Pullback attractors for fractional lattice systems with delays in weighted space

: This article deals with the asymptotic behavior of fractional lattice systems with time-varying delays in weighted space. First, we establish some su ﬃ cient conditions for the existence and uniqueness of solutions. Subsequently, we demonstrate the existence of pullback attractors for the considered fractional lattice systems.


Introduction
In this article, we investigate the limiting dynamics for retarded lattice system with fractional discrete Laplacian as follows: is a time dependent term, and 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][2][3][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 .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][32][33][34][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][8][9][10], while studies have also been conducted on systems with time-varying delays in [11][12][13][14][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 ,0 , 2 that can be extended to a solution in ([ ,0 , η 2 .Furthermore, we demonstrate that the solution generates a nonautonomous two-parameter semigroup or process { ( ) In Section 3, First, we construct a closed and bounded absorbing set for the process { ( )} ≥ t τ Ψ , t τ .Subsequently, we establish asymptotic tail estimates for the process { ( )} ≥ t τ Ψ , t τ , which will lead to the attainment of asymptotic compactness and the existence of pullback attractors.

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 1, and a sequence of positive weights ( ) ∈ η i i , a weighted sequence space ℓ η q is defined by Particularly, ℓ η 2 is a Hilbert space with the inner product and norm given by For any ∈ i , = η 1 i , 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.
For < < p 1 1and ∈ u j , the fractional discrete Laplacian ( ) −Δ d p is defined with the semigroup method [37] as follows: where Γ is the Gamma function with ( ) ( ) is the solution for the semidiscrete heat system , in 0, , 0 , on . 2) The solution of system (2.2) can be expressed using the semidiscrete Fourier transform: where the semidiscrete heat kernel ( ) and I m is the modified Bessel function of order m.Then, the pointwise formula for ( ) −Δ d p has been presented as follows.
where the discrete kernel K ˜p is given by In addition, there exist positive constants ≤ c c ˇp p such that for any By using the aforementioned notation, we can rewrite system (1.1) in ℓ 2 as follows: where In particular, we obtain that, for The subsequent lemma will be recurrently employed in diverse estimations of solutions to system (1.1).
Then for every ( ) , 2 , and ( ) m is continuous and uniformly bounded with .
Hereafter, for ∈ t , u t is defined by Denote 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) 2 , there exists a positive constant L such that Proof.By (H 2 )-(H 5 ), we find that is well defined.Given ( ) By the boundedness of the sequence (2.9) which along with (2.5), (2.8), and ( ) H 5 implies the continuity of .The Lipschitz condition can be proven in a similar manner.□ 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 and As demonstrated below, this local solution is defined for all ≥ t τ.
Lemma 2.5.Suppose ( ) , and ≥ t τ, the solution u of system (2.4) satisfies Proof.Take the inner product of system (2.4) with u in ℓ 2 , we obtain (2.10) By Lemma 2.2, we obtain By ( ) H 5 and Young's inequality, we have (2.12) For the delay term of (2.10), we obtain Pullback attractors for fractional lattice systems  5 By ( ) H 3 , ( ) H 5 , and Young's inequality, we obtain and Integrating the aforementioned inequality over ( ) τ t , , we obtain (2.17) First, we estimate the time-varying delay term of (2.17).By ( ) H 4 , we have Next, we proceed to analyze the double integral term in (2.17), which along with ( ) (2.20)This completes the proof.□ The solutions to system (2.4) in ℓ 2 are continuous in their initial data, and using the uniqueness of solution, they fulfill the two-parameter semigroup property The subsequent lemma establishes the Lipschitz continuity of these solutions in Lemma 2.6.Suppose ( ) 2 .Let u 1 and u 2 be the solutions of system (2.4) with g replaced by g 1 and g 2 , respectively.Then for every ∈ τ and > T 0, there exists a positive constant , , .
Pullback attractors for fractional lattice systems  7 Proof.Let ( ) . By system (2.4), we obtain Taking the inner with u ¯in ℓ η 2 , we obtain By ( ) H 2 , we obtain ) ¯2 ¯¯d ¯.
, integrating the aforementioned inequality over ( ) τ t , , we obtain As to the time-varying delay term in (2.25), by ( ) H 4 , we have  which along with (2.25) and (2.26) implies that (2.27) By (2.27) and Gronwall's inequality, we have (2.28) where (2.29) On the other hand, if which along with (2.29) implies the desired result.This completes the proof.□ Now, the main theorem of this section is hereby presented.
Theorem 2.1.Suppose ( ) H 1 -( ) H 6 hold.For every ∈ τ and ≥ t τ, the solution u of system (2.4) generates a two- Proof.Given ∈ τ and > T 0, by Lemma 2.6, there exists a continuous mapping such that where ( ) ⋅ u τ ϕ g , , , t is the unique solution to system (2.4) Since 2 , the mapping can be uniquely extended to a mapping where ( ) For every ∈ τ and ≥ t τ, define mapping ( )( ) (2.30) The continuity of the solution mapping u in its initial data, as proven by Lemma 2.6, implies that the mapping Ψ is also continuous in both τ and ϕ.Therefore, Ψ can be regarded as a nonautonomous two- parameter semigroup on ([ ] ℓ ) − C h, 0 , η 2 .This completes the proof.□

Existence of pullback attractors
In this section, we investigate the pullback dynamics of the process { ( )} ≥ t τ Ψ , t τ defined by equation (2.30) in Theorem 2.1.First, we will construct a pullback absorbing set for the process { ( )} ≥ t τ Ψ , t τ .Subsequently, we will Pullback attractors for fractional lattice systems  9 establish asymptotic tail estimates for the process { ( )} ≥ t τ Ψ , t τ , which will lead to the attainment of asymptotic compactness and the existence of pullback attractors.Lemma 3.1.Suppose ( ) H 1 -( ) H 6 hold.Then the process { ( )} ≥ t τ Ψ , t τ associated with system (2.4) has a closed and bounded pullback absorbing set 2 , which is positively invariant.
Proof.Take the inner product of system (2.4) with u in ℓ η 2 , we obtain .
For the delay term of (3.1), by the similar argument as in (2.14)-(2.16),we have Integrating the aforementioned inequality over ( ) τ t , , we obtain   (3.4) For the time-varying delay term in (3.4), by ( ) H 4 , we have Next, we analyze the last term in (3.4),   (3.7) (3.8) For every > ε 0, there exists ( ) > T ε 0 Set = ε 1, we define the closed and bounded set of . By (3.9), the set Q ρ η , is an absorbing set for nonautonomous two-parameter semigroup { ( ) This completes the proof.□ The next step involves obtaining a uniform estimate on the tails of solutions.To achieve this, we select a smooth function ( ) ϑ s that satisfies Pullback attractors for fractional lattice systems  11 Moreover, given ( ) ∈ p 0, 1 , by Lemma 3.3 of [16], we obtain that for all ∈ i and ∈ k , Lemma 3.2.Suppose ( ) H 1 -( ) H 6 hold.Then, for any > ε 0, there exist ( ) > T ε 0 and Proof.Taking the inner product of system (2.4) with ( ( ) ) .
For the last term of (3.11), we obtain 4 .
(3.  which implies that     , such that for all − ≥ t τ T 2 , which along with (3.7) implies that By ( ) Author contributions: All the authors contributed equally to this work.All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
[19]a 3.3.Suppose ( ) H 1 -( ) H 6 hold.Then the process { ( )} ≥ t τ Ψ , t τ defined by (2.30) associated with system (2.4) is asymptotically compact..By using (3.26) and Lemma 2.4, we obtain that there exists a positive constant c 3 , which implies the property of equicontinuity.By the Ascoli-Arzelà theorem, we can deduce that As an immediate consequence of Lemmaa 3.1, 3.3, and Theorem 2.7 in[19], we obtain the main result of this section as follows: This work was supported by the Start-up Research Fund for High-Level Talents of Liupanshui Normal University (LPSSYKYJJ202311) and the Scientific Research and Cultivation Project of Liupanshui Normal University (LPSSY2023KJYBPY14).
Next, we will show that the convergence in (3.22) is strong.Given > ε 0, by Lemma 3.2, there exist ( ) > n(3.27)This completes the proof.□