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.
1 Introduction
In this paper, we consider the existence of positive ω-periodic solutions for the nonlinear delay differential equation of the form
With respect to (1) throughout the paper we will assume the following conditions:
p, qi ∈ C([t0, ∞), (0, ∞)), i = 1;:::;n, f ∈ C1(R, R), f(x) > 0 for x >0,
τi ∈ C([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 [1–3], 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, 4–9] 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 [7–9, 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 t ≥ T, i = 1; : : : ; n.
Suppose that there exist functions ki ∈ C([T, ∞), (0, ∞)), i = 1; : : : ; n such that
Then the function
is ω-periodic.
Proof. For t ≥ T , we obtain
Thus, the function w(t) is ω-periodic. □
Suppose that there exist functions ki ∈ C([T, ∞), (0, ∞)), i = 1; : : : ; n such that (2) holds and
Then, (1) has a positive ω-periodic solution.
Proof. Let X = {x ∈ C([t0, ∞), R)} be the Banach space with the norm ||x|| = supt≥t0 |x(t)|. With regard to Lemma 2.1 for the function
we get 0 < m ≤ w(t) ≤ M, t ≥ T , where
We now define a closed, bounded and convex subset Ω of X as follows:
Define the operator S : Ω → X as follows:
We will show that for any x ∈ Ω, we have Sx ∈ Ω. For every x ∈ Ω and t ≥ T , we get
Furthermore, for t ≥ T and x ∈ Ω, we obtain
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
Finally, we will show that for x ∈ Ω; t ≥ T the function (Sx)(t) is ω-periodic. For x ∈ Ω; t ≥ T and according to (2), we obtain
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 t ≥ T , we get
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])
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 t ≥ T and x ∈ Ω:
For t ∈ [t0, T] and x ∈ Ω, we obtain:
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≤i≤n{inft≥Tτi(t)}. We denote x(t; T, φ), t ≥ r, φ ∈ C([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
By the mean value theorem, we obtain
Assume that |f′(x)| ≤ a, x ∈ (0, ∞), t – τi(t) ≤ b, t ≥ T; i = 1; : : : ; n and
Then there exists λ ∈ (0, 1] such that
Proof. Define a continuous function H(u) by
By hypothesis, we get
According to the continuity of H(u) and H(0) < 0, there exists λ ∈ (0, 1] such that H(λ) < 0, that is
We have achieved the desired result. □
Next we will assume that the function
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,x1and λ > 0 such that for every solution x(t; T, φ) of (1)
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, ∞), t – τi(t) ≤ b, t ≥ T; i = 1; : : : ; n and
Then there exists λ ∈ (0, 1] such that
where Kφ,x1 = maxt∈[r, T]eλT |y(t)| + 1.
Proof. We consider the Lyapunov function
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
For t = t* and applying Lemma 3.1, we get
which is a contradiction. Therefore we obtain
The proof is complete. □
Suppose that |f′(x)| ≤ a, x ∈ (0, ∞), t – τi(t) ≤ b, t ≥ T, i = 1; : : : ; n;
and there exist functions ki ∈ C([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
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:
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; t – τ(t) ≤ b, t ≥ T,
and there exists function k ∈ C([T, ∞), (0, ∞)) such that
Then (6) has a positive ω-periodic solution which is exponentially stable.
Consider the nonlinear delay differential equation
where γ > 0, τ(t) = t – π,
We choose
Then for conditions (8), (9) and ω = 2π, we get
Therefore:
The conditions (8), (9) of Theorem 4.1 are satisfied and (10) has a positive ω = 2π periodic solution
If we put γ = 0.4,
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, 198910.1007/978-3-642-61317-3Search in Google Scholar
[2] Agarwal R.P., et al., Nonoscillation Theory of Functional Differential Equations with Applications, New York, Dortrecht, Heidelberg, London, Springer, 2010, ISBN 1461434558Search in 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 0-7923-5504-010.1007/978-94-017-1965-0Search in 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, 2709-271710.1016/j.amc.2010.03.116Search in 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 0096-300310.1016/j.amc.2014.09.086Search in Google Scholar
[6] Erbe L.H., Kong Q.K., Zhang B.G., Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995Search in Google Scholar
[7] Graef J.R., Kong L., Periodic solutions of first order functional differential equations, Appl. Math. Lett., 2011, 24, 1981–198510.1016/j.aml.2011.05.020Search in Google Scholar
[8] Jin Z., Wang H., A note on positive periodic solutions of delayed differential equations, Appl. Math. Lett. 2010, 23, 581–58410.1016/j.aml.2010.01.015Search in Google 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, 527–53510.1016/j.jmaa.2011.06.003Search in 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/2014/1/17410.1186/s13661-014-0174-7Search in Google Scholar
[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, 922–92810.1016/j.amc.2014.01.104Search in 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, 38–4210.1016/j.aml.2012.02.034Search in 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, 212–22110.1016/j.jmaa.2013.10.049Search in Google Scholar
[14] Wang H., Positive periodic solutions of functional differential equations, J. Differential Equations, 2004, 202, 354–36610.1016/j.jde.2004.02.018Search in Google Scholar
[15] Schauder J., Der Fixpunktsatz in Functionalraümen, Studia Math., 1930, 2, 171–18010.4064/sm-2-1-171-180Search in Google Scholar
[16] Zhou Y., Existence for nonoscillatory solutions of second-order nonlinear differential equations, J. Math. Anal. Appl., 2000, 331, 91–9610.1016/j.jmaa.2006.08.048Search in 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, 23–40Search in Google Scholar
© 2016 Dorociaková and Olach, published by De Gruyter Open
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.