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Stability of analytic and numerical solutions for differential equations with piecewise continuous arguments. (English) Zbl 1244.34097
Summary: The asymptotic stability of the analytic and numerical solutions for differential equations with piecewise continuous arguments is investigated by using Lyapunov’s method. In particular, linear equations with variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the $\theta$-methods are obtained. Some examples are illustrated.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations 34K06 Linear functional-differential equations
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##### References:
 [1] J. Wiener, “Differential equations with piecewise constant delays,” in Trends in the Theory and Practice of Nonlinear Differential Equations, V. Lakshmikantham, Ed., pp. 547-552, Marcel Dekker, New York, NY, USA, 1983. · Zbl 0531.34059 [2] J. Wiener, “Pointwise initial value problems for functional-differential equations,” in Differential Equations, I. W. Knowles and R. T. Lewis, Eds., pp. 571-580, North-Holland, New York, NY, USA, 1984. · Zbl 0552.34061 [3] K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays,” Journal of Mathematical Analysis and Applications, vol. 99, no. 1, pp. 265-297, 1984. · Zbl 0557.34059 · doi:10.1016/0022-247X(84)90248-8 [4] S. M. Shah and J. Wiener, “Advanced differential equations with piecewise constant argument deviations,” International Journal of Mathematics and Mathematical Sciences, vol. 6, no. 4, pp. 671-703, 1983. · Zbl 0534.34067 · doi:10.1155/S0161171283000599 · eudml:45283 [5] J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scientific, Singapore, 1993. · Zbl 0874.34054 [6] M. Z. Liu, M. H. Song, and Z. W. Yang, “Stability of Runge-Kutta methods in the numerical solution of equation u$^{\prime}$(t)=au(t)+a0u([t]),” Journal of Computational and Applied Mathematics, vol. 166, no. 2, pp. 361-370, 2004. · Zbl 1052.65070 · doi:10.1016/j.cam.2003.04.002 [7] M. H. Song, Z. W. Yang, and M. Z. Liu, “Stability of \theta -methods for advanced differential equations with piecewise continuous arguments,” Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1295-1301, 2005. · Zbl 1082.65078 · doi:10.1016/j.camwa.2005.02.002 [8] Z. Yang, M. Liu, and M. Song, “Stability of Runge-Kutta methods in the numerical solution of equation u$^{\prime}$(t) = au(t) + a0u([t]) + a1u([t - 1]),” Applied Mathematics and Computation, vol. 162, no. 1, pp. 37-50, 2005. · Zbl 1063.65070 · doi:10.1016/j.amc.2003.12.081 [9] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer, New York, NY, USA, 1993. · Zbl 0787.34002 [10] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002 [11] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, The Mathematical Society of Japan, Tokyo, Japan, 1966. · Zbl 0144.10802