A.M.Mahmoud,D.A.M.Bakhit

(Dept.of Math.,Faculty of Science,New Valley Branch,Assiut University,El-Khargah 72511,Egypt)

Abstract Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained.Our results improve and complement some earlier results.Two examples are given to illustrate the importance of the topic and the main results obtained.

Keywords delay differential equations;asymptotic behaviour;stability;third-order differential equations;Lyapunov functional

1 Introduction

Differential equations(DEs)are used as tools for mathematical modeling in many fields of life science.When a model does not incorporate a dependence on its past history,it generally consists of so-called ordinary differential equation(ODEs).Models incorporating past history generally include delay differential equations(DDEs)or functional differential equations(FDEs).In applications,the future behaviour of many phenomena is assumed to be described by the solutions of an DDEs,which implies that the future behaviour is uniquely determined by the present and independent phenomena of the past.In FDEs,the past exerts its in fluence in a signi ficant manner upon the future.Many phenomena are more suitable to be described by DDEs than ODEs.In many processes including physical,chemical,political,economical,biological,and control systems,time-delay is an important factor.In particular the third-order delay differential equations usually describe the phenomena in various areas of applied mathematics and physics,for instance de flection of bucking beam with a variable cross-section,electromagnetic waves,gravity driven flows,etc.

As we know the study of qualitative properties of solutions,such as stability and boundedness,is very important in the theory of differential equations.Since it is difficult to solve solutions to DEs,Lyapunov method is usually used to study the stability and boundedness of the equations.

Many good results have been obtained on the qualitative behaviour of solutions to some kinds of third-order DDEs by Zhu[23],Sadek[14–16],Abou-El-Ela et al.[1],Tun¸c[18–22],Ademola et al.[2,3],Afuwape and Omeike[4],Bai and Guo[5],Shekhare et al.[17],Remili et al.[11],and the references therein.

Numerous authors have obtained some very interesting results about the asymptotic behaviour of solutions to third-order DDEs,for example,Chen and Guan[7],Mahmoud[8],Remili et al.[10,12,13],etc.

In 2016,Remili and Oudjedi[12]studied the ultimate boundedness and the asymptotic behaviour of solutions to a third-order nonlinear DDE of the form

where f,g,h,Ω and p are continuous functions in their respective arguments with g(x,0)=h(0)=0.

In 2017,Remili et al.[13]investigated the stability and ultimate boundedness of solutions to a kind of third-order DDE as follows

where r>0 is a fixed delay;e,f,g,h and ϕ are continuous functions in their respective arguments with f(0)=0.

The main objective of this research is to study the asymptotic stability and the boundedness of solutions to a nonlinear third-order DDE

where h,p,g,f and e are continuous functions with g(0)=f(0)=0,and the derivativesandexist and are also continuous.

We can take

Remark In equation(1.1),if h(x(t))=1 and p(x(t))=a,then equation(1.1)is reduced to the equation in Sadek[14].

2 Stability Result

To prove the stability result,we shall give some important theorems about the stability of solutions to DDEs.

Consider the general autonomous DDE

where f:CH→Rnis a continuous mapping,f(0)=0.We suppose that f maps closed bounded sets into bounded sets of Rn.Here(C,∥·∥)is the Banach space of continuous functions ϕ :[−r,0]→ Rnwith supremum norm,r>0;CHis an open ball of radius H in C;CH:={ϕ ∈ (C[−r,0],Rn):∥ϕ∥

Theorem 2.1[6]Let V:CH→R be a continuous functional satisfying a local Lipschitz condition,V(0)=0,such that

(i)W1(|ϕ(0)|) ≤ V(ϕ)≤ W2(∥ϕ∥),where W1(r)and W2(r)are wedges;

(ii)˙V(2.1)(ϕ)≤0,for ϕ∈CH.

Then the zero solution to(2.1)is uniformly stable.

Theorem 2.2[6]If there are a Lyapunov functional for(2.1)and wedges Wi(i=1,2,3),such that

(i)W1(|ϕ(0)|) ≤ V(ϕ)≤ W2(∥ϕ∥);

(ii)˙V(2.1)(ϕ)≤−W3(|ϕ|).

Then the zero solution to(2.1)is uniformly asymptotically stable.

Now,we shall give the main theorem and its proof.

Theorem 2.3 Suppose that there are positive constants a0,a1,a2,a3,b1,b2,L1,L2,γ and β which satisfy the following conditions:

where

Then the zero solution to(1.1)with e=0 is uniformly asymptotically stable.

Proof When e=0,equation(1.1)is equivalent to

The Lyapunov functional of the above system could be defined as:

From conditions(i)-(iii)and using the mean-value theorem,we get

Considering r(t)≤γ in(iv),we obtain

Then there exists a positive constant D0such as

It follows that

Then we have

There exists a positive constant D1such that

Now,differentiating both sides of(2.3)along the solution to system(2.2)and from(1.2),we have

By conditions(i)-(iii),we obtain

It follows from condition(iv)that

Take

so the above equation becomes

where

Thus,we find

where

From inequality(2.5),taking G(t)=|θ1|+|θ2|,we obtain

Then

So we can write the time derivative of Lyapunov functional V(xt,yt,zt)as

provided that

Now,we define the continuously differentiable functional W as follows

where

with α1=min{x(0),x(t)}and α2=max{x(0),x(t)}.

Taking the derivative of this equation,from(2.6)there is

Hence,from(2.7),W3(∥X∥)= α(y2+z2)is a positive definite function and from inequalities(2.4),(2.5),the Lyapunov functional V(xt,yt,zt)satisfies all conditions of Theorem 2.2.

Therefore we conclude that the zero solution to equation(1.1)is uniformly asymptotically stable.

Thus the proof of Theorem 2.3 is now finished.

3 Boundedness of Solutions

In this case e(t)≠0,equation(1.1)is equivalent to the following system

We can obtain the following theorem.

Theorem 3.1 In addition to conditions(i)-(vi),we assume that

where max(q(t))<∞ and q(t)∈L1(0,∞)with L1(0,∞)being a space of integrable Lebesgue function.Then there exists a finite positive constant D,such that the solution x(t)defined by the initial functions

satisfies the inequalities

Proof By the conditions of Theorem 3.1,and inequality(2.6)for system(3.1),we can write

Accordingly,the last inequality becomes

If we recall the inequalities|y|<1+y2and|z|<1+z2,then we get

for a positive constant c.From inequalities(2.5)and(3.2),we obtain x2+y2+z2≤then we conclude

Thus,we have

The proof of Theorem 3.1 is now finished.

4 Examples

Example 4.1 In this example we shall study the stability of a third-order nonlinear DDE of the following form

It is obvious that

and

Then,it follows that

Also,there are

and

Hence,we get

Note that

Then,we have

and

Also,the function

and

Therefore,we get

so

Hence,we have

So all conditions of Theorem 2.3 hold,therefore the zero solution to(4.1)is uniformly asymptotically stable.

Example 4.2 In this example we shall study the boundedness of a third-order nonlinear DDE of the following form

Note that

then

so q(t)∈ L1(0,∞).It follows that

where

Integrating(4.3)from 0 to t,using the fact thatwe obtain

Hence,we can conclude that all solutions to equation(4.2)are bounded.