# Exponential Decay for a System of Equations with Distributed Delays

Authors: Nasser-Eddine Tatar
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

We prove convergence of solutions to zero in an exponential manner for a
system of ordinary differential equations. The feature of this work is that
it deals with nonlinear non-Lipschitz and unbounded distributed delay terms
involving non-Lipschitz and unbounded activation functions.


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## Body

## 1. Introduction

Of concern is the following system(1)xi′t=-aitxit+∑j=1mbijtfij∫-∞tKijt-sgjxjsds+cit,for i=1,…,m and t>0 with xj(t)=x0j(t) continuous on (-∞,0]. Here ai(t)≥0, bij(t), fij(t), gj(t), Kij(t)≥0, and ci(t), i,j=1,…,m, are continuous functions and are subject to other conditions that will be specified below.Similar forms of this system arise, for instance, in Neural Network Theory [1–28] (see also the “Applications” section below). There, the functions fij(t) and gj(t) are much simpler. Usually, fij(t) are equal to the identity and gj(t) (called the activation functions) are assumed to be Lipschitz continuous. The integral terms represent the distributed delays. When the kernels are replaced by the delta distribution we recover the well-known discrete delays. The functions ci(t) account for the input functions. The first terms in the right hand side of (1) may be looked at as dissipative terms.Different methods have been used by many authors to study the well-posedness and the asymptotic behavior of solutions of these systems [1–3, 6, 9–17, 19–21, 25, 27]. In particular, a lot of efforts are devoted to improving the conditions on the different coefficients involved in the system as well as on the class of activation functions. Regarding the latter issue, the early assumptions of boundedness, monotonicity, and differentiability have been all relaxed to merely a global Lipschitz condition. Since then, it seems that this assumption has not been weakened further considerably. It has been pointed out that there are many activation functions which are continuous but not necessarily Lipschitz continuous in applications [29]. A slightly weaker condition: xigi(xi)>0,xi≠0, and there exist λi>0 such that λi=supxi≠0(gixi/xi), where gi(xi)=fi(xi)-fi(xi∗) and xi∗ is the equilibrium, which has been used in [4, 19, 26, 28] (see also [22–24]). Finally, we cite [5] where the authors consider non-Lipschitz continuous but bounded activation functions. There are also many works on discontinuous activation functions.Here we assume that the functionsfij and gj are continuous monotone nondecreasing functions that are not necessarily Lipschitz continuous and they may be unbounded (like power type functions with powers bigger than one). We prove that, for sufficiently small initial data, solutions decay to zero exponentially.We could not find similar interesting works on (continuous but) non-Lipschitz continuous activation functions to compare our results with. Our treatment is in fact concerned with a doubly non-Lipschitz continuous system.Using standard techniques and the Gronwall-type lemma below we may prove local existence of solutions. The global existence follows from the estimation in our theorem below. The uniqueness, however, is delicate and does not hold in general.In the next section we present and prove our result and illustrate it by an example.


## 2. Exponential Convergence

In this section we state and prove our exponential convergence result. Before that we need to present a lemma due to Bainov and Simeonov [30].LetI⊂R, and let g1,g2:I→R∖{0}. We write g1∝g2 if g2/g1 is nondecreasing in I.Lemma 1.
Leta(t) be a positive continuous function in J≔α,β. Assume that kj(t,s), j=1,…,n, are nonnegative continuous functions for α≤s≤t<β which are nondecreasing in t for any fixed s, gj(u), j=1,…,n, are nondecreasing continuous functions in R+, with gj(u)>0 for u>0, and u(t) is a nonnegative continuous function in J. If g1∝g2∝⋯∝gn in (0,∞), then the inequality(2)ut≤at+∑j=1n∫αtkjt,sgjusds,t∈J,implies that(3)ut≤ωnt,α≤t<β0,where ω0(t)≔sup0≤s≤ta(s),(4)ωjt≔Gj-1Gjωj-1t+∫αtkjt,sds,j=1,…,n,Gju≔∫ujudxgjx,u>0uj>0,j=1,…,n,and β0 is chosen so that the functions ωj(t), j=1,…,n, are defined for α≤t<β0.In order to shorten the statement of our result we define, fori,j=1,…,m,(5)b~ijt≔exp⁡∫0taσdσbijt,c≔∑i=1m∫0∞exp⁡∫0saσdσcisds,xt=∑j=1mxjt,t>0,x0t=∑j=1mx0jt,t≤0,ω0t=x00+c+∑i,j=1m∫-∞0Kij-σgjx0σdσ=const,ω~0t=ω00+c+∑i,j=1m∫0∞Kij′s∫-s0gjx0σdσds,Hju≔∫ujudxhjx,u>0uj>0,j=1,…,n=m2+m,ωjt≔Hj-1Hjωj-1t+∫αtλjsdsfor some λj(s) to be determined.Theorem 2.
Assume thatfij and gj are continuous monotone nondecreasing functions i,j=1,…,m, in R+ that can be relabelled as hk, k=1,…,n=m2+m, with h1∝h2∝⋯∝hn, and their corresponding coefficients b~ij(t) and ∑i=1mKij(0) are relabelled as λk(t). Assume further that fij(u)>0, gj(u)>0 for u>0, ai(t)≥0, bij(t), ci(t), i,j=1,…,m, are continuous functions on R+, Kij are continuously differentiable functions, and ∫0∞exp⁡∫0sa(σ)dσcisds<∞  (a(t)≔min1≤i≤mai(t))∫-∞0Kij(-σ)gjx0(σ)dσ<∞, i,j=1,…,m, c<∞. Then, there exists a positive constant β0 such that(a)
ifKij′≤0, i,j=1,…,m, we have(6)xt≤ωntexp⁡-∫0tasds,0≤t<β0,(b)
if in additionl≔∑i,j=1m∫0∞Kij′(s)·∫-s0gjx0(σ)dσds<∞ and Kij′(t) and tKij′(t) are summable and of arbitrary signs, then the conclusion in (a) holds with ω~0(t)=ω0(t)+l, Kij(0)+∫0∞Kij′(σ)dσ instead of Kij(0) in the new λ~k and the corresponding ω~n(t).Proof.
From (1) we entail that for t>0 and i=1,…,m(7)D+xit≤-aitxit+∑j=1mbijtfij∫-∞tKijt-sgjxjsds+cit,or, by summation, we get(8)D+xt≤-min1≤i≤m⁡aitxt+∑i,j=1mbijtfij∫-∞tKijt-sgjxjsds+∑i=1mcit,t>0,where D+ denotes the right Dini derivative. Therefore,(9)D+xt≤-atxt+∑i,j=1mbijtfij∫-∞tKijt-sgjxjsds+∑i=1mcit,t>0and clearly(10)D+xtexp⁡∫0tasds≤exp⁡∫0tasds·∑i,j=1mbijtfij∫-∞tKijt-sgjxsds+exp⁡∫0tasds∑i=1mcit,t>0.It follows that (see [29])(11)x~t≤x0+c+∑j=1m∫0t∑i=1mb~ijsfij∫-∞sKijs-σgjxσdσds,where(12)x~t≔xtexp⁡∫0tasds,and b~ij(s) and c are as defined before the theorem. Set(13)Fijt≔∫0tb~ijsfijGijsds,t>0,where(14)Gi,js≔∫-∞sKijs-σgjxσdσ,s≥0,i,j=1,…,m.
Define(15)L1t≔x0+c+∑i,j=1mFijt+Gi,jt,t>0.Clearly, from (11) and (15) we have x~(t)≤L1(t), t>0, and(16)L1′t=∑i,j=1mb~ijtfijGi,jt+Kij0gjxt+∫-∞tKij′t-σgjxσdσ≤∑i,j=1mb~ijtfijL1t+Kij0gjL1t,t>0.(a)  Kij′≤0,  i,j=1,…,m. By integration we see that(17)L1t≤L10+∑i,j=1m∫0tb~ijsfijL1s+Kij0gjL1sds,t>0with(18)L10=x0+c+∑i,j=1m∫-∞0Kij-σgjx0σdσ.According to our hypotheses we can relabel the terms in (17) so that it may be written as(19)L1t≤L10+∑k=1n∫0tλkshkL1sds,t>0with h1∝h2∝⋯∝hn.
Applying Lemma1 we obtain(20)L1t≤ωnt,0≤t<β0and hence(21)x~t≤ωnt,0≤t<β0,where ω0(t)≔L1(0) and(22)ωjt≔Hj-1Hjωj-1t+∫0tλjsds,j=1,…,n,Hju≔∫ujudxhjx,u>0uj>0,j=1,…,n,and β0 is chosen so that the functions ωj(t), j=1,…,n, are defined for 0≤t<β0.
(b)  Kij′ of Arbitrary Signs. Define, for t>0,(23)L2t≔x0+c+∑i,j=1mFijt+Gi,jt+∑i,j=1m∫0∞Kij′s∫t-stgjxσdσds.That is,(24)L2t=L1t+∑i,j=1m∫0∞Kij′s∫t-stgjxσdσds,t>0.We have from (16) and (24) that(25)L2′t=L1′t+∑i,j=1m∫0∞Kij′sgjxt-gjxt-sds≤∑i,j=1mb~ijtfijL2t+Kij0gjL2t+∑i,j=1mgjL2t∫0∞Kij′sds≤∑i,j=1mb~ijtfijL2t+Kij0+∫0∞Kij′sdsgjL2t,t>0and by integration we find, for t>0,(26)L2t≤L20+∑i,j=1m∫0tb~ijsfijL2s+Kij0+∫0∞Kij′σdσgjL2sds,with(27)L20=x0+c+∑i,j=1m∫-∞0Kij-σgjx0σdσ+∫0∞Kij′s∫-s0gjx0σdσds.Next, we proceed as in Case (a) with the new functional L2(t) (24), the constant L2(0) (27), and Kij(0)+∫0∞‍Kij′(σ)dσ instead of Kij(0) in the new λ~k. The proof is complete.Corollary 3.
If∫0∞λk(s)ds≤∫ωk-1∞(dz/hkz), k=1,…,n (in (a) and λ~k, ω~k in (b)), then solutions of (1) are global in time. Moreover, if ∫0ta(s)ds→∞ and ωn(t) (resp., ω~n(t)) grows up at most polynomially, then the decay is exponential.Remark 4.
We have judged it useful to treat case (a) separately even though it is covered by case (b) for the simple reason that this case arises in real applications as it corresponds to the “fading memory” situation. Same for the caseKij′(s)≤kKij(s), s>0 for some k>0 too looks unnecessary to study separately as it is covered by the second case in the proof but in fact it is also quite interesting. Indeed, in this case, from (16) we have(28)L1′t=∑i,j=1mb~ijtfijGi,jt+Kij0gjxt+∫-∞tKij′t-σgjxσdσ≤∑i,j=1mb~ijtfijGi,jt+Kij0gjxt+k∫-∞tKijt-σgjxσdσ≤kL1t+∑i,j=1mb~ijtfijL1t+Kij0gjL1t,t>0.Therefore,(29)L1te-kt′≤e-kt∑i,j=1mb~ijtfijL1t+Kij0gjL1t,t>0and thus(30)L1te-kt≤L10+∑i,j=1m∫0te-ksb~ijsfijL1s+Kij0gjL1sds.At this point we must point out that, unlike in the proof of the theorem, we cannot pass to L1se-ks inside the arguments of fij and gj (in (30)). However, if the functions fij and gj belong to the class H, that is, there exist ξij and ζj such that fijαu≤ξijαfiju and gjαu≤ζjαgju, α>0, u≥0, then for t>0(31)L1te-kt≤L10+∑i,j=1m∫0te-ksb~ijsfijeksL1se-ks+Kij0gjeksL1se-ksds≤L10+∑i,j=1m∫0te-ksb~ijsξijeksfijL1se-ks+Kij0ζjeksgjL1se-ksds.At this stage we may apply Lemma 1 (with the new coefficients b~ij(s)ξij(eks) and Kij(0)ζj(eks)) to get a bound Q(t) for the function L1te-kt and thereafter for x~(t)e-kt=xtexp⁡∫0ta(s)-kds. If ∫0ta(s)-kds→∞ as t→∞ and that bound Q(t) does not grow faster than exp∫0ta(s)-kds we will get an exponential decay. This decay rate is to be compared with the general one obtained in the second case of our result.


## 3. Applications

This system appears in Neural Network Theory. For a basic one the reader is referred to [7, 8].A Neural Network is designed in order to mimic the human brain. It is formed by a number of “neurons” with interconnections between them. In general there are an input layer, some (one or more) hidden layers, and an output layer. The input neurons feed the neurons in the hidden layers which perform a transformation of the signal and fires it to the output neurons (or other neurons). They are widely used for solving optimization problems, analyzing, classifying, and evaluating many things. They have the advantage (over traditional computers) of forecasting, predicting, and making decisions.There are numerous applications of which we cite the following: economic indicator, data compression, complex mapping, biological systems analysis, optimization, process control, time series analysis, stock market, diagnosis of hepatitis, engineering design, soil permeability, speech processing, pattern recognition, and so on.Most of the existing papers in this theory deal with the constant coefficients case. The few papers on variable coefficients treat mainly the existence of periodic solutions. In the constant coefficients case the system will have the form(32)xi′t=-aixit+∑j=1mbijfij∫-∞tKijt-sgjxjsds+cit,where ai and bij are constants. Our theorem gives the estimate(33)xt≤ωnte-at,0≤t<β0for some β0>0, where a≔min1≤i≤mai in case (a) and a similar estimation with ω~n(t) in case (b). The corollary provides sufficient conditions ensuring global existence. In this case we have exponential decay provided that ωn(t)(ω~n(t)) does not grow too fast.Example 5.
Consider the functionsgjx=xαj, fij(y)=yβij, αj,βij>1, i,j=1,…,m. The order h1∝h2∝⋯∝hk means ordering αj and βij in a nondecreasing manner n1≤n2≤⋯≤nk. Therefore, Hj(x)=x1-nj/1-nj-x01-nj/1-nj, Hj-1(z)=x01-nj-(nj-1)z-1/nj-1, and(34)ωjt=ωj-11-njt-nj-1∫0tλjsds-1/nj-1,j=1,…,k,0<t<T∗.The value T∗ will be the largest value of t for which(35)ωj-1nj-1t∫0tλjsds<1nj-1for all j=1,…,k. For the asymptotic behavior we need this T∗ to be infinity. In particular we need a smallness condition on ω0(t).



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