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Almost Periodicity in Shifts Delta(+/-) on Time Scales and Its Application to Hopfield Neural Networks

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Almost Periodicity in Shifts Delta(+/-) on Time Scales and Its Application to Hopfield Neural Networks Engineering Letters, 29:3, EL_29_3_06 ______________________________________________________________________________________ Almost Periodicity in Shifts Delta(+/-) on Time Scales and Its Application to Hopfield Neural Networks Meng Hu, and Lili Wang Abstract—In this paper, we first give three definitions with the definitions and the properties of shifts δ±, some basic results aid of the shift operators δ, that is, almost periodicity in shifts on almost periodic differential equations in shifts δ± on time δ, almost periodic function in shifts δ and ∆-almost periodic scales are established. Moreover, we applying the obtained function in shifts δ, then by using the theory of calculus on time scales and some mathematical methods, the existence and results to study a class of Hopfield neural networks on time scales as follows uniqueness theorem of solution of linear dynamic system on 8 almost periodic time scale in shifts δ is obtained. Finally, Pn > ∆ − we applying the obtained results to study the existence and > yi (t) = ai(t)yi(t) + cij(t)gj(yj(t)) exponential stability of the almost periodic solution in shifts < j=1 Pn δ of a class of Hopfield neural networks with delays, several (1) > + dij(t)fj(yj(δ−(τij; t))) + Ii(t); examples and numerical simulations are given to illustrate and > j=1 reinforce the main results. : yi(s) = ϕi(s); s 2 [δ−(^τ; t0); t0]T; i 2 N; Index Terms—Almost periodic time scale in shifts δ; Expo- nential dichotomy; Hopfield neural networks; Almost periodic where t 2 T, T is an almost periodic time scale in shifts solution in shifts δ; Global exponential stability. δ±, N = f1; 2; : : : ; ng, the integer n corresponds to the number of units in (1); yi(t) corresponds to the state of the I. INTRODUCTION ith unit at time t; ai(t) > 0 represents the passive decay rate; c and d weight the strength of jth unit on the ith T is well known that almost periodicity plays an im- ij ij unit at time t; I (t) is the input to the ith unit at time t from portant role in dynamic systems both theoretical study i I outside the networks; g and f denote activation functions and practical applications, see [1-6]. In recent years, with i i of transmission; the function δ−(·; t) is a delay function the development of the theory of time scales (see [7,8]), generated by the backward shift operator δ− on time scale the existence and stability of almost periodic solutions of T, τ corresponds to the signal transmission delay along the dynamic equations on time scales received many researchers’ ij axon of the jth unit which is nonnegative and bounded, and special attention; see, for example, [9-12]. In these works, τ^ = maxfτijg. the almost periodic time scale T satisfies the condition i;j2N “t ± s 2 T; 8t 2 T, s is the translation constant”. Under this condition all almost periodic time scales are unbounded II. ALMOST PERIODICITY IN SHIFTS δ± below and above. However, there are many time scales such T R as qZ = fqn : q > 1 is constant and n 2 Zg [ f0g which Let is a nonempty closed subset (time scale) of . The σ; ρ : T ! T is neither closed under the operation t ± s nor unbounded forward and backward jump operators and the µ T ! R+ below. Therefore, almost periodicity on time scales need to graininess : are defined, respectively, by be explored further. σ(t) = inffs 2 T : s > tg; ρ(t) = supfs 2 T : s < tg; In [13-16], Adıvar et al. defined two shift operators, i.e. µ(t) = σ(t) − t: δ+ (forward shift operator) and δ− (backward shift operator), which does not oblige the time scale to be closed under the A point t 2 T is called left-dense if t > inf T and ρ(t) = t, ± operations t s. The applications of the shift operators δ± left-scattered if ρ(t) < t, right-dense if t < sup T and σ(t) = on time scales; see, for example, [17-19]. Motivated by the t, and right-scattered if σ(t) > t. If T has a left-scattered above works, the main purpose of this paper is to define maximum m, then Tk = Tnfmg; otherwise Tk = T. If a new almost periodicity concept with the aid of the shift T has a right-scattered minimum m, then Tk = Tnfmg; operators δ±. We first give three definitions, that is, almost otherwise Tk = T. periodicity in shifts δ±, almost periodic function in shifts A function f : T ! R is right-dense continuous provided δ± and ∆-almost periodic function in shifts δ±. Under the it is continuous at right-dense point in T and its left-side T Manuscript received January 25, 2021; revised April 26, 2021. This work limits exist at left-dense points in . If f is continuous at was supported in part by the Key Scientific Research Project of Colleges and each right-dense point and each left-dense point, then f is Universities of Henan Province (No.21A110001) and the National Natural said to be a continuous function on T. Science Foundation of China (No.11801011). T ! R Meng Hu is an associate professor in the School of Mathematics and A function p : is called regressive provided 1 + Statistics, Anyang Normal University, Anyang, Henan 455000, China (Cor- µ(t)p(t) =6 0 for all t 2 Tk. The set of all regressive and responding author; e-mail: [email protected]). rd-continuous functions p : T ! R will be denoted by R = Lili Wang is a lecturer in the School of Mathematics and Statistic- R T R R+ R+ T R f 2 R s, Anyang Normal University, Anyang, Henan 455000, China (e-mail: ( ; ). We define the set = ( ; ) = p : ay [email protected]). 1 + µ(t)p(t) > 0; 8 t 2 Tg. Volume 29, Issue 3: September 2021 Engineering Letters, 29:3, EL_29_3_06 ______________________________________________________________________________________ l(") If r is a regressive function, then the generalized expo- that in any interval of length δ± (·), there exists at least a nential function er is defined by p 2 Ef"; fg such that { Z } t p ∗ jf(δ±(t)) − f(t)j < "; 8t 2 T ; er(t; s) = exp ξµ(τ)(r(τ))∆τ s p where δ± := δ±(p; t), p is called the "-translation constant for all s; t 2 T, with the cylinder transformation of f, l(") is called the inclusion length of Ef"; fg. { Log(1+hz) 6 2 1 ∗ h ; if h = 0; Remark 2. In Definition 2, if there exists a ! [!T; + )T ξh(z) = ! ∗ z; if h = 0: and ! 2 Ef"; fg such that f(δ±(t)) = f(t); 8t 2 T , the smallest constant ! is called the period of f. Let p; q : T ! R are two regressive functions, define p p Remark 3. If T = R or T = Z, then the operator δ±(t)) = p ⊕ q = p + q + µp q; ⊖p = − ; p ⊖ q = p ⊕ (⊖q): ± 1 + µp t p associated with the initial point t0 = 0, then Definition 2 is converted to the traditional definition of almost periodic T ! R Lemma 1. ([7]) Assume that p; q : are two regressive function on R or Z. functions, then (i) e0(t; s) ≡ 1 and ep(t; t) ≡ 1; Now, we give two examples to illustrate the Definition 2 (ii) ep(σ(t); s) = (1 + µ(t)p(t))ep(t; s); is more generality. 1 T R (iii) ep(t; s) = = e⊖p(s; t); Example 1. Let = and t0 = 1. The operators ep(s;t) { (iv) ep(t; s)ep(s; r) = ep(t; r); t ; if t ≥ 0; ∆ ⊖ δ−(p; t) = p for p 2 [1; +1) (v) (Re⊖p(t; s)) = ( p)(t)e⊖p(t; s); b pt; if t < 0; (vi) p(t)ep(c; σ(t))∆t = ep(c; a) − ep(c; b). a and { δ± A comprehensive review on the shift operators on time pt; if t ≥ 0; δ (p; t) = for p 2 [1; +1) scales can be found in [13-16]. + t ; if t < 0; Let T∗ is a non-empty subset of the time scale T and p ∗ ∗ t0 2 T is a fixed constant, define operators δ± :[t0; +1)× are backward and forward shift operators (on the set R = ∗ ∗ ∗ T ! T . The operators δ+ and δ− associated with t0 2 T R − f0g) associated with the initial point t0 = 1. (called the initial point) are said to be forward and backward Consider the function ∗ ( ) shift operators on the set T , respectively. The variable s 2 ln jtj f(t) = cos π ; t 2 R∗: [t0; +1)T in δ±(s; t) is called the shift size. The values 1 ∗ ln( 2 ) δ+(s; t) and δ−(s; t) in T indicate s units translation of the 2 T∗ By direct computation, term t to the right and the left, respectively. The sets { ±1 ≥ D f 2 1 × T∗ 2 T∗g p f(tp ); if t 0; ± := (s; t) [t0; + )T : δ±(s; t) f(δ±(t)) = t f( ±1 ); if t < 0; ( p ) are the domains of the shift operator δ±, respectively. Here- j j ± 1 ∗ ln t ln( ) after, T is the largest subset of the time scale T such that = cos p π 1 × T∗ ! T∗ ln( 1 ) the shift operators δ± :[t0; + ) exist. ( 2 ) Next, we give three definitions with the aid of the shift ln jtj ln( 1 ) = cos π ± p π : operators δ±, that is, almost periodicity in shifts δ±, almost 1 1 ln( 2 ) ln( 2 ) periodic function in shifts δ± and ∆-almost periodic function 2n 2 Z in shifts δ±.
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