Existence and multiplicity of solutions for some three-point nonlinear boundary value problems.

Xian, Xu; O'Regan, Donal

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In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation $x^{(n)} (t) = f (t, x (t), x^{'} (t), \dots, x^{(n - 1)} (t)) + e (t), 0 < t < 1, (*)$ and the following multi-point boundary value conditions $\begin{matrix} 1 * - 1 x^{(i)} (0) & = 0 f o r i = 0, 1, \dots, n - 3, \\ x^{(n - 1)} (0) & = α x^{(n - 1)} (ξ), x^{(n - 2)} (1) = \sum_{i = 1}^{m} β_{i} x^{(n - 2)} (η_{i}) . * * \end{matrix}$ Sufficient conditions for the existence of at least one solution of the BVP $(*)$ and $(* *)$ at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002),...

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