Periodic Solutions to Matrix Delay Difference Systems Under Exponential Dichotomy Conditions

Abstract

This paper studies the existence and uniqueness of periodic solutions to a class of nonlinear neutral matrix difference systems with multiple delays. The analysis is based on the construction of a suitable Green operator combined with fixed-point methods under exponential dichotomy assumptions. The existence of periodic solutions is established using Krasnoselskii’s fixed-point theorem, while uniqueness is demonstrated under a natural contraction condition via Banach’s principle. The results extend previous contributions on neutral difference systems and provide discrete analogues of related differential models. Examples are included to illustrate the applicability of the theory.

1. Introduction

Neutral functional difference equations constitute an important class of discrete dynamical systems in which the forward increment in the state depends not only on delayed values of the solution but also on the increment in a delayed nonlinear function [1,2,3]. Such systems arise naturally in numerical approximations of neutral delay differential equations [4], in population dynamics and mathematical biology [5], in engineering models with memory effects [2], in neural network theory [6], and in nonlinear economic structures with after-effect phenomena [7]. Their analysis is considerably more delicate than that of retarded systems, since the neutral term may affect the dominant part of the dynamics and is capable of altering stability and oscillatory behavior [1,2]. Recent advances in delay neural networks have further highlighted the importance of delayed and neutral-type dynamics in modeling complex systems with memory effects; see, for instance, references [8,9,10] and the references therein.

Over the past few decades, extensive work has been devoted to the study of the qualitative properties of nonlinear difference systems, including stability, asymptotic behavior, boundedness, periodicity, and almost periodicity; see, for instance, Coppel [11], Elaydi and Janglajew [12], Aulbach and Minh [13], Megan and Sasu [14], Latushkin et al. [15], and Dragičević [16]. A fundamental tool in these investigations is the notion of exponential dichotomy, which provides a robust framework for analyzing non-autonomous linear systems and constructing Green operators that allow nonlinear systems to be reformulated as fixed-point problems. Recent developments in the theory of delay and neutral-type systems can be found in [17,18,19,20], where various aspects, such as stability, exponential dichotomy, and periodic solutions, are investigated.

In the neutral setting, periodic solutions play a central role in describing recurrent and oscillatory dynamics. Various approaches have been employed to establish their existence, including Banach’s contraction principle, Krasnoselskii’s fixed-point theorem, and Schauder’s theorem. A discrete framework for neutral equations with functional delays was developed by Mesmouli, Ardjouni, and Djoudi [21], who studied a system of nonlinear neutral functional difference equations with two delayed arguments and discovered periodic solutions using Krasnoselskii’s theorem. More recently, the authors of [22] investigated a pair of nonlinear neutral differential systems under dichotomy assumptions and established the existence of periodic solutions in the continuous setting

$${\left(y\left(\zeta \right)-q\left(\zeta ,y\left(\zeta -\tau \left(\zeta \right)\right)\right)\right)}^{\prime }=A\left(\zeta \right)y\left(\zeta \right)+f\left(\zeta ,y\left(\zeta -{\sigma }_1\left(\zeta \right)\right),\dots ,y\left(\zeta -{\sigma }_m\left(\zeta \right)\right)\right),$$

(1)

in which $y:\mathbb{R}\to {\mathbb{R}}^n$, $\tau \left(\zeta \right),{\sigma }_i\left(\zeta \right),i=1,\dots ,m$, are real continuous T-periodic functions on $\mathbb{R}$, $T>0$. $A\left(\zeta \right)$ is a $n\times n$ real continuous matrix T-periodic function defined on $\mathbb{R}$. The functions $q\left(\zeta ,u\right)$ and $f\left(\zeta ,u_1,\dots ,u_m\right)$ are real continuous vector functions defined on $\mathbb{R}\times \mathbb{R}$ and $\mathbb{R}\times {\mathbb{R}}^m$, respectively, and periodic with respect to $\zeta$.

Motivated by these developments and by the need for a discrete analogue of the results in [22], the present work investigates the existence and uniqueness of periodic solutions for a general nonlinear neutral difference system with multiple delays of the form

$$\Delta X\left(n\right)=A\left(n\right)X\left(n\right)+\Delta Q\left(n,X(n-g(n\left)\right)\right)+G\left(n,X(n-{\sigma }_1\left(n\right)),\dots ,X(n-{\sigma }_m\left(n\right))\right).$$

(2)

Here, $X\left(n\right)\in {\mathbb{R}}^d$ for all $n\in \mathbb{Z}$, $A\left(n\right)\in {\mathbb{R}}^{d\times d}$ is a T-periodic matrix-valued function, while $Q:\mathbb{Z}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ and $G:\mathbb{Z}\times {\left({\mathbb{R}}^d\right)}^m\to {\mathbb{R}}^d$ are nonlinear T-periodic functions. The delay functions $g\left(n\right)$ and ${\sigma }_j\left(n\right)$, $j=1,\dots ,m$, are integer-valued and T-periodic.

The main contributions of this paper can be summarized as follows. We introduce a general class of multi-delay nonlinear neutral difference systems in matrix form and construct the associated Green operator under exponential dichotomy assumptions for the corresponding linear system. Within an appropriate periodic sequence space, we establish the existence of T-periodic solutions by applying Krasnoselskii’s fixed-point theorem to a closed, convex, and bounded invariant subset. Furthermore, by imposing a natural contraction condition that links the Lipschitz constants of the nonlinear terms with the dichotomy bounds, we derive a uniqueness criterion via Banach’s fixed-point theorem. Our results therefore provide a discrete analogue of the continuous-time neutral systems studied in [22] and significantly extend the discrete neutral framework developed in [21] to the setting of multiple delays and matrix-valued systems.

This paper is organized as follows. Section 2 contains preliminary material on exponential dichotomy, the Green matrix, and the fixed-point representation of the system. Section 3 presents the main existence and uniqueness results. Section 4 provides illustrative examples. Section 5 contains concluding remarks and directions for future research.

2. Preliminaries

We work in the Euclidean space ${\mathbb{R}}^d$ with its usual norm. For any sequence $X:\mathbb{Z}\to {\mathbb{R}}^d$, we use the forward difference operator:

$$\Delta X\left(n\right)=X(n+1)-X\left(n\right).$$

All coefficients and delays are T-periodic for some fixed integer $T>0$. A T-periodic solution is sought in the Banach space

$$\mathsf{\Omega }=\{X:\mathbb{Z}\to {\mathbb{R}}^d:X(n+T)=X\left(n\right)\},{\parallel X\parallel }_{\infty }=\underset{n\in \mathbb{Z}}{sup}\parallel X\left(n\right)\parallel ,$$

where $\parallel X\left(n\right)\parallel ={\sum }_{i=1}^d{x}_i\left(n\right)$.

• Exponential dichotomy: Throughout this work, we consider the linear system

$$Z_{n+1}=B\left(n\right)Z_n,$$

(3)

where

$$B\left(n\right)=A\left(n\right)+I_d.$$

Let $\mathsf{\Phi }\left(n\right)$ be the fundamental matrix satisfying $\mathsf{\Phi }\left(0\right)=I_d$. We say that (3) admits an exponential dichotomy with respect to $\mathbb{Z}$ if the constants $K>0$ and $\alpha >0$ exist along with a projection P such that

$$\begin{array}{c}\parallel \mathsf{\Phi }\left(n\right)P{\mathsf{\Phi }}^{-1}\left(k\right)\parallel \le K{e}^{-\alpha (n-k)},n\ge k,\end{array}$$

(4)

$$\begin{array}{c}\parallel \mathsf{\Phi }\left(n\right)(I-P){\mathsf{\Phi }}^{-1}\left(k\right)\parallel \le K{e}^{-\alpha (k-n)},n\le k.\end{array}$$

(5)

The associated Green matrix is

$$G(n,k)=\left\{\begin{array}{cc}\mathsf{\Phi }\left(n\right)P{\mathsf{\Phi }}^{-1}\left(k\right),\hfill & n\ge k,\hfill \\ -\mathsf{\Phi }\left(n\right)(I-P){\mathsf{\Phi }}^{-1}\left(k\right),\hfill & n<k,\hfill \end{array}\right.$$

satisfying

$$\underset{n\in \mathbb{Z}}{sup}\sum _{k=-\infty }^{\infty }\parallel G(n,k)\parallel <\infty .$$

(6)

For the non-homogeneous linear equation

$$Z_{n+1}=B\left(n\right)Z_n+H\left(n\right),$$

variation of the constant formula gives

$$Z\left(n\right)=\sum _{k=-\infty }^{\infty }G(n,k)H\left(k\right).$$

(7)

To facilitate the analysis, we introduce an auxiliary sequence that transforms the neutral system into a nonhomogeneous linear form, allowing the application of exponential dichotomy and the construction of the associated Green operator.

$$W\left(n\right)=X\left(n\right)-Q(n,X(n-g\left(n\right)\left)\right).$$

Substitution into (2) yields

$$W(n+1)-W\left(n\right)=A\left(n\right)W\left(n\right)+H\left(n\right),$$

or, equivalently,

$$W(n+1)=B\left(n\right)W\left(n\right)+H\left(n\right),$$

(8)

since $B\left(n\right)=A\left(n\right)+I_d$, where

$$H\left(n\right)=A\left(n\right)Q(n,X(n-g\left(n\right)))+G\left(n,X(n-{\sigma }_1\left(n\right)),\dots ,X(n-{\sigma }_m\left(n\right))\right).$$

Applying (7) yields

$$W\left(n\right)=\sum _{k=-\infty }^{\infty }G(n,k)H\left(k\right),$$

and therefore

$$X\left(n\right)=Q(n,X(n-g\left(n\right)))+\sum _{k=-\infty }^{\infty }G(n,k)H\left(k\right).$$

(9)

Define the operators

$$\left({\mathsf{\Gamma }}_{1}X\right)\left(n\right)=Q(n,X(n-g\left(n\right))),$$

$$\left({\mathsf{\Gamma }}_{2}X\right)\left(n\right)=\sum _{k=-\infty }^{\infty }G(n,k)\left[A\left(k\right)Q(k,X(k-g\left(k\right)))+G\left(k,X(k-{\sigma }_1\left(k\right)),\dots ,X(k-{\sigma }_m\left(k\right))\right)\right].$$

Thus, the neutral system reduces to

$$X={\mathsf{\Gamma }}_{1}X+{\mathsf{\Gamma }}_{2}X.$$

(10)

• Lipschitz conditions: We assume that the nonlinearities satisfy

$${\parallel Q(n,U)-Q(n,V)\parallel }_{\infty }\le L_1{\parallel U-V\parallel }_{\infty },$$

$$\parallel G(n,U_1,\dots ,U_m)-G(n,V_1,\dots ,V_m){\parallel }_{\infty }\le L_2\sum _{j=1}^m{\parallel U_j-V_j\parallel }_{\infty },$$

where $U,V,U_j,V_j\in {\mathbb{R}}^d$ and that the linear system (2) admits an exponential dichotomy.

We finally work in the closed, convex, bounded set

$$K=\{X\in \mathsf{\Omega }:\parallel X{\parallel }_{\infty }\le M\},$$

which will later be shown to satisfy

$${\mathsf{\Gamma }}_{1}K+{\mathsf{\Gamma }}_{2}K\subseteq K.$$

Fixed-point tools: To ensure completeness, we present the two fixed-point theorems that will be used in the main results.

Theorem 1

(Banach Contraction Principle [23]). Let $(E, \parallel ·\parallel )$ be a Banach space and let $T:E\to E$ be a contraction; i.e., there exists $0<\kappa <1$ such that

$$\parallel T\left(x\right)-T\left(y\right)\parallel \le \kappa \parallel x-y\parallel \mathrm{for}\mathrm{all}x,y\in E.$$

Then, T has a unique fixed point in E.

Theorem 2

(Krasnoselskii’s Fixed Point Theorem [23]). Let E be a Banach space; let $K\subset E$ be a closed, convex, nonempty set; and let $T_1,T_2:K\to K$ be two operators satisfying the following conditions:

• $T_1$ is a contraction on K;
• $T_2$ is completely continuous;
• $T_{1}x+T_{2}y\in K$ for all $x,y\in K$.

Then, the equation

$$x=T_{1}x+T_{2}x$$

has at least one fixed point in K.

3. Main Results

In this section, we establish the existence and uniqueness of T-periodic solutions of the neutral system (2). We use the fixed-point formulation

$$X={\mathsf{\Gamma }}_{1}X+{\mathsf{\Gamma }}_{2}X$$

${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ were introduced in Section 2.

Let

$$K=\{X\in \mathsf{\Omega }:\parallel X{\parallel }_{\infty }\le M\},$$

with $M>0$ determined later to guarantee invariance.

We verify that the hypotheses of Krasnoselskii’s fixed-point theorem are satisfied.

Theorem 3.

Assume that the nonlinearities Q and G satisfy the Lipschitz conditions stated in Section 2, the linear system (3) admits an exponential dichotomy, and that $L_1<1$. Then, the neutral system (2) admits at least one T-periodic solution.

Proof. 

The proof is divided into several steps.

• Step 1: ${\mathsf{\Gamma }}_1$ is a contraction.

For $X,Y\in \mathsf{\Omega }$,

$${\parallel {\mathsf{\Gamma }}_{1}X-{\mathsf{\Gamma }}_{1}Y\parallel }_{\infty }=\underset{n\in \mathbb{Z}}{sup}\parallel Q(n,X(n-g\left(n\right)))-Q(n,Y(n-g\left(n\right)))\parallel .$$

Using the Lipschitz property of Q,

$$\parallel {\mathsf{\Gamma }}_{1}X-{\mathsf{\Gamma }}_1{Y\parallel }_{\infty }\le L_1{\parallel X-Y\parallel }_{\infty }.$$

Thus, ${\mathsf{\Gamma }}_1$ is a contraction on $\mathsf{\Omega }$ and hence on K.

• Step 2: Boundedness of ${\mathsf{\Gamma }}_2$ on K.

Let $X\in K$. Since ${\parallel X\parallel }_{\infty }\le M$, the terms

$$A\left(k\right)Q(k,X(k-g\left(k\right)))\mathrm{and}G(k,X(k-{\sigma }_1\left(k\right)),\dots ,X(k-{\sigma }_m\left(k\right)))$$

are uniformly bounded. Denote this boundness using $C_0>0$.

Then,

$$\parallel \left({\mathsf{\Gamma }}_{2}X\right)\left(n\right)\parallel \le \sum _{k=-\infty }^{\infty }\parallel G(n,k)\parallel C_0.$$

Using the Green matrix estimate (6),

$$\parallel \left({\mathsf{\Gamma }}_{2}X\right)\left(n\right){\parallel }_{\infty }\le C_0{M}_G.$$

Thus, ${\mathsf{\Gamma }}_2$ maps bounded sets into bounded sets.

• Step 3: Complete continuity of ${\mathsf{\Gamma }}_2$.

The series defining ${\mathsf{\Gamma }}_{2}X$ converges uniformly in n due to the summability of $G(n,k)$. Uniform convergence, combined with the boundedness and continuity of the nonlinear terms, implies that ${\mathsf{\Gamma }}_2$ maps bounded sets into equicontinuous sets in $\mathsf{\Omega }$.

Therefore, as per the Arzel–Ascoli theorem in the discrete setting, ${\mathsf{\Gamma }}_2$ is completely continuous.

• Step 4: Invariance of K.

We estimate for all $X\in K$

$$\parallel {\mathsf{\Gamma }}_{1}X\left(n\right)\parallel \le L_{1}M,$$

$$\parallel {\mathsf{\Gamma }}_{2}X\left(n\right)\parallel \le C_0{M}_G.$$

Hence, by selecting

$$M\ge L_{1}M+C_0{M}_G,$$

we obtain

$${\mathsf{\Gamma }}_{1}K+{\mathsf{\Gamma }}_{2}K\subseteq K.$$

• Step 5: Application of Krasnoselskii’s theorem.

Since 1. ${\mathsf{\Gamma }}_1$ is a contraction, 2. ${\mathsf{\Gamma }}_2$ is completely continuous, and 3. ${\mathsf{\Gamma }}_{1}K+{\mathsf{\Gamma }}_{2}K\subseteq K$,

Krasnoselskii’s fixed-point theorem guarantees that

$$X={\mathsf{\Gamma }}_{1}X+{\mathsf{\Gamma }}_{2}X$$

has at least one fixed point in K. This fixed point is a T-periodic solution to (2). □

We now give detailed steps of arriving at the uniqueness proof.

Theorem 4.

Assume the hypotheses for Theorem 3 are true and that

$$L_1+M_G(L_1\underset{k}{sup}\parallel A\left(k\right)\parallel +L_{2}m)<1.$$

(11)

Then, the neutral system (2) admits a unique T-periodic solution.

Proof. 

The operator $\mathsf{\Gamma }={\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_2$ will be shown to be a contraction.

Using the Lipschitz continuity of Q,

$$\parallel {\mathsf{\Gamma }}_{1}X-{\mathsf{\Gamma }}_1{Y\parallel }_{\infty }\le L_1{\parallel X-Y\parallel }_{\infty }.$$

For any $X,Y\in K$,

$$\parallel {\mathsf{\Gamma }}_{2}X\left(n\right)-{\mathsf{\Gamma }}_{2}Y\left(n\right)\parallel \le \sum _{k=-\infty }^{\infty }\parallel G(n,k)\parallel (\parallel A\left(k\right)\parallel ·L_1{\parallel X-Y\parallel }_{\infty }+L_2\sum _{j=1}^m{\parallel X-Y\parallel }_{\infty }).$$

Thus,

$$\parallel {\mathsf{\Gamma }}_{2}X-{\mathsf{\Gamma }}_2{Y\parallel }_{\infty }\le M_G(L_1\underset{k}{sup}\parallel A\left(k\right)\parallel +L_{2}m){\parallel X-Y\parallel }_{\infty }.$$

Combining both bounds,

$${\parallel \mathsf{\Gamma }X-\mathsf{\Gamma }Y\parallel }_{\infty }\le [L_1+M_G(L_1\underset{k}{sup}\parallel A\left(k\right)\parallel +L_{2}m)]{\parallel X-Y\parallel }_{\infty }.$$

If condition (11) holds, then the bracketed expression is strictly less than 1.

Therefore, $\mathsf{\Gamma }$ is a contraction on K.

By the Banach contraction principle, the fixed-point equation

$$X=\mathsf{\Gamma }X$$

admits a unique solution in K, which gives a unique T-periodic solution to (2). □

4. Examples

We present two examples illustrating the applicability of the main theorems. Both systems are two-dimensional ($d=2$), and the nonlinearities Q and G are nonlinear polynomial-type functions. The first example satisfies only the hypotheses in Theorem 3 (existence), whereas the second example satisfies both Theorems 3 and 4 (existence and uniqueness).

Example 1 (Existence Without Uniqueness). 

Consider the nonlinear neutral difference system

$$\Delta X\left(n\right)=A\left(n\right)X\left(n\right)+\Delta Q(n,X(n-1\left)\right)+G(n,X(n-1),X(n-2\left)\right),$$

(12)

where $X\left(n\right)={(x\left(n\right),y\left(n\right))}^T\in {\mathbb{R}}^2$ and

$$A\left(n\right)=\left(\begin{array}{cc}0.05& 0.02\\ -0.03& 0.04\end{array}\right),T=2.$$

The nonlinear neutral term is defined by

$$Q(n,X)=\left(\begin{array}{c}0.08x^2\\ 0.06y^2\end{array}\right),$$

and the nonlinear perturbation is given by

$$G(n,U,V)=\left(\begin{array}{c}0.04sin\left(n\right)u_1^2+0.03v_1\\ 0.03cos\left(n\right)u_2^2+0.35v_2\end{array}\right),$$

where $U={(u_1,u_2)}^T$ and $V={(v_1,v_2)}^T$.

• Computation of the Lipschitz constants:

Let $X={(x,y)}^T$ and $Y={(\tilde{x},\tilde{y})}^T$ with ${\parallel X\parallel }_{\infty }, {\parallel Y\parallel }_{\infty }\le M$. Using the identity $a^2-b^2=(a-b)(a+b)$ and the bound $\left|x\right|, |\tilde{x}|, |y|, |\tilde{y}|\le M$, we obtain

$$|x^2-{\tilde{x}}^2|\le 2M|x-\tilde{x}|,|y^2-{\tilde{y}}^2|\le 2M|y-\tilde{y}|.$$

Hence,

$${\parallel Q(n,X)-Q(n,Y)\parallel }_{\infty }\le 0.16M{\parallel X-Y\parallel }_{\infty },$$

which yields

$$L_1=0.16M.$$

Let $U,\tilde{U},V,\tilde{V}\in {\mathbb{R}}^2$ with norms bounded by M. Using similar estimates together with $|sin(n\left)\right|, |cos(n\left)\right|\le 1$, we obtain

$${\parallel G(n,U,V)-G(n,\tilde{U},\tilde{V})\parallel }_{\infty }\le (0.08M+0.38)\left({\parallel U-\tilde{U}\parallel }_{\infty }+{\parallel V-\tilde{V}\parallel }_{\infty }\right),$$

and therefore

$$L_2=0.08M+0.38.$$

Setting $M=1$, we obtain $L_1=0.16<1$.

Hence, all the assumptions of Theorem 3 are satisfied, and system (12) admits at least one T-periodic solution.

Since ${sup}_n\parallel A\left(n\right)\parallel \approx 0.07$ and $m=2$, we compute

$$\begin{array}{cc}\hfill L_1+M_G(L_1\underset{n}{sup}\parallel A\left(n\right)\parallel +L_{2}m)& =0.16+M_G(0.0112+2\left(0.46\right))\hfill \\ \hfill & =0.16+0.9312M_G.\hfill \end{array}$$

Even for $M_G=1$,

$$1.0912>1.$$

Therefore, Banach’s contraction condition is not satisfied, and uniqueness of the periodic solution is not guaranteed.

• Conclusion: System (12) admits at least one periodic solution, while the contraction requirement ensuring uniqueness fails.

Example 2 (Existence and Uniqueness). 

Consider the nonlinear neutral difference system

$$\Delta X\left(n\right)=A\left(n\right)X\left(n\right)+\Delta Q(n,X(n-1\left)\right)+G(n,X(n-1),X(n-2\left)\right),$$

(13)

where $X\left(n\right)={(x_1\left(n\right),x_2\left(n\right))}^T\in {\mathbb{R}}^2$ and

$$A\left(n\right)=\left(\begin{array}{cc}0.05& 0.02\\ -0.01& 0.04\end{array}\right),T=3.$$

The nonlinear neutral term is defined by

$$Q(n,X)=\left(\begin{array}{c}0.03x_1^2+0.02x_1{x}_2\\ 0.025x_2^2\end{array}\right).$$

The nonlinear perturbation is given by

$$G(n,U,V)=\left(\begin{array}{c}0.03u_1^2+0.02v_1^2\\ 0.02u_2^2+0.015v_2^2\end{array}\right),$$

where $U={(u_1,u_2)}^T$ and $V={(v_1,v_2)}^T$.

Let ${\parallel X\parallel }_{\infty }, {\parallel Y\parallel }_{\infty }\le M$. Using the identity $a^2-b^2=(a-b)(a+b)$ and the bound $|x_i|,|y_i|\le M$, we obtain

$${\parallel Q(n,X)-Q(n,Y)\parallel }_{\infty }\le 0.10M{\parallel X-Y\parallel }_{\infty }.$$

Hence,

$$L_1=0.10M.$$

Similarly, for ${\parallel U\parallel }_{\infty }, {\parallel V\parallel }_{\infty }\le M$,

$$\parallel G(n,U,V)-G(n,\tilde{U},\tilde{V}){\parallel }_{\infty }\le 0.10M\left(\parallel U-\tilde{U}{\parallel }_{\infty }+{\parallel V-\tilde{V}\parallel }_{\infty }\right),$$

and therefore

$$L_2=0.10M.$$

Setting $M=1$, we obtain

$$L_1=0.10<1,L_2=0.10.$$

Since

$$\underset{n}{sup}\parallel A\left(n\right)\parallel \approx 0.07,m=2,$$

we compute

$$L_1+M_G(L_1\underset{n}{sup}\parallel A\left(n\right)\parallel +L_{2}m)=0.10+M_G(0.007+0.20).$$

Thus,

$$=0.10+0.207M_G.$$

In particular, for $M_G=1$,

$$0.307<1.$$

Therefore, Banach’s contraction condition holds, and system (13) admits a unique T-periodic solution.

Remark 1.

It is well known that nonlinear neutral difference systems with multiple delays often exhibit severe numerical instability when simulated by direct forward iteration. This phenomenon arises from the presence of the neutral term $\Delta Q(n,X(n-g\left(n\right)\left)\right)$, which amplifies numerical errors even when all theoretical hypotheses for existence and uniqueness are satisfied. Therefore, numerical simulations are not included in this section. The examples provided illustrate the validity of the analytical results (the existence theorem and the uniqueness theorem), but they are not intended for numerical approximation. This behavior is consistent with the classical theory of neutral functional differential and difference equations.

5. Conclusions and Future Work

In this study, we investigated a general class of nonlinear neutral matrix difference systems with multiple delays. By introducing an appropriate neutral transformation and employing the exponential dichotomy of the associated linear system, we derived a Green operator representation that allowed us to reduce the problem to a fixed-point equation in a Banach space of periodic sequences.

Using Krasnoselskii’s fixed-point theorem, we established sufficient conditions for the existence of at least one periodic solution. Furthermore, by deriving an explicit contraction boundary involving the nonlinear Lipschitz constants and the dichotomy parameters, we proved the uniqueness of the periodic solution under a natural smallness condition. Our results unify and extend previous contributions on discrete neutral systems, such as [21], and provide a discrete analogue to the differential systems considered in [22]. The examples included in Section 4 demonstrate how the proposed criteria can be applied to systems of practical interest.

The results presented here provide several natural directions for future research:

• Almost-periodic and pseudo-almost-periodic solutions: Extending the framework to almost-periodic coefficients or perturbations would generalize the theory to more realistic non-autonomous models.
• Impulsive and discontinuous neutral systems: Introducing impulsive effects or discontinuities in the neutral term would pose additional analytical challenges, especially regarding the robustness of exponential dichotomy.
• Neutral systems with state-dependent delays: Allowing the delays to depend on the solution itself requires new tools for establishing well-posedness and periodicity.
• Stability and global attractivity of periodic solutions: While this work focuses on existence and uniqueness, the asymptotic behavior of solutions near the periodic orbit remains an important open problem.
• Numerical methods preserving dichotomy structure: Constructing difference schemes that maintain exponential dichotomy could lead to robust numerical approximations for neutral delay differential equations.

The techniques developed in this paper provide a flexible analytical framework that can be applied to a broad class of nonlinear models. We expect that the combination of exponential dichotomy with fixed-point theory will continue to play an important role in the qualitative analysis of neutral difference equations and their applications.