New sufficient conditions for global asymptotic stability of a kind of nonlinear neutral differential equations

New sufficient conditions for global asymptotic stability of a kind of nonlinear neutral differential equations Mimia Benhadria, Tom´as Caraballob aFaculty of Sciences, Department of Mathematics, University 20 August 1955, P.O. Box 26, Skikda 21000, Algeria. emails: [email protected] bDepartamento de Ecuaciones Difererenciales y An´alisis Num´erico, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla (Spain) email: [email protected] February 28, 2021 Abstract This paper addresses the stability study for nonlinear neutral differential equations. Thanks to a new technique based on the fixed point theory, we find some new sufficient conditions ensuring the global asymptotic stability of the solution. In this work we extend and improve some related results presented in recent works of literature. Two examples are exhibited to show the effectiveness and advantage of the proved results. AMS Subject Classifications: 34K20, 34K13, 92B20 Keywords: Contraction mapping principle; Asymptotic stability; Neutral differential equations. 1 Introduction It is well-known that the theory of neutral functional differential equations has attracted many types of research due to its wide and great applications in many fields of mathematical science and engineering such as neural networks, population dynamics, control theory, and many other phenomena. For appropriate literature we can refer to the books [17]-[20]. A neutral delay differential equation is a kind of delay differential equation where the delay argument occurs in the highest order derivative of the state, which can be used to describe many real-world phenomena that arise in the areas for example, lossless transmission lines, theory of automatic control and others,we refer the reader to references 1 Brayton [10], Hale [18], Kuang [19], Kolmanovskii and Myshkis, [20], and the sources related. Up today, significant progress has been made in the qualitative theory (e.g. oscillation theory, periodicity, stability, the existence of a periodic solution, asymptotic behavior, boundedness, instability, and so on) of neutral delay differential equations. For more details, we can refer to ( [1]-[5] and [10]-[16],[23]-[31] ), and the references of these sources. One of the most qualitative concepts in mathematic theory is determining the stability of a given model. The theory of stability was created at the end of the 19th century by Lyapunov. This method is now known as the Lyapunov’s direct method or Lyapunov function. For decades, Lyapunov developed a method for determining stability in many areas of differential equations without solving the equations themselves. This theory has been proven significantly its effectiveness over a century and it has been achieved wide applications in various fields of physics and mathematical sciences. Unfortunately, when we try to carry over the principles of the Lyapunov stability theory to special problems, we face a large number of difficulties and it appears that new methods are needed to overcome those obstacles (see [5]-[7]). Luckily, Burton and many authors have used fixed point theory as an alternative to studying the stability of deterministic or stochastic systems, where some of these problems in Lyapunov functions have been solved. In the current study, we use this method to address a kind of nonlinear neutral differential equations (see [8], [10], [15], [21]). In [16], Jin and Luo studied the asymptotic stability in the space C0of the scalar nonlinear neutral differential equation of the form u0(t) = −a(t)u(t) + c(t)u0(t−τ(t)) −b(t)u(t−τ(t)), t ≥0.(1.1) The work of Jin and Luo in [16] requires that the delay τis twice differentiable, and τ0(t)6= 1 for t≥0 and cis differentiable. However, there are many interesting examples where these conditions are not satisfied. It is our purpose in this paper to remove these restrictive conditions by studying the global stability in the space C1. As it is known, there are a few papers [1]- [3] and [22],[28] have discussed the global asymptotic stability of solutions of neutral differential equations in C1. For example, Liu and Yang in [22] was the first to establish necessary and sufficient conditions for the asymptotic stability in C1for the equation u0(t) = −a(t)u(t) + c(t)u0(t−τ1(t)) + Q(t, u (t), u(t−τ2(t))) ,(1.2) where Qis a Lipschitz continuous function in u. Liu and Yang have been able, in their work, to avoid the derivative of the coefficient cand they also do not need that the delay τis twice differentiable, and τ0(t)6= 1 for t≥0. Otherwise, a good contribution of their work was obtained to relax the conditions on the coefficient cand the delay 3c4. Recently, by the same method of Liu et Yang [22], Ardjouni and Djoudi [1] have 2 adressed general form than (1.2) as follows u0(t) = −a(t)u(t) + g(t, u0(t−τ1(t), u0(t−τ2(t), ..., u0(t−τn(t)) +f(t, u(t−τ1(t), u(t−τ2(t), ..., u(t−τn(t)),(1.3) where f(t, u1, ..., un),g(t, u1, ..., un) are continuous and satisfy Lipschitz condition in u1, ..., un,respectively. However, it still remains unexplored the case in which one considers all terms of the equation (1.3) are nonlinear, what is the main reason for the analysis we will perform in the current paper. In 2020 Zaid et al. [31] obtained stability results in C0about the zero solution of the standard form of the totally nonlinear delay differential equation u0(t) = − N X i=1 ai(t, ut)u(t) + f(t, ut), t ≥t0.(1.4) In the case N= 1, Eq. (1.4) reduces to the one in [13]. With the previous motivation, in this paper, we extend the results in [31] to the totally nonlinear neutral differential equation represented in (2.1),(see below). More precisely, we will study the stability in the space C1(as described in more details below) which is a stronger concept of stability than the usual one in C0. The study of stability in C1is therefore much richer than the classical stability in C0. By applying the fixed point theory, we will state new and more applicable stability criteria in C1. The sufficient conditions obtained are quite practicable and we will no longer need the delay to be twice differentiable or coefficients are differentiable, which is required in some previous relevant works [3],[4],[12], [16],[30]. It is this new feature that makes the asymptotic behavior in C1more important and more useful as well. Our work extends and improves the results in [1],[13],[16],[22],[31]. In addition, two examples are given to test the feasibility and advantage of the proved results. 2 Notations and preliminaries Let R,R+,and R−denote (−∞,+∞),[0,+∞),(−∞,0] respectively. In the current paper, we aim to discuss the asympotic stabilty in C1for standard form of neutral differential equations as follows, u0(t) = − N X i=1 ai(t, ut)u(t) + g(t, u0 t) + f(t, ut), t ≥t0,(2.1) where f, g ∈C(R+×B, R) and ai∈C(R+×B, R),i= 1, N,with, B=φ∈CR−,R:φbounded, with the norm kφk◦:= sup θ∈(−∞,0] |φ(θ)|.Define also CL={ξ∈C:kξk◦≤L}and C1 L0=ξ∈C1: ξ0 ◦≤L0. 3 If u∈C1(R,R) is bounded and for t≥0 is a fixed number, we let ut, u0 t∈C be defined by ut(θ) = u(t+θ) and u0 t(θ) = u0(t+θ) for θ∈R−.(2.2) We define kxk[s,t]:= sup ξ∈[s,t] |x(ξ)|, for a function x:R→R. Before starting the main result of this paper, we impose the following assumptions: (A1) there exists a constant L > 0 and a function b1∈C(R,R+) such that, for all φ, ψ ∈CLand for all t≥0, |f(t, φ)−f(t, ψ)|≤|b1(t)| kφ−ψk◦.(2.3) (A2) there exists a constant L0>0 and a function b2∈C(R,R+) such that, for all φ, ψ ∈C1 L0and for all t≥0, gt, φ0−gt, ψ0≤ |b2(t)| φ0−ψ0 ◦(2.4) (A3) ∀ε > 0 and t1≥0,there exists a t2> t1such that [t≥t2, ut∈CL],imply |f(t, ut)|≤|b1(t)|ε+kuk[t1,t].(2.5) (A4) ∀ε > 0 and t1≥0,there exists a t3> t1such that t≥t3, u0∈C1 L0, imply |g(t, u0 t)|≤|b2(t)|ε+ku0k[t1,t].(2.6) (A5) there exists α1, α2∈C(R,R),(α2is bounded) such that α1(t)≤ N X i=1 ai(t, ut)≤α2(t). (A6) Assume furthermore that, f(t, 0) = g(t, 0) = 0 for all t≥t0.(2.7) which guarantees that (2.1) possesses a trivial solution u(t)=0. For each t0∈[0,∞),denote C1 t0=C1(]−∞, t0],R) with the norm defined by |u|t0:= max t∈(−∞,t0]{|u(t)|,|u0(t)|} , for u∈C1 t0=C1((−∞, t0],R). In addition, denote Φt0,where Φt0=(ϕ∈C1 t0:ϕ0 −(t0) = − N X i=1 ait0, ϕt0ϕ(t0) + gt0, ϕ0 t0+ft0, ϕt0). 4 For each t0∈[0,∞), we choose initial functions for equation (2.1) of the type ϕ∈Φt0. The definitions of stability in C1,as well as the necessary notation for our study are borrowed from the paper [22], but the nonlinearities in our model and the fact that we are considering a neutral term, make our study nontrivial and meaningful. We now recall some basic informations. Definition 1.1. For each initial value (t0, ϕ)∈[0,∞)×Φt0,uis called a solution of (2.1) with (t0, ϕ) if u∈C1((−∞,+∞),R) satisfies equation (2.1) for all most t≥t0and u=ϕfor t≤t0.Such a solution will be denoted by u(t) = u(t, t0, ϕ). We now prepare definitions in order to prove asymptotic stability in C1for (2.1). Definition 1.2. i) The trivial solution of (2.1) is: i) stable in C1, if for any ε > 0 and t≥t0,there is a scalar δ=δ(ε, t0)>0, such that for any initial function ϕ∈Φt0satisfying |ϕ|t0< δ, we have for the corresponding solution that max s∈(−∞,t]{|u(s, t0, ϕ)|,|u0(s, t0, ϕ)|} < ε for t≥t0. ii) asymptotically stable in C1,if u(t) is stable in C1,and for any initial function ϕ∈Φt0we have for the corresponding solution that lim t→∞u(t, t0, ϕ) = lim t→∞u0(t, t0, ϕ)=0. At the light in definition 1.1. For the problem of the initial value of the equation (2.1), sensible conditions are imposed. Since our model (2.1) involves nonlinear term N X i=1 ai(t, ut)u(t), so that it is more complex and different than those of the above literature [1], [13], [16],[22], [31] which also implies some difficulties in mathematical analysis. That means we study how the asymptotic behavior property in C1will be when (1.4) is added to the perturbed nonlinear neutral term g(t, u0 t). Motivated by the previously cited literature related to fixed point approch [5], [11], [12], [13], [15], [15]. As the main tool, it used Banach’s fixed point to obtain some new sufficient conditions ensuring the global asymptotic stability results in C1to Eq. (2.1). Finally, two examples are given to illustrate the real interest and importance of the proposed results. 3 Stability by contraction mapping It is well known that studying the stability of an equation by Banach’s fixed point method based on three essential points: a complete metric space, a variation of parameters, and the formulation of an appropriate contraction mapping. 5 The advantage of this method is that the fixed point argument leads, the existence, uniqueness, boundedness, and stability of the equation, all at once. Up till now, no work has concerned the equation (2.1) to establish sufficient conditions for the global asymptotic behavior in C1.Let us begin to explore this world of stability. In this section we shall discuss the asymptotic stability in C1for equation (2.1) . Theorem 3.1. Assume hypotheses (A1)–(A6) hold, and for any t≥t0,if there exists η∈(0,1 2)such that, lim inf t→∞ Zt t0 α1(s)ds > −∞,(3.1) and Zt t0 e−Rt sα1(z)dz (|b1(s)|+|b2(s)|)ds ≤η, (3.2) |α2(t)|Zt t0 e−Rt sα1(z)dz (|b1(s)|+|b2(s)|)ds + (|b1(t)|+|b2(t)|)≤η, (3.3) and Zt 0 α1(s)ds → ∞ as t→ ∞.(3.4) Then the equation (2.1) has a unique trivial solution. Moreover, it is asymptotic stable in C1. Proof. First, suppose that Rt 0α1(s)ds → ∞ as t→ ∞. For each t0∈[0,∞), let ϕ∈C((−∞, t0],R) be a fixed initial function. We define Sas the following space S=nu∈C1(R,R) : lim t→∞u(t) = lim t→∞u0(t) = 0o, with the metric defined by kuk:= max t∈R{|u(t)|,|u0(t)|} . Then Sis a complete metric space. Next, we define for any ϕ∈Φt0, Dl ϕ=u∈S:ut0=ϕand max t≥t0 {kutk◦,ku0 tk◦} ≤ l, Obviously, we know Dl ϕis a closed convex and bounded subset of S, where l= max {L, L0}. We can use the variation of parameter formula for writing the equation (2.1) as an integral equation suitable for Banach’s fixed point theorem. The 6 application Pis given in literature without details of the proof. It can be deduced from [22]. Hence, we omit it. Define the application P(u) : R→Rwith (Pu) (t) = ϕ(t) for t∈(−∞, t0], and (Pu) (t) = e−Rt t0 N P i=1 ai(s,us)dsϕ(t0) + Zt t0 e−Rt s N P i=1 ai(z,uz)dz [g(s, u0 s) + f(s, us)] ds, (3.5) for t≥t0.It is not difficult to see that P(u) : R→Ris continuous. Initially, we show that, P:Dl ϕ→Dl ϕ. In view of (3.5), we can derive, (Pu)0(t) = −ϕ(t0) N P i=1 ai(t, ut)e−Rt t0 N P i=1 ai(s,us)ds +g(t, u0 t) + f(t, ut) − N P i=1 ai(t, ut)Zt t0 e−Rt s N P i=1 ai(z,uz)dz [g(s, u0 s) + f(s, us)] ds =− N P i=1 ai(t, ut) (Pu) (t) + g(t, u0 t) + f(t, ut),(3.6) for t≥t0. By the definition of Φt0,(3.6) yields (Pu)0 +(t0) = − N P i=1 ai(t0, ut0)ϕ(t0) + gt0, u0 t0+f(t0, ut0) = ϕ0 −(t0). Hence, Pu∈C1(R) for u∈Dl ϕ. Next, we verify that max t≥t0 (Pu)0 t ◦,k(Pu)tk◦< l. Let A= sup t≥t0 {|α2(t)|} and K= sup t≥t0 e−Rt t0α1(s)ds. By conditions (3.4),(3.1), K, A ∈[0,∞).For a given small bounded initial function ϕwith |ϕ|t0< δ0,where δ0>0 satisfies δ0< l min 1,1−η K,1−2η KA .(3.7) Let u∈Dl ϕ, then max t≥t0 {ku0 tk◦,kutk◦} ≤ l. Du the conditions (2.3) ,(2.4) ,(3.7), and (3.2) ,we can get |(Pu) (t)|≤|ϕ(t0)|e−Rt t0α1(s)ds +Zt t0 e−Rt sα1(z)dz |b2(s)| ku0 sk◦ds +Zt t0 e−Rt sα1(z)dz |b1(s)| kusk◦ds ≤Kδ0+ηl < l. 7 Now, (3.6) and (2.3) ,(2.4) ,(2.7) and (3.2) ,(3.3) ,(3.7) imply that (Pu)0(t)≤ |ϕ(t0)| N P i=1 |ai(t, ut)|e−Rt t0 N P i=1 ai(s,us)ds +|g(t, u0 t)|+|f(t, ut)| + N P i=1 |ai(t, ut)|Zt t0 e−Rt s N P i=1 ai(z,uz)dz [|g(s, u0 s)|+|f(s, us)|]ds ≤KAδ0+|g(t, u0 t)−g(t, 0)|+|f(t, ut)−f(t, 0)| +l|α2(t)|Zt t0 e−Rt sα1(z)dz (|b1(s)|+|b2(s)|)ds ≤KAδ0+l(|b1(t)|+|b2(t)|) + ηl ≤KAδ0+ 2ηl < l, by the choice of δ0.This implies, max t≥t0|(Pu) (t)|,(Pu)0(t)< l. We will now show (Pu) (t) approaches zero as t→ ∞. Du the condition (3.4), we have lim t→∞e−Rt t0α1(z)dz = 0. Therefore, it is obvious that the first term of (Pu) (t) tends to zero as t→ ∞ because of condition (3.4). Next, we will show that the last term of (Pu) (t) tends to zero too. Since lim t→∞u(t) = lim t→∞u0(t) = 0,we can find T1> t0such that ∀t≥T1, max {|u(t)|,|u0(t)|} < ε, and the fact u∈Dl ϕimplies that ∀t≥t0, max {kutk◦,ku0 tk◦}< l. It is therefore follows from (2.5) and (2.6) that we can find t2> T1such that |f(t, ut)|≤|b1(t)|ε+kuk[T1,t], and |g(t, u0 t)|≤|b2(t)|ε+ku0k[T1,t], for t≥t2. 8 Hence for t≥t2,we have Zt t0 e −Rt s N P i=1 ai(z,uz)dz [g(s, u0 s) + f(s, us)] ds ≤Zt2 t0 e−Rt sα1(z)dz |g(s, u0 s) + f(s, us)|ds +Zt t2 e−Rt sα1(z)dz |g(s, u0 s) + f(s, us)|ds ≤Zt2 t0 e−Rt sα1(z)dz [|b1(s)|+|b2(s)|] max s≥t0 {ku0 sk◦,kusk◦}ds +Zt t2 e−Rt sα1(z)dz |b1(s)|ε+ku0k[T1,s]ds +Zt t2 e−Rt sα1(z)dz |b2(s)|ε+kuk[T1,s]ds, since max nkuk[T1,t],ku0k[T1,t]o≤εfor t≥t2. Then, ≤Zt2 t0 e−Rt sα1(z)dz [|b1(s)|+|b2(s)|] max s≥t0 {|u0 s|,|us|} ds +2εZt t2 e−Rt sα1(z)dz [|b1(s)|+|b2(s)|]ds ≤lZt2 t0 e−Rt2 sα1(z)dze−Rt t2α1(z)dz [|b1(s)|+|b2(s)|]ds + 2ηε. By using condition (3.4),we can find T≥t2such that for t≥T, we get le−Rt Tα1(z)dz Zt2 t0 e−RT sα1(z)dz [|b1(s)|+|b2(s)|]ds ≤ε. This yields lim t→∞ (Pu) (t) = 0 for u∈Dl ϕ. Moreover, for each u∈Dl ϕ,lim t→∞u(t) = lim t→∞u0(t)=0,given ε > 0 there exists T2> t0such that ∀t≥T2, max {|u(t)|,|u0(t)|} < ε . By conditions (2.5),(2.6) we can find a T0> T2such that, for t≥T0, we have |g(t, u0 t)| ≤ |b1(t)|ε+ku0k[T2,t], and |f(t, ut)| ≤ |b2(t)|ε+kuk[T2,t]. 9 and Zt t0 e−Rt sa(z)dz (|b1(s)|+|b2(s)|)ds ≤η, (3.18) |a(t)|Zt t0 e−Rt sa(z)dz (|b1(s)|+|b2(s)|)ds + (|b1(t)|+|b2(t)|)≤η. (3.19) Then equation (2.1) has a unique trivial solution and, it is an asymptotic stable in C1if only if Zt t0 a(s)ds → ∞ as t→ ∞. Remark 3.2. Theorem 3.1 remains true if the conditions (3.2),(3.3) are fulfilled for all t≥tσfor some tσ∈R+. 4 Remarks and illustrative Examples Let us discuss two examples for ullistration. Example 4.1. Given a nonlinear neutral differential equation u0(t) = −a(t, u(t−τ(t))) u(t) + f(t, u(t−τ(t))) + g(t, u0(t−τ(t))) ,(4.1) t≥0,where a(t, u(t−τ(t))) = 1 1 + t1 + |sin t| 1 + u2(t−τ(t)), and g(t, u0(t−τ(t))) = 0.1 1 + tsin u0(t−τ(t)) 10 , f(t, u(t−τ(t))) = 0.4 ln 1 + |u(t−τ(t))| 10 (1 + t). One can take α1(t) = 1 1 + tand α2(t) = 2|sin t| 1 + t,then α1(t)≤a(t, ut)≤α2(t). It is easy to check |α2(t)|<2,∀t∈[0,∞) , Zt 0 α1(s)ds → ∞ as t→ ∞. By straightforward computations, we can check that conditions (2.2) and (2.3) in Theorem 3.1 hold true, where τ∈C(R+,R+), and δ∈C(R+,R+) with t−τ(t)→ ∞ and t−δ(t)→ ∞ as t→ ∞.(4.2) 16 Assume that b1(t) = 0.1 2 (1 + t)and b2(t) = 0.5 10 (1 + t). Then (2.3),(2.4) hold. Also assume that η= 1/3,then for t∈[0,∞) Zt 0 e−Rt sα1(z)dz (|b1(s)|+|b2(s)|)ds ≤Zt 0 e−Rt s 1 1 + zdz 1 10 (1 + s)ds =1 10 ≤η, (4.3) and |α2(t)|Zt 0 e−Rt sα1(z)dz (|b1(s)|+|b2(s)|)ds + (|b1(t)|+|b2(t)|) ≤2Zt 0 e−Rt s 1 1 + zdz 1 10 (1 + s)ds +1 10 (1 + t) =3 10 ≤η. (4.4) Hence, all the conditions in Theorem 3.1 are verified. Therefore, the zero solution of equation (4.1) is asymptotic stable in C1. Example 4.2. Consider the following equation in the form of (2.1), u0(t) = − 2 X i=1 ai(t, u(t−τ(t))) u(t) + f(t, u(t−τ1(t)), u(t−τ2(t))) +g(t, u0(t−τ1(t)), u0(t−τ2(t))) .(4.5) By taking a1(t, u) = 0.5et 1 + et 1 + |cos t| 1 + e−u2!, a2(t, u)) = 0.5et 1 + et1 + |sin (u)| 2, τ∈C(R+,R+), and τi∈C(R+,R+) satisfy t−τi(t)→ ∞ as t→ ∞, i = 1,2.(4.6) By calculation, we have α1(t) := et 1 + et≤ 2 X i=1 ai(t, u(t−τ(t))) ≤1.75et 1 + et=: α2(t), and it is straightforward to check that |α2(t)|<1.75,∀t∈[0,∞) , and Zt 0 α1(s)ds → ∞ as t→ ∞. Let f(t, u1, u2) = ln 1 + 5 (|u1|+|u2|) 100 (1 + e−t), g (t, u1, u2) = 0.1 sin u1 5 (1 + e−t)+ 0.12 sin u2 4 (1 + e−t),then we obtain |f(t, u1, u2)−f(t, v1, v2)|≤|b1(t)| |u1−v1|+|b2(t)| |u2−v2|, 17 |g(t, u1, u2)−g(t, v1, v2)|≤|c1(t)| |u1−v1|+|c2(t)| |u2−v2|, where b1(t) = b2(t) = 5 100 (1 + e−t), and c1(t) = 0.02 1 + e−t, c2(t) = 0.03 1 + e−t. Then (A1)-(A6) hold. In addition, let η= 4/9,then for t∈[0,∞), Zt 0 e−Rt sα1(z)dz 2 X j=1 |bj(s)|+|cj(s)|ds <Zt 0 e−Rt s ez 1 + ezdz es 10 (1 + es)+0.05es (1 + es)ds < 0.15 < η, (4.7) and |α2(t)|Zt 0 e−Rt sα1(z)dz 2 X j=1 (|bj(s)|+|cj(s)|)ds + 2 X j=1 (|bj(t)|+|cj(t)|) <1.75 ×Zt 0 e−Rt s ez 1 + ezdz es 10 (1 + es)+0.05es (1 + es)ds +et 10 (1 + et)+0.05et (1 + et) <1.75 ×0.15 + et 10 (1 + et)+0.05et (1 + et) <1.75 ×0.15 + 0.15 = 0.413 ≤η. (4.8) Hence, (3.2) and (3.3) hold. According to Theorem 3.1, the zero solution of Eq. (4.5) is globally asymptotically stable in C1. Remark 4.1. Theorem 3.1 includes and generalizes the result of Ardjouni and Djoudi [1]. In fact, when we chose N= 1 and a1(t, ut) = a(t) (ais bounded), g(t, u0 t) = g(t, u0(t−τ1(t), u0(t−τ2(t), ..., u0(t−τn(t)) and f(t, ut) = f(t, u(t−τ1(t), u(t−τ2(t), ..., u(t−τn(t)),our conditions reduce to that of Ardjouni and Djoudi [1, Theorem 2.1]. Remark 4.2. It has been noted in [27] that a fading memory condition such as (2.5) ,(2.6) or (4.6) is necessary for the asymptotic behavior of a general neutral differential equation. This means that the equation representing a physical system must remember its past, but the memory must fade over time. Conclusion 1 Conclusion: In this work, a standard tottally nonlinear neutral differential equations have been studied. Based on Banach’s fixed point 18 theorem, some new sufficient conditions to guarantee that the trivial solution of equation (2.1) is globally asymptotic stable in C1have been achieved. The main contribution of this paper is confirming the importance and advantage of using the fixed point theory. The derived stability criteria are easily to apply in practice and do not need the differentiability of the delays or coefficients, which are required in [16].Moreover, we can easily see Theorem 3.1 and Corollaries cited above are independent of some restrictive conditions in reference [16]. Up now, the results derived here have not been published in the corresponding literature. An illustrative examples were given to show the efficiency of the results introduced. Hence, in futur, we would like to extend the application of this precious approach to more complex delay models such as the equations with damped stochastic perturbations and so on. We will leave this open problem for future research subjects. Acknowledgements. The research of T.C. has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-I00, and by Junta de Andaluc´ıa (Consejer´ıa de Econom´ıa y Conocimiento) under project US-1254251. References [1] Ardjouni, A., Djoudi, A., Global asymptotic stability of nonlinear neutral differential equations with variable delays. Nonlinear Stud. 23 (2016), no. 2, 157-166. [2] Ardjouni A., Djoudi A., Global asymptotic stability of nonlinear neutral differential equations with infinity delay. TJMM 9 (2017), no. 2, 125-133. [3] Ardjouni A., Djoudi A., Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis 74 (2011), 2062- 2070. 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