An Asymptotic Result for neutral differential equations

  • 1 Department of Mathematics, Faculty of Arts and Sciences, 12000, Bingol
Emel Bicer
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  • Department of Mathematics, Faculty of Arts and Sciences, Bingol University, 12000, Bingol
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Abstract

We obtain asymptotic result for the solutions of neutral differential equations. Our technique depends on characteristic equations.

1 Introduction

Neutral differential equations (NDEs) describe a certain form of delay differential equations. Recently, these equations have received much interest, because of they play important part in the mathematical modeling of natural phenomena. (NDEs) emerge in many fields of engineering and mathematical science. Most of solutions of (NDEs) can not be obtained in closed form. For this reason, researching the qualitative behaviour of solutions is an effective option. Until today, many authors have investigated qualitative behaviour of solutions of (NDEs). Particularly, for more results on the qualitative behaviour of solutions of (NDEs) see [1]–[14] and references therein.

There are several methods to investigate asymptotic behaviour of solutions, such as characteristic equations, fixed point methods, and Lyapunov functionals. While studying (NDEs), each of these methods has its advantages and disadvantages. In the present paper, we use characteristic equations to investigate asymptotic properties of solutions of a (NDE).

In [1], Ardjouni and Djoudi deal with the first order (NDE),

x(t)=j=1Nbj(t)x(tτj(t))+j=1Ncj(t)x(tτj(t)).

Applying the fixed point methods, they obtained asymptotic results to solutions of above (NDE).

In [8], Dix et al. obtained asmptotic results of solutions to first order linear (NDEs).

Motivated by the results of references therein, we investigate asymptotic properties of solutions to first order (NDE)

y(z)a(z)y(z)=i=1mbi(z)y(zgi(z))+j=1ncj(z)y(zhj(z))=0,zz0
with initial condition (IC)
y(z)=θ(z),forinf{zgi(z),zhj(z):i=1,2,,m,j=1,2,,n},
where a,bi,c jC(ℝ + ,ℝ ) and gi,hjC(ℝ + ,ℝ + ).

2 Main results

We denote

h=sup{hj(z):j=1,2,n}l=sup{gi(z),hj(z):i=1,2,,m,j=1,2,,n}.

Let C([z0l,z0] ,ℝ ) represent the set of continuous real-valued functions on [z0l,z0].

By the considered (NDE) (1), we combine the following equation:

λ(z)a(z)=i=1mbi(z)ezzgi(z)λ(t)dt+j=1ncj(z)λ(zhj(z))ezzhj(z)λ(t)dt,forzz0λ(z)=λ(z0),forz[z0l,z0].

Lemma

For each (IC) (2), there exists a solution of (NDE) (1).

Proof

Firstly, we will show that the characteristic equation has a unique solution. Let l1 = {infgi(z),hj(z), for i = 1,2,...,m and j = 1,2,...,n. Let

ζ(z)=ez0zλ(t)dt,forzz0l
and
β(z)=ez0zλ(t)dt,forz0lzz0.

So, from the characteristic equation (3), we get

ζ(z)=λ(z)ζ(z)=a(z)ζ(z)+i=1mbi(z)β(zgi(z))+j=1ncj(z)λ(zhj(z))β(zhj(z))
for z0zz0 + l1. Then, we obtain the solution of above equation as follows:
ζ(z)={ζ(z0)+z0z(i=1mbi(t)ez0tgi(t)λz0(s)ds+j=1ncj(t)λz0(thj(t))ez0thj(t)λz0(s)ds)ez0ta(s)dsdt}ez0za(s)ds.

From this, we can define λ (z) as follows:

λ(z)=ζ(z)ζ(z)
on [z0,z0 + l1]. Now, let β (z) = ζ (z) on [z0l,z0 + l1]. Then, for z ∈ [z0 + l1,z0 + 2l1] , we get
ζ(z)=λ(z)ζ(z)=a(z)ζ(z)+i=1mbi(z)β(zgi(z))+j=1ncj(z)λ(zhj(z))β(zhj(z)).

Similarly, from the solution of above equation, we can define ζ (z) on [z0 + l1,z0 + 2l1]. So, we define λ (z) for all z ≥ z0l.

For existing of solution of (NDE) (1)(2), we will consider two case:

Case 1. Let, θ (z) does not have zeros on [z0l,z0]. Let

θ(z)=θ(z0)ez0zλz0(t)dt.

That is

λz0(z)=θ(z)θ(z).

So, the characteristic equation has solution such that

y(z)=θ(z0)ez0zλz0(t)dt
is a solution of (1)(2) on [z0l,∞). Especially, equation (1) by constant θ (z) = c ≠ 0 has a solution.

Case 2. Let, θ (z) has zeros on [z0l,z0]. Since θ is continuous on [z0l,z0], there is a constant c ≠ 0. So, we can write

ω(z)=θ(z)+c>0
for [z0l,z0]. Then, equation (1) has a solution u by initial condition ω. Thus,
y(z)=u(z)yc(z)
is a solution of (1)(2). Here yc is a solution of (1) with θ (z) = c.

Theorem 2. Suppose that

supzz0+lh{i=1m|bi(z)||gi(z)|ezzgi(z)λ(t)dt+j=1n|cj(z)||1hj(z)||λ(zhj(z))|ezzhj(z)λ(t)dt}<1

Then for each solution y of (NDE) (1)(2), there exists a K (constant), such that

y(z)ez0zλ(t)dtK,forz
and
{y(z)ez0zλ(t)dt}0,forz.

Proof

For solutions y of (1)(2) and λ of (3), we set

γ(z)=y(z)ez0zλ(t)dt,zz0l.

Applying (1) and (3), we obtain

γ(z)=(y(z)y(z)λ(z))ez0zλ(t)dt=(i=1mbi(z)y(zgi(z))y(z)i=1mbi(z)ezzgi(z)λ(t)dt+j=1ncj(z)y(zhj(z))y(z)j=1ncj(z)λ(zhj(z))ezzhj(z)λ(t)dt)ez0zλ(t)dt.

Since y(z)=γ(z)ez0zλ(t)dt, and y(z)=(γ(z)+γ(z)λ(z))ez0zλ(t)dt, from above equality, we obtain

γ(z)=i=1mbi(z)[γ(zgi(z))γ(z)]ezzgi(z)λ(t)dt+j=1ncj(z)[γ(zhj(z))+(γ(zhj(z))γ(z))λ(zhj(z))]ezzhj(z)λ(t)dt,
for zz0. From this, we get
γ(z)=i=1mbi(z)zzgi(z)ezzgi(z)λ(t)dtγ(s)ds+j=1ncj(z)[γ(zhj(z))+λ(zhj(z))zzhj(z)γ(s)ds]ezzhj(z)λ(t)dt
for zz0 + l − h.

If all bi’s and cj’s are equal zero on [z0 + l − h,∞) , from (5), γ is constant and γ′ = 0 on [z0 + l − h,∞) which would complete the proof. Thus, we suppose that bi ≠ 0 or cj ≠ 0 on [z0 + l − h,∞). Let

σ=supzz0+lh{i=1m|bi(z)||gi(z)|ezzgi(z)λ(t)dt+j=1n|cj(z)||1hj(z)||λ(zhj(z))|ezzhj(z)λ(t)dt}.

From (4),

0<σ<1.

Let

M=max{|γ(z)|:z[z0h,z0+lh]}.

We shall prove that

|γ(z)|Mforallzz0h,
on [z0h,∞).

Conversely, suppose that there exist ɛ > 0 and z ≥ z0h such that |γ′(z)| > M + ɛ. Since |γ′(z)| ≤ M and from the continuity of γ′, there exists z* > z0 + lh such that

|γ(z)|<M+ɛ,forz[z0h,z*)
and
|γ(z)|=M+ɛ,
for z0hzz0 + l − h.

From the definition of σ,(5) and (6), we get

M+ɛ=|γ(z*)|i=1m|bi(z*)|z*z*gi(z*)ez*z*gi(z*)λ(t)dt|γ(s)|ds+j=1n|cj(z*)|[|γ(z*hj(z*))|+|λ(z*hj(z*))|z*z*hj(z*)|γ(s)|ds]ez*z*hj(z*)λ(t)dt(M+ɛ){i=1m|bi(z*)||gi(z*)|ez*z*gi(z*)λ(t)dt}+j=1n|cj(z*)||1hj(z*)||λ(z*hj(z*))|ez*z*hj(z*)λ(t)dt}(M+ɛ)σ<M+ɛ.

So, we obtain a contradiction. Thus, inequality (7) holds. If M = 0 , from (7), γ is constant and γ′ = 0 on [z0h,∞). Thus, we suppose that M > 0.

In view of, (5) and (7),

|γ(z)|i=1m|bi(z)|zzgi(z)|γ(s)|dsezzgi(z)λ(t)dt+j=1n|cj(z)|[|γ(zhj(z))|+|λ(zhj(z))|zzhj(z)|γ(s)|ds]ezzhj(z)λ(t)dtM{i=1m|bi(z)|gi(z)ezzgi(z)λ(t)dt+j=1n|cj(z)||1hj(z)||λ(zhj(z))|ezzhj(z)λ(t)dt}Mσ,forzz0+lh.

Applying above inequality, we can show from induction,

|γ(z)|Mσn,forzz0+nlh(n=0,1,).

For an arbitrary zz0h, we define n=zz0+hl.

So, zz0 + nl − h and zz0+hl1<n . Therefore, from (6) and (8),

|γ(z)|MσnMσzz0+hl1.

We get n → ∞, as z → ∞ , and from (6), σn → 0. Thus, from (9),

limzγ(z)=0.

To prove that limγ(z) exists, as z → ∞ , we benefit from the Cauchy convergence criterion. For z > Zz0−h, by (9) , we get

|γ(z)γ(Z)|Zz|γ(t)|dt=ZzMσsz0+hl1ds=Mllnσ[μsz0+hl1]s=Zs=z=Mllnσ[μzz0+hl1μzz0+hl1].

We have z → ∞ , as Z → ∞ , and from (6), the right sides of above inequality tends zero. Thus,

lim|γ(z)γ(Z)|=0
which implies that the existence of limγ(z), as z → ∞. The proof is completed.

References

  • [1]

    Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear Analysis. 74, 2062–2070 (2011).

  • [2]

    Bellman, R., Cooke K. L.: Differential-Difference Equations. Academic Press, New York (1963).

  • [3]

    Driver, R. D.: A mixed neutral system. Nonlinear Anal. 8, 155–158 (1984).

  • [4]

    Burton, T. A., Frumochi T.: Fixed points and problems in stability theory for ordinary and functional differential equations. Dynam. Systems Appl. 10, 89–116 (2001).

  • [5]

    Burton, T. A.: Stability by fixed point theory for functional differential equations. Dover Publications. New York (2006).

  • [6]

    Burton, T. A., Frumochi T.: Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dynamic Systems and Applications. 11, 499–519 (2002).

  • [7]

    Dib, Y. M., Maroun, M. R., Raffoul Y. N.: Periodicity and stability in neutral nonlinear differential equations with functional delay. Electronic Journal od Differential Equations. 142, 1–11 (2005).

  • [8]

    Dix, JG., Philos, CG., Purnaras, IK.: Asymptotic properties of solutions to linear non-autonomous neutral differential equations. J. Math. Anal. Appl. 318, 296–304 (2006).

  • [9]

    Pinelas S.: Asymptotic Behavior of solutions to mixed type differential equations. Electronic Journal of Differential Equations. 210, 1–9 (2014).

  • [10]

    Chen G.: A fixed point approach towards stability of delay differential equations with applications to neural networks. Ph. D. Thesis, Leiden University, Leiden (2013).

  • [11]

    Hadeler K.P.: Neutral delay equations from and for population dynamics. Electron. J. Qual. Theory Differ. Equ. 11, 1–18 (2008).

  • [12]

    Philos, Ch. G., Purnaras I. K.: Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations. Electron. J. Differential Equations. 1–17 (2004).

  • [13]

    Philos, Ch. G., Purnaras, I. K.: Behavior of the solutions to second order linear autonomous delay differential equations. Electron. J. Differential Equations. 106, 1–35 (2007).

  • [14]

    Philos, Ch. G., Purnaras, I. K.: An asymptotic property of the solutions to second order linear nonautonomous delay differential equations, Math. Comput. Modelling. 49, 1350–1358 (2009).

  • [15]

    Tunç,. Stability and boundedness for a kind of non-autonomous differential equations with constant delay, Applied Mathematics and Information Sciences, vol. 7, no. 1, pp. 355–361, (2013).

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  • [1]

    Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear Analysis. 74, 2062–2070 (2011).

  • [2]

    Bellman, R., Cooke K. L.: Differential-Difference Equations. Academic Press, New York (1963).

  • [3]

    Driver, R. D.: A mixed neutral system. Nonlinear Anal. 8, 155–158 (1984).

  • [4]

    Burton, T. A., Frumochi T.: Fixed points and problems in stability theory for ordinary and functional differential equations. Dynam. Systems Appl. 10, 89–116 (2001).

  • [5]

    Burton, T. A.: Stability by fixed point theory for functional differential equations. Dover Publications. New York (2006).

  • [6]

    Burton, T. A., Frumochi T.: Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dynamic Systems and Applications. 11, 499–519 (2002).

  • [7]

    Dib, Y. M., Maroun, M. R., Raffoul Y. N.: Periodicity and stability in neutral nonlinear differential equations with functional delay. Electronic Journal od Differential Equations. 142, 1–11 (2005).

  • [8]

    Dix, JG., Philos, CG., Purnaras, IK.: Asymptotic properties of solutions to linear non-autonomous neutral differential equations. J. Math. Anal. Appl. 318, 296–304 (2006).

  • [9]

    Pinelas S.: Asymptotic Behavior of solutions to mixed type differential equations. Electronic Journal of Differential Equations. 210, 1–9 (2014).

  • [10]

    Chen G.: A fixed point approach towards stability of delay differential equations with applications to neural networks. Ph. D. Thesis, Leiden University, Leiden (2013).

  • [11]

    Hadeler K.P.: Neutral delay equations from and for population dynamics. Electron. J. Qual. Theory Differ. Equ. 11, 1–18 (2008).

  • [12]

    Philos, Ch. G., Purnaras I. K.: Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations. Electron. J. Differential Equations. 1–17 (2004).

  • [13]

    Philos, Ch. G., Purnaras, I. K.: Behavior of the solutions to second order linear autonomous delay differential equations. Electron. J. Differential Equations. 106, 1–35 (2007).

  • [14]

    Philos, Ch. G., Purnaras, I. K.: An asymptotic property of the solutions to second order linear nonautonomous delay differential equations, Math. Comput. Modelling. 49, 1350–1358 (2009).

  • [15]

    Tunç,. Stability and boundedness for a kind of non-autonomous differential equations with constant delay, Applied Mathematics and Information Sciences, vol. 7, no. 1, pp. 355–361, (2013).

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