, Volume 55, Issue 3, pp 639-652

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

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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.

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).