Boundedness and stability in third order nonlinear vector differential equations with multiple deviating arguments

Peer review under responsibility of University of Bahrain.

Abstract

In this paper, we shall establish sufficient conditions for the uniform asymptotic stability and boundedness of solutions of a certain third order vector nonlinear non-autonomous differential equation with multiple deviating arguments, by using a Lyapunov function as basic tool. In doing so we extend some existing results. Example is given to illustrate our results.

Keywords:

• Boundedness
• Stability
• Lyapunov functional
• Delay vector differential equations

1 Introduction

As is well known, differential equations with retarded argument are used to describe many phenomena of physical interest. In the last years, there has been increasing interest in obtaining the sufficient conditions for the stability/instability/boundedness/ ultimately boundedness etc. The problem of the boundedness and stability of solutions of vector differential equations has been widely studied by many authors, who have provided many techniques especially for delay differential equations, for some related contributions, we refer the reader to Graef et al. (2015a,b), Oudjedi et al. (2014), Remili and Beldjerd (2014), Remili and Oudjedi (2014a,b, 2016a,b,c,d,e), Remili et al. (2016), Tunç (2006a,b) , Tunç and Gözen (2014), Zhengxin et al. (2010a,b, 2015), Zhihui and Jinde (2005, 2007)". In the following, we provide some background details regarding the study of various classes of third differential equations.

Tunç (2009) studied the stability and boundedness of the following vector differential equation of third order without delay:(1.1) for and respectively.

Later, Omeike and Afuwape (2010) proved the ultimate boundedness of the same equation.

After that, Tunç and Mohammed (2014) established conditions under which all solutions of third order vector differential equation with delay of the form(1.2) tend to the zero solution as for and ultimate boundedness for .

Recently, Tunç (2017) adapted Tunç and Mohammed (2014) and used a suitable Lyapunov function to establish criteria which guarantee asymptotic stability of solution of nonautonomous delay differential equation of the third order that is bounded together with its derivatives on the real line, and boundedness under explicit conditions on the nonlinear terms of the equation(1.3)

This research is concerned with more general third order nonlinear vector multi-delay differential equations of the form(1.4) and(1.5) in which ,  ,  ,   and are continuous differentiable functions with () and H is twice differentiable, where , for all i, ( and are some positive constants, will be determined later, and the primes in (1.4)(1.4) and (1(1.5) .5) denote differentiation with respect to .

Finally, the continuity of the functions and C guarantee the existence of the solution of (1.4)(1.4) and (1(1.5) .5). In addition, we assume that the functions and satisfy a Lipschitz condition with respect to their respective arguments, like X and . In this case, the uniqueness of solutions of the Eq. (1.5)(1.5) is guaranteed.

The motivation of the present work comes from papers mentioned above and the references listed in this paper. It should be also noted that the equations studied here is more general than (1.1), (1.2), (1.3)(1.3) and that considered in Remili and Oudjedi (2016b).

2 Preliminaries

The symbol corresponding to any pair X and Y in stands for the usual scalar product , that is, , Thus .

Definition 2.1

We definite the spectral radius  of a matrix D by

Lemma 2.2

For any , we have the norm . If D is symmetric then

We shall note all the equivalents norms by the same notation for and for a matrix .

The following results will be basic to the proofs of Theorems.

Lemma 2.3

Afuwape (1983), Afuwape and Omeike (2004)", Ezeilo and Tejumola (1966), Ezeilo (1967)", Ezeilo and Tejumola (1975), Tiryaki (1999)"

Let D be a real symmetric positive definite matrix, then for any Xin , we havewhere are the least and the greatest eigenvalues of D, respectively.

Lemma 2.4

Afuwape (1983), Afuwape and Omeike (2004), Ezeilo and Tejumola (1966), Ezeilo (1967), Ezeilo and Tejumola (1975), Tiryaki (1999)"

Let be any two real commuting matrices, then

• (i) The eigenvalues of the product matrix QD are all real and satisfy

• (ii) The eigenvalues  of the sum of matrices Q and D are all real and satisfy.

Lemma 2.5

Ezeilo and Tejumola (1966), Ezeilo (1967), Ezeilo and Tejumola (1975), Tiryaki (1999), Mahmoud and Tunç (2016)"

Let be a continuous vector function with .

Lemma 2.6

Ezeilo and Tejumola (1966), Ezeilo (1967), Ezeilo and Tejumola (1975), Tiryaki (1999), Mahmoud and Tunç (2016)"Let be a continuous vector function with .

Lemma 2.7

Let be a continuous vector function and that then,where are the least and the greatest eigenvalues of (Jacobian matrix of H), respectively.

Definition 2.8

We definite the spectral radius  of a matrix A by

Lemma 2.9

For any , we have the norm  if A is symmetric then

We shall note all the equivalents norms by the same notation for and for a matrix .

3 Stability

The following notations (see Omeike (2015)) will be useful in subsequent sections. For is the norm of x. For a given ,

In particular, denotes the space of continuous functions mapping the interval into and for . will denote the set of such that . For any continuous function defined on , where , and , the symbol will denote the restriction of to the interval , that is, is an element of C defined by

Consider the functional differential equation(3.1) where is a continuous mapping, , , and for , there exists , with when .

Definition 3.1

Burton (1985) An element is in the set of , say , if is defined on and there is a sequence , as , with as where  .

Definition 3.2

Burton (2005) A set is an invariant set if for any , the solution of (3.1)(3.1) , , is defined on and for .

Lemma 3.3

Burton (1985) If is such that the solution of (3.1)(3.1) with is defined on and for , then is a non-empty, compact, invariant set and

Lemma 3.4

Burton (1985) Let   be a continuous functional satisfying a local Lipschitz condition., and such that:

(i)

where  .

(ii)

,

where, ( are wedges. Then the zero solution of (3.1)(3.1) is uniformly asymptotically stable.

4 Result and discussion

Let us introduce the temporary notation

Let . The first main problem of this paper is the following theorem.

Theorem 4.1

In addition to the basic assumptions imposed on the functions and , let us assume that the matrices and (Jacobian matrix of and ) are symmetric and positive definite, and that there exist positive constants such that the following conditions hold:

i)

,

ii)

,

iii)

,

iv)

; ,

v)

,

vi)

,

vii)

.

Then every solution of (1.4)(1.4) is uniformly asymptotically stable, provided thatwhereand is the bound on .

Note that for any matrix M symmetric invertible, we have

Proof

We write the Eq. (1.4)(1.4) as the following equivalent system(4.1)

Let and are positive constants which will be specified later in the proof. For the sake of brevity, we define(4.2)

Our main tool in the proof of the theorem just stated above is a Lyapunov function defined by(4.3) where(4.4)

From the definition of V in (4.4)(4.4) and by using Lemma 2.5, we observe that the above functional can be rewritten as follows

The conditions (i)–(iv) and (vi) of the theorem and from Definition 2.8, Lemmas (2.2, 2.4, 2.7) and (4.2), we have

Sincethen by using Lemma 2.6, the above estimate implies thatwhere

Thus, we can find a positive constant k, small enough such that(4.5)

It is easy to check that by (iii) and (vii), we havethis may be combined with (4.5)(4.5) to obtain

Therefore, we can find a continuous function with

The existence of a continuous function which satisfies the inequality.

, is easily verified.

For the time derivative of the functional , along the trajectories of the system (4.1)(4.1) , we havewhere

Consequently by Lemma 2.3, Lemma 2.7 and the hypotheses (i)-(vi) we get

Now consider the termby (v), we get

Using the Schwartz inequality , we obtain the followingandWe rearrangeIf we takethe last inequality becomes

Using (4.5), (4.3) and taking we obtain:(4.6) Therefore, ifthe inequality (4.6)(4.6) becomes

where , for some . It follows, by the conditions (i) and (iv), that if and only if in the system (4.1)(4.1) , and for . Thus, all the conditions of Lemma 3.4 are satisfied. This shows that every solution of (1.4)(1.4) is uniformly asymptotically stable. Hence the proof of Theorem 4.1 is complete. □

Example 4.2

As a special case of the following equation(4.7) wherewhereandwhere

Clearly, and are diagonal matrices, hence they are symmetric and commute pairwise. Then, by an easy calculation, we obtain eigenvalues of the matrices and as follows:

A simple computation gives

It is easy to see that a trivial verification shows that H is nonsingular matrix and we havewhere

Thus, using Definition 2.1 and Lemma 2.2, we getand

For , a straightforward calculation giveswhereand

By taking it follows easily that

If we take , then and  for .

Thus, all the assumptions (i) through (vii) are satisfied, we can conclude using Theorem 4.1 that every solution of (4.7) is uniformly asymptotically stable.

In the case . The second main problem of this paper is the following theorem.

Theorem 4.3

In addition to the assumptions of Theorem 4.1, if we assume that P is continuous, andwhere is the space of Lebesgue integrable functions. Then all solutions of the perturbed Eq. (1.5)(1.5) are bounded.

Proof

We consider the equivalent system of (1.5)(1.5) (4.8)

An easy calculation from (4.8)(4.8) and (4(4.3) .3) yields that

Since and noting that , thenwhere . Let , then

Multiplying each side of this inequality by the integrating factor , we get

Integrating by parts the left hand side of this inequality from 0 to t, we obtainthuswhere . Since for all , we have

Now, since the right-hand side is a constant, and sinceit follows that there exists such thatthus we can deduce □

5 Conclusions

Obtaining uniform asymptotic stability and boundedness of solutions for the nonlinear non-autonomous third order vector differential equations with multiple deviating arguments (1.4)(1.4) and (1(1.5) .5) respectively, using a Lyapunov function. it is worth noting that our study complement some well known results on the third order differential equations in the literature.

Conflict of interest

The authors have no conflict of interest.

Acknowledgements

The authors of this paper would like to express his sincere appreciation to the main editor and the anonymous referees for their valuable comments and suggestions which have led to an improvement in the presentation of the paper.

Notes

Peer review under responsibility of University of Bahrain.