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.

中文翻译：

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多点半正边值问题的正解的存在性

在本文中，我们使用圆锥上的不动点定理讨论了以下非线性多点半正边值问题的正解的存在，

$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-（Lu）（t）= \ lambda f（t，u）+ \ mu g（t，u），＆{} 0 \ le t \ le 1，\\ u'（0）= 0，u（1）= \ sum \ limits _ {i = 1} ^ k \ alpha _ {i} u（\ eta _ {i}）-\ sum \ limits _ {i = k + 1} ^ {m-2} \ alpha _ {i} u（\ eta _ {i}）。＆{} \\ \ end {array} \ right。\ end {aligned} $$

其中\（（Lu）（t）= {u}''（t）+ a（t）{u}'（t）\），非线性项f，g均为半正值。我们得出一个明确的间隔\（\ lambda \），\（\ mu \），使得对于该间隔中的任何\（\ lambda \），\（\ mu \），存在边值问题的正解在非线性项f，g均为超线性（亚线性）或一个为超线性，另一个为亚线性的条件下得到保证。