Existence of Positive Solutions for Multi-point Semi-positive Boundary Value Problems

Abstract

In this paper, we discuss the existence of positive solutions for the following nonlinear multi-point semi-positive boundary value problems using the fixed-point theorems in cones, $$\begin{aligned} \left\{ \begin{array}{ll} -(Lu)(t)=\lambda f(t, u)+\mu g(t, u),&{} 0\le t\le 1, \\ u'(0)=0, u(1)=\sum \limits _{i=1}^k \alpha _{i} u(\eta _{i}) -\sum \limits _{i=k+1}^{m-2} \alpha _{i} u(\eta _{i} ). &{} \\ \end{array}\right. \end{aligned}$$ where $$(Lu)(t)={u}''(t)+a(t){u}'(t)$$ , nonlinear term f, g are both semi-positive. We derive an explicit interval of $$\lambda $$ , $$\mu $$ such that for any $$\lambda $$ , $$\mu $$ in this interval, the existence of positive solutions to the boundary value problem is guaranteed under the condition that nonlinear term f, g are all super-linear(sub-linear), or one is super-linear, the other is sub-linear.