Zhang, Chengjian; Sun, Gen The discrete dynamics of nonlinear infinite-delay-differential equations. (English) Zbl 1001.65091 Appl. Math. Lett. 15, No. 5, 521-526 (2002). Nonlinear stability criteria are established for one-leg \(\theta\)-methods for nonlinear infinite-delay-differential equations of the form \(y'(t)=f(t,y(t),y(pt))\), \(t>0\), \(0<p<1\), \(y(0)=\eta\). Two theorems are proved: one concerning a global stability inequality, and a second one concerning the asymptotic stability of one-leg methods. Reviewer: Dana Petcu (Timişoara) Cited in 1 ReviewCited in 21 Documents MSC: 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) Keywords:delay differential equations; one-leg methods; nonlinear stability; global stability; asymptotic stability PDFBibTeX XMLCite \textit{C. Zhang} and \textit{G. Sun}, Appl. Math. Lett. 15, No. 5, 521--526 (2002; Zbl 1001.65091) Full Text: DOI References: [1] Feldstein, M. A., Discretization methods for retarded ordinary differential equations, (Ph.D. Dissertation (1964), University of California), 8-10 [2] Feldstein, M. A.; Grafton, C. K., Experimental mathematics: An application to retarded ordinary differential equations with infinite lag, (Proc. 1968 ACM National Conference (1968), Brandon Systems Press), 67-71 [3] Buhmann, M. D.; Iserles, A.; Nørsett, S. P., Runge-Kutta methods for neutral differential equations, WSSIAA, 2, 85-98 (1993) · Zbl 0834.65061 [4] Buhmann, M. D.; Iserles, A., On the dynamics of a discretized neutral equation, IMA J. Numer. Aual., 12, 339-363 (1992) · Zbl 0759.65056 [5] Buhmann, M. D.; Iserles, A., Stability of the discretized pantograph differential equation, Math. Comput., 60, 575-589 (1993) · Zbl 0774.34057 [6] Buhmann, M. D.; Iserles, A., Numerical analysis of delay differential equations with variable delays, Ann. Numer. Math., 1, 133-152 (1994) · Zbl 0828.65083 [7] Liu, Y., Stability analysis of θ-methods for neutral functional-differential equations, Numer. Math., 70, 473-485 (1995) · Zbl 0824.65081 [8] Liu, Y., On the θ-methods for delay differential equations with infinite lag, J. Comput. Appl. Math., 71, 177-190 (1996) · Zbl 0853.65076 [9] Bellen, A.; Guglielmi, N.; Torelli, L., Asymptotic stability properties of θ-methods for the pantograph equation, Appl. Numer. Math., 24, 279-293 (1997) · Zbl 0878.65064 [10] Torelli, L., Stability of numerical methods for delay differential equations, J. Comput. Appl. Math., 25, 15-26 (1989) · Zbl 0664.65073 [11] Bellen, A.; Zennaro, M., Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math., 9, 321-346 (1992) · Zbl 0749.65042 [12] Zennaro, M., Contractivity of Runge-Kutta methods with respect to forcing terms, Appl. Numer. Math., 10, 321-345 (1993) · Zbl 0774.65054 [13] Zhang, C. J.; Zhou, S. Z., Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations, J. Comput. Appl. Math., 85, 225-237 (1997) · Zbl 0904.65082 [14] Huang, C. M.; Fu, H. Y.; Li, S. F.; Chen, G. N., Stability analysis of Runge-Kutta methods for nonlinear delay differential equations, BIT, 39, 270-280 (1999) · Zbl 0930.65090 [15] Huang, C. M.; Li, S. F.; Fu, H. Y.; Chen, G. N., Stability and error analysis of one-leg methods for nonlinear delay differential equations, J. Comput. Appl. Math., 103, 263-279 (1999) · Zbl 0948.65078 [16] Zennaro, M., Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations, Numer. Math., 77, 549-563 (1997) · Zbl 0886.65092 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.