Positive solutions of multi-point boundary value problems

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 $0\leq \alpha \leq\infty$, $00$, $\gamma_i<0$ ($i=1,2,\dots, m$). By using the fixed point index theory, we establish the existences of at least one positive solution and at least two positive solutions.

Keywords

• Boundary value problems
• multi-point boundary conditions
• second-order nonlinear differential equations
• positive solutions
• fixed point index