Dynamic behaviors for inertial neural networks with reaction-diffusion terms and distributed delays

A class of inertial neural networks (INNs) with reaction-diffusion terms and distributed delays is studied. The existence and uniqueness of the equilibrium point for the considered system is obtained by topological degree theory, and a sufficient condition is given to guarantee global exponential stability of the equilibrium point. Finally, an example is given to show the effectiveness of the results in this paper.

and Kang [10] studied a class of parabolic systems by using a semi-discrete method and a Lyapunov functional method. They obtained existence and several criteria of stability of the global generalized solutions for the considered systems. Li and Cao [11] investigated stochastic Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms by using the Halanay inequality technique and the Lyapunov method. The criteria of delay-independent exponential stability were established for the above system. Carpenter [12] studied a singular perturbation problem with application to nerve impulse equations by the use of a geometric approach. In [13], the authors studied the exponential periodicity and stability of neural networks with Lipschitz continuous activation functions. The considered system in [13] contains reaction-diffusion terms and mixed delays.
Motivated by the above discussions, in this paper we are devoted to studying the existence and global asymptotic stability of the equilibrium point for a class reaction-diffusion neural networks with distributed delays subject to Dirichlet boundary conditions. By using the Lyapunov functional method, some new stability criteria are obtained which also guarantee the network will be asymptotically convergent to the equilibrium point. The theoretical methods developed in this paper have universal significance and can be easily extended to investigate many other types of neural networks with distributed delays. Our main contributions are summarized as follows.
(i) In this paper, we firstly study a new class of INNs which contains reaction-diffusion terms and distributed delays. In generally, the diffusion effect cannot be avoided in the neural networks. So we must consider that the space is varying with the time. (ii) Due to influence of reaction-diffusion terms and distributed delays, we cannot easily obtain global exponential stability. By using the innovative mathematical analysis skills, we overcome the above difficulty. (iii) It is nontrivial to establish a unified framework to handle reaction-diffusion terms and distributed time delays. Our method provides a useful reference for studying more complex systems. Throughout this paper, let S = {1, 2, . . . , n}. For any f = (f 1 , . . . , f n ) ∈ R n , denote the norm f 1 = n k=1 |f | k . For any u(t, x) = (u 1 (t, x), . . . , u n (t, x)) ∈ R n , define u i (t, The rest of this paper is organized as follows. In Sect. 2, the reaction-diffusion neural networks with distributed delays are presented. In Sect. 3, the existence and global exponential stability of the equilibrium point for the considered model are studied. An example is presented to illustrate our theoretical results in Sect. 4. Finally, conclusions are drawn in Sect. 5.

Problem formulation
The classic INNs with distributed delays are described as follows: which has been extensively studied, see e.g. [4][5][6] and related references. By the use of the variable transformation function system (2.1) can be rewritten as follows: Add the reaction-diffusion terms in (2.2), then where i ∈ S, x = (x 1 , . . . , x l ) ∈ ⊂ R l , is a bounded compact set and has smooth boundary, u i (t, x) denotes the state of ith neuron at time t and in space x, I i is the external input on the ith neurons, f j (·) is the activation function which is a continuous function, is a delay kernel, a i , b i , D ik , p ij , and q ij are constants, a i > 0 is the damping coefficient, b i > 0 denotes the strength of different neuron, D ik = D ik (t, x, u) ≥ 0 is the transmission diffusion operator, p ij and q ij are connection weights. The boundary conditions and initial conditions of system (2.3) are given by (2.5)

Main results
We need the following assumptions for activation functions and delay kernels functions: (H 1 ) There exists positive constant l i such that Then the equilibrium point of system (2.3) is the solution of the following system: Define a bounded open set and a homotopic map H(λ, u, v) as follows: Hence, we get From the second equation of system (3.3) and assumption (H 2 ), we have and u * =ũ , v * =ṽ . Thus the equilibrium point of system (2.3) is unique.
The second equation of system (2.3) does not contain the term v i which is not a matching system. For obtaining exponential stability of system (2.3), we consider the following generalized system: where c i > 0, i ∈ S, whose boundary conditions and initial conditions are (2.4) and (2.5), respectively.
is an equilibrium point of (3.4) and (u , v ) is any solution of (3.4) satisfying We call (u * , v * ) globally exponentially stable.

Theorem 3.2 Assume that system (3.4) has a unique equilibrium point. Assume further
Then the unique equilibrium point of system (3.4) is globally asymptotically stable. Proof Suppose that system (3.4) has the unique equilibrium point (u * , v * ) , and (u , v ) is any solution of system (3.4). Rewrite system (3.4) as the following form: (3.5) Multiply both sides of the first equation of (3.5) by u iu * i and integrate them (3.7) By (3.6) and (3.7), we have Multiply both sides of the second equation of (3.5) by v iv * i and integrate them (3.10) In view of (3.9), (3.10), assumption (H 1 ), and Hölder's inequality, we have (3.12) where i ∈ S. By assumption (H 3 ), we have ψ i (0) = c i -1 > 0 for i ∈ S. By condition (3.1), It is easy to see that φ i (ξ i ) and ψ i (η i ) are continuous for ξ i , η i ∈ [0, ∞), and φ i (ξ i ) < 0 and ψ i (η i ) < 0 i.e. φ i (ξ i ) and ψ i (η i ) are strictly decreasing functions on [0, ∞). Hence, there exist constants (3.14) Construct the following Lyapunov functional: Calculating the upper right Dini derivative of V (t) along the solutions of system (3.4), in view of assumption (H 3 ), we have and The proof is completed.
Remark 3.1 We really want to obtain global exponential stability for system (2.3). However, system (2.3) has strong non match, which is different from some reaction-diffusion systems in [11][12][13]. For system (2.3), constructing a proper Lyapunov functional is very difficult because of its strong non match. Hence, we study system (3.4) which is similar to system (2.3) and obtain global exponential stability of system (3.4) by constructing a proper Lyapunov functional and some mathematical analysis technique. We hope that some authors will solve the global exponential stability problem of system (2.3) by the innovative approach.
Remark 3.2 In this paper, Lemma 1.

Numerical example
This section presents an example that demonstrates the validity of our theoretical results. Consider the following two-dimensional neural networks with distributed delays and reaction-diffusion terms: where i, j, k = 1, 2, D ik = 1 > 0, Obviously, Thus, all assumptions of Theorem 3.1 hold and (4.1) has a unique equilibrium point. The solution of system (4.1) is shown in Fig. 1.
Remark 4.1 For all we know, the INNs with reaction-diffusion terms and distributed delays are a new model in the present paper. Using topological degree theory and the mathematical analysis technique, we get some brand new results on the existence, uniqueness, and global exponential stability of solution of INNs. We can confirm the truth of the proposed methods, for example, in [4-6, 12, 13] cannot be generalized to the problems studied in this article. Remark 4.2 In [13], the authors gave some numerical simulations of a periodic solution for a class of neural networks with reaction-diffusion terms and both variable and unbounded delays. Numerical simulations in the present paper show the solution properties of second-order system, but neural networks in [13] are a first-order system. Hence, numerical simulations in the present paper are more complicated than the existing ones. For more similar numerical simulations of a first-order system, see e.g. [10][11][12].

Conclusions and discussions
In this paper we study the dynamic behaviors of solutions for inertial neural networks with reaction-diffusion terms and distributed delays. First, by applying topological degree theory to the system, we get a set of sufficient conditions for guaranteeing the existence and uniqueness of solutions. Then, the global exponential stability of the equilibrium point is obtained by using the Lyapunov functional. The efficacy of the obtained results has been demonstrated by numerical simulations. It is important to note that the practical implementation of INNs is typically encountered with certain type of uncertainties such as interval parameters. These results can be applied to design globally exponentially stable networks and thus have important significance in both theory and applications. Extending the results of this paper to INNs with interval uncertainties proves to be an interesting problem.