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Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral Differential Equations with Variable Delays

By Benhadri Mimia

Submitted: September 9th 2020Reviewed: January 15th 2021Published: March 15th 2021

DOI: 10.5772/intechopen.96040

Downloaded: 82


In this paper, we consider a neutral differential equation with two variable delays. We construct new conditions guaranteeing the trivial solution of this neutral differential equation is asymptotic stable. The technique of the proof based on the use of Krasnoselskii’s fixed point Theorem. An asymptotic stability theorem with a necessary and sufficient condition is proved. In particular, this paper improves important and interesting works by Jin and Luo. Moreover, as an application, we also exhibit some special cases of the equation, which have been studied extensively in the literature.


  • fixed points theory
  • stability
  • neutral differential equations
  • integral equation
  • variable delays

1. Introduction

For more than one hundred years, Liapunov’s direct method has been very effectively used to investigate the stability problems of a wide variety of ordinary, functional, and partial differential, integro-differential equations. The success of Liapunov’s direct method depends on finding a suitable Liapunov function or Liapunov functional. Nevertheless, the applications of this method to problems of stability in differential and integro-differential equations with delays have encountered serious difficulties if the delays are unbounded or if the equation has unbounded terms (see [1, 2, 3]). Therefore, new methods and techniques are needed to address those difficulties. Recently, Burton and his co-authors have applied fixed point theory to investigate the stability, which shows that some of these difficulties vanish when applying fixed point theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. It turns out that the fixed point method is becoming a powerful technique in dealing with stability problems for indeterministic scenes (see for instance [16, 17, 21, 23]).

For example, Luo [16] studied the mean-square asymptotic stability for a class of linear scalar neutral stochastic differential equations by means of Banach’s fixed point theory. The author did not use Lyapunov’s method; he got interesting results for the stability even when the delay is unbounded. The author also obtained necessary and sufficient conditions for the asymptotic stability. Moreover, it possesses the advantage that it can yield the existence, uniqueness, and stability criteria of the considered system in one step.

Neutral delay differential equations are often used to describe the dynamical systems which not only depend on present and past states but also involve derivatives with delays, (see [24, 25, 26, 27, 28]). It has been applied to describe numerous intricate dynamical systems, such as population dynamics [18], mathematical biology [27], heat conduction, and engineering [28], etc.

In particular, qualitative analysis for neutral type equations such as stability and periodicity, oscillation theory, has been an active field of research, both in the deterministic and stochastic cases. We can refer to [6, 7, 13, 15, 16, 17, 19, 20, 21, 23, 29, 30, 31], and the references cited therein.

With this motivation, in this paper, we aim to discuss the boundedness and stability for neutral differential equations with two delays (1). It is worth noting that our research technique is based on Krasnoselskii’s fixed point theory. We will give some new conditions to ensure that the zero solution is asymptotically stable. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. Our findings generalize and improve some results that can be found in the literature. In our result, the delays can be unbounded and the coefficients in the equations can change their sign. This paper is organized as follows. In Section 1 we present some basic preliminaries and the form of the neutral functional differential equations which will be studied. In Section 2, we present the inversion of the equation and we state Krasnoselskii’s fixed point theorem. The boundedness and stability of the neutral differential Eq. (1) are discussed in Section 3 via Krasnoselskii’s fixed point theory. Finally, in Section 4 an example is given to illustrate our theory and our method, also to compare our result by using the fixed point theory with the known results by Ardjouni and Djoudi [6].

In this work, we consider the following class of neutral differential equations with variable delays,


denote xtRthe solution to (1) with the initial condition


where ψCmt0t0R,σ01is a quotient with odd positive integer denominator. We assume that a,bCR+R,cC1R+Rand τiCR+R+satisfy tτitas t,i=1,2and for each t00,


Special cases of Eq. (1) have been recently considered and studied under various conditions and with several methods. Particularly, in the case σ=1/3, and ct=0,in [14] Jin and Luo using the fixed point theorem of Krasnoselskii obtained boundedness and asymptotic stability for the following equation:


More precisely, the following result was established.

Theorem A (Jin and Luo [14]).Letτ1be differentiable and suppose that there existsα0.1,k1,k2>0,and a functionhCm0R+such that fort1t21,




while fort0,


Then there is a solutionxt0ψof(4) onR+withxt0ψ1.

Notice that when ct=0in the second term on the right-hand side of (1), then (1) reduces to (4). On the other hand, in the case, τ1t=τ1, a constant, Eq. (4) reduces to the one in [9]. Therefore, we considered the more general system than in [9, 14].

Very recently, by the same method of Jin and Luo [14], Ardjouni and Djoudi [6] improved the results of Jin and Luo [14] to the generalized nonlinear neutral differential equation with variable delays of the form


where G:RRis locally Lipschitz continuous in x. That is, there is an L>0so that if x,y1then


We note that due to the presence of the term ctxtτ1t, once the equation is inverted then once will face with the term ct1τ1txtτ1t, (where, τ1t1for t0) which produces a restrictive condition for the stability of (8) (as described in more detail below).

Theorem B(Ardjouni and Djoudi [6]).Letτ1be twice differentiable and suppose thatτ1t1for all tm0and suppose that there are constantsα0.1,k1,k2>0,and a functionhCm0R+such that fort1t21,




while fort0,




Then there is a solutionxt0ψof(8) onR+withxt0ψ1.

By letting ct=0and Gxσtτ2t=xσtτ2tin (8), the present authors [14] have studied, the asymptotic stability and the stability by using Krasnoselskii’s fixed point theorem, under appropriate conditions, of the Eq. (4) and improved the results claimed in [9]. Consequently, Theorem B improves and generalizes Theorem A. Following the technique of Jin and Luo [14], Ardjouni and Djoudi [6] studied the stability properties of (8). However, the condition (11) in Ardjouni and Djoudi [6] is restrictive. By employing two auxiliary functions pand gfor constructing a fixed point mapping argument, the alternative condition (21) in Theorem 3.1 is obtained. Note that the condition


for some constant α01, is not needed in Theorem 3.1. In the present paper, we also adopt Krasnoselskii’s fixed point theory to study the boundedness and stability of (1). A new criteria for asymptotic stability with a necessary and sufficient condition is given. The considered neutral differential equations, the results and assumptions to be given here are different from those that can be found in the literature and complete that one. These are the contributions of this paper to the literature and its novelty and originality. In addition, an example is provided to illustrate the effectiveness and the merits of the results introduced.


2. Inversion of equation

In this section, we use the variation of parameter formula to rewrite the equation as an integral equation suitable for the Krasnoselskii theorem. The technique for constructing a mapping for a fixed point argument comes from an idea in [21]. In our consideration we suppose that:

A1)Let τ1be twice differentiable and suppose that τ1t1for all tmt0.

A2)There exists a bounded function p:mt00with pt=1for tmt0t0such that ptexists for all tmt0.

Let yt=ψton tmt0t0,and let


Make substitution of (12) into (1) to show


then it can be verified that xsatisfies (1).

We now re-write Eq. (13) in an equivalent form. To this end, we use the variation of parameter formula and rewrite the equation in an integral from which we derive a Krasnoselskii fixed point theorem. Besides, the integration by parts will be applied.

We need the following lemma in our proof of the main theorem.

Lemma 2.1.Leth:mt0R+be an arbitrary continuous function and suppose that (A1) and (A2) hold. Thenyis a solution of(13) if and only if






Proof.Let ytbe a solution of Eq. (13). Rewrite (13) as


Multiply both sides of (17) the previous equality by et0thsdsand then integrate from t0to t,we have


Performing an integration by parts, we can conclude, for tt0,




where μ¯sand β¯sare defined in (15) and (16), respectively. The proof is complete.

Below we state Krasnoselskii’s fixed point theorem which will enable us to establish a stability result of the trivial solution of (1) For more details on Krasnoselskii’s captivating theorem, we refer to smart [20] or [3].

Theorem 2.1.(see, [Kranoselskii’s fixed point theorem, [20]]). Suppose a Banach space andMis a bounded, convex, and closed subset ofX.Suppose further that there exist, two operators, A,BMintoXsuch that:

  1. Ax+ByMfor allx,yM;

  2. Ais completely continuous;

  3. Bis a contraction mapping.

ThenA+Bhas a fixed point inM.

3. Stability by Krasnoselskii fixed point theorem

From the existence theory, which can be found in Hale [26] or Burton [3], we conclude that for each t0ψR+×Cmt0t0R, a solution of (1) through t0ψis a continuous function x:mt0t0+ρRfor some positive constant ρ>0such that xsatisfies (1) on t0t0+ρand x=ψon mt0t0. We denote such a solution by xt=xtt0ψ. We define ψ=maxψt:mt0tt0.

As we mentioned previously, our results in this section extend and improve the work in [14] by considering more general classes of neutral differential equations presented by (1). Our main results in this part can be applied to the case when


which improve [14]. In other words, we will establish and prove a necessary and sufficient condition ensuring the boundedness of solutions and the asymptotic stability of the zero solution to Eq. (1). However, the mathematical analysis used in this research to construct the mapping to employ Krasnoselskii’s fixed point theorem is different from that of [14].

The results of this work are news and they extend and improve previously known results. To the best of our knowledge from the literature, there are few authors who have used the fixed point theorem to prove the existence of a solution and the stability of trivial equilibrium of several special cases of (1) all at once [9, 14].

Let us know to recall the definitions of stability that will be used in the next section. For stability definitions, we refer to [3].

Definition 3.1.The zero solution of (1) is said to be:

  1. stable, if for any ε>0and t00,there exists a δ=δεt0>0such that ψCmt0t0Rand ψ<δimply xtt0ψ<εfor tt0.

  2. asymptotically stable, if the zero solution is stable and for any ε>0and t00,there exists a δ=δεt0>0such that ψCmt0t0Rand ψ<δimply xtt0ψ0as t..

Now, we can state our main result.

Theorem 3.1. Suppose that assumptions (A1) and (A2) hold, and that there are constantsα01,k1,k2>0,and an arbitrary continuous functionhCmt0R+such that fort1t21,




while fortt0


whereμ¯sandβ¯sare defined in(15) and(16), respectively. Ifψis a given continuous initial function which is sufficiently small, then there is a solutionxtt0ψof(1) onR+withxtt0ψ1.

We are now ready to prove Theorem 3.1.

Proof.We start with some preparation:

Let X.gbe the Banach space of continuous φ:mt0Rwith


For each t00and ψCmt0t0Rfixed, we define Xψas the following space


It is easy to check that Xψis a complete metric space with metric induced by the norm .g.

We note that to apply Krasnoselskii’s fixed point theorem we need to construct two mappings; one is contraction and the other is compact. Therefore, we use (14) to define the operator H:XψXψby


where A,B:XψXψare given by




If we are able to prove that Hpossesses a fixed point φon the set Xψ,then ytt0ψ=φtfor tt0,ytt0ψ=ψton mt0t0,ytt0ψsatisfies (13) when its derivative exists and ytt0ψ<1for tt0.That Amaps Xψinto itself can be deduced from condition (21).

For α01, we choose δ>0such that


Let ψ:mt0t0Rbe a given continuous initial function with ψ<δ.Let g:mt01be any strictly increasing and continuous function with gmt0=1,gsas s,such that


Now we split the rest of our proof into three steps.

First step:We now show that φ,ϕXψimplies that Aφ+BϕXψ.Now, let .be the supremum norm on mt0of φXψif φis bounded. Note that if φ,ϕXψthen


By applying (24), we see that Aφt+Bϕt1for tmt0.

We see that also Bmaps Xψinto itself by letting φ=0in the preceding sum.

Second step:Next, we will show that AXψis equicontinuous and Ais continuous. We first show that AXψis equicontinuous. If φXψand if 0t1<t2with t2t1<1,then


by (19)–(21). In the space X.g, the set AXψis uniformly bounded and equicontinuous. Hence by Ascoli-Arzela theorem AXψresides in a compact set.

Next, we need to show that Ais continuous. Let ε>0be given and let φ,ϕXψ. Now yσ,is uniformly continuous on 1+1so for a fixed T>0with 4/gT<εthere is an η>0such that y1y2<ηgTimplies y1σy2σ<ε/2.Thus for φtϕt<ηgtand for t>Twe have


Third step:Finally, we show that Bis a contraction with respect to the norm .gwith constant α.Let Bbe defined by (23). Then for ϕ1,ϕ2Xψwe have


αϕ1ϕ2g,by (22).

Since α01,we can conclude that Bis a contraction on Xψ.g.

The conditions of Krasnoselskii’s theorem are satisfied with M=Xψ. Hence, we deduce that H:XψXψhas a fixed point yt, which is a solution of (13) with ys=ψson smt0,t0]and ytt0ψ1for tmt0. Since there exists a bounded function p:mt00with pt=1for tmt0t0, by hypotheses (12) and from the above arguments we deduce that there exists a solution xof (1) with x=ψon mt0t0satisfies xtt0ψ1for all tmt0. The proof is complete.

Letting σ=1/3,and ct=0in Theorem 3.1. Then we have the following corollary.

Corollary 3.1.Let(19) and(20) hold, and(21) be replaced by


Then there is a solutionxtt0ψof(4) onR+withxtt0ψ1.

Remark 3.2:When pt=1,then Corollary 3.1 reduces to Theorem A, which was recently stated in Jin and Luo [14]. Therefore, the paper (Jin and Luo [14]) is a particular case of ours.

For the next Theorem, we manipulate function spaces defined on infinite t-intervals. So, for compactness, we need an extension of the Arzelà-Ascoli theorem. This extension is taken from ([3], Theorem 1.2.2 p. 20).

Theorem 3.2. Let(19)–(21) hold and assume that




Ifψis given continuous initial function which is sufficiently small, then(1) has a solutionxtt0ψ0astif and only if


Proof.We set


by (28), Kis well defined. Suppose that (29) holds.

Since pis bounded, it remains to prove that the zero solution of (1) is asymptotically stable.

All of the calculations in the proof of Theorem 3.1 hold with gt=1when .gis replaced by the supremum norm ..



where qt0as tby (27).

Add to Xψthe condition that φXψimplies that φt0as t. We can see that for φXψthen Aφt0as tby (31), and Bφt0as tby (29).

Since AXψhas been shown to be equicontinuous, Amaps Xψinto a compact subset of Xψ. By Krasnoselskii’s theorem, there is yXψwith Ay+By=y. As yXψ,ytt0ψ0as t.By condition (12), it is very easy to show that there exists a solution xXψof (1) with xtt0ψ0as t.

Conversely, we suppose that (29) fails. From (28) there exists a sequence tnwith tnas nsuch that limnt0tnhudu=ξfor some ξR+.We may also choose a positive constant Jsatisfying


for all n1.To simplify the expression, we define


for all s0.By (21), we have


This yields


The sequence t0tne0shuduωsdsis bounded, hence there exists a convergent subsequence. Without loss of generality, we can assume that


for some θR+.Let mbe an integer such that


for all nm, where δ0>0satisfies 2δ0KeJ+α1.

We now consider the solution yt=yttmψof (1) with ψtm=δ0and ψsδ0for stm.We may choose ψso that yt1for ttmand


In follows from (22) and (23) with yt=Ayt+Bytthat for nm


On the other hand, if the zero solution of (13) yt=yttmψ0as t,since tnτitnas t,i=1,2,and (21) holds, we have


as t,which contradicts (32). Hence condition (29) is necessary for the asymptotic stability of the zero solution of (13), and hence the zero solution of (1) is asymptotically stable. The proof is complete.

For the special case ct=0and σ=13, we can get.

Corollary 3.2.Let(19), (20) and(27) hold and(21) be replaced by


Then the zero solutionxtt0ψof(4) with a small continuous functionψtis asymptotically stable if only if


Remark 3.3.The method in this paper can be applied to more general nonlinear neutral differential equations than Eq. (1).

Remark 3.4.Theorem 3.1 is still true if condition (21) is satisfied for ttρwith some tρR+.

4. Example

In this section, we now give an example to show the applicability of Theorem 3.1.

Example.Let us consider the following neutral differential equation of first order with two variable delays, which is a special case of (1):


for t0where τ2t=0.5t,τ1t=0.05t,and atsatisfies


where μ¯tand β¯tare defined in (15) and (16), respectively. Choosing ht=1.5t+1and pt=1t+1. By straightforward computations, we can check that condition (21) in Theorem 3.1 holds true. As t,we have



0testhudubspσsτ2spsds0.4,and since 0thsdsas t,pt1.Let α=0.263+0.026+0.026+0.2+0.4. It is easy to see that all the conditions of Theorem 3.1 hold for α0.915<1.Thus, Theorem 3.1 implies that the zero solution of (33) is asymptotic stable.

However, for the asymptotic stable of the zero solution of (33), the corresponding conditions used by the fixed point theory in Ardjouni and Djoudi [6] are


This implies that condition (11) does not hold. So it is clear that the reduction of the conservatism by our method is quite significant when compared to Ardjouni and Djoudi [6].

Remark 4.1. It is an open problem whether the zero solution of (1) is uniform asymptotically stable, perseverance, and so on.

5. Conclusion

This work is a new attempt at applying the fixed point theory to the stability analysis of neutral differential equations with variable delays, several special cases of which have been studied in [9, 14]. Some of the results, like Theorem B, is mainly dependent on the constraint


But in many environments, the constraint does not hold. So by employing two auxiliary continuous functions gand pto define an appropriate mapping, and present new criteria for asymptotic stability of Eq. (1) which makes stability conditions more feasible and the results in [14] are improved and generalized. From what has been discussed above, we see that Krasnoselskii’s fixed point theorem is effective for not only the investigation of the existence of solution but also for the boundedness and the stability analysis of trivial equilibrium. We introduce an example to verify the applicability of the results established. In the future, we will continue to explore the application of other kinds of fixed point theorems to the stability research of fractional neutral systems with variable delays.


Additional classifications

AMS Subject Classifications: 34K20, 34K30, 34B40

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Benhadri Mimia (March 15th 2021). Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral Differential Equations with Variable Delays, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.96040. Available from:

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