Pullback attractors for fractional lattice systems with delays in weighted space

1 Introduction

In this article, we investigate the limiting dynamics for retarded lattice system with fractional discrete Laplacian as follows:

where u i ∈ R , i ∈ Z , ( − Δ d ) p is the fractional discrete Laplacian, p ∈ ( 0 , 1 ) , τ ∈ R , t > τ , λ is positive number, f ( t , u t ) = ( f i ( t , u i t ) ) i ∈ Z is a nonlinear function satisfying certain conditions, g ( t ) = ( g i ( t ) ) i ∈ Z is a time dependent term, and ϕ = ( ϕ i ) i ∈ Z ∈ C ( [ − h , 0 ] , ℓ 2 ) . The fractional discrete Laplacian ( − Δ d ) p simplifies to the discrete Laplacian − Δ d if p = 1 .

It is well known that lattice systems arise from spatial discretizations of partial differential equations. Lattice differential equations with standard discrete Laplacian have been extensively studied in the literature. The traveling wave solutions of such equations were investigated by [1–4]. The chaotic properties of solutions were examined by [5,6] and the references therein. For the asymptotic behavior of lattice systems, we refer the reader to [7–30]. Of those, the pullback and forward attractors of a nonautonomous lattice system with discrete Laplacian operator have been established in the weighted space in [19], and the pullback attractors of a nonautonomous retarded lattice dynamical systems have been investigated in [11,14].

The fractional discrete Laplacian, which represents the fractional powers of the discrete Laplacian, has been extensively studied in previous works [31–35]. The discrete diffusion equations with fractional discrete Laplacian were performed in [35], where the pointwise nonlocal formula and some properties regarding this operator were derived, as well as Schauder estimates in discrete Hölder spaces and the existence and uniqueness of solutions for the considered problem. By employing theories of analytic semigroups and cosine operators, the existence, and uniqueness of solutions for the heat, wave and Schrödinger equations driven by the fractional discrete Laplacian were successfully established in [36]. The relationship between the fractional powers of the discrete Laplacian and the fractional derivative, as defined by Liouville, was studied in [33]. The existence, uniqueness, and upper semi-continuity of random attractors of fractional stochastic lattice systems with linear multiplicative noise and nonlinear multiplicative noise have been recently investigated in [30,31].

Time delays are a common occurrence in various systems and can result in instability, oscillation, and other alterations in system dynamics. The increasing theoretical and practical significance of time-varying systems have prompted a growing number of scholars to engage in their study. The global attractors of discrete lattice systems with fixed delays have been obtained in recent research [7–10], while studies have also been conducted on systems with time-varying delays in [11–15]. However, as far as we know, there are few literatures about the existence of pullback attractors for fractional lattice systems with time-varying delays in weighted space.

This article has been organized as follows. In Section 2, we first establish conditions on the weights for the sequence space and state fundamental assumptions regarding the nonlinearity and forcing term of the lattice system (1.1). We also present some valuable lemmas and properties that greatly facilitate the analysis throughout this article, along with necessary preliminaries on the process formulation of lattice system (1.1). In addition, we express the lattice system (1.1) as an ordinary differential equation on ℓ 2 and ℓ η 2 , subsequently establish the existence and uniqueness of solutions to the resulting ordinary differential equation on C ( [ τ , ∞ ) , C ( [ − h , 0 ] , ℓ 2 ) ) that can be extended to a solution in C ( [ τ , ∞ ) , C ( [ − h , 0 ] , ℓ η 2 ) ) . Furthermore, we demonstrate that the solution generates a nonautonomous two-parameter semigroup or process { Ψ ( t , τ ) } t ≥ τ on C ( [ − h , 0 ] , ℓ η 2 ) . In Section 3, First, we construct a closed and bounded absorbing set for the process { Ψ ( t , τ ) } t ≥ τ . Subsequently, we establish asymptotic tail estimates for the process { Ψ ( t , τ ) } t ≥ τ , which will lead to the attainment of asymptotic compactness and the existence of pullback attractors.

2 Existence of two-parameter semigroup

In this section, we will discuss the existence of two-parameter semigroup generated by the fractional retarded lattice system (1.1) in weighted space.

Given q ∈ [ 1 , ∞ ) and a sequence of positive weights ( η i ) i ∈ Z , a weighted sequence space ℓ η q is defined by

with norm ‖ u ‖ q , η q = ∑ i ∈ Z η i ∣ u i ∣ q . Particularly, ℓ η 2 is a Hilbert space with the inner product and norm given by

For any i ∈ Z , η i = 1 , then weighted space ℓ η q transforms into normal space ℓ q .

In the rest of this article, we choose the weights that satisfy the following assumption.

( H 1 ) η i > 0 for all i ∈ Z and ∑ i ∈ Z η i ≔ η Σ < ∞ .

For 0 ≤ p ≤ 1 , define ℓ p by

Obviously, ℓ m ⊂ ℓ n ⊂ ℓ p if 1 ≤ m ≤ n ≤ ∞ and 0 ≤ p ≤ 1 .

For i ∈ Z , the discrete Laplacian − Δ d is given by

For 1 < p < 1 and u j ∈ R , the fractional discrete Laplacian ( − Δ d ) p is defined with the semigroup method [37] as follows:

where Γ is the Gamma function with Γ ( − p ) = ∫ 0 ∞ ( e − r − 1 ) d r r 1 + p < 0 and v j ( t ) = e t Δ d u j is the solution for the semidiscrete heat system

The solution of system (2.2) can be expressed using the semidiscrete Fourier transform:

where the semidiscrete heat kernel G ( i , t ) = e − 2 t I i ( 2 t ) and I m is the modified Bessel function of order m . Then, the pointwise formula for ( − Δ d ) p has been presented as follows.

By using the aforementioned notation, we can rewrite system (1.1) in ℓ 2 as follows:

where u = ( u i ) i ∈ Z , f ( t , u t ) = ( f i ( t , u i t ) ) i ∈ Z , and g ( t ) = ( g i ( t ) ) i ∈ Z .

It follows from Lemma 2.1 that the fractional discrete Laplacian ( − Δ d ) p u is a nonlocal operator on Z and ( − Δ d ) p u is well defined bounded function wherever u ∈ ℓ q ( 1 ≤ q ≤ ∞ ) . In particular, we obtain that, for 0 < p < 1 and u ∈ ℓ 2 ,

The subsequent lemma will be recurrently employed in diverse estimations of solutions to system (1.1).

Furthermore, we assume that the nonlinear term f i ( t , u i t ) in (1.1) includes delay terms as follows:

Regarding the functions f n , i ( n = 1 , 2 ) , ξ i , ρ ( t ) and g ( t ) in (2.6) and (1.1), we assume that

( H 2 ) There exist positive constants L m ( m = 1 , 2 , 3 ) , such that

for all i ∈ Z and w n ∈ R , n = 1 , 2 .

( H 3 ) There exist positive constants α m ( m = 1 , 2 , 3 ) , and nonnegative functions γ m , i ( t ) ( m = 1 , 2 , 3 ) such that

for all i ∈ Z , w ∈ R and s ∈ [ − h , 0 ] . We set

( H 4 ) ρ ∈ C 1 ( R , [ 0 , h ] ) with ∣ ρ ′ ( t ) ∣ ≤ ρ ∗ < 1 .

( H 5 ) g ( t ) ∈ C b ( R , ℓ 2 ) , γ 1 ( t ) = ( γ 1 , i ( t ) ) i ∈ Z ∈ C b ( R , ℓ 1 ) , γ 2 ( t ) = ( γ 2 , i ( t ) ) i ∈ Z ∈ C b ( R , ℓ 2 ) , and γ ˜ 3 ( t ) = ( γ ˜ 3 , i ( t ) ) i ∈ Z ∈ C b ( R , ℓ 2 ) , where C b ( R , ℓ m ) = { ζ ( t ) = ( ζ i ( t ) ) i ∈ Z : R → ℓ m is continuous and uniformly bounded with sup t ∈ R ‖ ζ ( t ) ‖ ℓ m < ∞ } , m = 1 , 2 .

( H 6 ) 2 α 1 − α 2 − 8 α 3 2 h 2 λ e λ h − α 2 1 − ρ * e λ h ≥ 0 .

Hereafter, for t ∈ R , u t is defined by

Denote by

For any i ∈ Z and η i = 1 , weighted space C ( [ − h , 0 ] , ℓ η 2 ) become normal space C ( [ − h , 0 ] , ℓ 2 ) . We define the map P : R × C ( [ − h , 0 ] , ℓ 2 ) by

Then, system (2.4) can be rewritten as the following functional equation in ℓ 2 ,

The two lemmas presented below are adequate to guarantee the existence of local solutions for system (2.4).

By Lemma 2.3, it is easy to obtain the following lemma.

By Lemmas 2.3 and 2.4, and the standard theory of the functional equations, one can deduce that for τ ∈ R and ϕ ∈ C ( [ − h , 0 ] , ℓ 2 ) , there exists T > 0 such that system (2.4) has a unique solution u t ( ⋅ , τ , ϕ ) ∈ C ( [ τ , τ + T ] , C ( [ − h , 0 ] , ℓ 2 ) ) . As demonstrated below, this local solution is defined for all t ≥ τ .