Numerical oscillations for first-order nonlinear delay differential equations in a hematopoiesis model

Qi Wang 0 Jiechang Wen 0 Shenshan Qiu 2 Cui Guo 1 0 School of Applied Mathematics, Guangdong University of Technology , Guangzhou, 510006 , China 1 College of Science, Harbin Engineering University , Harbin, 150001 , China 2 CSIB Software Technology Center, Administrative Commission of Guangzhou Tianhe Software Park , Guangzhou, 510635 , China In this paper, we consider the oscillations of numerical solutions for the nonlinear delay differential equations in a hematopoiesis model. Using two -methods, namely the linear -method and the one-leg -method, several conditions, under which the numerical solutions oscillate, are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to an equilibrium point of the original system. Some numerical experiments are provided to support the theoretical results. MSC: 65L05; 65L20 1 Introduction In recent years, there has been much research activity concerning the oscillatory behavior of solutions of difference equations [, ], differential equations with piecewise continuous arguments (EPCA) [, ], dynamic equations [, ] and partial differential equations [, ]. Among these investigations, oscillations of solutions of delay differential equations (DDEs) have also been the subject of many recent studies []. The strong interest in this study is motivated by the fact that it has many useful applications in some mathematical models such as biology, ecology, spread of some infectious diseases in humans and so on. For more information on this study, the reader can see [, ] and the references therein. Relative to the investigation of the oscillations of the analytic solutions, much research has been focused on the corresponding behavior of the numerical solutions for DDEs. In [, ], oscillations of numerical solutions in -methods and Runge-Kutta methods for a linear EPCA x (t) + ax(t) + ax([t ]) = were considered, respectively. Wang et al. [] studied numerical oscillations of alternately advanced and retarded EPCA, the conditions of oscillations for the -methods are obtained. To the best of our knowledge, until now very few results dealing with the oscillations of the numerical solutions for nonlinear DDEs have been reported except for []. Different from [], in our paper, we consider another nonlinear DDEs in a hematopoiesis model and get some new results. In this paper, we consider the following equation: x (t) + px(t) with the conditions Equation () is one of the models proposed by Nazarenko [] to study the control of a single population of cells. Here x(t) is the size of the population of cells, and cells are lost at a rate p, and the function is the flux function, which depends on the size of cells x(t) and x(t ) at times t and t , respectively, and is the time of maturation. The model () has been recently investigated by several researchers. Kubiaczyk and Saker [] considered Equation () and gave a sufficient condition for oscillations of all solutions about the positive unique equilibrium point K and proved that every non-oscillatory positive solution of Equation () tends to K as t . Following up the investigation in [], Saker and Agarwal [] studied the oscillations and global attractivity of Equation () with time periodic coefficients. Song et al. [] considered the existence of local and global Hopf bifurcations of Equation (). Up to now, few results on the properties of numerical solutions for Equation () were established. In the present paper, we investigate some sufficient conditions under which the numerical solutions are oscillatory. We also consider the asymptotic behavior of non-oscillatory numerical solutions. The remainder of this paper is organized as follows. In the next section, some necessary concepts and results for oscillations of the analytic solutions are given. In Section , we obtain a recurrence relation by applying the -methods to the simplified form which comes from making an invariant oscillation transformation on Equation (). Moreover, the oscillations of the numerical solutions are discussed and conditions under which the numerical solutions oscillate are obtained. In Section , we investigate the asymptotic behavior of non-oscillatory solutions, and the results of some numerical experiments are given in Section . Finally, conclusions are drawn in Section . 2 Preliminaries Definition . A solution x(t) of Equation () is said to oscillate about K if x(t) K has arbitrarily large zeros. Otherwise, x(t) is called non-oscillatory. When K = , we say that x(t) oscillates about zero or simply oscillates. Definition . A sequence {xn} is said to oscillate about {yn} if {xn yn} is neither eventually positive nor eventually negative. Otherwise, {xn} is called non-oscillatory. If {yn} = {y} is a constant sequence, we simply say that {xn} oscillates about {y}. When {yn} = {}, we say that {xn} oscillates about zero or simply oscillates. Definition . We say Equation () oscillates if all of its solutions are oscillatory. Theorem . (see []) Consider the difference equation qjan+j = , assume that k, l N and qj R for j = k, . . . , l. Then the following statements are equivalent: (i) Every solution of Equation () oscillates; (ii) The characteristic equation + jl=k qjj = has no positive roots. Theorem . (see []) Consider the difference equation where n = , , , . . . , R and k Z. Then every solution of Equation () oscillates if and only if one of the following conditions holds: (i) k = and ; (ii) k = and ; (iii) k {. . . , , } {, , . . .} and (k + )k+/kk > . Lemma . For x > and x = , we have ln( + x) > x/( + x). Lemma . For x < and x = , we have ex < /( x). Lemma . (see []) For all m M, (i) ( + a/(m a))m ea if and only if / for a > , () for a < ; (ii) ( + a/(m a))m < ea if and only if < / for a < , () for a > , where (x) = /x /(ex ) and M is a positive constant. Theorem . (see []) Assume that condition () holds, then every solution of Equation () oscillates about K if and only if K = then () can be written as is the unique positive equilibrium point of Equation (). Then x(t) oscillates about K if and only if z(t) oscillates. 3.2 The difference scheme Applying the linear -method and the one-leg -method to Equation (), we obtain the same recurrence relation qK qK zn+ = zn h (r + K ) f (zn+m) h( ) (r + K ) f (znm), where , h = /m, m is a positive integer. zn+ and zn+m are approximations to z(t) and z(t ) of Equation () at tn+, respectively. Let zn = ln(xn/K ), then we have 3 Oscillations of numerical solutions 3.1 Transformation We associate an initial condition of the form with Equation (), where C([ , ], (, )), () > . According to the corresponding method in [], let us introduce an invariant oscillation transformation x(t) = Kez(t), then Equation () can be reduced to Definition . We call the iteration formula () the exponential -method for Equation (), where [, ], h = /m, m N = {, , . . .}, xn+ and xn+m a (...truncated)