Global Asymptotical Stability of Neutral-Type Neural Networks with D− Operator and Mixed Delays

where σ > 0 is a constant delay, C � diag c1, . . . , cn 􏼈 􏼉 with |ci|< 1; α(x(t)) � (Ax)(t) denotes the behaved function; the functions f(x(·)) � (f1(x1(·)), . . . , fn(xn(·))) ⊤, g(x(·)) � (g1(x1(·)), . . . , gn(xn(·))) ⊤, and h(x(·)) � (h1(x1(·)) , . . . , hn(xn(·))) ⊤ denote the neuron activations; τ(t) and μ(t) are the discrete time-varying delay and the distributed time-varying delay, respectively, which satisfy (0≤ τ(t)≤ τ, 0≤ μ(t)≤ μ), where τ and μ are positive constants; and ρ � max σ, τ, μ 􏼈 􏼉, for φ ∈ C([− ρ, 0],R), and the norm is defined by ‖φ‖ρ � sup− ρ≤s≤0|φ(s)|. ,e matrix B � (bij)n×n> 0 is a constant matrix, and K � (kij)n×n> 0, U � (uij)n×n, andW � (wij)n×n are some unknown constant matrices. I(t) denotes the exogenous inputs at time t.

Lemma 1 (see [2,3]). When |c| ≠ 1, then A has a unique continuous bounded inverse A − 1 satisfying When the neutral-type operator A is defined on continuous function space C(R, R), we have the following lemma: Lemma 2 (see [3]). If |c| < 1, then the inverse of difference operator A denoted by A − 1 exists, and Proof. e proof of Lemma 2 is similar to the proof in [3]. For the convenience of the reader, we provide a detailed proof. Let

exists, and
□ Remark 2. Since the matrices K, U, and W in (1) are uncertain matrices, system (1) is interval neural networks. In the real world, the dynamic systems are often destroyed by various inevitable factors, such as parameter fluctuation, external disturbance, and random disturbance. Hence, the research for interval neural networks has important practical application value. ere are lots of results for interval neural networks, e.g., [4,5]. Let us recall the research for the stability problems of neural networks. Zhang [6] studied the stability problem of periodic solutions for a time-varying recurrent cellular neural network with time delays and impulses. en, Zhang and Wang [7] further investigated dynamic properties of periodic solutions for high-order Hopfield neural networks with time delays and impulses. e models in [6,7] are nonneutral-type models which are different from the model in the present paper. In [8], periodic oscillation problems of discrete-time bidirectional associative memory neural networks have been studied by employing the theory of coincidence degree and Halanay-type inequality technique. In the present paper, we obtain the existence, uniqueness, and global asymptotical stability of periodic solutions by using LMI approach, Lyapunov function, and a blend of matrix theory which are different from the methods in [8]. Zhu and Cao [9,10] studied two different types of stochastic neural networks which are not neutral-type neural networks and are different from the model in the present paper. In [11,12], the authors considered the following two kinds of neutral-type neural system: Liu et al. [13] studied a class of Markovian jumping neutral-type neural networks with mode-dependent mixed time delays: h(x(s))ds.

(8)
By the above work, we note that the neutral character in neural networks shows by the nonlinear term g(t, x ′ (t − τ(t))) or differential operator (x ′ (t − τ(t))). In this paper, we will study the neutral-type neural networks when the neutral term has the D− operator form which is shown by For more references about neutral-type neural networks with mixed delays, see [14,15].
To best knowledge of the authors, few authors have studied the global asymptotical stability problems of periodic solutions to neutral-type neural networks with mixed delays. e main areas of challenge are as follows: (1) since system (1) contains neutral-type operator A, constructing a viable Lyapunov-Krasovskii functional seems to be very difficult; (2) as the mixed delays exist in system (1), the corresponding LMI approach becomes more complicated since the LMI approach is required to reflect mixed delay's influence; and (3) it is extraordinary to establish a unified framework to dispose of the uncertain matrices, the neutral-type terms, and mixed delays' influence. erefore, the main purpose of this article is to try for the first time to address the challenges listed.
In this paper, we will study global asymptotical stability of interval neural networks for a neutral-type neural network with mixed delays. Note that system (1) contains uncertain matrices, the neutral-type terms, and mixed delays that are all dependent on the properties of the neutral operator.

Remark 3.
e main purpose of this paper is to obtain some sufficient conditions for guaranteeing asymptotic stability of system (1). We develop LMI approach to answer the addressed challenges. A simulation example is given to show the usefulness of the main results in the present paper. ere are three aspects to the contribution of this paper: (1) For using the LMI method, we have to take into account the properties of the neutral-type operator. (2) Unlike the most existing results, we develop a new unified framework to deal with global asymptotical stability of interval neural networks by LMI approach, Lyapunov function, and a blend of matrix theory which may be of independent interest. It is worth pointing out that our main methods are also important for the case of nonneutral system with constant delays. (3) is article uses some new inequality techniques. In particular, using the properties of the neutraltype operator, we construct an appropriate Lyapunov-Krasovskii functional to handle the considered system. e following sections are organized as follows: Section 2 gives some preliminaries including some useful lemmas and definitions. In Section 3, we obtain some sufficient conditions for existence, uniqueness, and global asymptotic stability of periodic solution to system (1). In Section 4, a numerical example verifies the accuracy of the results in the present paper.

Some Preliminaries
roughout the paper, Λ � 1, 2, . . . , n { }, and R n and R n×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. e superscript "T" represents the matrix transposition. A > 0 (or A < 0) denotes that A is a symmetric and positive definite (or negative definite) matrix. |z| denotes the Euclidean norm of a vector z and ‖A‖ denotes the induced norm of the matrix A, that is, where λ max (·) means the largest eigenvalue of A. If their dimensions are not explicitly stated, they are assumed to be compatible for algebraic operations.
Lemma 3 (see [16]). For any vectors with x, y ∈ R n , the inequality holds, where M is any n × n matrix with M > 0.
Lemma 4 (see [17]). Let p, q, c, τ, and σ be the positive constants, and the function f ∈ PC(R, R + ) satisfies the scalar impulsive differential inequality: where is a continuous function on [0, ∞), and the results of Lemma 4 also hold.
roughout this paper, the following assumptions are needed: (H 1 ) For i ∈ Λ, the neuron activation functions in (1) satisfy where

Main Results
Assume that the unknown constant matrices K, U, and W satisfy where Mathematical Problems in Engineering 3 with K � k ij n×n , Let where e i ∈ R n and i ∈ Λ denotes the column vector with ith element to be 1 and others to be 0. en, system (1) can be rewritten as

Mathematical Problems in Engineering
i.e., Now, we present the main results of the present paper.

Mathematical Problems in Engineering
We calculate every term of D + V(t). It follows from the assumption (H 1 ), Lemma 2, and Lemma 3 that Using the definition of Ψ z (t), assumption (H 1 ), and Lemma 2, we have By (29), we have Adding the terms on the right-hand side of (26)-(28) and (30) to (25), and using condition (20), we get which together with Remark 4 yields where V ρ � sup s∈[− ρ,0] V(s) and λ > 0 satisfies λ < ε 1 − ε 2 e λτ − ε 3 (e λμ /λ). us, By (33) and Lemma 2, we have Mathematical Problems in Engineering i.e., We show that X m (t) is uniformly convergent. For s ∈ Z + , by (35), we have It is obvious that the function sequence X m (t) ∞ m�1 is uniformly convergent by the Cauchy convergence criterion. In addition, Hence, x(t + mT, ϕ) is also uniformly convergent, which implies that Hence, we can deduce that x 0 (t) is a T− periodic solution of system (1) since x(t + mT, ϕ) is a solution of system (1). 8 Mathematical Problems in Engineering Now, we prove that x 0 (t) is a unique T− periodic solution of system (1). Let x 1 (t) � x 1 (t, ϕ 1 ) and x 2 (t) � x 2 (t, ϕ 2 ) be different T− periodic solutions of system (1), where ϕ 1 , ϕ 2 ∈ C([− ρ, 0], R n ). It follows by (35) that us, In the proof of eorem 1, since z(t) is not a continuous T− periodic solution, Lemma 1 cannot be used for the proof of eorem 1. We generalize the results of Lemma 1 to the continuous function space which can be used for the proof of eorem 1. However, when the parameter |c| ≥ 1 in (2), we cannot prove the result of eorem 1. We hope that subsequent researchers can solve the above problems. Hence, it follows by (35) and (43) that and the periodic solution of system (1) is global exponential stability.

Numerical Example
In this section, a simulation example is presented for illustrating the usefulness of our main results. Consider a 2neuron neural network (1) with the following parameters: By simple calculation, Let ε 1 � 6, ε 2 � 2, and ε 3 � 4.2. We can obtain the following feasible solutions to LMIs in eorem 1: Hence, all the conditions in eorem 1 hold, and the 2neuron neural networks (47) have a stationary oscillation with T period.
Particularly, let the concrete parameter matrices in (47) be (50) e corresponding numerical simulations for different T are presented in Figures 1-4. Figure 1 shows the state trajectories of system (47) with the period T � 10. Figure 2 shows the phase plots of system (47) with the period T � 10. Figure 3 shows the state trajectories of system (47) with the period T � 0.2. Figure 4 shows the phase plots of system (47) with the period T � 0.2. Remark 7. In [4,5,8,9,11], the authors studied the stability problems of some nonneutral interval neural networks with delays. However, there are few results for neutral-type interval neural networks with impulsive delays which is the problem we will solve in the future.

Conclusions
In this paper, we had investigated the stability problems of neutral-type neural networks with D− operator and mixed delays. It is interesting and challenging to extend our results to the stochastic delay system [18], impulsive stochastic delay differential systems [19], semi-Markov switched stochastic systems [20], and stochastic systems with Lévy noise [21].

Data Availability
e data used to support the findings of this study are included in this paper.