.LDSs occur in a wide variety of applications, ranging from image processing and pattern recognition [2][3][4] to electrical engineering [5], chemical reaction theory [6,7], laser systems [8], material science [9], biology [10], and so forth.
Very recently, Zhou and Han [30] presented some sufficient conditions for the existence of the pullback exponential attractor for the continuous process on Banach spaces and weighted spaces of infinite sequences.Also, they applied their results to study the existence of pullback exponential attractors for first-order nonautonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and timedependent external terms in weighted spaces.
In this paper, we will use the abstract theory of [30] to study the pullback exponential behavior of solutions for the following nonautonomous lattice systems:  ż  − ()  +   +     =   () ,  ∈ Z,  > , (1) with initial conditions: where  is a linear operator defined as () ∈ C,   () ∈ R, C and R are the sets of complex and real numbers, respectively, Z is the set of integer numbers,  = √ −1 is the unit of the imaginary numbers, and , , ], and  are positive constants. (5) Equations ( 5) describe the interaction of a scalar nucleon interacting with a neutral scalar meson through Yukawa coupling [31], where  and  represent the complex scalar nucleon field and the real meson field, respectively, and the complex-valued function (, ) and the real-valued function (, ) are the time-dependent external sources.There are many works concerning the Cauchy problem and the initial boundary value problem of the continuous model of KGS equations or its related version, see [32][33][34][35][36] and references therein.
We want to mention that the lattice KGS equations have been studied by [28,37].In [37], the authors first presented some sufficient conditions for the existence of the uniform exponential attractor for a family of continuous processes on separable Hilbert spaces and the space of infinite sequences.Then, they studied the existence of uniform exponential attractors for the dissipative nonautonomous KGS lattice system and the Zakharov lattice system driven by quasiperiodic external forces.In [28], the authors first proved the existence of compact kernel sections and gave an upper bound of the Kolmogorov -entropy for these kernel sections.Also they verified the upper semicontinuity of the kernel sections.Articles in [28,37] use the same transformation of the variable  = (  ) ∈Z .
The aim of the present paper is to use the abstract result of [30] to prove the existence of pullback exponential attractors for the LDSs (1)- (3).When verifying the discrete squeezing property (see Lemma 5(II)) of the generated process, we encounter the difficulty coming from the nonlinear terms     and |  | 2 in the coupled lattice equations.To overcome this difficulty, we make a proper transformation of the variable  = (  ) ∈Z and use the technique of cutoff functions.We want to remark that the idea concerning the transformation of the variable  originates from articles in [28,37], but our transformation is other than that of [28,37].
The rest of the paper is organized as follows.In Section 2, we first introduce some spaces and operators.Then, we recall some results on the existence, uniqueness, and some estimates of solutions.Section 3 is devoted to proving the existence of the pullback exponential attractor for the process associated to the lattice KGS equations.
For convenience, we will express ( 1)-( 3) as an abstract Cauchy problem of first-order ODE with respect to time  in   .To this end, we put ) ∈Z , and   = ( , ) ∈Z ,  1 = ( 1, ) ∈Z .Then, we rewrite (1)-(3) in a vector form as Set Then, ( 16)-( 18) can be written as Let C  (R, ) be the set of continuous and bounded functions from R into , then for each () ∈ C  (R, ), we have sup We next recall some results of solutions to ( 21)- (22).
where  0 ,  0 , and  0 are positive constants independent of  and .
Proof.Using Lemmas 2.3 and 3.1 and Theorem 2 of [30], we obtain the result.
Remark 7. The spectrum of Lyapunov exponents is the most precise tool for identification of the character of motion of a dynamical system [38].There are some works on the estimation of the dominant Lyapunov exponent of nonsmooth systems by means of synchronization method, one can refer to the articles of Stefański et al. [38][39][40].In [38], Stefański and Kapitaniak presented a method to estimate the value of largest Lyapunov exponent both for discrete dynamical systems of known difference equations and also for discrete maps reconstructed from the time evolution of the given system.Following this clue, we can ask naturally the problem that whether the method presented in [38] could be applied to estimate Lyapunov exponents for the trajectories on the pullback attractor {A()}.It is an interesting and challenging issue for us to investigate.