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R E S E A R C H

Open Access

Existence of solutions converging to zero

for nonlinear delayed differential systems

Josef Rebenda

1

and Zden ˇek Šmarda

1,2*

*Correspondence: [email protected] 1CEITEC BUT, Brno University of Technology, Technicka 3058/10, Brno, 61600, Czech Republic 2Department of Mathematics, Brno University of Technology, Brno, Czech Republic

Abstract

We present a result about an interesting asymptotic property of real two-dimensional delayed differential systems satisfying certain sufficient conditions. We employ two previous results, which were obtained using a Razumikhin-type modification of the Wa˙zewski topological method for retarded differential equations and the method of a Lyapunov-Krasovskii functional. The result is illustrated by a nontrivial explanatory example.

MSC: 34K12; 34K20

Keywords: differential system with delays; stability of solutions

1 Introduction

Various properties of solutions of differential equations with delay were extensively stud-ied recently. Among others we mention [–] and the references therein. The results con-tained in this paper are a generalization of previous research published in [–] and [].

Our aim here is to study the asymptotic behavior of solutions of the following system of differential equations:

x(t) =A(t)x(t) +

m

k=

Bk(t)xθk(t)

+ht,x(t),xθ(t)

, . . . ,xθm(t)

, ()

wheretθk(t)≥ are bounded nonconstant delays satisfyinglimt→∞θk(t) =∞,θk(t) are

real functions,

h(t,x,y) =h(t,x,y, . . . ,ym),h(t,x,y, . . . ,ym)

is a real vector function, wherex= (x,x),yk= (yk,yk), and

A(t) =aij(t)

, Bk(t) =

bijk(t)

, i,j= , ;k= , . . . ,m,

are real square matrices.

In this paper, we introduce an interesting result, which is a combination of two theorems presented in [], one regarding the instability of solutions, the other one dealing with the existence of bounded solutions.

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It is supposed that the functionhsatisfies the Carathéodory conditions on [t,∞)×

R(m+), the functionsb

ijkare locally Lebesgue integrable on [t,∞), and the functionsθk,

aijare locally absolutely continuous on [t,∞).

Since we study two-dimensional systems, we use a transformation into complex vari-ables to simplify the system () into one equation with complex coefficients.

The complex variables are defined asz=x+ix,w=y+iy, . . . ,wm=ym+iym. Using this transformation we get

z(t) =a(t)z(t) +b(t)z(t) +¯

Ak,BkLloc(J,C)(i.e.locally Lebesgue integrable complex-valued functions onJ) for k= , . . . ,m,

θkACloc(J,R)(i.e.locally absolutely continuous real-valued functions onJ) for k= , . . . ,m,

a,bACloc(J,C)(i.e.locally absolutely continuous complex-valued functions onJ),gK(J×Cm+,C)(i.e.a complex-valued function which satisfies the Carathéodory

conditions onJ×Cm+).

Obviously, the functiongis in general dependent onz¯as well as on everyz(¯θk). However,

the fact that the functiongsatisfies the Carathéodory conditions enables us to significantly simplify the notation by using onlyz, since the validity of the Carathéodory conditions is not violated by composing with continuous functions¯z,w¯k, andθk.

The relations between the functions are the following:

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bk(t) =Im

Ak(t) +Bk(t)

, bk(t) =Re

Ak(t) –Bk(t)

, h(t,x,y, . . . ,ym) =Reg(t,x+ix,y+iy, . . . ,ym+iym), h(t,x,y, . . . ,ym) =Img(t,x+ix,y+iy, . . . ,ym+iym),

equation () can be written in the real form () as well.

The equivalence of the dynamical invariants and asymptotic properties of the solutions of the real system () and the complex equation () is shown in [] for the simple case covering ordinary differential equations.

In this paper we consider () in the case when

lim inf

t→∞ a(t)b(t) >  ()

and study the behavior of the solutions of () under this assumption, which generally means thatdetA(t) >  fortsufficiently large. This situation corresponds to the case when the equilibrium point  of the autonomous homogeneous system

x= Ax, ()

where A is supposed to be a regular constant matrix, is a center, a focus or a node. Such a situation has some geometrical aspects, which are used in an analysis of the transformed equation (). See [] for more details.

Further, we suppose that () satisfies the uniqueness property of solutions.

2 Preliminaries

Throughout this paper we will assume that

lim inf

t→∞ a(t)b(t) > , tθk(t)≥trfortt+r, () wherer>  is a constant, which means that the delaysθkare bounded. This is the same

case as considered in []. Similar results for a case different from () were obtained in []. Then there are numbersTt+randμ>  such that

a(t) > b(t) +μ fortT. ()

Denote

c(t) = a(t)b(t)¯

|a(t)| , γ(t) = a(t) + a(t)

b(t)  ()

and

ϑ(t) = Re(γ(t)γ

(t) –¯c(t)c(t)) –|γ(t)c(t) –γ(t)c(t)|

γ(t) –|c(t)| ,

α(t) =  – b(t) a(t)

sgn Rea(t).

()

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(i) The numbersTt+randμ>  satisfy condition ().

(ii) There are functions,κ,κk: [T,∞)→R, whereis continuous on [T,∞), such that γ(t)g(t,z,w, . . . ,wm) +c(t)g(t,¯ z,w, . . . ,wm)

≤κ(t) γ(t)z+c(t)¯z +

m

k=

κk(t) γ

θk(t)

wk+c

θk(t)

¯

wk +(t)

fortT,z,wk∈C(k= , . . . ,m).

(iin) There are numbersRn≥ and functionsκn,κnk : [T,∞)→Rsatisfying the

in-equality

γ(t)g(t,z,w, . . . ,wm) +c(t)g(t,¯ z,w, . . . ,wm)

≤κn(t) γ(t)z+c(t)¯z + m

k=

κnk(t) γ

θk(t)

wk+c

θk(t)

¯ wk

fortτnT,|z|+ m

k=|wk|>Rn.

(iii) The functionβACloc([T,∞),R–) is such that

θk(t)β(t)≤–λk(t) a.e. on [T,∞), ()

whereλkis given fortTby

λk(t) =κk(t) + Ak(t) + Bk(t)

γ(t) +|c(t)|

γ(θk(t)) –|c(θk(t))|

. ()

(iiin) The functionβnACloc([T,∞),R–) is such that

θk(t)βn(t)≤–λnk(t) a.e. on [τn,∞), ()

whereλnkis given fortT by

λnk(t) =κnk(t) + Ak(t) + Bk(t)

γ(t) +|c(t)|

γ(θk(t)) –|c(θk(t))|

. ()

(ivn) The functionΛn is real locally Lebesgue integrable and the inequalitiesβn(t)≥ Λn(t)βn(t), Θn(t)≥Λn(t) are satisfied for almost all t∈[τn,∞), where Θn is given by Θn(t) =α(t)Rea(t) +ϑ(t) –κn(t) +mβn(t).

Furthermore, denote

Θ(t) =α(t)Rea(t) +ϑ(t) –κ(t). ()

3 Main results

First of all, we recall the two results from [].

Lemma  Let the assumptions(i), (ii), (iii), (iv)be fulfilled for someτ≥T.Suppose there exist t≥τandν∈(–∞,∞)such that

inf

ttt

t

Λ(s)ds–ln

γ(t) + c(t)

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If z(t)is any solution of ()satisfying

min θ(t)≤st

z(s) >R, Δ(t) >Reν, ()

where

θ(t) = min

k=,...,mθk(t),

Δ(t) =γ(t) – c(t) z(t) +β(t) max θ(t)≤st

z(s)

m

k=

t

θk(t)

γ(s) + c(s) ds,

then

z(t)Δ(t)

γ(t) +|c(t)|exp

t

t

Λ(s)ds

()

for all ttfor which z(t)is defined.

Lemma  Let the conditions(i), (ii), (iii)be fulfilled andΛ,θk(k= , . . . ,m)be continuous functions such that inequalityΛ(t)≤Θ(t)holds a.e.on[T,∞),whereΘis defined by(). Suppose thatξ: [T–r,∞)→Ris a continuous function such that

Λ(t) +β(t)

m

k=

θk(t)exp

t

θk(t)

ξ(s)ds

ξ(t) >(t)C–exp

t

T ξ(s)ds

()

for t∈[T,∞]and some constant C> .Then there exists a t>T and a solution z(t)of ()satisfying

z(t) ≤ C

γ(t) –|c(t)|exp

t

T ξ(s)ds

()

for tt.

If we combine the previous two results, we are able to prove the following theorem, which is the fundamental result of this paper.

Theorem  Assume that the hypotheses(i), (ii), (iin), (iii), (iiin), (ivn)are valid for Tτn,

where <Rn,n∈N,infn∈NRn= .Suppose thatθk,Λare continuous functions such that

inequalityΘ(t)≥Λ(t)is satisfied almost everywhere on[T,∞),whereΘ(t) =α(t)Rea(t) +

ϑ(t) –κ(t).Letξ: [T–r,∞)→Rbe a continuous function satisfying the inequality

(t)C–exp

t

T ξ(s)ds

<Λ(t) +β(t)

m

k=

θk(t)exp

t

θk(t)

ξ(s)ds

ξ(t) ()

for some constant C> and t∈[T,∞).Assume

inf τnst<∞

t

s

Λn(σ)dσ–ln

γ(t) + c(t)

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lim sup ()with the property

lim

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The inequalities () and () give the estimation

forτt, which is in contradiction to (). The proof is complete.

Remark  Theorem  covers more general situations than Theorem  in [], where the different fundamental assumptionlim inft→∞(|Ima(t)|–|b(t)|) >  is supposed to hold.

Indeed, if we take for examplea(t)≡ +iandb(t)≡i, then condition () in this paper is satisfied but the conditionlim inft→∞(|Ima(t)|–|b(t)|) >  is not valid.

The following nontrivial example was constructed to illustrate an application of the the-oretical result presented in Theorem .

Example  Consider the two-dimensional system of the nonlinear delayed differential equations

This system can be written in matrix form (), where

A(t) =

Following our approach, we use a transformation into the complex plane and obtain the delayed differential equation () with complex-valued coefficients,

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Further,

Thus conditions (i) and (ii) are fulfilled withκ(t)≡ √

Condition (iii) holds with

λk(t) Condition (iiin) is satisfied for

βn(t) = –

We get condition (ivn) by setting

n(t) =Θn(t) =

Then condition () holds for

ξ(t) = 

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conclusion that

βn=O

e–t, exp

s

T

ξ(σ)dσ

=Oe.t

and

m

k=

t

θk(t)

γ(s) + c(s) ds=O

m

k= ekt

=Oe–t.

Consequently, the product of these factors is asymptotically equal toO(eδt), whereδ> 

and thus condition () is satisfied. All assumptions of Theorem  are fulfilled and we can conclude that there existt>  and a solutionz(t) of () satisfying () fortt.

Remark  This result is slightly surprising. However, it is in good agreement with the well-known fact that introducing delay into an unstable system without delay can cause a change of behavior of the system. Such situations are described and corresponding results are formulatede.g.in [].

4 Conclusion

We proved an interesting result about the stability of two-dimensional systems with bounded delays. Sufficient conditions for the stability of an originally unstable system were presented. The result is in perfect agreement with the results stated and proved in the es-tablished literature. An example showed how this result can be used in practice.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

The authors have made contributions of the same significance. All authors read and approved the final manuscript. Acknowledgements

The work of the authors was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund and the second author was supported by the project FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This support is gratefully acknowledged.

Received: 15 September 2015 Accepted: 5 November 2015

References

1. Baculíková, B, Džurina, J, Rogovchenko, YV: Oscillation of third order trinomial delay differential equations. Appl. Math. Comput.218(13), 7023-7033 (2012)

2. Berezansky, L, Braverman, E: On nonoscillation and stability for systems of differential equations with a distributed delay. Automatica48(4), 612-618 (2012)

3. Berezansky, L, Diblík, J, Svoboda, Z, Šmarda, Z: Simple uniform exponential stability conditions for a system of linear delayed differential equations. Appl. Math. Comput.250(1), 605-614 (2012)

4. Braverman, E, Berezansky, L: New stability conditions for linear differential equations with several delays. Abstr. Appl. Anal.2011, Article ID 178568 (2011)

5. Burton, TA: Stability by Fixed Point Theory for Functional Differential Equations, 348 pp. Dover, New York (2006) 6. Olach, R: Positive periodic solutions of delay differential equations. Appl. Math. Lett.26, 1141-1145 (2013) 7. Šmarda, Z, Rebenda, J: Asymptotic behaviour of a two-dimensional differential system with a finite number of

nonconstant delays under the conditions of instability. Abstr. Appl. Anal.2012, Article ID 952601 (2012)

8. Stevi´c, S: Onq-difference asymptotic solutions of a system of nonlinear functional differential equations. Appl. Math. Comput.219(15), 8295-8301 (2013)

9. Stevi´c, S: Solutions converging to zero of some systems of nonlinear functional differential equations with iterated deviating argument. Appl. Math. Comput.219(8), 4031-4035 (2012)

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11. Kalas, J: Asymptotic properties of an unstable two-dimensional differential system with delay. Math. Bohem.131, 305-319 (2006)

12. Kalas, J: Asymptotic properties of a two-dimensional differential system with a bounded nonconstant delay under the conditions of instability. Far East J. Math. Sci.: FJMS29, 513-532 (2008)

13. Kalas, J, Rebenda, J: Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay in an unstable case. Electron. J. Qual. Theory Differ. Equ.2012, 8 (2012)

14. Kalas, J, Rebenda, J: Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability. Math. Bohem.136(2), 215-224 (2011)

15. Ráb, M, Kalas, J: Stability of dynamical systems in the plane. Differ. Integral Equ.3, 127-144 (1990)

References

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