Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$begin{aligned} u^{prime prime }(t)+ a(t) f(u(t) )=0,quad 0

subject to the boundary conditions

$$begin{aligned} u(0)=sum _{i=1}^{m-2} a_i u (xi _i) ,quad u^{prime } (1) = sum _{i=1}^{m-2} b_i u^{prime } (xi _i), end{aligned}$$

where ( xi _i in (0,1) ) with ( 0< xi _1 and (a_i,b_i ) satisfy   (a_i,b_iin [0,infty ),~~ 0< sum _{i=1}^{m-2} a_i <1,) and ( sum _{i=1}^{m-2} b_i <1. ) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$begin{aligned} displaystyle inf _{0 le t le 1} u(t) ge gamma Vert uVert _infty . end{aligned}$$