Oscillations of Numerical Solutions for Nonlinear Delay Differential Equations in the Control of Erythropoiesis
We consider the oscillations of numerical solutions for the nonlinear delay differential equations in the control of erythropoiesis. The exponential -method is constructed and some conditions under which the numerical solutions oscillate are presented. Moreover, it is proven that every nonoscillatory numerical solution tends to the equilibrium point of the continuous system. Numerical examples are given to illustrate the main results.
The oscillatory and asymptotic behavior of solutions of delay differential equations has been the subject of intensive investigations during the past decades. A large number of articles has appeared in the literature, we refer to [1–4] and the references therein. 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, spread of some infectious diseases in humans, and so on. For more information on this investigation, the reader can see [5, 6] and the references therein.
By contrast with the research on the oscillations of the analytic solutions, much studies have been focused on the oscillations of the numerical solutions for delay differential equations. In [7, 8], oscillations of numerical solutions in -methods and Runge-Kutta methods for a linear differential equation with piecewise constant arguments (EPCA, a special type of Delay Differential Equations) were considered, respectively. More recently, Wang et al.  studied numerical oscillations of alternately advanced and retarded linear EPCA, the conditions of oscillations for the -methods are obtained. To the best of our knowledge, until now less attention had been paid for the oscillations of the numerical solutions for nonlinear delay differential equations except for . Differently from , in our paper, we will investigate another nonlinear delay differential equation in the control of erythropoiesis and obtain some new results.
Consider the following nonlinear delay differential equation: with conditions Mackey and Glass  have proposed (1) as model of hematopoiesis (blood cell production). In (1), 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 monotonic decreasing function of . Equation (1) has been recently studied by many authors. Gopalsamy et al.  obtained sufficient and also necessary and sufficient 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. Using the linearization method, Zaghrout et al.  considered (1) and gave a sufficient condition for oscillations of all solutions about the positive steady state and proved that every nonoscillatory positive solution of (1) tends to as . For more details of (1), we refer to Mackey and Milton  and Mackey . Up to now, few results on the properties of numerical solutions for (1) were obtained. In the present paper, our main goal is to investigate some sufficient conditions under which the numerical solutions are oscillatory. We also consider the asymptotic behavior of nonoscillatory numerical solutions.
The contents of this paper are as follows. In Section 2, some necessary definitions and results for oscillations of the analytic solutions are given. In Section 3, we obtain a recurrence relation by applying the -methods to the simplified form which comes from making two transformations on (1). Moreover, the oscillations of the numerical solutions are discussed and conditions under which the numerical solutions oscillate are obtained. In Section 4, we study the asymptotic behavior of nonoscillatory solutions. In Section 5, we present numerical examples that illustrate the theoretical results for the numerical methods. Finally, Section 6 gives conclusions and issues for future research.
In this section, we start by introducing some definitions, lemmas and theorems that will be employed throughout the work.
Definition 1. A function of (1) is said to oscillate about if has arbitrarily large zeros. Otherwise, is called nonoscillatory. When , we say that oscillates about zero or simply oscillates.
Definition 2. A sequence is said to oscillate about if is neither eventually positive nor eventually negative. Otherwise, is called nonoscillatory. If is a constant sequence, we simply say that oscillates about . When , we say that oscillates about zero or simply oscillates.
Definition 3. We say (1) oscillates if all of its solutions are oscillatory.
Theorem 4 (see ). Consider the difference equation assume that and for . Then the following statements are equivalent:(i)every solution of (3) oscillates;(ii)the characteristic equation has no positive roots.
Lemma 6. The inequality holds for and .
Lemma 7. The inequality holds for and .
Lemma 8 (see ). For all ,(i) if and only if for , for ;(ii) if and only if for , for ,where and is a positive constant.
3. Oscillations of Numerical Solutions
For (1), we take an initial condition of the form where .
In order to simplify (1), we introduce a similar method in . The change of variables transforms (1) to the delay differential equation where . One can see that (8) has a unique equilibrium and that The following result concerning oscillations of the analytic solution of (8) is given in .
Theorem 9. Assume that then every positive solution of (8) oscillates about its positive equilibrium .
Therefore, we obtain the following corollary naturally.
Corollary 10. Assume that the condition holds, then every positive solution of (1) oscillates about its positive equilibrium .
Next, we also introduce an invariant oscillation transformation , then (8) can be written as where Then oscillates about if and only if oscillates about zero. Moreover, for the sake of brevity, let then (12) becomes For convenience, denote then the inequality (11) yields
3.2. The Difference Scheme
In this subsection we consider the adaptation of the -methods. Let be a given step size with integer . The adaptation of the linear -method and the one-leg -method to (15) leads to the same numerical process of the following type: where , and are approximations to and of (15) at , respectively.
Let , and take into account of the expressions of and we have
The following theorem gives the convergence of exponential -method. We can easily prove it by the method of steps which is used in .
Theorem 12. The exponential -method (19) 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 (19), we only need to consider the oscillations of (18). The following conditions which are taken from  will be used in the next analysis: For (18), its linearized form is given by which is equivalent to It follows from  that (18) oscillates if (23) oscillates under the condition (21).
Definition 13. Equation (19) is said to be oscillatory if all of its solutions are oscillatory.
Definition 14. We say that the exponential -method preserves the oscillations of (1) if (1) oscillates, then there is a or , such that (19) oscillates for . Similarly, we say that the exponential -method preserves the nonoscillations of (1) if (1) nonoscillates, then there is a or , such that (19) nonoscillates for .
In the following, we will study whether the exponential -method preserves the oscillations of (1). That is, when Corollary 10 holds, we will investigate the conditions under which (19) is oscillatory.
Lemma 15. The characteristic equation of (22) is given by
Proof. Let in (22), we have that is which is equivalent to In view of , we know that the stability function of the -method is thus the characteristic equation of (22) is given by (24). This completes the proof of the lemma.
Lemma 16. If , then the characteristic equation (24) has no positive roots for .
Proof. Let . By Lemma 8, we have
Now we will prove that for . Suppose the opposite, that is, there exists a such that , then we have , and
Multiplying both sides of the inequality (30) by , we obtain
therefore we have the following two cases.
Case 1. If , then , which contradicts the condition .
Case 2. If , then in view of Lemma 7, we get that is, so , which is also a contradiction to .
Combining both the cases, by (29) we obtain that for which implies that the characteristic equation (24) has no positive roots. The proof of the lemma is complete.
Without loss of generality, in the case of , we assume that .
Lemma 17. If and , then the characteristic equation (24) has no positive roots for , where
Proof. Since is an increasing function of when , then for and
Next, we will prove that the inequality
holds under certain conditions.
From (38), it follows that where so we only need to prove for . It is not difficult to know that is the characteristic polynomial of the following difference scheme In view of Theorems 4 and 5, we have 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 , then (43) holds true.
Case 2. If and then according to Lemma 6 we have Therefore the inequality (38) holds for , where So we get that the following inequality: holds for and , which implies that the characteristic equation (24) has no positive roots. This completes the proof.
Remark 18. By inequality (43) and condition , we have that thus is meaningful.
4. Asymptotic Behavior of Nonoscillatory Solutions
From the relationship between (8) and (12), we know that the nonoscillatory solution of (12) satisfies if Lemma 20 holds. Furthermore, is also obtained. Next, we will prove that the numerical solution of (1) can inherit this property.
Lemma 21. Let be a nonoscillatory solution of (18), then .
Proof. Without loss of generality, we may assume that for sufficiently large . Then by condition (21) we know that the following inequalities holds true: for sufficiently large . Moreover, it is can be seen from (18) that which gives Thus then the sequence is decreasing, and therefore Now we will prove that . If , then there exists and such that for , . Hence and . So inequality (52) yields which implies that , where Thus as , which is a contradiction to (54). This completes the proof.
Therefore, the second main theorem of this paper is as follows.
Theorem 22. Let be a positive solution of (19), which does not oscillate about , then .
5. Numerical Experiments
In this section, we will give some numerical examples to illustrate our results.
Firstly, we consider the equation with initial value for . In (57), it is easy to see that condition (11) holds true and . That is, the analytic solutions of (57) are oscillatory. In Figures 1–3, we draw the figures of the analytic solutions and the numerical solutions of (57), respectively. The parameters , in Figure 2 and , in Figure 3. From the two figures, we can see that the numerical solutions of (57) oscillate about , which are in agreement with Theorem 19.
Secondly, we consider with initial value for . In (58), it is not difficult to see that condition (11) is fulfilled. That is, the analytic solutions of (58) are oscillatory. In Figures 4–7, we draw the figures of the analytic solutions and the numerical solutions of (58), respectively. The parameters , in Figure 5, , in Figure 6 and , in Figure 7. We can see from the three figures that the numerical solutions of (58) oscillate about , which are consistent with Theorem 19. On the other hand, by direct calculation, we get . We notice that and in Figures 6 and 7, respectively, so the stepsize is not optimal.
Thirdly, we consider another equation, with initial value for . For (59), it is easy to see that , so the condition (11) is not satisfied. That is, the analytic solutions of (59) are nonoscillatory. In Figures 8–10, we draw the figures of the analytic solutions and the numerical solutions of (59), respectively. In Figure 8, we can see that as . From Figures 9 and 10, we can also see that the numerical solutions of (59) satisfy as . That is, the numerical method preserves the asymptotic behavior of nonoscillatory solutions of (59), which coincides with Theorem 22.
All the above numerical examples confirm our theoretical findings.
In this paper, we discuss the oscillations of the numerical solutions of a nonlinear delay differential equation in the control of erythropoiesis. The convergent exponential -method, namely the linear -method and the one-leg -method in exponential form, is constructed. We establish some conditions under which the numerical solutions oscillate in the case of oscillations of the analytic solutions. We also prove that nonoscillatory numerical solutions can inherit the corresponding properties of analytic solutions. It is pointed out that the stepsize in Lemma 17 is not optimal. Therefore, our future work will be devoted to investigating this problem.
The authors would like to thank Professor Mingzhu Liu and D. Zhanwen Yang for their useful suggestions. Q. Wang’s work is supported by the National Natural Science Foundation of China (no. 11201084).
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