Existence results for some fourth-order multi-point boundary value problem

In this paper, we consider the following fourth-order multi-point boundary value problem { u ( i v ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) , u ‴ ( t ) ) , 0 < t < 1 , u ( 0 ) = 0 , u ′ ( 1 ) = ∑ i = 1 m − 2 a i u ′ ( ξ i ) , u ″ ( 0 ) = 0 , u ‴ ( 1 ) = ∑ i = 1 m − 2 b i u ‴ ( ξ i ) , ξ i ∈ ( 0 , 1 ) , where nonlinear f is depending on all lower-order derivatives of u . By using the method of upper and lower solutions and Schauder’s fixed point theorem, existence result is obtained.