Triple solutions for nonlinear singular m-point boundary value problem

In this paper, we study the existence of three solutions to the following nonlinear m-point boundary value problem  u′′(t) + β2u(t) = h(t)f (t, u(t)), 0 t 1,   u′(0) = 0, u(1) = m−2 ∑ i=1 αiu(ηi), where 0 β π , f ∈ C([0, 1] × R+, R+). h(t) is allowed to be singular at t = 0 and t = 1. The arguments are based only upon the Leggett-Williams fixed point theorem. We also prove nonexist results.