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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

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Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations

Božena Dorociaková
  • Corresponding author
  • Department of Applied Mathematics, Faculty of Mechanical Engineering University of Žilina, Univerzitná 1, 010 26 Žilina, Slovak Republic Slovakia
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  • De Gruyter OnlineGoogle Scholar
/ Rudolf Olach
  • Department of Applied Mathematics, Faculty of Mechanical Engineering University of Žilina, Univerzitná 1, 010 26 Žilina, Slovak Republic Slovakia
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Published Online: 2016-06-02 | DOI: https://doi.org/10.1515/math-2016-0033

Abstract

The paper deals with the existence of positive ω-periodic solutions for a class of nonlinear delay differential equations. For example, such equations represent the model for the survival of red blood cells in an animal. The sufficient conditions for the exponential stability of positive ω-periodic solution are also considered.

Keywords: Positive periodic solution; Delay differential equation; Nonlinear; Exponential stability; Red blood cells; Banach space

MSC 2010: 34K13

1 Introduction

In this paper, we consider the existence of positive ω-periodic solutions for the nonlinear delay differential equation of the form x˙(t)=p(t)x(t)+i=1nqi(t)f(x(τi(t))),tt0.(1)

With respect to (1) throughout the paper we will assume the following conditions:

  1. p, qiC([t0, ∞), (0, ∞)), i = 1;:::;n, fC1(R, R), f(x) > 0 for x >0,

  2. τiC([t0, ∞), (0, ∞)), τi(t) < t and limtt→∞ τi(t) = ∞, i = 1; : : : ; n.

In the last several years, the problem of the existence of positive periodic solutions for the nonlinear delay differential equations received a considerable attention. It is due to the fact that such equations have found a variety of applications in several fields of natural sciences. They have been proposed as models for physiological, ecological and physical processes, neural interactions [13], etc.

One important question is whether these equations can support the existence of positive periodic solutions. Such question has been studied extensively by a number of authors. For example the authors in [2, 49] studied the existence, multiplicity and nonexistence of positive periodic solutions for the nonlinear delay differential equations. Periodic properties of solutions of some special types of differential equations are discussed in [10, 11]. Zhang, Wang and Yang [12] and Lin [13] studied the existence and exponential stability of positive periodic solutions.

In this paper, we will obtain existence criteria for the positive !-periodic solution of (1) and sufficient conditions for the exponential stability of such solution. The existence results in the literature are largely based on the assumption that the functions p(t), qi(t), i = 1; : : : ; n are ω-periodic. It is interesting to know if there is a positive periodic solution of (1) when the periodicity conditions for the functions p(t), qi(t), i = 1; : : : ; n are not satisfied. This substantially extends and improves the results in [79, 14] where the exponential stability for the positive periodic solution is not studied.

The following fixed point theorem will be used to prove the main results in the next section.

(Schauder’s Fixed Point Theorem [6, 15]). Let Ω be a closed, convex and nonempty subset of a Banach space X. Let S : Ω → Ω be a continuous mapping such that S(Ω) is a relatively compact subset of X. Then S has at least one fixed point in Ω, that is, there exists an x ∈ Ω such that Sx = x.

The remaining of this paper is organized as follows. In Section 2, we consider the existence of positive periodic solutions. In Section 3, the exponential stability of such solution is treated and in Section 4, the obtained results are applied to the model for the survival of red blood cells and illustrated with an example.

2 Existence of positive periodic solutions

In this section, we will study the existence of positive ω-periodic solutions of (1). We choose T sufficiently large so that τi(t) ≥ t0 for tT, i = 1; : : : ; n.

Suppose that there exist functions kiC([T, ∞), (0, ∞)), i = 1; : : : ; n such that tt+ω[p(s)+i=1nqi(s)ki(s)]ds=0,tT.(2)

Then the function w(t)=exp(Tt[p(s)+i=1nqi(s)ki(s)]ds),tT,

is ω-periodic.

Proof. For tT , we obtain

w(t+ω)=exp(Tt+ω[p(s)+i=1nqi(s)ki(s)]ds)=exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)exp(Tt+ω[p(s)+i=1nqi(s)ki(s)]ds)=w(t).

Thus, the function w(t) is ω-periodic. □

Suppose that there exist functions kiC([T, ∞), (0, ∞)), i = 1; : : : ; n such that (2) holds and f(exp(Tτi(t)[p(s)+i=1nqi(s)ki(s)]ds))×exp(Tt[p(s)i=1nqi(s)ki(s)]ds)=kj(t),τj(t)T,j=1,...,n.(3)

Then, (1) has a positive ω-periodic solution.

Proof. Let X = {xC([t0, ∞), R)} be the Banach space with the norm ||x|| = suptt0 |x(t)|. With regard to Lemma 2.1 for the function

w(t)=exp(tt[p(s)+i=1nqi(s)ki(s)]ds),tT,

we get 0 < mw(t) ≤ M, tT , where

m=mint[T,){exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)},M=maxt[T,){exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)}.(4)

We now define a closed, bounded and convex subset Ω of X as follows:

Ω={xX:x(t+ω)=x(t),tT,mx(t)M,tT,ki(t)x(t)=f(x(τi(t))),i=1,...,n,tT,x(t)=1,t0tT}.

Define the operator S : Ω → X as follows:

(Sx)(t)={exp(Tt[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds)1,t0tT.,tT,

We will show that for any x ∈ Ω, we have Sx ∈ Ω. For every x ∈ Ω and tT , we get

(Sx)(t)=exp(Tt[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds)=exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)M.

Furthermore, for tT and x ∈ Ω, we obtain

(Sx)(t)=exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)m.

For t ∈ [t0, T] we have (Sx)(t) = 1. By hypothesis (3) for every x ∈ Ω and τj(t) ≥ T, j = 1; : : : ; n, we get

f((Sx)(τj(t)))=f(exp(Tτj(t)[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds))=f(exp(Tτj(t)[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds))×exp(Tt[p(s)i=1nqi(s)f(x(τi(s)))x(s)]ds)×exp(Tt[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds)=f(exp(Tτj(t)[p(s)+i=1nqi(s)ki(s)]ds))×exp(Tt[p(s)i=1nqi(s)ki(s)]ds)(Sx)(t)=kj(t)(Sx)(t),j=1,...,n.

Finally, we will show that for x ∈ Ω; tT the function (Sx)(t) is ω-periodic. For x ∈ Ω; tT and according to (2), we obtain

(Sx)(t+ω)=exp(Tt+ω[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds)=exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)×exp(Tt+ω[p(s)+i=1nqi(s)ki(s)]ds)=(Sx)(t).

Thus (Sx)(t) is ω-periodic on [T, ∞). Therefore we have proved that Sx ∈ Ω for any x ∈ Ω.

We now show that S is completely continuous. At first we will show that S is continuous. Let xk = xk(t) ∈ Ω be such that xk(t) → x(t) ∈ Ω as k → ∞. For tT , we get

|(Sxk)(t)(Sx)(t)|=|exp(Tt[p(s)+i=1nqi(s)f(xk(τi(s)))xk(s)]ds)exp(Tt[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds)|.

Since f(xk(τi(t)))/xk(t) → f(x(τi(t)))/x(t) as k → ∞ for i = 1; 2; : : : ; n, by applying the Lebesgue dominated convergence theorem, we obtain that (cf. [13, p.66], [16, p.95])

limk(Sxk)(t)(Sx)(t)=0.

For t ∈ [t0, T] the relation above is also valid. This means that S is continuous.

Now, we will show that S(Ω) is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions {Sx : x ∈ Ω} is uniformly bounded and equicontinuous on every finite subinterval of [t0, ∞). The uniform boundedness follows from the definition of Ω. It remains to prove the equicontinuity. Using (4), we get for tT and x ∈ Ω:

|ddt(Sx)(t)|=|p(t)+i=1nqi(t)f(x(τi(t)))x(t)|exp(Tt[p(s)+i=1nqi(s)f(x(τi(s)))x(s)]ds)=|p(t)+i=1nqi(t)ki(t)|exp(Tt[p(s)+i=1nqi(s)ki(s)]ds)M1,M1>0.

For t ∈ [t0, T] and x ∈ Ω, we obtain:

|ddt(Sx)(t)|=0.

This shows the equicontinuity of the family S(Ω) and, therefore, S is completely continuous (cf. [6, p.265]). Hence S(Ω) is relatively compact. By Theorem 1.1, there is an x0 ∈ Ω such that Sx0 = x0. Therefore, by the definition of S, we have that x0(t) is a positive ω-periodic solution of (1). The proof is complete. □

3 Stability of positive periodic solution

In this section, we consider the exponential stability of the positive periodic solution of (1). Let r = min1≤in{inftT τi(t)}. We denote x(t; T, φ), tr, φC([r, T], (0, ∞)) for a solution of (1) satisfying the initial condition x(t; T, φ) = φ(t), t ∈ [r, T], where T is the initial point. Let x(t) = x(t; T, φ), x1(t) = x(t; T, φ1) and y(t) = x(t) – x1(t), t ∈ [r, ∞). By (1), we get

y˙(t)=p(t)y(t)+i=1nqi(t)[f(x(τi(t)))f(x1(τi(t)))],tT.

By the mean value theorem, we obtain

y˙(t)=p(t)y(t)+i=1nqi(t)f(xi*)[x(τi(t))x1(τi(t))],f(x)=df(x)dx,y˙(t)=p(t)y(t)+i=1nqi(t)f(xi*)y(τi(t)),tT.(5)

Assume that |f(x)| ≤ a, x ∈ (0, ∞), tτi(t) ≤ b, tT; i = 1; : : : ; n and suptT{p(t)+ai=1nqi(t)}<0.

Then there exists λ ∈ (0, 1] such that p(t)+λ+aeλbi=1nqi(t)<0fortT.

Proof. Define a continuous function H(u) by

H(u)=suptT{p(t)+u+aeubi=1nqi(t)},u[0,1].

By hypothesis, we get

H(0)=suptT{p(t)+ai=1nqi(t)}<0.

According to the continuity of H(u) and H(0) < 0, there exists λ ∈ (0, 1] such that H(λ) < 0, that is

p(t)+λ+aeλbi=1nqi(t)<0fortT.

We have achieved the desired result. □

Next we will assume that the function

F(t,x,x1,...,xn)=p(t)x(t)+i=1nqi(t)f(xi(t)),tr,

satisfies Lipschitz-type condition with respect to x, xi > 0, i = 1; : : : ; n.

Let x1(t) be a positive solution of (1). If there exist constants Tφ,x1 ;Kφ,x1 and λ > 0 such that for every solution x(t; T, φ) of (1) |x(t;T,φ)x1(t)|<Kφ,x1eλtforallt>Tφ,x1.

Then x1(t) is said to be exponentially stable.

In the next lemma, we establish sufficient conditions for the exponential stability of the positive solution x1(t) = x(t; T, φ1) of (1).

Suppose that |f′(x)| ≤ a, x ∈ (0, ∞), tτi(t) ≤ b, tT; i = 1; : : : ; n and suptT{p(t)+ai=1nqi(t)}<0.

Then there exists λ ∈ (0, 1] such that |x(t;T,φ)x(t;T,φ1)|<Kφ,x1eλt,t>T,

where Kφ,x1 = maxt∈[r, T] eλT |y(t)| + 1.

Proof. We consider the Lyapunov function

L(t)=|y(t)|eλt,tr,λ(0,1].

We claim that L(t) < Kφ,x1 for t > T . In order to prove it, suppose that there exists t* > T such that L(t*) = Kφ,x1 and L(t) < Kφ,x1 for t ∈ [r, t*). Calculating the upper left derivative of L(t) along the solution y(t) of (5), we obtain

D(L(t))p(t)|y(t)|eλt+eλti=1nqi(t)f(xi*)|y(τi(t))|+λ|y(t)|eλt,tT.

For t = t* and applying Lemma 3.1, we get

0D(L(t*))[λp(t*)]|y(t*)|eλt+aeλt*i=1nqi(t*)|y(τi(t*))|=[λp(t*)]|y(t*)|eλt+ai=1nqi(t*)|y(τi(t*))|eλτi(t*)eλ(t*τi(t*))=[λp(t*)]Kφ,x1+ai=1nqi(t*)L(τi(t*))eλ(t*τi(t*))<[λp(t*)+aeλbi=1nqi(t*)]Kφ,x1<0,

which is a contradiction. Therefore we obtain

L(t)=|y(t)|eλt<Kφ,x1fort>Tandforsomeλ(0,1].

The proof is complete. □

Suppose that |f′(x)| ≤ a, x ∈ (0, ∞), tτi(t) ≤ b, tT, i = 1; : : : ; n; suptT{p(t)+ai=1nqi(t)}<0

and there exist functions kiC([T, ∞), (0, ∞)), i = 1; : : : ; n such that (2), (3) hold. Then (1) has a positive ω-periodic solution which is exponentially stable.

Proof. The proof follows from the Theorem 2.2 and Lemma 3.3. □

4 Model for the survival of red blood cells

In this section, we consider the existence of positive ω-periodic solutions for the nonlinear delay differential equation of the form

x˙(t)=p(t)x(t)+q(t)eγx(τ(t)),tt0,(6)

which is a special case of (1), where q1(t) = q(t), qi(t) = 0, i = 2; : : : ; n, and f (x(τ1(t))) = exp(.γx(τ(t))), γ > 0. We will also establish the sufficient conditions for the exponential stability of the positive periodic solution.

The autonomous case of (6) is given by: x˙(t)=p(t)x(t)+qeγx(tτ),tt0,

and it has been used by Wazewska-Czyzewska and Lasota in [17] as a model for the survival of red blood cells in an animal. The function x(t) denotes the number of red blood cells at time t. The positive constants p, q and are related to the production of red blood cells per unit of time and τ is the time required to produce red blood cells.

Rewriting the Theorem 3.4 to the equation (6) we obtain the next result.

Suppose that γ > 0; tτ(t) ≤ b, tT, suptT{p(t)+γq(t)}<0(7)

and there exists function kC([T, ∞), (0, ∞)) such that tt+ω[p(s)+q(s)k(s)]ds=0,tT,(8) lnk(t)=tt[p(s)q(s)k(s)]dsγexp(Tτ(t)[p(s)+q(s)k(s)]ds),τ(t)T.(9)

Then (6) has a positive ω-periodic solution which is exponentially stable.

Consider the nonlinear delay differential equation x˙(t)=p(t)x(t)+q(t)eγx(τ(t)),tt0,(10)

where γ > 0, τ(t) = t – π, p(t)=14(4+et+sint),q(t)=14(4+et)exp(14(costcosT))exp(e0.25(cost+cosT)),T>0.

We choose k(t)=exp(14(costcosT))exp(γe0.25(cost+cosT)).

Then for conditions (8), (9) and ω = 2π, we get tt+ω[p(s)+q(s)k(s)]ds=14tt+2πsinsds=0.

Therefore: tt[p(s)q(s)k(s)]dsγexp(Ttπ[p(s)+q(s)k(s)]ds) =14Ttsinsdsγexp(14Ttπsinsds)=14(costcosT)γexp(14(cost+cosT))=lnk(t),tT+π.

The conditions (8), (9) of Theorem 4.1 are satisfied and (10) has a positive ω = 2π periodic solution x(t)=exp(Tt[p(s)+q(s)k(s)]ds)=exp(14Ttsinsds)=exp(14(costcosT)),tT.

If we put γ = 0.4, T=π2, then also the condition (7) is satisfied and solution x(t) is exponentially stable. The numerical simulation in Figure 1 supports the conclusion.

Numerical simulation of exponential stability
Fig. 1

Numerical simulation of exponential stability

Acknowledgement

The authors gratefully acknowledge the Scientific Grant Agency VEGA of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences for supporting this work under the Grant No. 1/0234/13.

References

  • [1]

    Levin S.A., Hallam T.G., Gross L.J., Applied Mathematical Ecology, Springer Verlag, New York, Berlin, Heidelberg, 1989 Google Scholar

  • [2]

    Agarwal R.P., et al., Nonoscillation Theory of Functional Differential Equations with Applications, New York, Dortrecht, Heidelberg, London, Springer, 2010, ISBN 1461434558 Google Scholar

  • [3]

    Kolmanovskii V., Myshkis A., MIA, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic publishers, Dordrecht, The Netherlands, 463, 1999, ISBN 0-7923-5504-0 Google Scholar

  • [4]

    Dix J.G., Padhi S., Existence of multiple positive periodic solutions for delay differential equation whose order is a multiple of 4, Appl. Math. Comput., 2014, 216, Issue 9, 2709-2717 Google Scholar

  • [5]

    Dorociaková B., Olach R., Existence of positive periodic solutions to nonlinear integro-differential equations, Appl. Math. Comput., 2015, 253, 287-293, ISSN 0096-3003Google Scholar

  • [6]

    Erbe L.H., Kong Q.K., Zhang B.G., Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995 Google Scholar

  • [7]

    Graef J.R., Kong L., Periodic solutions of first order functional differential equations, Appl. Math. Lett., 2011, 24, 1981–1985 Google Scholar

  • [8]

    Jin Z., Wang H., A note on positive periodic solutions of delayed differential equations, Appl. Math. Lett. 2010, 23, 581–584Google Scholar

  • [9]

    Ma R., Chen R., Chen T., Existence of positive periodic solutions of nonlinear first-order delayed differential equations, J. Math. Anal. Appl., 2011, 384, 527–535 Google Scholar

  • [10]

    Astashova I., On quasi-periodic solutions to a higher - order Emden - Fowler type differential equation, Boundary Value Problems, 2014 http://www.boundaryvalueproblems.com/content/2014/1/174 

  • [11]

    Diblík J., Iričanin B., Stević S., Šmarda Z., Note on the existence of periodic solutions of a class of systems of differential-difference equations, Appl. Math. Comput., 2014, 232, 922–928 Google Scholar

  • [12]

    Zhang H., Wang L., Yang M., Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis, Appl. Math. Lett., 2013, 26, 38–42 Google Scholar

  • [13]

    Lin B., Global exponential stability of positive periodic solutions for a delayed Nicholson’s blowflies model, J. Math. Anal. Appl., 2014, 412, 212–221 Google Scholar

  • [14]

    Wang H., Positive periodic solutions of functional differential equations, J. Differential Equations, 2004, 202, 354–366 Google Scholar

  • [15]

    Schauder J., Der Fixpunktsatz in Functionalraümen, Studia Math., 1930, 2, 171–180 Google Scholar

  • [16]

    Zhou Y., Existence for nonoscillatory solutions of second-order nonlinear differential equations, J. Math. Anal. Appl., 2000, 331, 91–96 Google Scholar

  • [17]

    Wazewska-Czyzewska M., Lasota A., Mathematical problems of the dynamics of the red blood cells system, Annals Polish Math. Society, Applied Mathematics, 1988, 17, 23–40 Google Scholar

About the article

Received: 2015-11-04

Accepted: 2016-04-08

Published Online: 2016-06-02

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0033.

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