Upper and lower solution method for fourth-order four-point boundary value problems

This paper is concerned with the fourth-order four-point boundary value problem {u(4)(t)+f(t,u,u″)=0,t∈[0,1]=I,u(0)=0,u(1)=au(η),u″(0)=0,u″(1)=bu(ξ), where η,ξ∈(0,1)η,ξ∈(0,1) and a,b≥0a,b≥0. The upper and lower solution method and a new maximum principle are employed to establish existence results and we release the increasing condition imposed on f(t,u,v)f(t,u,v).