Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations

: This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear di ff erential equations and the fixed point theory of contraction mapping. The su ffi cient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting di ff erential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the e ff ectiveness of the results obtained in the paper


Introduction
This paper considers the existence and global exponential stability of piecewise pseudo almost periodic solutions for interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. The mixed time-varying delays include leakage delays and time-varying delays: s ji (t) f j (x j (t), y j (t − τ ji (t))) + c i (t), t > 0, t t k , i = 1, 2, · · · , m, t > 0, t t k , j = 1, 2, · · · , m, ∆y j (t) = y j (t + ) − y j (t − ) = J k (y j (t)), t = t k , k ∈ Z + , where Z + is the set of nonnegative integers; the numbers of neurons in layers X and Y are denoted by m. x i (t) represents the state variables of the i-th neurons at time t. y j (t) represents the state variables of the j-th neurons at time t. a i (t) > 0, b j (t) > 0 are continuous functions and represent the decay rates of neurons in different layers, respectively. s ji (t), t i j (t) are the connection weights. f j (·, ·), g i (·, ·) represent the activation functions of the j-th and i-th units, respectively. c i (t), d j (t) represent the external inputs of the i-th neuron and the j-th neuron acting on different layers at time t, respectively. α i (t) > 0, β j (t) > 0 are time-varying leakage delays. τ ji (t) > 0, δ i j (t) > 0 are time-varying delays respectively satisfying 1 − τ ′ ji (t) > 0, 1 − δ ′ i j (t) > 0. The sequence {t k } has no finite accumulation point. I k (·) : R m → R and J k (·) : R m → R, k ∈ Z + .
Compared with the activation functions f j (·), g i (·) in BAM neural networks [10,11,[13][14][15]17,19,[21][22][23][24], the activation functions f j (x j , y j ), g i (x j , y j ) in the interval general BAM neural networks [16,17,[20][21][22]24] more clearly show the connections between different layers of neurons in the networks. Ding and Huang [16] studied the existence and global robust exponential stability of equilibrium points for the following a class of interval general BAM neural networks with constant delays by using fixed-point theory and constructing suitable Lyapunov functions: s ji f j (x j (t), y j (t − τ ji )) + c i , (1.3) By variable transformations, fixed point theory and constructing suitable delay differential inequalities, Xu et al. [17] studied the existence, uniqueness and global exponential stability of the equilibrium point of system (1.3) with proportional delays. Given a i = a i (t), b j = b j (t), s ji = s ji (t) and t i j = t i j (t) in the system (1.3), the authors of [20] studied the existence and global exponential stability of periodic solutions for the system (1.3) on time scales by using fixed point theory and constructing suitable Lyapunov functions. Given a i = a i (t), b j = b j (t), s ji = s ji (t), t i j = t i j (t), c i = c i (t) and d j = d j (t) in the system (1.3), Duan [22] studied the existence and global exponential stability of pseudo almost periodic solutions for system (1.3) by using exponential dichotomy, fixed point theory and inequality techniques. Given a i = a i (t), b j = b j (t), s ji = s ji (t), t i j = t i j (t), c i = c i (t) and d j = d j (t) in the system (1.3), the authors of [24] studied the existence and exponential stability of almost periodic solutions for system (1.3) with leakage delays by using exponential dichotomy, fixed point theory and inequality techniques.
However, in the existing literature, we have not found any research on the dynamic behavior of interval general BAM neural networks with impulses and leakage delays. The stability of the interval general BAM neural networks may be destroyed by external perturbations and leakage delays. Therefore, the effects of leakage delays and impulsive perturbations on the dynamic behavior of interval general BAM neural networks are worth exploring.
Motivated by the above discussions, this paper studies the existence and global exponential stability of piecewise pseudo almost periodic solutions for the interval general BAM neural network described by (1.1) with mixed time-varying delays and impulsive perturbations. The mixed time-varying delays include leakage delays and time-varying delays.
Throughout this paper, let e + i = sup For {t k } k∈Z + ⊂ T , assume that {t j k : t j k = t k+ j − t k , k, j ∈ Z + } are equipotentially almost periodic and t 0 = 0; it can be easily proved that the sequence {t j k } satisfies t j k+i − t j k = t i k+ j − t i k and t j k − t i k = t j−i k+i . And suppose that the following conditions hold.
and there are two positive constants I, J satisfying 0 < I, J < 1 such that Remark 1.1. In the existing literature on BAM neural networks, in addition to the condition (H 1 ), the activation functions f j (·) and g i (·) are required to satisfy f j (0) = 0 and g i (0) = 0. In this paper, the activation functions f j (0, 0) = 0, g i (0, 0) = 0 are not required, as it only needs to meet the condition (H 1 ). In this paper, first, by using the exponential dichotomy of linear differential equations and the fixed point theory for contraction mapping, the existence of piecewise pseudo almost periodic solutions for system (1.1) satisfying the initial value conditions given by (1.2) is studied. And then by using mathematical methods of induction and inequality techniques, the global exponential stability of piecewise pseudo-almost periodic solutions for system (1.1) satisfying the initial value conditions given by (1.2) will be discussed. Remark 1.2. Compared with the systems studied in [11,16,17,[20][21][22]24,25], the system (1.1) studied in this paper is more general, as some classical Hopfield neural networks and BAM neural networks are special cases of the system (1.1). Duan [22] studied the effects of constant delays in the activation functions on the dynamic behavior of the system, but they did not consider the effects of leakage delays and impulsive perturbations on the dynamic behavior of the system. The mixed time-varying delays studied in this paper include not only the time-varying delays in the activation functions but also the leakage delays; the influences of impulsive perturbations on the dynamic behavior of the system are also considered. This paper not only considers the norm ∥ · ∥ ∞ but also ∥(·) ′ ∥ ∞ . Therefore, the results obtained in [21,22,24] are special cases of the research results in this paper.

Preliminaries
This section mainly gives the necessary definitions, lemmas and notations to prove the existence and global exponential stability of the piecewise pseudo almost periodic solutions of the system described by (1.1)-(1.2). C 1 (R, R n ) is the set of continuous functions with a continuous derivative, and BC(R, R n ) is the set of bounded continuous functions, and (BC(R, R n ), ∥ · ∥ ∞ ) is a Banach space. Let PC(R, R n ) be the set of all piecewise continuous functions ϕ : R → R n such that ϕ is countinuous at t for t {t k : k ∈ Z + } and that The set of all almost periodic functions is represented by AP(R, R n ).

Definition 2.2. [35]
A sequence {u n } is called almost periodic if for ∀ϵ > 0, there is a natural number L(ϵ) such that for k ∈ Z, there is at least one number q in [k, k + L(ϵ)] for which the inequality ∥u n+q − u n ∥ < ϵ holds for all n ∈ N. The set of all almost periodic sequences is represented by AP(Z, R n ).

Definition 2.3. [36]
A function ϕ ∈ PC(R, R n ) is called piecewise almost periodic if the following conditions are satisfied.
• (1) {t j k }, t j k = t k+ j − t k , k, j ∈ Z are equipotentially almost periodic, that is to say, for any ϵ > 0, there is a relatively dense set in R of ϵ-almost periods that are common for any of the sequences {t j k }. • (2) For any ϵ > 0, there is a constant δ(ϵ) > 0 such that, if t ′ and t ′′ belong to the same interval of continuity of ϕ and |t ′ − t ′′ | < δ, then ∥ϕ(t ′ ) − ϕ(t ′′ )∥ < ϵ.
Denote by AP T (R, R n ) the set of piecewise almost periodic functions; UPC(R, R n ) is the set of the function f ∈ PC(R, R n ) such that f satisfies the condition (2) in Definition 2.3. Let , and given t ∈ R, |t − t i | > ϵ.
Definition 2.5. [37] Sequence {u n } n∈Z ∈ L ∞ (Z, R n ) is called pseudo almost periodic if there exist u 1 n ∈ AP(Z, R n ) and u 2 n ∈ PAP 0 (Z, R n ) such that u n = u 1 n + u 2 n . The set of pseudo almost periodic sequences is denoted by PAP(Z, R n ).
Definition 2.6. [38] A function ϕ ∈ PC(R, R n ) is known as piecewise pseudo almost periodic if it can be disintegrated into ϕ = ϕ 1 + ϕ 2 , where ϕ 1 ∈ AP T (R, R n ) and ϕ 2 ∈ PAP 0 T (R, R n ). The set of piecewise pseudo almost periodic functions is denoted by PAP T (R, R n ).
Lemma 2.1. [38] Let the sequence of vector-valued functions {I i } i∈Z be pseudo almost periodic and there be a constant L > 0 such that Lemma 2.5. [40] Assume that f ∈ PAP T (R × Ω 1 × Ω 2 , X) and the following conditions hold: Definition 2.7.
[34] Let x ∈ R n and A(·) be an n × n continuous matrix defined on R. The linear system x ′ (t) = A(t)x(t) is said to admit an exponential dichotomy on R if there exist positive constants k and α and a projection P such that the fundamental solution matrix X(t) of x ′ (t) = A(t)x(t) satisfies the following: .., −a n (t))x(t) admits an exponential dichotomy on R.

Existence of piecewise pseudo almost periodic solutions
This section establishes the conditions of the existence for piecewise pseudo almost periodic solutions of the system described by (1.1)-(1.2).

Global exponential stability of piecewise pseudo almost periodic solutions
This section states and proves the sufficient conditions for the global exponential stability of piecewise pseudo almost periodic solutions of the system described by (1.1)-(1.2). (H 5 ) For i, j = 1, 2, · · · , m, Then, the piecewise pseudo almost periodic solutions of the system described by (1.1)-(1.2) is globally exponentially stable.

Conclusions
The pseudo almost periodic behaviors have been applied to the qualitative theory of differential equations. Piecewise pseudo almost periodic solutions of BAM neural networks are generalizations of almost periodic solutions, which have obvious scientific significance and application value in many fields such as signal processing, pattern recognition, associative memory, image processing and optimization problems.
This paper investigated the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By employing the exponential dichotomy of linear differential equations, fixed-point theory for contraction mapping, differential inequality techniques and mathematical methods of induction, the effective conditions for the existence and global exponential stability of piecewise pseudo almost periodic solutions of system (1.1) have been established. From Theorems 3.4 and 4.1, the delays contained in the activation functions do not affect the existence and global exponential stability of piecewise pseudo almost periodic solutions of system (1.1). The existence and global exponential stability of the piecewise pseudo almost periodic solutions of the system (1.1) are determined by the negative feedback terms, leakage delays, connection weights, impulsive perturbations and the Lipschitz constants of the activation functions.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.