Electron. J. Differential Equations, Vol. 2016 (2016), No. 231, pp. 1-9.

Positive solutions of multi-point boundary value problems

Youyuan Yang, Qiru Wang

Abstract:
This article concerns the boundary value problem consisting of the nonlinear differential equation
$$ u''+g(t)f(t,u(t))=0,\quad t\in(0, 1) $$
and the multi-point boundary conditions
$$\displaylines{ u(0)=\alpha u'(0), \cr u(1)=\sum_{i=1}^m\beta_iu(\eta_i)+\sum_{i=1}^m\gamma_iu'(\eta_i), }$$
where $0\leq \alpha \leq\infty$, $0<\eta_1<\eta_1<\eta_{2}<\dots<\eta_m<1$, $\beta_i>0$, $\gamma_i<0$ ($i=1,2,\dots, m$). By using the fixed point index theory, we establish the existences of at least one positive solution and at least two positive solutions.

Submitted December 30, 2015. Published August 24, 2016.
Math Subject Classifications: 34B10, 34B18.
Key Words: Boundary value problems; multi-point boundary conditions; second-order nonlinear differential equations; positive solutions; fixed point index.

An addendum was posted on February 7, 2017. It points out some mistakes in this article; see the last page.

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Youyuan Yang
School of Mathematics
Sun Yat-Sen University
Guangzhou 510275, China
email: yangyouyuan2016@163.com
Qiru Wang
School of Mathematics
Sun Yat-Sen University
Guangzhou 510275, China
email: mcswqr@mail.sysu.edu.cn

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