Positive solutions of m-point boundary value problems

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The second-order m-point boundary value problem ϕ″(x)+h(x)f(ϕ(x))=0, 0<x<1, ϕ(0)=0, ϕ(1)= ∑ i=1 m−2 a iϕ(ξ i), is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where ξ i ∈(0,1) with 0< ξ 1< ξ 2<⋯< ξ m−2 <1, a i ∈[0,∞) with ∑ i=1 m−2 a i <1. h( x) is allowed to be singular at x=0 and x=1. The existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory. Similarly conclusions hold for some other m-point boundary value conditions.