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The Existence and Asymptotic Estimates of Solutions for a Third-Order Nonlinear Singularly Perturbed Boundary Value Problem

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Abstract

In this paper, we consider a class of third-order nonlinear differential equation with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a boundary layer at one endpoint. By using the Schauder fixed point theorem, Green’s function and the method of upper–lower solutions, we first establish an existence result of corresponding boundary value problem without perturbation. Furthermore, by constructing an appropriate lower solution-upper solution pair, as well as analysis technique, the existence and asymptotic estimates of the solutions for the singularly perturbed boundary value problems are obtained.

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Acknowledgements

The authors express their sincere thanks to the anonymous referees for their valuable comments and corrections of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Xiaojie Lin, Jiang Liu & Can Wang

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  1. Xiaojie Lin
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  2. Jiang Liu
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  3. Can Wang
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Correspondence to Xiaojie Lin.

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This work is supported by the Natural Science Foundation of China (Grant Nos. 11771185 and 11871251).

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Lin, X., Liu, J. & Wang, C. The Existence and Asymptotic Estimates of Solutions for a Third-Order Nonlinear Singularly Perturbed Boundary Value Problem. Qual. Theory Dyn. Syst. 18, 687–710 (2019). https://doi.org/10.1007/s12346-018-0307-y

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