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

This paper is concerned with the fourth-order ordinary differential equation u ( 4 ) ( t ) = f ( t , u ( t ) , u ″ ( t ) ) , 0 < t < 1 with the four-point boundary conditions u ( 0 ) = u ( 1 ) = 0 , au ″ ( ξ 1 ) - bu ‴ ( ξ 1 ) = 0 , cu ″ ( ξ 2 ) + du ‴ ( ξ 2 ) = 0 , where ξ i ∈ [ 0 , 1 ] ( i = 1 , 2 ) and a , b , c , d are nonnegative constants satisfying ad + bc + ac ( ξ 2 - ξ 1 ) > 0 . Some new existence results are obtained by developing the upper and lower solution method and the monotone iterative technique.