New criteria for global asymptotic stability of linear neutral differential equations by a fixed point approach
Abstract
New criteria ensuring global asymptotic stability of the zero solution for a class of linear neutral differential equations in C 1 are proved, by using two auxiliary functions on a contraction condition. Necessary and sufficient conditions for the stability of our equation which also improves recent results on this field are shown. Finally, an example is provided to illustrate the feasibility and advantage of our results
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Noname manuscript No. (will be inserted by the editor) New criteria for global asymptotic stability of linear neutral differential equations by a fixed point approach Mimia Benhadri ·Tom´as Caraballo Received: date / Accepted: date Abstract New criteria ensuring global asymptotic stability of the zero solution for a class of linear neutral differential equations in C1are proved, by using two auxiliary functions on a contraction condition. Necessary and sufficient conditions for the stability of our equation which also improves recent results on this field are shown. Finally, an example is provided to illustrate the feasibility and advantage of our results. Keywords Contraction mapping principle ·Asymptotic stability ·Neutral differential equations Mathematics Subject Classification (2010) 34K20 ·34K13 ·92B20 1 Introduction Theory and applications of functional differential equations with delay have been studied by many authors (see, for example, [19,20] and the references therein). More recently, researchers have paid special attention to the study of equations in which the delay argument occurs in the derivative of the state variable as well as in the independent variable, the so-called neutral differential equations, see for instance [18,19]. Practical examples of neutral delay differential systems include biological models of single species growth [24], distributed networks containing lossless transmission lines [11], population ecology [20], and other engineering systems [21]. In particular, qualitative analysis such as M. Benhadri LAMAHIS Lab, Faculty of Sciences, Department of Mathematics, Univ. Skikda, P.O. Box 26, Skikda 21000, Algeria. E-mail: [email protected] T. Caraballo Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain. E-mail: [email protected]
2 Mimia Benhadri, Tom´as Caraballo stability of solutions of neutral differential equations has received much attention over the last few decades. We refer to [1–4,13,14,17,22,23,25] for some recent work on the subject of stability of neutral equations. Lyapunov’s direct method has been successfully used to investigate stability properties of a wide variety of differential equations. Nevertheless, the application of this method to problems of stability in differential and integrodifferential equations with delays has encountered serious obstacles when the delay is unbounded or when the equation has unbounded terms [5]-[7]. In recent years, several investigators have analyzed stability by using a new technique. Particularly, Burton and other researchers studied stability using various fixed point theorems which overcame the difficulties encountered in the study of stability by means of Lyapunov’s direct method. We can refer to [5,15,25,26] and [28]-[32] for more details. It turns out that the fixed point method is becoming a powerful technique in dealing with stability problems for stochastic differential equations with delays [12,16,22,32]. Previously, almost all scholars, who used fixed point theory to study the asymptotic stability of zero solutions of nonlinear neutral differential equations with variable delays, imposed that cmust be differentiable and τtwice differentiable and τ0(t)6= 1 for t≥0.As distinguished from this line, in [17], Jin and Luo stated a sufficient and necessary condition for the asymptotic stability in the space C0of the following equation (1.1), by using a fixed point method of continuous functions: x0(t) = −a(t)x(t) + c(t)x0(t−τ(t)) −b(t)x(t−τ(t)), t ≥0.(1.1) In [26], Raffoul obtained sufficient conditions for the asymptotic stability of the zero solution, under appropriate conditions, to the following equation x0(t) = −a(t)x(t) + c(t)x0(t−τ(t)) + Q(x(t), x(t−τ(t))) , t ≥0.(1.2) by using the contraction mapping theorem. On the other hand, Liu and Yang [23] were the first to establish a necessary and sufficient condition for the global asymptotic stability in C1of the zero solutions of the following nonlinear neutral functional differential equation x0(t) = −a(t)x(t) + c(t)x0(t−τ1(t)) + Q(t, x (t), x(t−τ2(t))) ,(1.3) by the fixed point theory, where Qis a Lipschitz continuous function with the respect to x. Very recently, by the same method of Jin and Luo [23], Ardjouni and Djoudi [1] improved the results of Liu et al. [23] to the generalized nonlinear neutral differential equation with variable delays of the form x0(t) = −a(t)x(t) + f(t, x(t−τ1(t)), ..., x(t−τN(t))) +h(t, x0(t−τ1(t)), ..., x0(t−τN(t))) ,(1.4)
Title Suppressed Due to Excessive Length 3 where f(t, 0, ..., 0) = h(t, 0, ..., 0) = 0 and there exist bounded functions bi, ci∈C([0,∞),(0,∞)), a∈C(R+,R), such that |f(t, x1, ..., xN)−f(t, y1, ..., yN)| ≤ N X i=1 bi(t)|xi−yi|,(1.5) |h(t, x0 1, ..., x0 N)−h(t, y0 1, ..., y0 N)| ≤ N X i=1 ci(t)|x0 i−y0 i|,(1.6) for all xi, yi∈R, i = 1, ..., N. More precisely, the following result was established. Theorem A. (Ardjouni and Djoudi [1]) Suppose that assumptions (1.5),(1.6) hold, and there exists a constant η∈(0,1) such that for t≥t0, lim inf t→∞ Zt t0 a(s)ds > −∞,(1.4) and Zt t0 e−Rt sa(u)du N X i=1 (|bi(s)|+|ci(s)|)ds ≤η, (1.5) |a(t)|Zt t0 e−Rt sa(u)du N X i=1 (|bi(s)|+|ci(s)|)ds + N X i=1 (|bi(t)|+|ci(t)|)≤η. (1.6) Then the zero solution of equation (1.4) is globally asymptotically stable in C1if and only if Zt 0 a(s)ds → ∞ as t→ ∞.(1.7) By using the contraction mapping principle, the authors established some new conditions to ensure that the zero solution of equation (1.4) is globally asymptotically stable in C1. Unlike most research methods, these conditions do not require a quadratic differentiability of delay τand τ0(t)6= 1 for t≥0. In addition, in [1,2,28], they all studied the global asymptotic stability in C1. Inspired by the application of the fixed point method mentioned above, in this paper, we will state some new conditions, which make stability conditions more feasible and the results in [1,17,23,26] are improved. By using two auxiliary functions gand pto construct a contraction mapping on a complete metric space S, defined below, which may depend on the initial condition ϕ, we obtain Theorem 1.3 which will be proved in Section 2. Namely, a necessary and sufficient condition ensuring the global asymptotic stability in C1is proved. In addition, an example is eventually analyzed to illustrate the effectiveness of the proved results.
4 Mimia Benhadri, Tom´as Caraballo Notice that the condition (1.6) in Theorem A is mainly dependent of the constraint N X i=1 |bi(t)|+ N X i=1 |ci(t)|<1. However, there are some interesting examples where the constraint is not satisfied. It is our aim in this paper to remove this constraint condition and consider the global stability in C1of the special case of (1.4) when f(t, x(t−τ1(t)), ..., x(t−τN(t))) = N X i=1 bi(t)x(t−τi(t)), and h(t, x0(t−τ1(t)), ..., x0(t−τN(t))) = N X i=1 ci(t)x0(t−τi(t)). In particular, we introduce two auxiliary continuous functions gand pto define an appropriate mapping, and present new criteria for the global asymptotic stability of equation (1.2) which can be applied to the case N X i=1 |bi(t)|+ N X i=1 |ci(t)| ≥ 1, as well. 2 Preliminaries Let us consider the following class of neutral differential equations with variable delays, x0(t) = −a(t)x(t) + N X i=1 bi(t)x0(t−τi(t)) + N X i=1 ci(t)x(t−τi(t)) , t ≥t0, (2.1) denote x(t)∈Rthe solution to (2.1) with the initial condition x(t) = ϕ(t) for t∈[m(t0), t0], where ϕ∈C([m(t0), t0],R).We assume that a, bi, ci∈C(R+,R), τi∈ C(R+,R+) satisfy t−τi(t)→ ∞ as t→ ∞, i = 1,2, ..., N, (2.2) and for each t0≥0,mi(t0) = inf{t−τi(t), t ≥t0}, m(t0) = min{mi(t0), i = 1,2, ..., N}. For each t0∈[0,∞),denote C1 t0=C1([m(t0), t0],R) with the norm defined by |x|t0:= max t∈[m(t0),t0]{|x(t)|,|x0(t)|} ,
Title Suppressed Due to Excessive Length 5 for x∈C1 t0=C1([m(t0), t0],R). In addition, denote Φt0,where Φt0: = (ϕ∈C1 t0:ϕ0 −(t0) = −a(t)ϕ(t0) + N X i=1 bi(t0)ϕ0(t0−τi(t0)) + N X i=1 ci(t0)ϕ(t0−τi(t0))). For each t0∈[0,∞), we choose initial functions for equation (2.1) of the type ϕ∈Φt0. Let us recall the definitions of stability that will be used in the next section. Definition 1.1. For each initial value (t0, ϕ)∈[0,∞)×Φt0,xis said to be a solution of equation (2.1) through (t0, ϕ)if x∈C1([m(t0),∞),R)satisfies equation (2.1)on [t0,∞)and x(t) = ϕ(t)for t∈[m(t0), t0]. Such a solution will be denoted by x(t) = x(t, t0, ϕ). Definition 1.2. i) The Zero solution of equation (2.1) is said to be stable in C1if, for any t0∈[0,∞),ε > 0,there is a δ=δ(ε, t0)such that ϕ∈Φt0and |ϕ|t0< δ implies max s∈[m(t0),t0]{|x(s, t0, ϕ)|,|x0(s, t0, ϕ)|} < ε for t≥t0. ii) The zero solution of equation (2.1) is said to be globally asymptotically stable in C1if it is stable in C1,and for any t0∈[0,∞), ϕ ∈Φt0implies lim t→∞x(t, t0, ϕ) = lim t→∞x0(t, t0, ϕ)=0. At light of the previous definition of solution of equation (2.1), it is clear that the conditions imposed on the initial functions are sensible. 3 Stability by contraction mapping As we mentioned previously, the results of this work extend and improve previously known results. More exactly, we will consider a linear scalar neutral delay differential equation with variable delays and give new conditions to ensure that the zero solution is global asymptotically stable in C1by means of fixed point theory. By weakening the assumptions on the neutral coefficient ciand delays τi, τ0 i(t)6= 1,∀t≥0,and by obtaining some criteria, easier to check in applications, which does not satisfy the constraint N X i=1 |bi(t)|+ N X i=1 |ci(t)|<1. However, the mathematical analysis used in this research to construct the mapping to employ fixed point theorem is different from that of [1]. The results of this article are new and they extend and improve previously known results.
6 Mimia Benhadri, Tom´as Caraballo To the best of our knowledge, there are few authors who have used the fixed point theorem to prove the existence and uniqueness of solution and the stability of trivial equilibrium of several special cases of (2.1) all at once [3, 29,31]. In our study, as we are mainly concerned with the stability analysis of our model, we will assume that there exists a unique solution of (2.1) globally defined in time. Theorem 3.1.Consider the neutral delay differential equation (2.1)and suppose the following conditions are satisfied: H1) Suppose there exists a bounded function p: [m(t0),∞[→(0,∞)with p(t) = 1 for t∈[m(t0), t0]such that p0(t)exists for all t∈[m(t0),∞[, H2) there exists an arbitrary bounded continuous function g∈C([m(t0),∞[,R+) and lim inf t→∞ Zt t0 g(s)ds > −∞,(3.1) H3) there exists a constant η∈(0,1 2)such that for t≥t0, r1(t) := Zt t0 e−Rt sg(s)ds g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s)#ds ≤η, (3.2) and r2(t) := |g(t)|Zt t0 e−Rt sg(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s))ds + g(t)−a(t) + p0(t) p(t) + N X i=1 bi(t)p(t−τi(t)) p(t) + N X i=1 bi(t)p0(t−τi(t)) p(t) + N X i=1 ci(t)p(t−τi(t)) p(t) ≤η.(3.3) Then the zero solution of equation (2.1) is globally asymptotically stable in C1if and only if Zt 0 g(s)ds → ∞ as t→ ∞.(3.4)
Title Suppressed Due to Excessive Length 7 Proof. (⇐:) First, suppose that Rt 0g(s)ds → ∞ as t→ ∞. For each t0∈[0,∞),we define Sas the following space S=nz∈C1([m(t0),∞),R) : lim t→∞z(t) = lim t→∞z0(t)=0o, with the metric defined by kzk:= max t∈[m(t0),∞){|z(t)|,|z0(t)|} . Then Sis a complete metric space. For any initial function ϕ∈Φt0, let Dl ϕ=z∈S:z(t) = ϕ(t) for t∈[m(t0), t0] and max t≥t0 {|z(t)|,|z0(t)|} ≤ l, which is a nonempty, closed convex subset of S. The technique for constructing a contraction mapping comes from an idea in [31]. Indeed, let z(t) = ϕ(t) on t∈[m(t0), t0] and for t≥t0 x(t) = p(t)z(t).(3.5) Replacing (3.5) into (2.1),we have z0(t) = −a(t) + p0(t) p(t)z(t) + N X i=1 bi(t)p(t−τi(t)) p(t)z0(t−τi(t)) + N X i=1 ci(t)p(t−τi(t)) + bi(t)p0(t−τi(t)) p(t)z(t−τi(t)) .(3.6) If zsatisfies (3.6) then it can be verified that xsatisfies (2.1). Since pis a positive bounded function, to obtain global asymptotic stability of the zero solution of (2.1), it remains to prove that the zero solution of (3.6) is globally asymptotically stable in C1. Multiplying both sides of (3.6) by eRt 0g(u)du and integrating from t0to t, Zt t0heRs 0g(u)duz(s)i0 ds = t Zt0 eRs 0g(u)du g(s)−a(s) + p0(s) p(s)z(s)ds + t Zt0 eRs 0g(u)du N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) ds + t Zt0 eRs 0g(u)du N X i=1 ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s)) p(s)z(s−τi(s)) ds.
8 Mimia Benhadri, Tom´as Caraballo As a consequence, we arrive at z(t)eRt 0g(u)du =ϕ(t0) + t Zt0 eRs 0g(u)du g(s)−a(s) + p0(s) p(s)z(s)ds + t Zt0 eRs 0g(u)du N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) ds + t Zt0 eRs 0g(u)du N X i=1 ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s)) p(s)z(s−τi(s)) ds. Dividing both sides of the above equation by eRt 0g(s)ds, we obtain z(t) = e−Rt t0g(s)dsϕ(t0) + Zt t0 e−Rt sg(u)du g(s)−a(s) + p0(s) p(s)z(s)ds +Zt t0 e−Rt sg(u)du N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) ds +Zt t0 e−Rt sg(u)du N X i=1 ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s)) p(s) ×z(s−τi(s)) ds. Clearly, Ψ(z) : R→Ris continuous with (Ψz) (t) = ϕ(t) for t∈[m(t0), t0], and for t≥t0, (Ψz) (t) = e−Rt t0g(s)dsϕ(t0) + Zt t0 e−Rt sg(u)du g(s)−a(s) + p0(s) p(s)z(s)ds +Zt t0 e−Rt sg(u)du N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) ds +Zt t0 e−Rt sg(u)du N X i=1 ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s)) p(s) ×z(s−τi(s)) ds. (3.7)
Title Suppressed Due to Excessive Length 9 Initially, we show that, Ψ:Dl ϕ→Dl ϕ. In view of (3.7), we can derive, for t≥t0, (Ψz)0(t) = −ϕ(t0)g(t)e−Rt t0g(s)ds +g(t)−a(t) + p0(t) p(t)z(t) + N X i=1 bi(t)p(t−τi(t)) p(t)z0(t−τi(t)) + N X i=1 ci(t)p(t−τi(t)) + bi(t)p0(t−τi(t)) p(t)z(t−τi(t)) −g(t)Zt t0 e−Rt sg(u)du g(s)−a(s) + p0(s) p(s)z(s)ds −g(t)Zt t0 e−Rt sg(u)du N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) ds −g(t)Zt t0 e−Rt sg(u)du N X i=1 ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s)) p(s) ×z(s−τi(s)) ds. Thus (Ψz)0(t) = −g(t) (Ψz) (t) + g(t)−a(t) + p0(t) p(t)z(t) + N X i=1 bi(t)p(t−τi(t)) p(t)z0(t−τi(t)) (3.8) + N X i=1 ci(t)p(t−τi(t)) + bi(t)p0(t−τi(t)) p(t)z(t−τi(t)) . By the definition of Φt0,(3.8) yields (Ψz)0 +(t0) = −g(t0)ϕ(t0) +g(t0)−a(t0) + p0(t0) p(t0)z(t0) + N X i=1 bi(t0)p(t0−τi(t0)) p(t0)z0(t0−τi(t0)) + N X i=1 ci(t0)p(t0−τi(t0)) + bi(t0)p0(t0−τi(t0)) p(t0) ×z(t0−τi(t0)) =ϕ0 −(t0).
16 Mimia Benhadri, Tom´as Caraballo implies that the zero solution of equation (3.6) is globally asymptotically stable in C1.This shows that the zero solution of (2.1) is asymptotically stable if (3.4) holds. (:⇒) Assume that the zero solution of equation (2.1) is globally asymptotically stable in C1. Now, we prove that (3.4) holds. If not, let us assume that (3.1) does not hold. Otherwise, set J= lim inf t→∞ Zt 0 g(s)ds 0 , K = sup t≥t0 e−Rt t0g(s)ds and 0 A= sup t≥t0 {|g(t)|} . Thus, it follows from (3.1) that J∈(−∞,∞), 0 K, 0 A∈[0,∞). Therefore, there exists an increasing sequence {tn} ⊂ [0,∞) such that lim n→∞ tn=∞and lim n→∞ Ztn 0 g(s)ds =J, n = 1,2, ... (3.13) Denote In=Ztn 0 eRs 0g(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s)!ds, n= 1,2, ... From (3.2), it follows that In=eRtn 0g(u)du Ztn 0 eRs 0g(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s)!ds ≤ηeRtn 0g(u)du < eJ. This, together with (3.13), implies that the sequence {In}is bounded. Furthermore, there exists a convergent subsequence. For brevity of notation, we still assume that {In}is convergent. Therefore, there exists a positive
Title Suppressed Due to Excessive Length 17 integer msuch that for any integer n > m, Ztn tm eRs 0g(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s)!ds <1−η 8B(e−J+ 1),(3.14) and e−Rtn tmg(u)du >1 2, e−Rtn 0g(u)du < e−J+ 1, eRtm 0g(u)du < eJ+ 1,(3.15) where B= max 0 KeJ+ 1,0 K0 AeJ+ 1,1. For any δ0>0, consider the solution z(t) = z(t, tm, ϕ) of equation (3.6) with |ϕ|tm< δ0and |ϕ(tm)|>δ0 2.It follows from (3.7) ,(3.8) ,(3.15) and (3.1) −(3.3) ,that for t∈[tm,∞), |z(t)| ≤ δ0e−Rt tmg(s)ds +Zt tm e−Rt sg(u)du g(s)−a(s) + p0(s) p(s) |z(s)| + N X i=1 bi(s)p(s−τi(s)) p(s) |z0(s−τi(s))| +"N X i=1 ci(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s)#|z(s−τi(s))|!ds ≤0 KeJ+ 1δ0+kzktmZt tm e−Rt sg(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) +"N X i=1 ci(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s)#!ds ≤Bδ0+ηkzktm,
18 Mimia Benhadri, Tom´as Caraballo and |z0(t)|≤|z(tm)| |g(t)|e−Rt tmg(s)ds + g(t)−a(t) + p0(t) p(t) |z(t)| + N X i=1 bi(t)p(t−τi(t)) p(t) |z0(t−τi(t))| +"N X i=1 ci(t)p(t−τi(t)) p(t) + N X i=1 bi(t)p0(t−τi(t)) p(t)#|z(t−τi(t))| +|g(t)|Zt tm e−Rt sg(u)du g(s)−a(s) + p0(s) p(s) |z(s)| + N X i=1 bi(s)p(s−τi(s)) p(s) |z0(s−τi(s))| + N X i=1 ci(s)p(s−τi(s)) p(s) + bi(s)p0(s−τi(s)) p(s)|z(s−τi(s))|ds ≤0 K0 AeJ+ 1δ0 +kzktm|g(t)|Zt tm e−Rt sg(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s) + N X i=1 bi(s)p0(s−τi(s)) p(s)# + g(t)−a(t) + p0(t) p(t) + N X i=1 ci(t)p(t−τi(t)) p(t) + N X i=1 bi(t)p0(t−τi(t)) p(t) ≤Bδ0+ηkzktm. Hence, kzktm≤Bδ0+ηkzktm,thus we have kzktm≤B 1−ηδ0,for all t≥tm.(3.16)
Title Suppressed Due to Excessive Length 19 It follows from (3.7) ,(3.14) −(3.16) that ,for any n > m, |z(tn)| ≥ |ϕ(tm)|e−Rtn tmg(s)ds −Ztn tm e−Rtn sg(u)du g(s)−a(s) + p0(s) p(s)z(s) + N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) + N X i=1 ci(s)p(s−τi(s)) p(s)z(s−τi(s))#ds ≥δ0e−Rtn tmg(u)du −Ztn tm e−Rtn sg(u)du g(s)−a(s) + p0(s) p(s)z(s) + N X i=1 bi(s)p(s−τi(s)) p(s)z0(s−τi(s)) + N X i=1 ci(s)p(s−τi(s)) p(s)z(s−τi(s))#ds ≥δ0e−Rtn tmg(u)du − kzktme−Rtn 0g(u)du Ztn tm eRs 0g(u)du g(s)−a(s) + p0(s) p(s) + N X i=1 bi(s)p(s−τi(s)) p(s) + N X i=1 ci(s)p(s−τi(s)) p(s)#ds ≥1 4δ0−δ0B 1−ηe−J+ 11−η 8B(e−J+ 1) =1 4δ0. The facts that lim n→∞tn=∞and the zero solution of equation (3.6) is globally asymptotically stable in C1imply lim n→∞z(t, tn, ϕ) = lim n→∞z0(t, tn, ϕ) = 0,which is in contradiction with (3.17) .Hence condition (3.4) is necessary in order that (3.6) has a solution asymptotically stable in C1. Thus, the zero solution of (3.6) is asymptotically stable, and hence the zero solution of (2.1) is asymptotically stable in C1. The proof is complete. Remark 3.1. When f(t, x(t−τ1(t))), ..., x(t−τN(t)))) = N X i=1 bi(t)x(t−τi(t))
20 Mimia Benhadri, Tom´as Caraballo and h(t, x(t−τ1(t))), ..., x(t−τN(t)))) = N X i=1 ci(t)x0(t−τi(t)), with g(t)≡a(t) and p(t) = 1, Theorem 3.1 reduces to Theorem A. Remark 3.2. It follows from the first part of the proof of Theorem 3.1 that the zero solution of (2.1) is globally asymptotically stable in C1under (3.1),(3.2), and (3.3). Moreover, Theorem 3.1 still holds true if (3.2),(3.3) are satisfied for t≥tσfor some tσ∈R+. 4 An Example In this section, we analyse an example to illustrate two facts. On the one hand, we will show how to apply our main result in this paper, Theorem 3.1. On the other hand and most importantly, we will highlight the real interest and importance of our result because the previous theory developed by Ardjouni and Djoudi [1] cannot be applied to this example. Example 4.1. Consider the following linear neutral delay differential equation x0(t) = −a(t)x(t) + b1(t)x0(t−τ1(t)) + c1(t)x(t−τ1(t)),(4.1) for t≥0,corresponding to equation (2.1) when N= 1, τ1(t) = π 2and a(t)=1−6 sin tcos t 1 + sin2t, c1(t) = sin6t, b1(t) = 1 10 sin6t. By choosing g(t)=1 and p(t) = 1 + sin2t3in Theorem 3.1, we obtain that g(t)−a(t) + p0(t) p(t) = 0. By straightforward computations, we have p(t−τ1(t)) p(t)c1(t) =2−sin2t3 1 + sin2t3sin6t ≤0.1539, and p(t−τ1(t)) p(t)b1(t) =2−sin2t3 1 + sin2t3 sin6t 10 ≤0.1539 10 ≤0.0153. Since |sin tcos t|<1 for t∈R,then we deduce p0(t−τ1(t)) p(t)b1(t) =6 10 2−sin2t2 1 + sin2t3sin6t |sin tcos t| ≤6 10 ×0.1286 × |sin tcos t| ≤0.0771.
Title Suppressed Due to Excessive Length 21 According to the definition of r1, r2under conditions (3.2) and (3.3) of Theorem 3.1, for t∈[0,∞),we can calculate and obtain r1(t) := 0.1539 + 0.0153 + 0.0771 = 0.2463, and r2(t) := 2 (0.1539 + 0.0153 + 0.0771) = 0.4926, then, we have r1(t)<1/2 and r2(t)<1/2. Hence, since t−π 2→ ∞ as t→ ∞,and it is easy to verify that Rt 0g(s)ds → ∞ as t→ ∞,|p(t)| ≤ 2, then assumptions of Theorem 3.1 are fulfilled. Therefore, the zero solution of (4.1) is globally asymptotically stable in C1thanks to Theorem 3.1. Note that |b1(t)|+|c1(t)|=1 10 sin6t+sin6t= 1.1 when t=kπ+π 2for k= 0,1,2, .... Thus, Theorem A cannot be applied to equation (4.1), when f(t, x(t−τ1(t)))) = b1(t)x(t−τ1(t)) and h(t, x(t−τ1(t))) = c1(t)x0(t−τ1(t)) in (1.4). Remark 3.3: The method in this paper can be applied to more general neutral differential systems than Eq. (2.1). Conflict of interest The authors declare that they have no conflict of interest. Acknowledgements This work has been partially supported by FEDER and Ministerio de Ciencia, Innovaci´on y Universidades of Spain (Grant PGC2018-096540-B-I00), and Junta de Andaluc´ıa, Spain (Grant US-1254251) References 1. Ardjouni A. and Djoudi A., Global asymptotic stability of nonlinear neutral differential equations with variable delays. Nonlinear Stud. 23, 2 (2016), 157-166. 2. Ardjouni A.and Djoudi A., Global asymptotic stability of nonlinear neutral differential equations with infinity delay. TJMM 9 No. 2(2017), 125-133. 3. Ardjouni A. and Djoudi A., Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis 74 (2011), 2062-2070. 4. Ardjouni A. and Djoudi A., Stability in nonlinear neutral differential with variable delays using fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations, No. 43 (2011), pp.1-11. 5. Burton T.A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, (2006). 6. Burton T.A., Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Studies 9 (2001), 181-190. 7. Burton T.A., Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4 (2003), 15–32. 8. Burton T. A., A fixed point theorem of Krasnoselskii. App. Math. Lett. 11 (1998), 85–88.
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