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The Method of Upper and Lower Solutions for a Lidstone Boundary Value Problem

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Abstract

In this paper we develop the monotone method in the presence of upper and lower solutions for the 2nd order Lidstone boundary value problem

$$\begin{array}{*{20}c} {u^{(2n)} (t) = f(t,\;u(t),\;u''(t),...,u^{(2(n - 1))} (t)),\quad 0 < t < 1,} \\ {u^{(2i)} (0) = u^{(2i)} (1) = 0,\quad 0 \leqslant i \leqslant n - 1,} \\ \end{array}$$

where f : [0, 1] × ℝn → ℝ is continuous. We obtain sufficient conditions on f to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.

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Authors and Affiliations

  1. College of Science, Hebei University of Science and Technology, Shijiazhuang, Hebei, 050018, P.R. China

    Yanping Guo

  2. College of Physical and Environmental Oceanography, Ocean University of China, Qingdao, 266003, P. R. China

    Yanping Guo

  3. Department of Mathematics, Yanbei Normal Institute, Datong, Shanxi, 037000, P. R. China

    Ying Gao

Authors
  1. Yanping Guo
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  2. Ying Gao
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Additional information

The project is supported by the Natural Science Foundation of China (10371030), by the Science and Technology Research development foundation for Universities of Shanxi Province (20051254), and by the Doctoral Program Foundation of Hebei Province (B2004204).

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Guo, Y., Gao, Y. The Method of Upper and Lower Solutions for a Lidstone Boundary Value Problem. Czech Math J 55, 639–652 (2005). https://doi.org/10.1007/s10587-005-0051-8

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  • DOI: https://doi.org/10.1007/s10587-005-0051-8

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