Some remarks on three-point and four-point BVP's for second-order nonlinear differential equations

Man Kam Kwong, Hong Kong Polytechnic University, Hong Kong, SAR, China
James S. W. Wong, University of Hong Kong, Hong Kong, SAR, China

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 20., pp. 1-18.

Abstract: We are interested in the existence of a positive solution to the four-point boundary value problem
$$\begin{cases} y''(t)+a(t)f(y(t)) = 0, \quad 0<t<1, y(0)=\alpha y(\xi), y(1)=\beta y(\eta), \end{cases} \eqno (*) $$ where $ 0<\xi \leq \eta <1 $, $ 0<\alpha <1/(1-\xi) $, $ 0<\beta <1/\eta $ and $ \alpha \beta (1-\beta)+(1-\alpha)(1-\beta \eta)>0 $. A result of B. Liu is improved with an alternative, simplified proof. The same method is used to obtain extensions of earlier results by Ma, Liu, Liu and Yu, and others, on three-point boundary value problems, i.e, with $\alpha=0$ in (*).

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