Existence of solutions of three-point boundary value problems in Banach spaces

In this paper, we study the second-order three-point boundary value problem u ″ ( t ) + f ( t , u ( t ) , u ′ ( t ) ) = θ , 0 < t < 1 , u ′ ( 0 ) = θ , u ( 1 ) = α u ( η ) in a Banach space E , where θ is the zero element of E , 0 < α < 1 , 0 < η < 1 . By using the Sadovskii fixed point theorem, we get the existence of at least one (positive) solution. As an application, we give an example to demonstrate our results.