# Square-Mean Pseudo Almost Periodic Solutions for Quaternion-Valued Stochastic Neural Networks with Time-Varying Delays

Authors: Yuanyuan Hou; Lihua Dai
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we are concerned with a class of quaternion-valued stochastic neural networks with time-varying delays. Firstly, we cannot explicitly decompose the quaternion-valued stochastic systems into equivalent real-valued stochastic systems; by using the Banach fixed point theorem and stochastic analysis techniques, we obtain some sufficient conditions for the existence of square-mean pseudo almost periodic solutions for this class of neural networks. Then, by constructing an appropriate Lyapunov functional and stochastic analysis techniques, we can also obtain sufficient conditions for square-mean exponential stability of the considered neural networks. All of these results are new. Finally, two examples are given to illustrate the effectiveness and feasibility of our main results.


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## Body

## 1. Introduction

It is well known that the dynamic research on neural network models has achieved fruitful results, and it has been widely used in the fields of pattern recognition, automatic control, signal processing, and artificial intelligence. However, most neural network models proposed and discussed in the literature are deterministic. It has the characteristics of simple and easy to analyze. In fact, for any actual system, there is always a variety of random factors. As we know, in real nervous systems and in the implementation of artificial neural networks, noise is unavoidable [1, 2] and should be taken into consideration in modelling. Stochastic neural network is an artificial neural network and is used as a tool of artificial intelligence. Therefore, it is of practical importance to study the stochastic neural networks. In 1996, Liao and Mao [3] studied stochastic effects to the stability property of a neural network. Subsequently, some scholars carried out a lot of research work and made some progress [4–7]. Due to the finite switching speed of neurons and amplifiers, time delays inevitably exist in biological and artificial neural network models. In recent years, the research on the stability of delay stochastic neural networks has become a hot spot in many scholars [8–11].On the one hand, the quaternion-valued neural network has been one of the most popular research hot spots, due to the storage capacity advantage compared to real-valued neural networks and complex-valued neural networks. It can be applied to the fields of robotics, attitude control of satellites, computer graphics, ensemble control, color night vision, and image impression [12, 13]. Because all of these applications rely heavily on their dynamics, the study of various dynamical behaviors for quaternion-valued neural networks has received much attention of many scholars, and some results have been obtained for the stability [14–16], dissipativity [17], and pseudo almost periodicity [18, 19] of quaternion-valued neural networks. In recent years, authors of [20, 21] considered the existence and global exponential stability of pseudo almost periodic solutions and pseudo almost automorphic solutions for quaternion-valued neural networks by direct method. It should be noted that most studies on quaternion-valued neural network dynamic behaviors are concerned with the quaternion-valued deterministic neural networks, and so far, only a few results consider the stochastic quaternion-valued neural networks via a decomposing method [22, 23]. For example, Sriraman et al. [22] considered the square-mean asymptotic stability for the discrete-time stochastic quaternion-valued neural networks; in [23], authors studied the square-mean exponential input-to-state stability for the continuous-time stochastic memristive quaternion-valued neural networks.On the other hand, pseudo almost periodicity is a natural generalization of almost periodicity. Therefore, for nonautonomous neural networks, pseudo almost periodic oscillation is a very important dynamic phenomenon. In the past few decades, many researchers have investigated various dynamical behaviors of stochastic differential equations such as the existence and stability of almost periodic solutions [24], almost automorphic solutions [25, 26], and pseudo almost periodic solutions [27, 28]. Besides, as we all know, there have been only few results about the dynamic behaviors of nonautonomous stochastic neural networks. Subsequently, some scholars have studied the existence and stability of periodic solutions and almost periodic solutions for stochastic neural networks [29–31], for example, Lu and Ma [29] dealt with the stability analysis and the existence of periodic solution problems for the stochastic neural networks; Huang and Yang [31] investigated the problems of existence of quadratic mean almost periodic and global exponential stability for stochastic cellular neural networks with delays. Compared with the previous results, rare results are available for pseudo almost periodic solutions of stochastic neural networks in the mean square sense.However, to the best of our knowledge, up to now, there is no paper published on the existence and square-mean exponential stability for square-mean pseudo almost periodic solutions of quaternion-valued stochastic neural networks. So, it is a challenging and important problem in theories and applications.Motivated by the above, in this paper, we consider the following quaternion-valued stochastic neural network:(1)dult=−αltult+∑k=1nblktfkukt+∑k=1nclktgkukt−υlkt+ℐltdt+∑k=1nδlkukt−ηlktdWkt,where l∈1,2,…,n≕T; n is the number of neurons in layers; ult∈ℚ is the state of the l-th neuron at time t; αlt>0 is the self-feedback connection weight; blkt and clkt∈ℚ are, respectively, the connection weight and the delay connection weight from neuron k to neuron l; υlkt and ηlkt are the transmission delays; fk,gk:ℚ⟶ℚ are the activation functions; ℐlt∈ℚ is an external input on the l-th unit at time t; Wt=W1t,W2t,…,WntT is an n-dimensional Brownian motion defined on a complete probability space; δlk:ℚ⟶ℚ is a Borel measurable function; A=δlkn×n is the diffusion coefficient matrix.The skew field of quaternion by(2)ℚ≔q=qR+iqI+jqJ+kqK,where qR, qI, qJ, and qK are real numbers; the three imaginary units i, j, and k obey the Hamilton’s multiplication rules:(3)ij=−ji=k,jk=−kj=i,ki=−ik=j,i2=j2=k2=ijk=−1.For everyu∈ℚ, the conjugate transpose of u is defined as u∗=uR−iuI−juJ−kuK, and the norm of u is defined as(4)uℚ=uu∗=uR2+uI2+uJ2+uK2.For everyu=u1,u2,…,unT∈ℚn, we define uℚn=maxl∈Tulℚ.LetW,ℱ,ℱtt≥0,P be a complete probability space with a natural filtration ℱtt≥0, satisfying the usual conditions (i.e., it is right continuous, and ℱ0 contains all P-null sets). Denote by BCℱ0−ν,0,ℚn the family of all bounded, ℱ0-measurable, C−ν,0,ℚn-valued random variables ϕ. Denote by ℒℱ02−ν,0,ℚn the family of all ℱ0-measurable, C−ν,0,ℚn-valued random variables ϕ, satisfying sups∈−ν,0Eϕs2<∞, where E denotes the expectation of stochastic process.For the convenience, we will adopt the following notation:(5)αl−=inft∈ℝαlt,blk+=supt∈ℝblktℚ,clk+=supt∈ℝclktℚ,υ+=maxl.k∈Tsupt∈ℝυlkt,η+=maxl,k∈Tsupt∈ℝηlkt,ν=maxυ+,η+.The initial conditions of system (1) are of the form(6)uls=ϕls,s∈−ν,0,where ϕl∈BCℱ0−ν,0,ℚ, l∈T.Remark 1.
Quaternion-valued stochastic system (1) includes real-valued stochastic systems and complex-valued stochastic systems as its special cases. In fact, in system (1),(i)
If all the coefficients and delays are functions fromℝ to ℝ, and all the activation functions are functions from ℝ to ℝ, then the state ult≡ulRt∈ℝ; in this case, system (1) is a real-valued stochastic system;(ii)
If the coefficients are functions fromℝ to ℂ, and all the activation functions are functions from ℂ to ℂ, then the state ult≡ulRt+iulIt∈ℂ; in this case, system (1) is a complex-valued stochastic system.With the inspiration from the previous research, in order to fill the gap in the research field of quaternion-valued stochastic neural networks, the work of this article comes from two main motivations. (1) In practical applications, pseudo almost periodic motion is an interesting and significant dynamical property for stochastic differential equations. In the past decade, many authors studied square-mean almost periodic oscillations and square-mean almost automorphic oscillations of stochastic differential equations [24–26]. Yet, few literatures considered square-mean pseudo almost periodic oscillation of stochastic differential equations [27, 28]. (2) Recently, a few literatures [22, 23] had studied the square-mean stability of quaternion-valued stochastic neural networks via a decomposing method. It is noteworthy that the scholars have not begun to consider the square-mean pseudo almost periodic oscillation for quaternion-valued stochastic neural networks; thus, it is worth studying square-mean pseudo almost periodic motion of quaternion-valued stochastic neural network models by direct method.Compared with the previous literatures, the distinct characteristics and main contributions of this article are narrated as follows:(1)
Firstly, to the best of our knowledge, this is the first time to investigate the existence and square-mean exponential stability of square-mean pseudo almost periodic solutions for quaternion-valued stochastic delayed neural networks.(2)
Secondly, the method that we use to quaternion-valued stochastic neural networks is different from that used in [22, 23], and we improve the norm.(3)
Thirdly, the techniques of this paper can be applied to study the square-mean pseudo almost periodic solutions for other types of quaternion-valued stochastic neural networks.(4)
Finally, examples and numerical simulations are given to verify the effectiveness of the conclusion.This paper is organized as follows: In Section2, we introduce some definitions and some preliminary results. In Section 3, we establish some sufficient conditions for the existence of square-mean pseudo almost periodic solutions of system (1). In Section 4, we obtain the square-mean exponential stability of system (1). In Section 5, we give two examples to demonstrate the feasibility of our results. Finally, we draw a conclusion in Section 6.


## 2. Preliminaries and Basic Knowledge

Assume thatℍ is a real separable Hilbert space, and L2P,ℍ stands for the space of all ℍ-valued random variables X such that EX2=∫ΩX2dP<∞. Then, L2P,ℍ is a Banach space with the norm X2=∫ΩX2dP1/2.Similar to the definition in [27], we give the following definitions.Definition 1.
A stochastic processX:ℝ⟶L2P,ℚn is said to be L2-continuous if(7)limt⟶sEXt−Xsℚn2=0.
It isL2-bounded if supt∈ℝEXtℚn2<∞.Definition 2.
AnL2-continuous stochastic process X:ℝ⟶L2P,ℚn is said to be square-mean almost periodic, if for any ε>0, there exists L=Lε>0, such that for any a∈ℝ, there exists τ∈a,a+L with(8)supt∈ℝEXt+τ−Xtℚn2≤ε.We denote bySAPℝ,L2P,ℚn the set of all square-mean almost periodic functions from ℝ to L2P,ℚn, and let BCℝ,L2P,ℚn be the set of all bounded continuous stochastic processes from ℝ to L2P,ℚn.Let(9)SPAP0ℝ,L2P,ℚn=X∈BCℝ,L2P,ℚn|limT⟶+∞12T∫−TTEXtℚn2dt=0.We give the following definition for quaternion.Definition 3.
AnL2-continuous stochastic process X:ℝ⟶L2P,ℚn is said to be square-mean pseudo almost periodic, if it can be decomposed as X=Y+Z, where(10)Y∈SAPℝ,L2P,ℚn,Z∈SPAP0ℝ,L2P,ℚn.We denote bySPAPℝ,L2P,ℚn the set of all square-mean pseudo almost periodic functions from ℝ to L2P,ℚn.Similar to the lemma in [27], one can easily show as follows.Lemma 1.
Iff,g∈SPAPℝ,L2P,ℚ, then f+g,fg∈SPAPℝ,L2P,ℚ.Lemma 2.
Ifϕ∈SPAPℝ,L2P,ℚ, θ∈SAPℝ,L2P,ℝ∩C1ℝ,ℝ with ϑ≔inft∈ℝ1−θ′t>0, then ϕ⋅−θ⋅∈SPAPℝ,L2P,ℚ.Throughout the rest of the paper, we assume that(H1)αl∈SAPℝ,ℝ+, υkl,ηlk∈SAPℝ,ℝ+∩C1ℝ,ℝ with γ=supt∈ℝυkl′t, β=supt∈ℝηlk′t, and blk,clk,ℐl∈SPAPℝ,ℚ, where l,k∈T.(H2)
There exist positive constantsLkf, Lkg, and Llkδ such that for any x,y∈ℚ,(11)fkx−fkyℚ≤Lkfx−yℚ,gkx−gkyℚ≤Lkgx−yℚ,δlkx−δlkyℚ≤Llkδx−yℚ,andfk0=gk0=δlk0=0, where l,k∈T.(H3)maxl∈T3Υl/αl−2≕ρ<1, where(12)Yl=∑k=1nblk+2∑k=1nLkf2+∑k=1nclk+2∑k=1nLkg2+αl−2∑k=1nLlkδ2.(H4)
Forl∈T, there exist positive constants λ and ξk such that(13)2ξlαl−>λξl+2ξl+∑k=1nξkbkl+2Lkf2+∑k=1nξkckl+2Lkg2eλυ+1−γ+∑k=1nξkLklδ2eλη+1−β.


## 3. Square-Mean Pseudo Almost Periodic Solutions

In this section, we will study the existence of square-mean pseudo almost periodic solutions of system (1).LetX=ϕ|ϕ∈SPAPℝ,L2P,ℚn with the norm ϕX=supt∈ℝEϕtℚn21/2, and then X is a Banach space.Setϕ0=ϕ10,ϕ20,…,ϕn0T, where ϕl0t=∫−∞te−∫stαlθdθℐlsds, t∈ℝ, and l∈T, and κ is a constant satisfying κ≥ϕ0X.Theorem 1.
LetH1–H3 hold. Then, system (1) has a unique square-mean pseudo almost periodic solution in the region X∗=ϕϕ∈Xϕ−ϕ0X≤κ.Proof.
Firstly, it is easy to see that ifut=u1t,u2t,…,untTt∈ℝ is a solution of the following stochastic integral equation(14)ult=∫−∞te−∫stαlθdθ∑k=1nblksfkuks+∑k=1nclksgkuks−υkls+ℐlsds+∫−∞te−∫stαlθdθ∑k=1nδlkuks−ηlksdWks,where l∈T, then u is also a solution of system (1).
We define a mappingΦ:X⟶X by setting(15)ϕ1,ϕ2,…,ϕn⟶u1ϕ,u2ϕ,…,unϕ.
We shall show thatΦ has a unique fixed point in X∗. For ϕ∈X∗, we have(16)ϕX≤ϕ0X+ϕ−ϕ0X≤2κ.
For anyϕ∈X∗, from (14), we consider the following stochastic integral equation:(17)ulϕt≕∫−∞te−∫stαlθdθFlsds+∫−∞te−∫stαlθdθ∑k=1nGlksdWks,l∈T,where(18)Fls=∑k=1nblksfkϕks+∑k=1nclksgkϕks−υkls+ℐls,Glks=δlkϕks−ηlks.
From Lemmas1 and 2, we have Fl,Glk∈SPAPℝ,L2P,ℚ, l,k∈T, that is, Fl,Glk can be rewritten as Fl=Fl1+Fl0, Glk=Glk1+Glk0, where Fl1,Glk1∈SAPℝ,L2P,ℚ, Fl0,Glk0∈SPAP0ℝ,L2P,ℚ. Hence,(19)ulϕt=∫−∞te−∫stαlθdθFl1sds+∫−∞te−∫stαlθdθ∑k=1nGlk1sdWks+∫−∞te−∫stαlθdθFl0sds+∫−∞te−∫stαlθdθ∑k=1nGlk0sdWks≕ℋl1t+ℋl0t,l∈T.
First, we will prove thatℋl1∈SAPℝ,L2P,ℚ, l∈T. For any ε>0, since αl∈SAPT,ℝ+, Fl1,Glk1∈SAPℝ,L2P,ℚ, it is possible to find a real number L=Lε>0; for any interval with length Lε, there exists a number τ=τε in this interval such that Eαls+τ−αls2<ε and(20)EFl1s+τ−Fl1sℚ2<ε,EGlk1s+τ−Glk1sℚ2<ε.
Forl∈T, then we consider(21)Eℋl1t+τ−ℋl1tℚ2=E∫−∞te−∫stαlθdθFl1s+τ−Fl1sds+∫−∞te−∫stαlθ+τdθ−e−∫stαlθdθFl1s+τds+∫−∞te−∫stαlθdθ∑k=1nGlk1s+τ−Glk1sdWks+∫−∞te−∫stαlθ+τdθ−e−∫stαlθdθ∑k=1nGlk1s+τdWksℚ2≤4E∫−∞te−∫stαlθdθFl1s+τ−Fl1sdsℚ2+4E∫−∞te−∫stαlθ+τdθ−e−∫stαlθdθFl1s+τdsℚ2+4E∫−∞te−∫stαlθdθ∑k=1nGlk1s+τ−Glk1sdWksℚ2+4E∫−∞te−∫stαlθ+τdθ−e−∫stαlθdθ∑k=1nGlk1s+τdWksℚ2≤8αl−2∑k=1nblk1+2∑k=1nLlf2+∑k=1nclk1+2∑k=1nLlg2+ℐl1+2+αl−2∑k=1nLlkδ2+4ε.
Thus, we proved thatℋl1 is square-mean almost periodic, l∈T.
On the other hand, we will prove thatℋl0∈SPAP0ℝ,L2P,ℚ. Hence, we only need to show(22)limT⟶+∞12T∫−TTEℋl0tℚ2dt=0,l∈T.
Thus, we obtain(23)12T∫−TTEℋl0tℚ2dt=12T∫−TTE∫−∞te−∫stαlθdθFl0sds+∫−∞te−∫stαlθdθ∑k=1nGlk0sdWksℚ2dt≤1T∫−TTE∫−∞te−∫stαlθdθFl0sdsℚ2dt+1T∫−TTE∫−∞te−∫stαlθdθ∑k=1nGlk0sdWksℚ2dt≕Ql1+Ql2.
It follows from the Cauchy–Schwarz inequality that(24)Ql1=1T∫−TTE∫−∞te−∫stαlθdθFl0sdsℚ2dt≤1T∫−TT∫−∞te−∫stαlθdθds∫−∞te−∫stαlθdθEFl0sℚ2dsdt≤1Tαl−∫−TT∫−Tte−∫stαlθdθEFl0sℚ2dsdt+1Tαl−∫−TT∫−∞−Te−∫stαlθdθEFl0sℚ2dsdt≕Ω1+Ω2.
Setζ=t−s, and using Fubini’s theorem, we obtain that(25)Ω1=1Tαl−∫−TT∫−Tte−∫stαlθdθEFl0sℚ2dsdt≤1Tαl−∫−TT∫−Tte−t−sαl−EFl0sℚ2dsdt≤1Tαl−∫−TT∫0+∞e−ζαl−EFl0t−ζℚ2dζdt≤2αl−∫0+∞e−ζαl−T+ζT12T+ζ∫−T−ζT−ζEFl0θℚ2dθdζ≤2αl−212T∫−TTEFl0sℚ2ds,since Fl0∈SPAP0ℝ,L2P,ℚ, thus Ω1⟶0 as T⟶+∞. Since Fl0 is bounded, as T⟶+∞, we have(26)Ω2=1Tαl−∫−TT∫−∞−Te−∫stαlθdθEFl0sℚ2dsdt≤1Tαl−∫−TT∫T+t+∞e−ζαl−EFl0t−ζℚ2dζdt≤1Tαl−3supt∈ℝEFl0tℚ2∫−TT∫T+t+∞e−ζαl−dζdt=1−e−2αl−TTαl−3supt∈ℝEFl0tℚ2⟶0.
Hence, forl∈T, we have that Ql1⟶0 as T⟶+∞. By the Itô isometry, one has(27)Ql2=1T∫−TTE∫−∞te−∫stαlθdθ∑k=1nGlksdWksℚ2dt≤1T∑k=1n∫−TT∫−Tte−2αl−t−sEGlksℚ2dsdt+1T∑k=1n∫−TT∫−∞−Te−2αl−t−sEGlksℚ2dsdt≤1αl−∑k=1n12T∫−TTEGlksℚ2ds+1−e−4αl−TTαl−2supt∈ℝ∑k=1nEGlktℚ2⟶0,asT⟶+∞.
Therefore, we have verified(28)limT⟶+∞12T∫−TTEℋl0tℚ2dt=0,l∈T.
Therefore, we haveulϕ∈SPAPℝ,L2P,ℚ.
Next, we show thatΦ is a self-mapping from X∗ to X∗. For ϕ∈X∗, by the Cauchy–Schwarz inequality and Itô isometry, we have(29)EΦϕt−ϕ0tℚn2=maxl∈TE∫−∞te−∫stαlθdθ∑k=1nblksfkϕks+∑k=1nclksgkϕks−υklsds+∫−∞te−∫stαlθdθ∑k=1nδlkϕks−ηlksdWksℚ2≤3maxl∈TE∫−∞te−∫stαlθdθ∑k=1nblksfkϕksdsℚ2+3maxl∈TE∫−∞te−∫stαlθdθ∑k=1nclksgkϕks−υklsdsℚ2+3maxl∈TE∫−∞te−∫stαlθdθ∑k=1nδlkϕks−ηlksdWksℚ2≤maxl∈T3αl−E∫−∞te−∫stαlθdθ∑k=1nblksℚLkfϕksℚ2ds+maxl∈T3αl−E∫−∞te−∫stαlθdθ∑k=1nclksℚLkgϕks−υklsℚ2ds+3maxl∈TE∑k=1n∫−∞te−2∫stαlθdθδlkϕks−ηlksℚ2ds≤maxl∈T3αl−2∑k=1nblktℚ2∑k=1nLkf2Eϕtℚn2+maxl∈T3αl−2∑k=1nKlk∗clktℚ2∑k=1nLkg2Eϕtℚn2+maxl∈T32αl−∑k=1nLlkδ2Eϕtℚn2≤maxl∈T3αl−2∑k=1nblktℚ2∑k=1nLkf2+∑k=1nclktℚ2∑k=1nLkg2+αl−2∑k=1nLlkδ2Eϕtℚn2.
It follows from (29) and H3 that(30)Φϕ−ϕ0X2=supt∈ℝEΦϕt−ϕ0tℚn2≤maxl∈T3αl−2∑k=1nblk+2∑k=1nLkf2+∑k=1nclk+2∑k=1nLkg2+αl−2∑k=1nLlkδ24κ2,which implies that Φϕ−ϕ0X≤κ; hence, we have Φϕ∈X∗. Next, we show that Φ is a contraction mapping in X∗. For any ϕ,ψ∈X∗, we have(31)EΦϕt−Φψtℚn2=maxl∈TE∫−∞te−∫stαlθdθ∑k=1nblksfkϕks−fkψksds+∫−∞te−∫stαlθdθ∑k=1nclksgkϕks−υkls−gkψks−υklsds+∫−∞te−∫stαlθdθ∑k=1nδlkϕks−ηlks−δlkψks−ηlksdWksℚ2≤maxl∈T3αl−2∑k=1nblk+2∑k=1nLkf2+∑k=1nclk+2∑k=1nLkg2+αl−2∑k=1nLlkδ2ϕt−ψtℚn2.
Thus, we have(32)supt∈ℝEΦϕt−Φψtℚn2≤maxl∈T3Ylαl−2supt∈ℝEϕt−ψtℚn2.
It follows from (32) that(33)Φϕ−ΦψX≤ρϕ−ψX.
In view ofH3, we see that Φ is a contraction mapping. Therefore, Φ has a unique fixed point in X∗, that is, (1) has a unique square-mean pseudo almost periodic solution in X∗. The proof is finished.


## 4. Square-Mean Exponential Stability

In this section, we will consider the square-mean exponential stability of system (1).Definition 4.
Letu=u1,u2,…,unT be a solution of (1) with the initial value ϕ=ϕ1,ϕ2,…,ϕnT. If there exist positive constants λ>0 and ℳ>0 such that for an arbitrary solution w=w1,w2,…,wnT with initial value, ψ=ψ1,ψ2,…,ψnT satisfies(34)Ewt−utℚn2≤ℳe−λtEψ−ϕ12,t>0,where(35)wt−utℚn=maxl∈Twlt−ultℚ,ψ−ϕ1=maxl∈Tsups∈−ν,0ψls−ϕlsℚ.
Then, the solutionut is said to be square-mean exponentially stable.Theorem 2.
Suppose that the assumptionsH1–H4 are satisfied, then for any initial value of dynamical system (1), there exists a square-mean pseudo almost periodic solution ut, which is square-mean exponentially stable.Proof.
From Theorem1, we see that system (1) has a square-mean pseudo almost periodic solution u=u1,u2,…,unT with initial value ϕ=ϕ1,ϕ2,…,ϕnT. Suppose that w=w1,w2,…,wnT is an arbitrary solution of system (1) with initial value ψ=ψ1,ψ2,…,ψnT and zt=wt−ut. Let δt=δlktn×n, where δlkt=δlkzkt−ηlkt. We consider the Lyapunov function as follows:(36)Vt,zt=maxl∈Teλtξlzl∗tzlt+ξlΓlt,zt,where(37)Γlt,zt=∑k=1nclk+2Lkg2eλυ+1−γ∫t−υklttzk∗szkseλsds+∑k=1nLlkδ2eλη+1−β∫t−ηlkttzk∗szkseλsds.
Then, by Itô formula, we have the following stochastic differential:(38)dVt,zt=ℒVt,ztdt+Vzt,ztδtdWt,where Vzt,zt=∂Vt,zt/∂z1,…,∂Vt,zt/∂zn, and ℒ is the weak infinitesimal operator such that(39)ℒVt,zt=maxl∈Tλeλtξlzl∗tzlt+eλtξlzlt−αltzl∗t+∑k=1nblktfkwkt−fkukt∗+∑k=1nclktgkwkt−υklt−gkukt−υklt∗+eλtξlzl∗t−αltzlt+∑k=1nblktfkwkt−fkukt+∑k=1nclktgkwkt−υklt−gkukt−υklt+ξl∑k=1nclk+2Lkg2eλυ+1−γzk∗tzkteλt−ξl∑k=1nclk+2Lkg2eλυ+1−γ1−υkl′t×zk∗t−υkltzkt−υklteλt−υkltds+ξl∑k=1nLlkδ2eλη+1−βzk∗tzkteλt−ξl∑k=1nLlkδ2eλη+1−β1−ηlk′tzk∗t−ηlktzkt−ηlkteλt−ηlkt+eλtξl∑k=1nδlkwkt−ηlkt−δlkukt−ηlkt∗δlkwkt−ηlkt−δlkukt−ηlkt≤maxl∈Teλtλξl−2ξlαltzl∗tzlt+eλtξl∑k=1nblktfkwkt−fkukt∗×blktfkwkt−fkukt+eλtξl∑k=1nclktgkwkt−υklt−gkukt−υklt∗clktgkwkt−υklt−gkukt−υklt+2eλtξlzl∗tzlt+eλt∑p=1nξl∑k=1nclk+2Lkg2eλυ+1−γzk∗tzkt−eλtξl∑k=1nclk+2×Lkg2eλυ+1−γ1−γzk∗t−υkltzkt−υklte−λυ++eλtξl∑k=1nLlkδ2eλη+1−βzk∗tzkt−eλtξl∑k=1nLlkδ2eλη+1−β1−βzk∗t−ηlkt×zkt−ηlkte−λη++eλtξl∑k=1nLlkδ2zk∗t−ηlktzkt−ηlkt≤maxl∈Teλtλξl+2ξl−2ξlαl−zl∗tzlt+eλtξl∑k=1nblk+2Lkf2zk∗tzkt+eλtξl∑k=1nclk+2Lkg2eλυ+1−γzk∗tzkt+eλtξl∑k=1nLlkδ2eλη+1−βzk∗tzkt=maxl∈Teλtλξl+2ξl−2ξlαl−+∑k=1nξkbkl+2Lkf2+∑k=1nξkckl+2Lkg2eλυ+1−γ+∑k=1nξkLklδ2eλη+1−βzl∗tzlt.
FromH4, we easily derive(40)ℒVt,zt≤0.
Now, similar to the previous literature, we define the stopping time (or Markov time)σk≔infs≥0:zs≥k, and by the Dynkin formula, we have(41)EVt∧σk,zt∧σk=EV0,z0+E∫0t∧σkℒVs,zsds.
Lettingk⟶∞ on both sides (41), from the monotone convergence theorem, we can get(42)EVt,zt≤EV0,z0=maxl∈TξlEzl∗0zl0+ξl∑k=1nclk+2Lkg2×eλυ+1−γ∫−υkl00Ezk∗szkseλsds+ξl∑k=1nLlkδ2eλη+1−β∫−ηlk00Ezk∗szkseλsds≤maxl∈Tξl+ξl∑k=1nbkl+2Lkg2υ+eλυ+1−γ+ξl∑k=1nLklδ2η+eλη+1−βEsups∈−ν,0ψls−ϕlsℚ2.
On the other hand, it follows from the definition ofVt,zt that(43)EVt,zt≥Eeλtmaxl∈Tξlzl∗tzlt≥eλtminl∈TξlEztℚn2.
Combining (42) and (43), the following holds:(44)Eztℚn2≤ℳe−λtEψ−ϕ12,where(45)ℳ=1minl∈Tξlmaxl∈Tξl1+∑k=1nckl+2Lkg2υ+eλυ+1−γ+∑k=1nLklδ2η+eλη+1−β,which together with Definition 4 verifies that square-mean pseudo almost periodic solution of system (1) which is square-mean exponentially stable. The proof is finished.Remark 2.
From the proofs of Theorems1 and 2, one can easily see that if all the coefficients of (1) are square-mean almost automorphic and square-mean almost periodic, respectively, then, similar to the proofs of Theorems 1 and 2 and under the same corresponding conditions, one can show that the similar results of Theorems 1 and 2 are still valid for both cases of the square-mean pseudo almost automorphy and square-mean weighted pseudo almost periodicity.


## 5. Illustrative Example

In this section, we give numerical examples to illustrate the feasibility and effectiveness of our results obtained in Sections3 and 4.Example 1.
Letn=2. Consider the following quaternion-valued stochastic neural network:(46)dult=−αltult+∑k=12blktfkukt+∑k=12clktgkukt−υlkt+ℐltdt+∑k=12δlkukt−ηlktdWkt,where l=1,2; the coefficients are as follows:(47)fkuk=gkuk=17tanhukR+i15ukI+j14sinukI+k19sinukJ+ukK,δlkuk=112sinukR+ukI+i18sinukI+j19ukJ+k15tanhukK,α1t=3+cos2t,α2t=6−2sin3t,blkt=0.5sin2t+i0.5sint+j0.8cost+k0.4cos2t,clkt=0.8cost+i1.2sin2t+j1.5sint+k0.9cos3t,ℐ1t=0.2sin3t+i0.5cos2t+j0.3cos2t+k0.3sin3t,ℐ2t=0.3cos2t+i0.4sin3t+j0.5sint+k0.2cos3t,υklt=12sint,ηlkt=45cost.
Through simple calculations, we have(48)α1−=3,α2−=4,Lkf=Lkg=14,Llkδ=15,υ+=12,η+=45,γ=12,β=45,blk+≤1.1402,clk+≤2.2672,Y1≈1.6901,Y2≈1.7701.
Then, we have(49)max3Y1α1−2,3Y2α2−2≈max0.5634,0.3319=0.5634=ρ<1.
Takeλ=0.1, ξ1=0.3, and ξ2=0.4, then we have(50)2ξ1α1−=1.8>λξ1+2ξ1+∑k=12ξkbk1+2Lkf2+∑k=12ξkck1+2Lkg2eλυ+1−γ+∑k=12ξkLk1δ2eλη+1−β≈0.9501,2ξ2α2−=3.2>λξ2+2ξ2+∑k=12ξkbk2+2Lkf2+∑k=12ξkck2+2Lkg2eλυ+1−γ+∑k=12ξkLk2δ2eλη+1−β≈1.5295.
We can check that other conditions of Theorems1 and 2 are satisfied. So, we know that square-mean pseudo almost periodic solutions of system (46) are square-mean exponentially stable. System (46) has the initial value u10=0.3−0.5i+0.6j+0.25k and u20=−0.2+0.4i−0.4j−0.35k. The results are verified by the numerical simulations in Figures 1–4, which shows the time response curve for the four parts between the variables u1t and u2t of system (46).Figure 1
The states response ofu1Rt and u2Rt.Figure 2
The states response ofu1It and u2It.Figure 3
The states response ofu1Jt and u2Jt.Figure 4
The states response ofu1Kt and u2Kt.Example 2.
Letn=3. Consider the following quaternion-valued stochastic neural network:(51)dult=−αltult+∑k=13blktfkukt+∑k=13clktgkukt−υlkt+ℐltdt+∑k=13δlkukt−ηlktdWkt,where l=1,2,3; the coefficients are as follows:(52)fkuk=114sinukR+i112sinukI+j115sinukI+ukJ+k110sinukK,gkuk=112sinukR+ukJ+i120sinukR+ukI+j115tanhukJ+k110tanhukK,δlkuk=115sinukR+i110sinukI+j18sinukR+ukJ+k112tanhukK,α1t=2+sin3t,α2t=5−2cos2t,α3t=7−3cos5t,blkt=0.4cos3t+i0.6sin3t+j0.7sint+k0.5cos2t,clkt=1.2sint+i0.9cos2t+jsin2t+k1.5cos3t,ℐ1t=0.2sin3t+i0.5cos2t+j0.3cos2t+k0.3sin3t,ℐ2t=0.3sin2t+e−t2cos2t+i0.4sin3t+j0.5sint+k0.2cos3t,ℐ3t=0.2sin2t+i0.3cos2t+j0.4cost+k0.3sin5t,υklt=12cos2t,ηlkt=45sin2t,l,k=1,2,3.
Through simple calculations, we have(53)α1−=2,α2−=3,α3−=4,Lkf=Lkg=110,Llkδ=18,υ+=12,η+=45,γ=12,β=45,blk+≤1.1225,clk+≤2.3452,Y1≈0.6553,Y2≈0.6787,Y3≈0.7021.
Then, we have(54)max3Y1α1−2,3Y2α2−2,3Y3α3−2≈max0.4915,0.2262,0.1316=0.4915=ρ<1.
Takeλ=0.1, ξ1=0.3, ξ2=0.4, and ξ3=0.5, then we have(55)2ξ1α1−=1.2>λξ1+2ξ1+∑k=13ξkbk1+2Lkf2+∑k=13ξkck1+2Lkg2eλυ+1−γ+∑k=13ξkLk1δ2eλη+1−β≈0.8704,2ξ2α2−=2.4>λξ2+2ξ2+∑k=13ξkbk2+2Lkf2+∑k=13ξkck2+2Lkg2eλυ+1−γ+∑k=13ξkLk2δ2eλη+1−β≈1.0804,2ξ3α3−=4>λξ3+2ξ3+∑k=13ξkbk3+2Lkf2+∑k=13ξkck3+2Lkg2eλυ+1−γ+∑k=13ξkLk3δ2eλη+1−β≈1.2904.
We can check that other conditions of Theorems1 and 2 are satisfied. So, we know that square-mean pseudo almost periodic solutions of system (51) are square-mean exponentially stable. System (51) has the initial values u10=0.3−0.3i+0.5j−0.3k, u20=−0.2+0.4i−0.4j−0.45k, and u30=0.1−0.1i+0.2j+0.35k. The results are verified by the numerical simulations in Figures 5–8, which shows that the time response curve for the four parts between the variables u1t, u2t, and u3t of system (51).Figure 5
The states response ofu1Rt, u2Rt, and u3Rt.Figure 6
The states response ofu1It, u2It, and u3It.Figure 7
The states response ofu1Jt, u2Jt, and u3Jt.Figure 8
The states response ofu1Kt, u2Kt, and u3Kt.


## 6. Conclusion

In this paper, we consider the existence and square-mean exponential stability of square-mean pseudo almost periodic solutions of the quaternion-valued stochastic neural networks. By using the Banach fixed point theorem and stochastic analysis techniques, we obtain some sufficient conditions for the existence of square-mean pseudo almost periodic solutions for the neural networks by direct method, and we improve the norm. Then, by constructing an appropriate Lyapunov functional, stochastic analysis theory and Itô formula, a novel sufficient condition has been derived to ensure the square-mean exponential stability for the considered stochastic neural networks. In order to demonstrate the usefulness of the presented results, some numerical examples are given. The works of this paper improve and extend the old results in literatures [22, 23] and propose a good research thinking to study square-mean pseudo almost periodic solutions and square-mean exponential stability of quaternion-valued stochastic neural networks with time-varying delays.



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