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Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space

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Abstract

This paper is concerned with the contractivity and asymptotic stability properties of the implicit Euler method (IEM) for nonlinear functional differential equations (FDEs). These properties are first analyzed for Volterra FDEs and then the analysis is extended to the case of neutral FDEs (NFDEs). Such an extension is particularly important since NFDEs are more general and have received little attention in the literature. The main result we establish is that the IEM with linear interpolation can completely preserve these stability properties of the analytical solution to such FDEs.

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Author notes

  1. Wansheng Wang

    Present address: School of Mathematics and Computational Science, Changsha University of Science and Technology, 410114, Changsha, China

Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074, Wuhan, China

    Wansheng Wang & Chengjian Zhang

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  1. Wansheng Wang
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  2. Chengjian Zhang
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Correspondence to Wansheng Wang.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant No. 10871078) and the China Postdoctoral Science Foundation Funded Project (Grant No. 20080440946, 200902437).

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Wang, W., Zhang, C. Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space. Numer. Math. 115, 451–474 (2010). https://doi.org/10.1007/s00211-009-0281-z

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  • DOI: https://doi.org/10.1007/s00211-009-0281-z

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