Periodic Solutions for Shunting Inhibitory Cellular Neural Networks of Neutral Type with Time-Varying Delays in the Leakage Term on Time Scales

A class of shunting inhibitory cellular neural networks of neutral type with time-varying delays in the leakage term on time scales is proposed. Based on the exponential dichotomy of linear dynamic equations on time scales, fixed point theorems, and calculus on time scales we obtain some sufficient conditions for the existence and global exponential stability of periodic solutions for that class of neural networks. The results of this paper are completely new and complementary to the previously known results even if the time scale T = R or Z. Moreover, we present illustrative numerical examples to show the feasibility of our results.


Introduction
As we know, shunting inhibitory cellular neural networks (SCINNs) have been applied in a wide range of practical fields such as psychophysics, speech, perception, robotics, adaptive pattern recognition, and image processing.Hence, they have been the object of intensive analysis by numerous authors in recent years.In particular, there have been extensive results on the problem of the existence and stability of periodic solutions and almost periodic solutions for SCINNs in the literature.For example, in [1][2][3], authors consider the existence and stability of almost periodic solutions for SCINNs; in [4,5], authors consider the existence and stability of periodic solutions for SCINNs; in [5], authors by using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions consider the periodic solution for SCINNs; in [6,7], authors obtained some sufficient conditions for the existence and stability of an equilibrium point.
Recently, another type of time delays, namely, neutral type time delays, which always appears in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar, and so forth, has recently drawn much research attention.There are some results on the stability and the existence of periodic solutions to delayed neural networks of neutral type, for example in [8][9][10][11][12][13], by using the Lyapunov functions and the linear matrix inequality approach, authors studied the asymptotic stability or exponential stability of the equilibrium point for delayed neural networks of neutral type and in [14,15], by using the theory of abstract continuation theorem of -set contractive operator, authors studied the existence of periodic solutions for delayed cellular neural networks of neutral type and Hopfield neural networks with neutral delays, respectively.In a recent paper [16], authors studied the existence and exponential stability for the following SICNN with continuously distributed delays of neutral type: where   is the cell at the (, ) position of the lattice, the  neighborhood   (, ) of   is defined as follows: (, ) = {  : max {| − | ,      −      } ≤ , where   represents the state of the cell   , the coefficient   () > 0 means the passive decay rate of the passive decay rate of the cell activity, and    () and    () describe the connection or coupling strength of postsynaptic activity of the cell   transmitted to the cell   .
Very recently, a leakage delay, which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks [17,18].However, so far, very little attention has been paid to neural networks with time delay in the leakage (or "forgetting") term [19][20][21][22][23][24][25].Such time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks.Also, it is well known that both continuous time and discrete time neural networks have equal importance in various applications.Moreover, the theory of calculus on time scales was initiated by Stefan Hilger [26] in his Ph.D. thesis in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his foundational work.For instance, in [27], the authors studied antiperiodic solutions to impulsive SICNNs with distributed delays on time scales.In [28], the authors studied almost periodic solutions of SCINNs on time scales.However, to the best of our knowledge, there is no paper published on the existence and stability of periodic solutions for SCINNs of neutral type with the time delay in the leakage term.
Motivated by the above discussions, in this paper, we are concerned with the following SCINNs of neutral type with time-varying delays in the leakage term on time scale T: where T is a periodic time scale and T + = T ∩ [0, +∞),   is the cell at the (, ) position of the lattice, the  neighborhood   (, ) of   is defined as follows: represents the state of the cell   , the coefficient   () > 0 means the passive decay rate of the passive decay rate of the cell activity, and    () and    () describe the connection or coupling strength of postsynaptic activity of the cell   transmitted to the cell   .
Our main purpose of this paper is to study the existence and global exponential stability of periodic solutions to (3) by using the exponential dichotomy of linear dynamic equations on time scales and some inequality technics.Our results of this paper are completely new and complementary to the previously known results even if the time scale T = R or Z.Our methods used in this paper are different from those used in [14,15,19] and can be used to study other types' delayed neural networks of neutral type with delays in the leakage term.
For convenience, we denote Throughout this paper, we assume that the following conditions hold: for all , V ∈ R,  = 1, 2, . . ., .
( 3 ) For  = 1, 2, . . ., ,  = 1, 2, . . ., , the delay kernels   ,   : [0,∞) ∩ T → R are continuous and integrable with This paper is organized as follows: in Section 2, we introduce some notations and definitions and state some preliminary results which are needed in later sections.In Section 3, we establish some sufficient conditions for the existence of periodic solutions of (3).In Section 4, we prove that the periodic solution obtained in Section 3 is globally exponentially stable.In Section 5, we give examples to illustrate the feasibility of our results obtained in previous sections.

Preliminaries
In this section, we introduce some definitions and state some preliminary results.
Definition 1 (see [29]).Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators ,  : T → T and the graininess  : T → R + are defined, respectively, by Definition 2 (see [29]).A point  ∈ T is called left-dense if  > inf T and () = , left-scattered if () < , right-dense if  < sup T and () = , and right-scattered if () > .If T has a left-scattered maximum , then T  = T \ {}; otherwise T  = T.If T has a right-scattered minimum , then T  = T \ {}; otherwise T  = T.
Definition 3 (see [30]).One says that a time scale T is periodic if there exists  > 0 such that if  ∈ T, then  ±  ∈ T. For T ̸ = R, the smallest positive  is called the period of the time scale.
Definition 9 (see [29]).If  ∈ T, sup T = ∞ and  is rdcontinuous on [, ∞), then one defines the improper integral by provided this limit exists, and one says that the improper integral converges in this case.If this limit does not exist, then one says that the improper integral diverges.
Definition 10 (see [29]).If  ∈ T, inf T = −∞, and  is rd-continuous on (−∞, ), then one defines the improper integral by provided this limit exists, and we say that the improper integral converges in this case.If this limit does not exist, then one says that the improper integral diverges.
Lemma 11 (see [29]).Let  ∈ T  and  ∈ T and assume that  : where  Δ denotes the derivative of  with respect to the first variable.Then Definition 12 (see [31]).Let  ∈ R  and () be a  ×  matrix-valued function on T, the linear system where Lemma 13 (see [31]).If (17) admits an exponential dichotomy, then the following -periodic system: has an -periodic solution as follows: where () is the fundamental solution matrix of (17).
Lemma 14 (see [31]).If () is a uniformly bounded continuous  ×  matrix-valued function on T and there is a  > 0 such that then ( 17) admits an exponential dichotomy on T.

Existence of Periodic Solutions
In this section, we will state and prove the sufficient conditions for the existence of periodic solutions of (3).

Exponential Stability of the Periodic Solution
In this section, we will discuss the exponential stability of the periodic solution of (3).