Positive solutions of fourth-order multi-point boundary value problems with bending term

In this paper, we study m-point boundary value problem for fourth-order ordinary differential equation with bending term x″x(4)(t)=g(t)f(t,x(t),x″(t)),t∈(0,1),x(0)=0,x(1)=∑i=1m-2aix(ξi),x″(0)=0,x″(1)=∑i=1m-2bix″(ξi),where 0=ξ0<ξ1<ξ2<⋯<ξm-2<ξm-1=10=ξ0<ξ1<ξ2<⋯<ξm-2<ξm-1=1. By constructing an available integral operator and combining fixed point index theorem, we establish sufficient conditions for the existence of positive solutions under some conditions concerning the first eigenvalue corresponding to the relevant linear operator. The interesting point of the results is that the nonlinear term g may be singular at t = 0 and (or) t = 1, moreover f(t,u,v)f(t,u,v) is also allowed to have singularity at u = 0 and (or) v = 0.