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

Authors: Zahra Eidinejad; Reza Saadati; Dušan D. Repovš
Journal: Journal of Function Spaces
 (2022)

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## Abstract

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-Leffler function, we obtain the existence of unique solutions for cellular neural equations and investigate the Mittag-Leffler stability and attractiveness of pseudo almost periodic functions. We also present numerical examples to illustrate the application of our results.


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## Body

## 1. 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 [1–4]. 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 [5–8], 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 theϕ-pseudo almost periodic type (for more details, see [9–11]). Assume that ϕ is a measure, η is a positive measurable function in ℝ, and ϕ1 is singular Lebesgue measure. Here, measure ϕ is defined by dϕy=ηydy+dϕ1y.The cellular neural system with mixed delay is described by(1)zp′y=−epyzpy+∑q=1nϑ¯pqyΘ¯qzqy+∑q=1nϑpqyΘqzqy−ζpq+∑q=1nhpqy∫0∞ψpqrΘ∧qzqy−rdr+Lpy,fory∈ℝ.This system with the initial value is expressed as follows:(2)zp′y=−epyzpy+∑q=1nϑ¯pqyΘ¯qzqy+∑q=1nϑpqyΘqzqy−ζℓj+∑q=1nhpqy∫0∞ψpqrΘ∧qzqy−rdr+Lpy,fory≥0,(3)zpy=ξpy,fory≤0.The parameters in this equation are as follows:(i)
zpy is the p-th neuron state(ii)
epy represents the rate of decay,(iii)
Real functionsΘ¯p,Θp,Θ∧p are activation functions of the p-th neuron(iv)
Lpy is the input(v)
ζpq are the delays that are constant(vi)
ψpq is the transmission delay kernelConsidering a special case of the stated measure, i.e.,dϕy=ηydy,ϕ1=0, the ϕ-pseudo almost periodic solutions of the above system are of the weighted pseudo almost periodic functions type.In the present paper, we shall derive some sufficient conditions for existence and uniqueness results for cellular neural equations [3, 12–14]. We first state the basic concepts and then obtain the unique solution for equation (1). In the sequel, we prove our main results, i.e., the Mittag-Leffler stability and attractiveness of ϕ-pseudo almost periodic solutions of equation (2), which improves upon and extends [11, 15–20].We conclude the introduction by describing the structure of the paper. In Section2, 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.


## 2. Preliminaries

We denote the space of all positive measures on Lebesgueψ-field A with N. If μ is a positive measure, then we have
(i)
ϕℝ=+∞(ii)
ϕℓ,j<∞, for all ℓ,j∈ℝℓ≤jConsideringBCℝ,Y as the space of all continuous and bounded functions, as well as the supremum norm g∞=supy∈ℝgy, we have a Banach space.Definition 1.
The Mittag-Leffler function is defined by(4)Eλg=∑ς=0∞gςΓςλ+1,where λ is a real number, λ≤0, and g is a complex variable. The generalization of Eλg is defined as
(5)Eλ,μg=∑ς=0∞gςΓςλ+μ,where λ,μ∈ℂ,Reλ>0,Reμ>0.Definition 2.
If0<λ≤1 and ν is a complex number, then
(6)coshλνxλ=∑ς=0∞ν2ςx2ςλΓ2ςλ+1is called the λ-order fractional hyperbolic cosine function and
(7)sinhλνxλ=∑ς=0∞ν2ς+1x2ς+1λΓ2ς+1λ+1is called the λ-order fractional hyperbolic sine function.Proposition 3.
Assume that0<λ≤1. Then,
(8)coshλνxλ=Eλνxλ+Eλ−νxλ2,sinhλνxλ=Eλνxλ−Eλ−νxλ2.Definition 4.
A continuous functiong:ℝ⟶Y is said to be almost periodic if gy+ζ−gy<ϖ, for all y∈ℝ,ϖ>0,ζ∈ℓ,j.Definition 5.
Letϕ∈N. A bounded continuous function g:ℝ⟶Y is said to be ϕ-ergodic if
(9)limu⟶+∞1ϕ−u,u∫−uugydϕy=0.Definition 6.
Suppose thatϕ∈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 byAPℝ,Y, the space of all ϕ-ergodic functions by Eℝ,Y,ϕ, and the space of all ϕ-pseudo almost periodic functions by PAPℝ,Y,ϕ. All these spaces, equipped with the supremum norm, are Banach spaces. Also, we have APℝ,Y⊂PAPℝ,Y,ϕ⊂BCℝ,Y; for more details, see [9].Definition 7.
Letz∗y=zp∗yp=1n be a solution of equation (2), with initial value zp∗y:y≤0. Suppose that for every solution zy=zpyp=1n of equation (2) with initial value ξ=ξpy, there exist constants γ>0 and Wξ>1 such that
(10)zpy−zy∗y≤Wξξ−z∗1∑ς=0∞−γyςΓςλ+1,for all y>0,p=1,2,3,⋯,n, where
(11)ξ−z∗1=sup−∞≤c≤0maxp=1,2,3,⋯,nξpc−zp∗c.Then, the property of Mittag-Leffler stability holds for z∗.We can derive the Mittag-Leffler attractiveness from the Mittag-Leffler stability; for more details, see [21–27].Definition 8.
Letz∗y=zp∗yp=1n be a solution of equation (2), with initial value zp∗y:y≤0. Suppose that there exists ρ>0 such that
(12)limy⟶+∞∑ς=0∞ϱyςΓςλ+1zy−z∗y=0,for any solution zy=zpyp=1n 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 zy:−∞<y≤0.Definition 9.
The convolution of functionsv and x from ℝ to ℝ, if any, is defined as follows:
(13)v⋆xy≔∫−∞+∞vrxy−rdr,where ϕ∈N and for p,q=1,2,3,⋯,n,ϑ¯pq,ϑpq,hpq,Lp∈PAPR,ℝ,ϕ, and ep∈APR,ℝ.Definition 10 (see [28]).
LetG,A be a Borel space. If ϕ and τ are measures on G,A, we say that ϕ and τ are mutually singular, if there exist disjoint sets R and Din A such that G=R∪D and τR=ϕD=0.Definition 11 (see [28]).
Assume thatϕ and τ are measures on the Borel space G,A. We say that τ is absolutely continuous relative to ϕ, provided that ϕR=0⇒τR=0, for each R∈A.Following Lebesgue-Radon-Nikodym [28], we assume that dϕy=ηydy+dϕ1. We impose the following assumptions for every p=1,2,3,⋯,n:
(I1)
Θ¯q,Θq,Θq are globaly Lipschitzian with Lipschitz constants TqΘ¯,TqΘ, and TqΘ∧, respectively(I2)
ψpq:ℝ+⟶ℝ is bounded and continuous(I3)
There existsn>0 such that
(14)ψpqy∑ς=0∞nyςΓςλ+1is integrable on ℝ+(I4)
For the bounded intervalL and all ζ∈ℝ, there exists ε>0 such that ϕR+ζ≤εϕR, when R∈A satisfies R∩L=∅(I5)
There existe∧p∈BCℝ,0,+∞,Op>0, such that
(15)∑ς=0∞−∫byeprdrςΓςλ+1≤Op∑ς=0∞−∫bye∧prdrςΓςλ+1,forally,b∈ℝ,y≥b,e∧p+=supy∈ℝe∧py,e∧p−=infy∈Re∧py>0(I6)
There existe∧p∈BCℝ,0,+∞,Op>0,κp>0 and σp>0 such that
(16)supr∈ℝ−e∧py+Opσp−1∑q=1nϑ¯pyTqΘ¯+ϑpqyTqσ+T^qΘ∧hpy∫0∞ψprdrσq≪−κp<0(I7)
There existe∧p∈BCℝ,0,+∞,Op>0,κp>0 and σp>0 such that
(17)0<maxp=1,2,3,⋯,ne∧p+e∧p−−κpe∧p+<1(I8)
For allℓ,j,i∈ℝ such that 0≤ℓ<j≤i, there exist ζ0≥0 and λ0>0 such that
(18)ζ≥ζ0⟹ϕℓ+ζ,j+ζ≥λ0ϕζ,i+ζHypothesisI4 implies hypothesis I8, whereas the converse is not true. Also, if hypothesis I8 holds, then PAPℝ,ℝn,ϕ,·∞ is a Banach space. If hypothesis I4holds, then for any g∈PAPℝ,ℝn,ϕ,ζ∈ℝ, gy−ζ∈PAPℝ,ℝn,ϕ,U∈Eℝ,ℝ,ϕ, and E∈L1ℝ, we have E⋆U∈Eℝ,ℝ,ϕ. The proofs can be found in [9].Theorem 12 (see [29]).
For any integrable functiong:ℝ⟶ℝsuch that g∈APℝ,ℝ, we have g⋆k∈APℝ,ℝ.Theorem 13 (see [30]).
For anyζ on the interval with positive length lϱ and any ϱ>0, we have
(19)gy−ζ−gy<ϱ,ky−ζ−ky<ϱ,where g,k∈APℝ,ℝ, for all y∈ℝ. In particular, kg∈APℝ,ℝ.Remark 14.
For a globally Lipschitzian mappingQ:Y⟶Zsuch that Y and Z are Banach spaces and every almost periodic functions ω, we have Q∘ω∈APℝ,Y, which means that Q∘ω is an almost periodic function.


## 3. Examples of Measures Satisfying HypothesesI4 and I8

Next, we shall introduce three examples of measures which satisfy hypothesesI4 and I8.Example 15 (see [9]).
We consider a measureϕ∈N which is not absolutely continuous and satisfies I8. This measure is defined as dϕy=dy+dx, where dy is a measure of the Lebesgue type. Also, x is the measure on ℝ,A, which in A is the ψ-field of the Lebesgue type. This measure is defined as follows:
(20)xR=cardR∩ℤ,ifR∩ℤisfinite,∞,ifR∩ℤisinfinite.Example 16.
Consider the following measure:(21)dϕψ,ϰy=∑ς=0∞ψyςΓςλ+1dy+ϰ∑n=−∞∞∑ς=0∞ψnςΓςλ+1δn,where ψ≥0,ϰ>0, and according to the integer n, δn is a Dirac measure (DM), and
(22)∑n=−∞∞∑ς=0∞ψnςΓςλ+1δnis a generalized Dirac comb (GDC). When ψ=0, this measure is called a Dirac comb (DC).
Sinceζ,b+ζ⊂ζ,b+1+b, we shall show that I8 is satisfied for b>0, such that b≥j:(23)ϕψ,ϰζ,b+ζ=∫ζi+ζ∑ς=0∞ψyςΓςλ+1dy+ϰ∑n∈ζ,b+ζ∑ς=0∞ψnςΓςλ+1≤∑ς=0∞ψςb+ζς+1/ς+1−ζς+1/ς+1Γςλ+1+ϰ∑n=ζb+ζ∑ς=0∞ψnςΓςλ+1=∑ς=0∞ψςb+ζς+1/ς+1−ζς+1/ς+1Γςλ+1+ϰ∑ς=0∞ψζςΓςλ+1∑z=0b∑ς=0∞ψzςΓςλ+1=∑ς=0∞ψςΓςλ+1b+ζς+1ς+1−ζζ+1+ϰζς∑z=0b∑ς=0∞ψzςΓςλ+1.
We also have that(24)ϕψ,ϰℓ+ζ,ȷ+ς≥∫ℓ+ζȷ+ζ∑ς=0∞ψyςΓςλ+1dy=∑ς=0∞ψςΓςλ+1ȷ+ςς+1ς+1−ℓ+ζς+1ς+1.
The conclusion follows withζ0=0 and
(25)ϱ0=∑ς=0∞ψ/Γςλ+1ȷ+ζς+1/ς+1−ℓ+ζς+1/ς+1∑ς=0∞ψ/Γςλ+1b+ζς+1/ζ+1−ζ/ζ+1+ϰζς∑z=0b∑ς=0∞ψzς/Γςλ+1e.
This means thatPAPℝ,ℝn,ϕψ,ù,.∞ is a Banach space.
The measureϕψ,ù does not satisfy I4. In the sequel, we shall prove this. Let
(26)D=∪n=−∞∞n−12−ϱn,n−12+ϱn,ϱ=1ψsinh−112n+1∑ς=0∞ψnς/Γςλ+1,ζ=12.
Then,D+ζ=∪n=−∞∞n−ρn,n+ρn contains ℤ and
(27)ϕψ,ϰD+ζ=∑n=−∞∞∫n−ϱnn+ϱn∑ς=0∞ψyςΓςλ+1dy+∑n=−∞∞∑ς=0∞ψnςΓςλ+1=∑n=−∞∞12n+1+∞=∞,provided that D∩ℤ=∅. Now,
(28)ϕψ,ùD=∑n=−∞∞∫n−1/2+ρnn+1/2+ρn∑ς=0∞ψyςΓςλ+1dy+0=∑ς=0∞ψ/2ςΓςλ+1∑n=−∞∞12n+1<∞.
Therefore, ifR=D/L, where L is a bounded interval, we obtain
(29)ϕψ,ϰR+ζ≥∑n∈ℤ/L∑ς=0∞ψnςΓςλ+1=∞,provided that ϕψ,ϰR≤ϕψ,ϰD<∞.Example 17.
We consider the following measure forϕ∈N,ψ⩾0,ϰ>0,(30)dϕψ,ϰy=∑ς=0∞ψyςΓςλ+1dy+ϰ∑n=1∞1n2δ1/n,where δ1/n is the Dirac measure at 1/n and satisfying I4.
Forζ∈ℝ, let L=−1−ζ,1+ζ. We can easily see that R∩L=∅, R∩0,1=∅ and also R+ζ∩0,1=∅. Then,
(31)ϕR+ζ=∫R+ζ∑ς=0∞ψyςΓςλ+1dy+0=∑ς=0∞ψςΓςλ+1∫R+ζyςdy=∑ς=0∞ψςΓςλ+1R+ζς+1ς+1=∑ς=0∞ψςΓςλ+1R+ζς+1ς+1R+ζς+1ζ−ςR−ς+1=∑ς=0∞ψζςΓςλ+1Rς+1ς+1=ϕR.
The conclusion now follows withϑ=1.


## 4. On the Integral Solution of Equation (34)

Proposition 18.
Assume thatI1 and I2 hold. If zq∈BCℝ,ℝ, then
(32)ϑpqyΘqzqy−ζpq∈BCℝ,ℝ,ϑ¯pqyΘ¯qzqy∈BCℝ,ℝ,hpqy∫0∞ψpqrΘ∧qzqy−rdr∈BCℝ,ℝ,for p,q=1,2,3⋯,n.
Assume further thatI1,I2, and I4 hold. If zq∈PAPℝ,ℝ,ϕ, then
(33)ϑpqyΘqzqy−ζpq∈PAPℝ,ℝ,ϕ,ϑ¯pqyΘ¯qzqy∈PAPℝ,ℝ,ϕ,hpqy∫0∞ψpqrΘ∧qzqy−rdr∈PAPℝ,ℝ,ϕ.Proof.
This follows from [15] (Theorem 4.1).
In the sequel, we setz¯py=σy−1zy. Then, equation (1) is transformed into the following system:
(34)z¯py=−epyz¯py+σp−1∑q=1nϑpyΘ¯qσz¯py+σp−1∑q=1nϑpqyΘqσz¯qy−ζpq+σp−1∑q=1nhpqy∫0∞ψpqvΘ∧qσpz¯qy−vdv+σp−1Lpy.
Now we show that the integral solutions of equation (34) are mappings of PAPℝ,ℝn,ϕ to itself.Theorem 19.
Assuming thatI1,I2, and I5 hold, we define the nonlinear mapping P on BCℝ,ℝn for p=1,2,3,⋯,n as follows:
(35)PUpy=∫0y∑ς=0∞−∫byeprdrςΓςλ+1σ−1∑q=1nϑ¯pqbΘ¯qσqUqb+σp−1∑q=1nϑpqbΘqσqUqb−ζq+σp−1∑p=1nhpqb∫0∞ψpqvΘ∧qσqUqb−vdv+σp−1Lpbdb.
If we assume conditionI4 along with the other three conditions, then P∈PAPℝ,ℝn,ϕ.Proof.
We havePU∈BCℝ,ℝn (see [29]). According to Proposition 18, for p=1,2,3,⋯,n, there exist Δp∈APℝ,ℝ and Ψp∈Eℝ,ℝ,ϕ such that
(36)σp−1∑q=1nϑ¯pqyΘ¯qσqUqy+σp−1∑q=1nϑpqyΘqσqUqy−ζpq+σp−1∑q=1nhpqy×∫0∞ψpqvΘ∧qσqUqy−vdv+σp−1Lpy=Δp+Ψp∈PAPℝ,ℝ,ϕ.(1)
We claim that(37)∫0y∑ς=0∞−∫byeprdrΓςλ+1Δpbdb∈APℝ,ℝ,p=1,2,3,⋯,n.
In fact,(38)Λpy≕∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpbdb,p=1,2,3,⋯,n.
According to Theorem13, and since Δp,ep∈APℝ,ℝ, for every ϖ>0, there exists a number such as ζ belonging to an interval of positive length lϖ such that Δpy+ζ−Δpy<ϖ, and
(39)∑ς=0∞−∫b+ζα+ζeprdrΓςλ+1epy+ζ−∑ς=0∞−∫αyeprdrΓςλ+1epy<ϖ,for all y∈ℝ (see [30]). Then,
(40)Λpy+ζ−Λpy=∫0y+ζ∑ς=0∞−∫by+ζeprdrςΓςλ+1Δpbdb−∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpbdb=∫0y∑ς=0∞−∫b+ζy+ζeprdrςΓςλ+1Δpb+ζdb−∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpbdb=∫0y∑ς=0∞−∫b+ζy+ζeprdrςΓςλ+1Δpb+ζdb−∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpb+ζdb+∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpb+ζdb−∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpbdb≤∫0y∑ς=0∞−∫b+ζy+ζeprdrςΓςλ+1−∑ς=0∞−∫byeprdrςΓςλ+1Δpb+ζdb+∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpb+ζ−Δprdr.
Let(41)J=∑ς=0∞−∫b+ϖy+ζeprdrςΓςλ+1−∑ς=0∞−∫byeprdrςΓςλ+1.
Since(42)∑ς=0∞−∫byeprdrςΓςλ+1y′=ς−∫byeprdrς−1−epr,ep∈APℝ,ℝ,using assumption I8, we obtain that
(43)J=∑ς=0∞−∫αyeprdrςΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrςΓςλ+1α+by=∫by∑ς=0∞−∫αyeprdrςΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrςΓςλ+1α′dα+∫st∑ς=0∞−∫αyeprdrςΓςλ+1α′∑ς=0∞−∫b+ζα+ζeprdrεΓςλ+1dα=∫by∑ς=0∞−∫αyeprdrςΓςλ+1∑ς=0∞ς−∫b+ζα+ζeprdrς−1−epα+ζΓςλ+1dα+∫by∑ς=0∞ς−∫αyeprdrς−1−epαΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrΓςλ+1dα=∫by∑ς=0∞−∫αyeprdrςΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrςΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrΓςλ+1epα+ζ−∑ς=0∞−∫αyeprdrΓςλ+1epαdα≤∫by∑ς=0∞−∫αyeprdrςΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrςΓςλ+1∑ς=0∞−∫b+ζα+ζeprdrΓςλ+1epα+ζ−∑ς=0∞−∫αyeprdrΓςλ+1epαdα≤Op2∈∫by∑ς=0∞−e∧p−y−bςΓςλ+1dα≤Op2ρ∑ς=0∞−e∧p−y−bςΓςλ+1y−b.
Then,Δp is a continuous and bounded function, given that Δp∈APℝ,ℝ. Now, by putting equation (39) in equation (38), we have
(44)Λpy+ζ−Λpy≤Op2ρ∫0y∑ς=0∞−e∧p−y−bςΓςλ+1y−bΔpb+ζdb+ρ∫0y∑ς=0∞−∫byeprdrςΓςλ+1db≤Op2ρΔp∞∫0y∑ς=0∞−e∧p−y−bςΓςλ+1y−bdb+Opρ∫−∞y∑ς=0∞−∫bye∧prdrςΓςλ+1db≤Op2ρΔp∞∫0y∑ς=0∞−e∧p−y−bςΓςλ+1y−bdb+Opρ∫0y∑ς=0∞−e∧p−y−bςΓςλ+1dλ≤Op2ρΔp∞∑ς=0∞e∧p−ςyς+2Γςλ+1ς+2+Opρ∑ς=0∞e∧p−ςyς+1Γςλ+1ς+1.
We obtain that(45)∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpbdb∈APℝ,ℝ.(2)
Let us show that(46)∫0y∑ς=0∞−∫byeprdrςΓςλ+1Ψpbdb∈Eℝ,ℝ,ϕ.
According to hypothesisI4, for Ψp∈Eℝ,ℝ,ϕ and for p=1,⋯,n, we have
(47)0=limu⟶+∞1ϕ−u,u∫−uu∫0y∑ς=0∞−e∧p−rςΓςλ+1Ψpy−rdrdϕy=limu⟶+∞1ϕ−uu∫−uu∫0y∑ς=0∞−e∧p−y−bςΓςλ+1Ψpbdbdϕy.
Then, byI5,
(48)0≤limu⟶+∞1ϕ−u,u∫−uu∫0y∑ς=0∞−∫byepξdξςΓςλ+1Ψpbdbdϕy≤Oplimu⟶+∞1ϕ−u,u∫−uu∫0y∑ς=0∞−e∧p−y−bςΓςλ+1Ψpbdbdϕy=0.
Hence,(49)∫0y∑ς=0∞−∫byeprdrςΓςλ+1Ψpbdb∈Eℝ,ℝ,ϕ,for p=1,2,3,⋯,n. Combined with (37), we have
(50)PUpy=∫0y∑ς=0∞−∫byeprdrςΓςλ+1Δpbdb+∫0y∑ς=0∞−∫byeprdrςΓςλ+1Ψpbdb∈PAPℝ,ℝ,ϕ.


## 5. Existence and Uniqueness ofϕ-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.Theorem 21.
1 Given assumptions I1,I2, and I5, there is a unique solution z∗∈BCℝ,ℝn for equation (1).
2 Given assumptions I1,I2, I5, I4,I6, and I7, we have z∗∈PAPℝ,ℝn,ϕ.Proof.
LetU,G∈BCℝ,ℝn (resp., U,G∈PAPℝ,ℝn,ϕ). Then, in view of Theorem 19, we have that PU,PG∈BCℝ,ℝn (resp. PU,PG∈PAPℝ,ℝn,ϕ).
LetS=PUpy−PGpy. Using I1 and I5,I6, we get that
(51)S=∫0y∑ς=0∞−∫byeprdrςΓςλ+1σp−1∑q=1nϑ¯pqbΘ¯qσqUqb−Θ¯qσqGqb+σp−1∑q=1nϑpqbΘqσqUqb−ζpq−ΘqσqGqb−ζpq+σp−1∑q=1nhpqb∫0∞ψpqvΘ∧qσqUqb−v−Θ∧qσqUqb−vdvdb≤Opσp−1∫0y∑ς=0∞−∫bye∧prdrςΓςλ+1∑q=1nϑ¯pqbTqΘ¯+ϑpqbTqΘ+TqΘ∧hpqs∫0∞ψpqrdrσqdbU−G∞≤∫0y∑ς=0∞−∫bye∧prdrςΓςλ+1e∧pb−κpdbU−G∞≤e∧p+∫0y∑ς=0∞−∫bye∧prdrςΓςλ+1db−κp∫0y∑ς=0∞−∫bye∧prdrςΓςλ+1dbU−G∞≤maxp=1,2,3,ne∧p+e∧p−−κpe∧p+U−G∞,so
(52)PUpy−PGpy≤maxp=1,2,3,⋯,ne∧p+e∧p−−κpe∧p+U−G∞.
Invoking conditionI7 and 0<maxp=1,2,3,⋯,ne∧++/ep−−κ1/ep+<1, we conclude that P∈BCℝ,ℝn is a contraction (since PAPℝ,ℝn,μ is a Banach space, according to condition I3 it is also a contraction on this space). Therefore, we conclude that z¯∗∈BCℝ,ℝn (or z¯∗∈PAPℝ,ℝn,ϕ is a unique fixed point for P. Also, given (34), z∗∈BCℝ,ℝn (or z∗∈PAPℝ,ℝn,ϕ) is a unique solution of type ϕ-PAP for equation (1).
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-PAP. First, we state Lemma 22.Lemma 22.
Assuming conditionsI3 and I6, we define Z=Zpp=1n:0,m⟶ℝnbyZpw=supy∈ℝZpw,y, where
(53)Zpw,y=w−e∧py+Opσp−1∑q=1nϑ¯pqyTqΘ¯+ϑpqyTqΘ∑ς=0∞wζpqςΓςλ+1+TqΘ∧hpqy∫0∞ψpqr∑ς=0∞wrςΓςλ+1drσq.
ThenZpν<0 for all 0<ν0<m and 0<ν<ν0.Proof.
According to conditionI3, function y↦Zpw,y is defined on the interval 0,m. Then according to condition I6, we have
(54)Zp0=supy∈ℝZp0,y=supy∈ℝ−e^py+Opσp−1∑q=1nϑ¯pqyTqΘ¯+ϑpqyTqΘ+TqΘ∧hpqy∫0∞ψpqrdrσq<−κp<0.
Next, we shall show that there exists0<ν0<m such that Zpν<0, for all 0<ν<ν0. Also, we have
(55)Zpw,y−Zp0,y=w+Opσp−1∑q=1nϑpqyTqΘ∑ς=0∞wζpqςΓςλ+1−1+TqΘhpqy∫0∞ψpqr∑ς=0∞wrςΓςλ+1dr−∫0∞ψpqrdrσq.
If we takeϑp and hp nonnegative numbers
(56)ϑp=supq=1nϑpqyTqΘ,hp=supq=1nhpqyTqΘ∧,then
(57)Zpw,y−Zp0,y≤w+Opσp−1ϑp∑q=1n∑ς=0∞wζpqςΓςλ+1−1σq+Oqσp−1hp∑q=1n∫0∞ψpqr∑ς=0∞wrςΓςλ+1dr−∫0∞ψpqrdrσq,for all w∈0,m and y∈ℝ. Now, for every ρ>0, by continuity, there exists 0<δp1<m such that the following holds:
(58)w<δp1⇒w+Opσp−1ϑp∑q=1n∑ς=0∞wζpqςΓςλ+1−1σq<ρ2.
In the sequel, invoking conditionI3, the Lebesgue dominated convergence theorem (LDCT), and the integrability of the function ψpqr∑ς=0∞ςrς/Γςλ+1 on the interval 0,∞, we get 0<δp2<m such that w<δp2 implies
(59)Opσp−1hp∑q=1n∫0∞ψpqr∑ς=0∞wrςΓςλ+1dr−∫0∞ψpqrdrσq<ρ2.
If we now take0<ν<minδp1,δp2 and ρ<κp/2, we can conclude that for every y∈ℝ,
(60)Zpν,y≤Zp0,y+Zpν,y−Zp0,y<−κp+ρ<−κ2.


## 6. On the Mittag-Leffler Stability and Attractiveness of Unique Solutions

Theorem 23.
If we assume conditionsI1,I3, I5 and I6, then the Mittag-Leffler stability for any solution z∗ of equation (2) is established by the initial condition z∗y:y≤0. If we add condition I2, then the Mittag-Leffler stability for the solution z∗∈BCℝ,ℝn of equation (2) is also established. Also, if we add condition I7 to conditions I1,I3, I5, I6, and I2, then the Mittag-Leffler stability for the solution z∗∈PAPℝ,ℝn,μ of equation (2) is also established.Proof.
Letzt=zpyp=1n be a solution of equation (2) with initial value ξy=ξpyp=1n. Set
(61)fy=fpyp=1n=σi−1zpy−zp∗yp=1n.
Then,(62)fp′b+epbfpb=σp−1∑q=1nϑ¯pqbΘ¯qzqb−Θ¯qzq∗b+σp−1∑q=1nϑ¯pqb×Θqzqb−ζpq−Θqzq′b−ζpq+σp−1∑q=1nhpqb×∫0∞ψpqvΘ∧qzqb−v−Θ∧qzq∗b−vdv.
Letξ−z∗σ=supy∈−∞,0maxp=1,2,3,⋯,nσp−1ξpy−zp∗y, and ℵ>0 be such that
(63)ℵ>∑p=1nOp+1.
Then, for ally∈−∞,0, we have
(64)fy≤ξ−z∗σ.
Givenν0 in Lemma 22, we assume that 0<ν<minminp=1,2,⋯,ne∧p+,ν0. Then for all y,ρ>0, we have
(65)fy<ℵξ−z∗σ+ρ∑ς=0∞−νyςΓςλ+1.
Otherwise, forp=1,2,3,⋯,n and ù>0, given the continuity, we have
(66)fpϰ=ℵξ−z∗σ+ρ∑ς=0∞−νϰςΓςλ+1,fy<ℵξ−z∗σ+ρ∑ς=0∞−νyςΓςλ+1,forally∈−∞,ϰ.
Now, first, we multiply both sides of (62) by
(67)∑ς=0∞∫0beprdrςΓςλ+1.
Then, we integrate the obtained equation with respect tob on 0,ù. Finally, we multiply by
(68)∑ς=0∞−∫0yeprdrςΓςλ+1.
Therefore, we have(69)fpϰ=fp0∑ς=0∞−∫0ϰeprdrςΓςλ+1+σp−1∫0ϰ∑ς=0∞−∫bϰeprdrςΓςλ+1∑q=1nϑ¯pqbΘ¯qzqb−Θ¯qzq∗b+ϑpqbΘqzqb−ζpq−Θqzq∗b−ζpq+hpqb∫0∞ψpqvΘ∧qzqb−v−Θ∧qzq∗b−vdvdb.
Next, byI1 and I5we obtain that
(70)fpϰ≤Opfp0∑ς=0∞−∫0ϰe∧prdrςΓςλ+1+σp−1∫0ϰ∑ς=0∞−∫bϰe∧prdrςΓςλ+1∑q=1nϑ¯pqbTqΘ¯σqfqb+ϑpqbTqΘσqfqb−ζpq+TqΘ∧hpqb∫0∞ψpqvσqfqb−vdvdb.
Now, by (64) and (66), we get
(71)fpϰ≤Opξ−z∗σ+ρ∑ς=0∞−∫0ϰe∧prdrςΓςλ+1+ℵσp−1∫0ϰ∑ς=0∞−∫bϰe∧prdrςΓςλ+1∑q=1nσqϑ¯pqbTqΘ¯∑ς=0∞−νbςΓςλ+1+ϑpqbTqΘ∑ς=0∞−νb−ζqςΓςλ+1+TqΘhpqb∫0∞ψpqr∑ς=0∞−νb−rςΓςλ+1drdb
Sinceb>0 and 0<ν<ν0, we have by virtue of Lemma22,
(72)fpϰ≤Opξ−z∗σ+ρ∑ς=0∞−∫0ϰe∧prdrςΓςλ+1+ℵOp−1∫0ù∑ς=0∞−∫bϰe∧prdrςΓςλ+1e∧pb−ν∑ς=0∞−νbςΓςλ+1=Opξ−z∗σ+ρ∑ς=0∞−νϰ−∫0ϰe∧pr−νdrςΓςλ+1+ℵOp−1−νϰς∫0ϰ∑ς=0∞−∫bϰe∧pr−νdrςΓςλ+1e∧pb−νdb=Opξ−z∗σ+ρ∑ς=0∞−νϰςΓςλ+1−∫0ϰe∧pr−νdrς+ℵOp−1∫0ϰ−∫bϰe∧pr−νdrςe∧pb−νdb=Opξ−z∗σ+ρ∑ς=0∞−νϰςΓςλ+1−∫0ϰe∧pr−νdrς+ℵOp−1−∫bϰe∧pr−νdrς0ϰ=Opξ−z∗σ+ρ∑ς=0∞−νϰςΓςλ+1−∫0ϰe∧pr−νdrς+ℵOp−10−−∫0ϰe∧pr−νdrς=ℵξ−z∗σ+ρ∑ς=0∞−νϰςΓςλ+1−1−Opℵ∑ς=0∞−∫0ϰe∧pr−νdrςΓςλ+1.
We recall thatν<e∧p−. This implies that e∧pr−ν>0 for all r∈0,∞. Hence, by (61), we see that
(73)fpϰ<ℵξ−z∗σ+ρ∑ς=0∞−νϰςΓςλ+1,which contradicts (66) and we can conclude that what was claimed in (65) is true.
We now assume thaty is constant and ρ tends to zero. Then we get
(74)fy≤ℵξ−z∗σ∑ς=0∞−νϰςΓςλ+1.so the proof is complete.Corollary 24.
If we assume that conditionsI1,I2,I3, I5, and I6 hold, then the Mittag-Leffler attractiveness for unique solution z∗∈BCℝ,ℝn of equation (2) holds.Corollary 25.
If we assume that conditionsI4, and I7 hold, then the Mittag-Leffler attractiveness for unique solution z∗∈PAPℝ,ℝn of equation (2) holds.


## 7. Applications

We shall provide two examples (see Figures1–3).Example 26.
Letn=2,p,q=1,2,
(P1)
We consider Lipschitz functionsΘ¯pr=0,Θpr=Θ∧pr=2/5arctanr, with the Lipschitz constants TpΘ¯=0,TΘ=TpΘ=2/5.(P2)
Then,ψpq is bounded and continuous,
(75)ψpqy=25∑ς=0∞−8yςΓςλ+1(P3)
Furthermore, the next sum is integrable on0,+∞ for n=1,(76)ψpq∑ς=0∞nyςΓςλ+1(P4)
(see [1]) If ϕ∈N, where dϕy=ηψydy, and
(77)ηψy=∑ς=0∞ψyςΓςλ+1,ψpqy=∑ς=0∞−8yςΓςλ+1,with ϑ0≔2−ψ>0,−u≤b<y≤u, then for ω=1 we have that L=∅.(P5)
Fore1y=2/51+3/2siny,e2y=2/51+7/6cosy,e∧py=2/5, we have
(78)Op=∑ς=0∞4/15ςΓςλ+1(P6)
Forσp=1,R=π,ζpq=3/p+q,ϑ¯pqy=0,ϑpqy=8/10siny,hpqt=9/10cos2y,ζℓ=9/100, we have
(79)supy∈ℝ−e∧py+Opσ−1∑qnϑ¯pqyTqΘ+ϑσyTqΘ+T^qΘ∧hpy∫0∞ψpqrdrσq,≤−κp≤0(P7)
Also,(80)0<maxp=1,2,3,n,ne∧p+e∧p−−κpe∧p+<1.
LetLpy=20+pcosy+Uy, where
(81)Ub=e−b,b≤01,b<0,and Lp∈PAPℝ,ℝ,ρ. Then, all solutions of (1) are in the Mittag-Leffler form and they converge to a unique solution of equation (1) such that z∗y∈PAPℝ,ℝ2,ϕ, when y⟶+∞ with convergence rate ν≈0.05<e∧p−.Example 27.
Assume conditionsP1 to P7 hold and consider functions from Example 26. For ψ>1,ν>0 and Dirac measure δ1/n, define the following measure:
(82)dϕψ,νy=∑ς=0∞ψyςΓςλ+1dy+ν∑n=1∞1n2δ1/n.
Consider the intervalI=−1−ζ,1+ζ, for ζ∈ℝ and ϑ=1. Then for all R∈A and ζ∈ℝ, there exist ϑ>0 and a bounded interval L such that ϕR+ζ≤ϑϕR and R∩L=∅.
LetLpy=20+pcosy+Uy, where
(83)Ub=e−b,b≤00,b>0,and Lp∈PAPℝ,ℝ,ρ. Then, all solutions of (1) are in the Mittag-Leffler form and they converge to a unique solution of equation (1) such that z∗y∈PAPℝ,ℝ2,ϕ, when y⟶+∞ with convergence rate ν≈0.05.Figure 1
Graphs related to numerical solutions of CNNs (1) for different values.
(a)
Numerical solutions of CNNs fory∈−20,30(b)
Numerical solutions of CNNs fory∈28,−56(c)
Numerical solutions of CNNs fory∈100,−40Figure 2
Graphs related to numerical solutions of CNNs (1) for different values.
(a)
Numerical solutions of CNNs fory∈−10/7,200(b)
Numerical solutions of CNNs fory∈−200/18,−300(c)
Numerical solutions of CNNs fory∈−10/6,10Figure 3
Graphs related to numerical solutions of CNNs (1) for different values.
(a)
Numerical solutions of CNNs fory∈20,−15(b)
Numerical solutions of CNNs fory∈15/6,−40/7(c)
Numerical solutions of CNNs fory∈10/6,−60/8


## 8. 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 isdϕ=ηydy+dϕ1. 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.



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