Mittag-Leffler Stability and Attractiveness of Pseudo Almost Periodic Solutions for Delayed Cellular Neural Networks

We consider a class of nonautonomous cellular neural networks (CNNs) with mixed delays, to study the solutions of these systems which are type pseudo almost periodicity. Using general measure theory and the Mittag-Le ﬄ er function, we obtain the existence of unique solutions for cellular neural equations and investigate the Mittag-Le ﬄ er stability and attractiveness of pseudo almost periodic functions. We also present numerical examples to illustrate the application of our results.


Introduction
Due to the many applications of neural cell networks in various fields, these systems have been extensively studied. Image processing, robotics, optimization, etc. are among the fields used by these differential systems [1][2][3][4]. Due to the importance of network systems, stability analysis and synchronization control for these systems have always been considered by many researchers who have studied these systems with different tools. For example, we can mention [5][6][7][8], where Lyapunov functions have been used as a tool for these synchronization analyses.
We shall introduce a neural cellular system and investigate the solutions of this differential equation, which are of the ϕ-pseudo almost periodic type (for more details, see [9][10][11]). Assume that ϕ is a measure, η is a positive measurable function in ℝ, and ϕ 1 is singular Lebesgue measure. Here, measure ϕ is defined by dϕðyÞ = ηðyÞdy + dϕ 1 ðyÞ: The cellular neural system with mixed delay is described by z p ′ y ð Þ = −e p y ð Þz p y ð Þ + 〠 n q=1 ϑ pq y ð Þ Θ q z q y ð Þ À Á + 〠 n q=1 ϑ pq y ð ÞΘ q z q y − ζ pq À Á À Á This system with the initial value is expressed as follows: z p y ð Þ = ξ p y ð Þ, for y ≤ 0: The parameters in this equation are as follows: (i) z p ðyÞ is the p-th neuron state (ii) e p ðyÞ represents the rate of decay, (iii) Real functions Θ p , Θ p , b Θ p are activation functions of the p-th neuron (iv) L p ðyÞ is the input (v) ζ pq are the delays that are constant (vi) ψ pq is the transmission delay kernel Considering a special case of the stated measure, i.e., dϕðyÞ = ηðyÞdðyÞ, ϕ 1 = 0, the ϕ-pseudo almost periodic solutions of the above system are of the weighted pseudo almost periodic functions type.
We conclude the introduction by describing the structure of the paper. In Section 2, we collect the preliminary information. In Section 3, we present several examples of interesting measures. In Section 4, we prove our first main result (Theorem 19). In Section 5, we prove our second main result (Theorem 21). In Section 6, we prove our third main result (Theorem 23). In Section 7, we present some applications.

Preliminaries
We denote the space of all positive measures on Lebesgue ψ-field A with N . If μ is a positive measure, then we have Considering BCðℝ, YÞ as the space of all continuous and bounded functions, as well as the supremum norm kgk ∞ = sup y∈ℝ kgðyÞk, we have a Banach space.

Definition 1. The Mittag-Leffler function is defined by
where λ is a real number, λ ≤ 0, and g is a complex variable.
The generalization of E λ ðgÞ is defined as where λ, μ ∈ ℂ, Re ðλÞ > 0, Re ðμÞ > 0: Definition 2. If 0 < λ ≤ 1 and ν is a complex number, then is called the λ-order fractional hyperbolic cosine function and is called the λ-order fractional hyperbolic sine function.
Definition 6. Suppose that ϕ ∈ N , k, and ω are almost periodic and ϕ-ergodic functions, respectively. Then, g : ℝ ⟶ Y is a ϕ-pseudo almost periodic function, provided that g = k + ω: We denote the space of all almost periodically functions by AP ðℝ, YÞ, the space of all ϕ-ergodic functions by Eðℝ, Y, ϕÞ, and the space of all ϕ-pseudo almost periodic functions by P AP ðℝ, Y, ϕÞ. All these spaces, equipped with the supremum norm, are Banach spaces. Also, we have AP ðℝ, YÞ ⊂ P AP ðℝ, Y, ϕÞ ⊂ BCðℝ, YÞ; for more details, see [9]. Definition 7. Let z * ðyÞ = fz * p ðyÞg n p=1 be a solution of equation (2), with initial value fz * p ðyÞ: y ≤ 0g. Suppose that for every 2 Journal of Function Spaces solution zðyÞ = fz p ðyÞg n p=1 of equation (2) with initial value ξ = fξ p ðyÞg, there exist constants γ > 0 and W ξ > 1 such that for all y > 0, p = 1, 2, 3, ⋯, n, where Then, the property of Mittag-Leffler stability holds for z * .
Definition 8. Let z * ðyÞ = fz * p ðyÞg n p=1 be a solution of equation (2), with initial value fz * p ðyÞ: y ≤ 0g. Suppose that there exists ρ > 0 such that for any solution zðyÞ = fz p ðyÞg n p=1 of equation (2). Then, the property of Mittag-Leffler attractiveness holds for z * .
If the Mittag-Leffler stability for any solution of equation (2) is established, then z depends on its initial value fzðyÞ: −∞<y ≤ 0g.
Definition 10 (see [28]). Let ðG, AÞ be a Borel space. If ϕ and τ are measures on ðG, AÞ, we say that ϕ and τ are mutually singular, if there exist disjoint sets R and D in A such that G = R ∪ D and τðRÞ = ϕðDÞ = 0: Definition 11 (see [28]). Assume that ϕ and τ are measures on the Borel space ðG, AÞ. We say that τ is absolutely continuous relative to ϕ, provided that ðϕðRÞ = 0Þ ⇒ ðτðRÞ = 0Þ, for each R ∈ A.
Theorem 12 (see [29]). For any integrable function g : ℝ ⟶ ℝsuch that g ∈ AP ðℝ, ℝÞ, we have g⋆k ∈ AP ðℝ, ℝÞ: Theorem 13 (see [30]). For any ζ on the interval with positive length l ϱ and any ϱ > 0, we have where g, k ∈ AP ðℝ, ℝÞ, for all y ∈ ℝ. In particular, kg ∈ AP ðℝ, ℝÞ: For a globally Lipschitzian mapping Q : Y ⟶ Zsuch that Y and Z are Banach spaces and every almost periodic functions ω, we have Q ∘ ω ∈ AP ðℝ, YÞ, which means that Q ∘ ω is an almost periodic function. Example 15 (see [9]). We consider a measure ϕ ∈ N which is not absolutely continuous and satisfies ðI 8 Þ. This measure is defined as dϕðyÞ = dy + dx, where dy is a measure of the Lebesgue type. Also, x is the measure on ðℝ, AÞ, which in A is the ψ-field of the Lebesgue type. This measure is defined as follows:

Examples of Measures Satisfying Hypotheses
Example 16. Consider the following measure: where ψ ≥ 0, ϰ > 0, and according to the integer n, δ n is a Dirac measure (DM), and is a generalized Dirac comb (GDC). When ψ = 0, this measure is called a Dirac comb (DC).
We also have that The conclusion follows with ζ 0 = 0 and

Journal of Function Spaces
This means that ðP AP ðℝ, ℝ n , ϕ ψ,u Þ, k:k ∞ Þ is a Banach space.
Example 17. We consider the following measure for ϕ ∈ N , where δ 1/n is the Dirac measure at 1/n and satisfying ðI 4 Þ.
In the sequel, we set z p ðyÞ = σ −1 y zðyÞ. Then, equation (1) is transformed into the following system: Now we show that the integral solutions of equation (34) are mappings of P AP ðℝ, ℝ n , ϕÞ to itself.

Existence and Uniqueness of ϕ-Pseudo Almost Periodic Solutions
Assuming that the solution of equation (1) is of the ϕ-pseudo almost periodic type, we shall prove the existence and uniqueness of these solutions.
In the sequel, we shall investigate the Mittag-Leffler stability and the Mittag-Leffler attractiveness for the unique solution of equation (2), which is of type ϕ-P AP . First, we state Lemma 22.

Conclusion
In this work, we considered differential systems of cellular neural networks (CNNs) with mixed delays. We also considered general measurement theory whose general form is dϕ = ηðyÞdy + dϕ 1 . We first investigated the existence of a unique solution of this system and proved that the solutions of equation (1) are ϕ-pseudo almost periodic. Then we studied the Mittag-Leffler stability and the Mittag-Leffler attractiveness of these solutions. We obtained our results by considering new conditions and using the fixed point contraction mapping theorem. Also, two examples were given to illustrate our results.

Data Availability
The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.