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

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.


Introduction
In this paper, we consider the existence of positive !-periodic solutions for the nonlinear delay differential equation of the form With respect to (1) throughout the paper we will assume the following conditions: (i) p; q i 2 C.OEt 0 ; 1/; .0; 1//; i D 1; : : : ; n; f 2 C 1 .R; R/; f .x/ > 0 for x > 0; (ii) i 2 C.OEt 0 ; 1/; .0; 1//; i .t / < t and lim t !1 i .t / D 1; i D 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][2][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][5][6][7][8][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 /; q i .t /; i D 1; : : : ; n are !-periodic. It is interesting to know if there is a positive periodic solution of (1) when the periodicity conditions for the functions p.t /; q i .t /; i D 1; : : : ; n are not satisfied. This substantially extends and improves the results in [7][8][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.
Theorem 1.1 (Schauder's Fixed Point Theorem [6,15]). Let be a closed, convex and nonempty subset of a Banach space X . Let S W ! be a continuous mapping such that S. / is a relatively compact subset of X . Then S has at least one fixed point in , that is, there exists an x 2 such that S x D 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.

Existence of positive periodic solutions
In this section, we will study the existence of positive !-periodic solutions of (1). We choose T sufficiently large so that i .t / t 0 for t T; i D 1; : : : ; n.
Then the function is !-periodic.
Proof. For t T , we obtain Thus, the function w.t / is !-periodic.
Proof. Let X D fx 2 C.OEt 0 ; 1/; R/g be the Banach space with the norm jjxjj D sup t t 0 jx.t /j. With regard to Lemma 2.1 for the function We now define a closed, bounded and convex subset of X as follows: : : : ; n; t T; Define the operator S W ! X as follows: .
We will show that for any x 2 , we have S x 2 . For every x 2 and t T , we get Furthermore, for t T and x 2 , we obtain For t 2 OEt 0 ; T we have .S x/.t / D 1. By hypothesis (3) for every x 2 and j .t / T; j D 1; : : : ; n, we get j D 1; : : : ; n: Finally, we will show that for x 2 ; t T the function .S x/.t / is !-periodic. For x 2 ; t T and according to (2), we obtain Thus .S x/.t / is !-periodic on OET; 1/. Therefore we have proved that S x 2 for any x 2 . We now show that S is completely continuous. At first we will show that S is continuous. Let For t 2 OEt 0 ; T the relation above is also valid. This means that S is continuous. Now, we will show that S. / is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions fS x W x 2 g is uniformly bounded and equicontinuous on every finite subinterval of OEt 0 ; 1/. The uniform boundedness follows from the definition of . It remains to prove the equicontinuity. Using (4), we get for t T and x 2 W For t 2 OEt 0 ; T and x 2 , we obtain:ˇd

dt .S x/.t /ˇD 0:
This shows the equicontinuity of the family S. / and, therefore, S is completely continuous (cf. [6, p.265]). Hence S. / is relatively compact. By Theorem 1.1, there is an x 0 2 such that S x 0 D x 0 . Therefore, by the definition of S , we have that x 0 .t / is a positive !-periodic solution of (1). The proof is complete.

Stability of positive periodic solution
In this section, we consider the exponential stability of the positive periodic solution of (1). Let r D min 1Äi Än finf t T i .t /g. We denote x.tI T; '/; t r; ' 2 C.OEr; T ; .0; 1// for a solution of (1) satisfying the initial condition x.tI T; '/ D '.t /; t 2 OEr; T , where T is the initial point. Let x.t / D x.tI T; '/; x 1 .t / D x.tI T; ' 1 / and y.t/ D x.t / x 1 .t /; t 2 OEr; 1/. By (1), we get By the mean value theorem, we obtain By hypothesis, we get According to the continuity of H.u/ and H.0/ < 0, there exists 2 .0; 1 such that H. / < 0, that is We have achieved the desired result.
Next we will assume that the function F .t; x; x 1 ; : : : ; satisfies Lipschitz-type condition with respect to x; x i > 0; i D 1; : : : ; n.
In the next lemma, we establish sufficient conditions for the exponential stability of the positive solution x 1 .t/ D x.tI T; ' 1 / of (1). where K ';x 1 D max t 2OEr;T e T jy.t /j C 1.
Proof. We consider the Lyapunov function L.t / D jy.t /je t ; t r; 2 .0; 1: We claim that L.t / < K ';x 1 for t > T . In order to prove it, suppose that there exists t > T such that L.t / D K ';x 1 and L.t / < K ';x 1 for t 2 OEr; t /. Calculating the upper left derivative of L.t / along the solution y.t/ of (5), we obtain For t D t and applying Lemma 3.1, we get which is a contradiction. Therefore we obtain L.t / D jy.t /je t < K ';x 1 for t > T and for some 2 .0; 1: The proof is complete.

Model for the survival of red blood cells
In this section, we consider the existence of positive !-periodic solutions for the nonlinear delay differential equation of the form which is a special case of (1), where q 1 .t / D q.t /; q i .t / D 0; i D 2; : : : ; n, and f .x.
x. .t///; > 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 is the time required to produce red blood cells. Rewriting the Theorem 3.4 to the equation (6) we obtain the next result.
and there exists function k 2 C.OET; 1/; .0; 1// such that Then (6) has a positive !-periodic solution which is exponentially stable. If we put D 0:4; T D 2 , then also the condition (7) is satisfied and solution x.t / is exponentially stable. The numerical simulation in Figure 1 supports the conclusion.