### Convergence and contractivity of boundary value methods for nonlinear delay differential equations

ZHANG Ru1, HAN Xu2, LIU Xiao-gang3

1. 1. School of Information Engineering, Northwestern Polytechnical University Mingde College, Xian 710124, Shaanxi, China;
2. Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China;
3. Department of Mathematics, Modern College of Northwest University, Xian 710130, Shaanxi, China
• Online:2019-08-20 Published:2019-07-03

Abstract: The boundary value methods are applied to the nonlinear delay differential equations. Under the assumptions of Lipschitz conditions, the convergence and the global contractivity, the weakly global contractivity of the boundary value methods are analyzed. Finally, some numerical experiments are carried out to illustrate the theoretical results.

CLC Number:

• O241.8
 [1] HUANG Chengming. Asymptotic stability of multistep methods for nonlinear delay differential equations[J]. Applied Mathematics and Computation, 2008, 203(2):908-912.[2] LI Dongfang, ZHANG Chengjian. Nonlinear stability of discontinuous Galerkin methods for delay differential equations[J]. Applied Mathematics Letters, 2010, 23(4):457-461.[3] TORELLI L. Stability of numerical methods for delay differential equations[J]. Journal of Computational and Applied Mathematics, 1989, 25(1):15-26.[4] IAVERNARO F, MAZZIA F. Convergence and stability of multistep methods solving nonlinear initial value problems[J]. SIAM Journal on Scientific Computing, 1997, 18(1):270-285.[5] ORTEGA J M, RHEINBOLDT W C. Iterative solution of nonlinear equations in several variables[M]. New York: Academic Press, 1970.[6] BRUGNANO Luigi, TRIGIANTE Donato. Solving differential problems by multistep initial and boundary value methods[M]. Amsterdam: Gordonand Breach Science Publishers, 1998.
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