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 ) = a u ( η ) , u ″ ( 0 ) = 0 , u ″ ( 1 ) = b u ( ξ ) , where η , ξ ∈ ( 0 , 1 ) and a , 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 ) .