Research Article | Open Access
Numerical Oscillations Analysis for Nonlinear Delay Differential Equations in Physiological Control Systems
This paper deals with the oscillations of numerical solutions for the nonlinear delay differential equations in physiological control systems. The exponential -method is applied to and it is shown that the exponential -method has the same order of convergence as that of the classical -method. Several conditions under which the numerical solutions oscillate are derived. Moreover, it is proven that every nonoscillatory numerical solution tends to positive equilibrium of the continuous system. Finally, the main results are illustrated with numerical examples.
The nonlinear delay differential equation where has been proposed by Mackey and Glass  as model of hematopoiesis (blood cell production). Here, denotes the density of mature cells in blood circulation, is the time delay between the production of immature cells in the bone marrow and their maturation for release in the circulating blood stream, and the production is a single-humped function of . Equation (1.1) has been recently studied by many authors. Mackey and Heidn  considered the local asymptotic stability of the positive equilibrium by the well-known technique of linearization. Gopalsamy et al.  obtained sufficient and also necessary conditions for all positive solutions to oscillate about their positive steady states. They also obtained sufficient conditions for the positive equilibrium to be a global attractor. For more details of (1.1), we refer to Mackey [4, 5], and Su et al. .
Our aim in this paper is to investigate the oscillations of numerical solutions for (1.1). The oscillatory and asymptotic behavior of solutions of delay differential equations has been the subject of intensive investigations during the past decades. The strong interest in this study is motivated by the fact that it has many useful applications in some mathematical models, such as ecology, biology, and spread of some infectious diseases in humans. The general theory and basic results for this paper have been thoroughly studied in [7, 8]. In recent years, much research has been focused on the oscillations of numerical solutions for delay differential equations [9–12]. Until now, very few results dealing with the corresponding behavior for nonlinear delay differential equations have been presented in the literature except for . In , the authors investigate the oscillations of numerical solutions for the nonlinear delay differential equation of population dynamics. Different from , in our paper, we will consider another nonlinear delay differential equation (1.1) in physiological control systems and obtain some new results. We not only investigate some sufficient conditions under which the numerical solutions are oscillatory but also consider the asymptotic behavior of nonoscillatory numerical solutions.
The structure of this paper is as follows. In Section 2, some necessary definitions and results for oscillations of the analytic solutions are given. In Section 3, we obtain the numerical discrete equation by applying the -methods to the simplified form which comes from making two transformations on (1.1). Moreover, the oscillations of the numerical solutions are discussed, and conditions under which the numerical solutions oscillate are obtained. In Section 4, we investigate the asymptotic behavior of nonoscillatory solutions. In Section 5, we present numerical examples that illustrate the theoretical results for the numerical methods.
Let us state some definitions, lemmas, and theorems that will be used throughout this paper.
Definition 2.1. A function of (1.1) is said to oscillate about if has arbitrarily large zeros. Otherwise, is called non-oscillatory. When , we say that oscillates about zero or simply oscillates.
Definition 2.2. A sequence is said to oscillate about if is neither eventually positive nor eventually negative. Otherwise, is called non-oscillatory. If is a constant sequence, we simply say that oscillates about . When , we say that oscillates about zero or simply oscillates.
Definition 2.3. We say that (1.1) oscillates if all of its solutions are oscillatory.
Theorem 2.4 (see ). Consider the difference equation and assume that and for . Then, the following statements are equivalent:(1)every solution of (2.1) oscillates;(2)the characteristic equation has no positive roots.
Lemma 2.6. The inequality holds for and .
Lemma 2.7. The inequality holds for and .
Lemma 2.8 (see ). For all , one has(1) if and only if for , for ;(2) if and only if for , for ,where and is a positive constant.
3. Oscillations of Numerical Solutions
3.1. Two Transformations
We introduce a similar method in . The change of variables turns (1.1) into the delay differential equation with positive equilibrium , which is denoted as The following theorem gives oscillations of the analytic solution of (3.3).
The following corollary is naturally obtained.
Next, we introduce an invariant oscillation transformation , and then (3.3) can be written as where Then, oscillates about if and only if oscillates about zero.
For our convenience, denote then the inequality (3.10) yields
3.2. The Difference Scheme
Let be a given stepsize with integer . The adaptation of the linear -method and the one-leg -method to (3.11) leads to the same numerical process of the following type: where and are approximations to and of (3.11) at , respectively.
Letting and using the expressions of and , we have
The following theorem, for the proof of which we refer to , allows us to obtain the convergence of exponential -method.
Theorem 3.4. The exponential -method (3.15) is convergent with order
3.3. Oscillation Analysis
It is not difficult to know that oscillates about if and only if is oscillatory. In order to study oscillations of (3.15), we only need to consider the oscillations of (3.14). The following conditions which are taken from  will be used in the next analysis: The linearized form of (3.14) is given by Then by (3.12), (3.18) gives It follows from  that (3.14) oscillates if (3.19) oscillates under the condition (3.17).
Definition 3.5. Equation (3.15) is said to be oscillatory if all of its solutions are oscillatory.
Definition 3.6. We say that the exponential -method preserves the oscillations of (1.1) if (1.1) oscillates, then there is a or , such that (3.15) oscillates for . Similarly, we say that the exponential -method preserves the nonoscillations of (1.1) if (1.1) non-oscillates, then there is a or , such that (3.15) nonoscillates for .
In the following, we will study whether the exponential -method inherits the oscillations of (1.1). Equivalently, when Corollary 3.2 holds, we will investigate the conditions under which (3.15) is oscillatory.
Lemma 3.7. The characteristic equation of (3.18) is given by
Proof. Letting in (3.18), we have that is, which is equivalent to In view of , we know that the stability function of the -method is By noticing (3.12), thus the characteristic equation of (3.18) is given by (3.20). The proof is completed.
Proof. Let . By Lemma 2.8, we know that
holds for and . In the following, we will prove that for . Suppose the opposite, that is, there exists a such that , then we get , and
Multiplying both sides of the inequality (3.26) by , we have
Thus, we have the following two cases.
Case 1. If , then , which contradicts the condition (3.13).
Case 2. If , then in view of Lemma 2.7, we obtain that is, so , which is also a contradiction to (3.13).
In summary, we have, for , which implies that the characteristic equation (3.20) has no positive roots. The proof is complete.
Without loss of generality, in the case of , we assume that .
Proof. Since is an increasing function of when , then, for and ,
In the following, we will prove that the inequality
holds under certain conditions.
The left side of inequality (3.33) can be rewritten as where so we only need to prove that for . It is easy to know that is the characteristic equation of the following difference equation By Theorems 2.4 and 2.5, we know that has no positive roots if and only if which is equivalent to We examine two cases depending on the position of : either or .
Case 1. If , by , (3.38) holds true.
Case 2. If and then by Lemma 2.6, we have Therefore, the inequality (3.33) holds for , where is defined in (3.31). So, we get that the following inequality holds for and , which implies that the characteristic equation (3.20) has no positive roots. This completes the proof.
Remark 3.10. By inequality (3.38) and condition , we have thus is meaningful.
4. Asymptotic Behavior of Nonoscillatory Solutions
Lemma 4.1 (see ). Assume that then every solution of the initial value problem is which is non-oscillatory about satisfies
From (3.3) and (3.7), we know that the non-oscillatory solution of (3.7) satisfies if Lemma 4.1 is satisfied. Furthermore, is also obtained. In the following, we will prove that the numerical solution of (1.1) can inherit this property.
Lemma 4.2. Let be a non-oscillatory solution of (3.14), then .
Proof. Without loss of generality, we assume that for sufficiently large . Then by condition (3.17), we have for sufficiently large . Moreover, it is can be seen from (3.14) that which gives thus we have then the sequence is decreasing, and, therefore, Next, we prove that . If , then there exist and such that for . Thus, and . So, inequality (4.6) yields which implies that , where Thus, as , which is a contradiction to (4.8). This completes the proof.
As a consequence, the second main theorem of this paper is as follows.
5. Numerical Examples
In order to verify our results, three numerical examples are examined in this section.
Example 5.1. Consider the following equation: subject to the initial condition In (5.1), it can be seen that condition (3.6) holds true and . That is, the analytic solutions of (5.1) are oscillatory. In Figures 1–3, we draw the figures of the analytic solutions and the numerical solutions of (5.1), respectively. Set , in Figure 2 and , in Figure 3, respectively. From the two figures, we can see that the numerical solutions of (5.1) oscillate about , which are in agreement with Theorem 3.11.
Example 5.2. Let us consider the equation with the initial value for . In (5.3), it is easy to see that condition (3.6) is fulfilled and . That is, the analytic solutions of (5.3) are oscillatory. In Figures 4–6, we draw the figures of the analytic solutions and the numerical solutions of (5.3), respectively. Set , in Figure 5 and , in Figure 6, respectively. We can see from the three figures that the numerical solutions of (5.3) oscillate about , which are consistent with Theorem 3.11. On the other hand, by direct calculation, we get , so the stepsize is not optimal.
Example 5.3. Consider the following equation: subject to the initial condition For (5.4), it is easy to see that , so the condition (3.6) is not satisfied. That is, the analytic solutions of (5.4) are non-oscillatory. In Figures 7–9, we draw the figures of the analytic solutions and the numerical solutions of (5.4), respectively. In Figure 7, we can see that as . From Figures 8 and 9, we can also see that the numerical solutions of (5.4) satisfy as . That is, the numerical method inherits the asymptotic behavior of non-oscillatory solutions of (5.4), which coincides with Theorem 4.3.
Q. Wang’s work was supported by the National Natural Science Foundation of China (no. 11201084). The authors would like to thank Professor Mingzhu Liu, Minghui Song, and Dr. Zhanwen Yang for their useful suggestions.
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Copyright © 2012 Qi Wang and Jiechang Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.