Positive solutions for some second-order four-point boundary value problems

Abstract

In this paper, the second-order four-point boundary value problem x ″ ( t ) + λ h ( t ) f ( t , x ( t ) ) = 0 , 0 < t < 1 , x ( 0 ) = a x ( ξ ) , x ( 1 ) = b x ( η ) , is studied, where 0 < ξ < η < 1 , 0 ⩽ a , b < 1 , and h : [ 0 , 1 ] → [ 0 , ∞ ) , f : [ 0 , 1 ] × [ 0 , ∞ ) → [ 0 , ∞ ) are nonnegative continuous functions. By the use of the property of the corresponding Green's function, fixed-point index theory, Leray–Schauder degree and upper and lower solution method, some existence, nonexistence, and multiplicity results of positive solutions are acquired.