# Oscillation of θ-methods for the Lasota-Wazewska model

### Abstract

The aim of this paper is to discuss the oscillation of numerical solutions for the Lasota-Wazewska model. Using two θ-methods (the linear θ-method and the one-leg θ-method), some conditions under which the numerical solutions oscillate are obtained for different range of θ. Furthermore, it is shown that every non-oscillatory numerical solution tends to the fixed point of the original continuous equation. Numerical examples are given.

### References

[2] B. Karpuz, “Sufficient conditions for the oscillation and asymptotic beaviour of higher-order dynamic equations of neutral type,” Appl. Math. Comput. 221 (2013) 453-462.

[3] Q.X. Ma, A.P. Liu, “Oscillation criteria of neutral type impulsive hyperbolic equations,” Acta. Math. Sci. 34 (2014) 1845-1853.

[4] B. Abdalla, T. Abdeljawad, “On the oscillation of Caputo fractional differential equations with Mittag-Leffler nonsingular kernel,” Chaos Soliton Fract. 127 (2019) 173-177.

[5] D. Efimov, W. Perruquetti, A. Shiriaev, “On existence of oscillations in hybrid systems,” Nonlinear Analysis: HS 12 (2014) 104-116.

[6] Y. Muroya, “New contractivity condition in a population model with piecewise constant arguments,” J. Math. Anal. Appl. 346 (2008) 65-81.

[7] E.M. Bonotto, L.P. Gimenes, M. Federson, “Oscillation for a second-order neutral differential equation with impulses,” Appl. Math. Comput. 215 (2009) 1-15.

[8] C.H. Zhang, R.P. Agarwal, T.X. Li, “Oscillation and asymptotic behavior of higher-order delay differential equations with p-Laplacian like operators,” J. Math. Anal. Appl. 409 (2014) 1093-1106.

[9] M.Z. Liu, J.F. Gao, Z.W. Yang, “Oscillation analysis of numerical solution in the -methods for equation ,” Appl. Math. Comput. 186 (2007) 566-578.

[10] M.Z. Liu, J.F. Gao, Z.W. Yang, “Preservation of oscillations of the Runge-Kutta method for equation ,” Comput. Math. Appl. 58 (2009) 1113-1125.

[11] Q. Wang, Q.Y. Zhu, M.Z. Liu, “Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type,” J. Comput. Appl. Math. 235 (2011) 1542-1552.

[12] J.F. Gao, M.H. Song, M.Z. Liu, “Oscillation analysis of numerical solutions for nonlinear delay differential equations of population dynamics,” Math. Model. Anal. 16 (2011) 365-375.

[13] M. Wazewska-Czyzewska, A. Lasota, “Mathematical problems of the dynamics of the red blood cells system,” Annals of the Polish Mathematical Society, Series III, Appl. Math. 17 (1988) 23-40.

[14] I. Gyori, G. Ladas, “Oscillation theory of delay differential equations with applications,” Oxford: Academic Press, 1993.

[15] G.R. Liu, A.M. Zhao, J.R. Yan, “Existence and global attractivity of unique positive periodic solution for a Lasota-Wazewska model,” Nonlinear Analysis: TMA 64 (2006) 1737-1746.

[16] J. Mallet-Paret, R.D. Nussbaum, “Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,” Ann. Mat. Pura Appl. 145 (1986) 33-128.

[17] H. Zhou, Z.F. Zhou, Q. Wang, “Positive almost periodic solution for a class of Lasota-Wazewska model with infinite delays,” Appl. Math. Comput. 218 (2011) 4501-4506.

[18] L. Wang, M. Yu, P.C. Niu, “Periodic solution and almost periodic solution of impulsive Lasota-Wazewska model with multiple time-varying delays,” Comput. Math. Appl. 64 (2012) 2383-2394.

[19] W.T. Li, S.S. Cheng, “Asymptotic properties of the positive equilibrium of a discrete survival model,” Appl. Math. Comput. 157 (2004) 29-38.

[20] M.H. Song, Z.W. Yang, M.Z. Liu, “Stability of -methods for advanced differential equations with piecewise continuous arguments,” Comput. Math. Appl. 49 (2005) 1295-1301.

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