Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in banach spaces

Bellen, A., Zennaro, M. Strong contractivity properties of numerical methods for ordinary and delay differential equations. Appl. Numer. Math., 9: 321–346 (1992)

Bellen, A., Jackiewicz, Z., Zennaro, M. Contractivity of vaveform relaxation Runge-Kutta iterations and related limit methods for dissipative systems in the maximum norm. SIAM J. Numer. Anal., 31: 499–523 (1994)

Bellen, A., Zennaro, M. Numerical methods for delay differential equations. Oxford University Press, Oxford, 2003

Bellen, A., Guglielmi, N., Zennaro, M. On the contractivity and asymptotic stability of systems of delay differential equations of neutral type. BIT, 39: 1–24 (1999)

Bellen, A., Guglielmi, N., Zennaro, M. Numerical stability of nonlinear delay differential equations of neutral type. J. Comput. Appl. Math., 125: 251–263 (2000)

Brunner, H., Vermiglio, R. Stability of solutions of delay functional integro-differential equations and their discretizations. Computing, 71: 229–245 (2003)

Gil’, M.I. Stability of finite and infinite dimensional systems. Kluwer Academic Publishers, Boston, 1998

Hale, J. Theory of functional differential equations. Springer-Verlag, New York, 1977

Huang, C.M., Fu, H.Y., Li, S.F., Chen, G.N. Stability analysis of Runge-Kutta methods for non-linear delay differential equations. BIT, 39: 270–280 (1999)

Huang, C.M., Li, S.F., Fu, H.Y., Chen, G.N. Nonlinear stability of general linear methods for delay differential equations. BIT, 42: 380–392 (2002)

Kolmanovskii, V.B., Myshkis, A. Introduction to the theory and applications of functional differential equations. Kluwer Academy, Dordrecht, 1999

Kraaijevanger, J.F.B.M. Contractivity of Runge-Kutta methods. BIT, 31: 482–528 (1991)

Kuang, J.X., Cong, Y.H. Stability of numerical methods for delay differential equations. Science Press, Beijing, 2005

Lancaster, P., Tismenetsky, M. The theory of matrices. Academic Press, Orlando, 1985

Li, S.F. Nonlinear stability of a class of linear multistep methods. Numer. Math. A J. of Chinese Univ., 9: 110–118 (1987)

Li, S.F. Theory of computational methods for stiff differential equations. Hunan Science and Technology Publisher, Changsha, 1997

Li, S.F. Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces. Sci. China (Series A), 35: 286–301 (2005)

Li, S.F. B-theory of Runge-Kutta methods for stiff Volterra functional differential equations. Science in China (Series A), 46: 662–674 (2003)

Li, S.F. B-theory of general linear methods for stiff Volterra functional differential equations. Appl. Numer. Math., 53: 57–72 (2005)

Li, S.F. High order contractive Runge-Kutta methods for Volterra functional differential equations. SIAM J. Numer. Anal., 47: 4290–4325 (2010)

Li, S.L., Wen, L.Z. Functional differential equations. Hunan Science and Technology Publisher, Changsha, 1985

Sand, J., Skelboe, S. Stability of backward Euler multirate methods and convergence of waveform relaxation. BIT, 32: 350–366 (1992)

Torelli, L. Stability of numerical methods for delay differential equations. J. Comput. Appl. Math., 25: 15–26 (1989)

Wang, W.S., Li, S.F. Stability analysis of nonlinear delay differential equations of neutral type. Math. Num. Sinica, 26: 303–314 (2004)

Wang, W.S., Li, S.F. On the One-leg Θ-methods for solving nonlinear neutral functional differential equations. Appl. Math. Comput., 193: 285–301 (2007)

Wang, W.S., Zhang Y., Li, S.F. Nonlinear stability of one-leg methods for delay differential equations of neutal type. Appl. Numer. Math., 58: 122–130 (2008)

Wang, W.S., Li, S.F., Su, K. Nonlinear stability of Runge-Kutta methods for neutral delay differential equations. J. Comput. Appl. Math., 214: 175–185 (2008)

Wang, W.S., Li, S.F., Su, K. Nonlinear stability of general linear methods for neutral delay differential equations. J. Comput. Appl. Math., 224: 592–601 (2009)

Wang, W.S., Wen, L.P., Li, S.F. Nonlinear stability of Θ-methods for neutral differential equations in Banach space. Appl. Math. Comput., 198: 742–753 (2008)

Wen, L.P., Yu, Y.X., Li, S.F. The test problem class of Volterra functional differential equations in Banach space. Appl. Math. Comput., 179: 30–38 (2006)

Zennaro, M. Contractivity of Runge-Kutta methods with respect to forcing terms. Appl. Numer. Math., 10: 321–345 (1993)

Zennaro, M. Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations. Numer. Math., 77: 549–563 (1997)

Zhang, C.J., Li, S.F. Dissipativity and exponential asymptotic stability of the solutions for nonlinear neutral functional differential equations. Appl. Math. Comput., 119: 109–115 (2001)

Zhang, C.J. Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations. J. Comput. Math., 20: 583–590 (2002)

Zhang, C.J., Vandewalle, S. Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization. J. Comput. Appl. Math., 164–165: 797–814 (2004)

Zhang, C.J., Vandewalle, S. Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J. Numer. Anal., 24: 193–214 (2004)

Zhang, C.J., Vandewalle, S. General linear methods for Volterra integro-differential equations with memory. SIAM J. Sci. Comput., 27: 2010–2031 (2006)

Zubik-kowal, B., Vandewalle, S. Waveform relaxation for functional-differential equations. SIAM J. Sci. Comput., 21: 207–226 (1999)