Positive solutions of a nonlinear four-point boundary value problems in Banach spaces

Abstract In this paper, by using the fixed points of strict-set-contractions, we study the existence of at least one or two positive solutions to the four-point boundary value problem y ″ ( t ) + a ( t ) f ( y ( t ) ) = θ , 0 t 1 , y ( 0 ) = μ y ( ξ ) , y ( 1 ) = β y ( η ) , in Banach space E, where θ is zero element of E, 0 ξ ⩽ η 1 , 0 μ 1 1 − ξ , 0 β 1 η and μ ξ ( 1 − β ) + ( 1 − μ ) ( 1 − β η ) > 0 . As an application, we also give one example to demonstrate our results.