Positive solutions to a new kind Sturm-Liouville-like four-point boundary value problem

Abstract In this paper, we considered the following four-point boundary value problem x ″ ( t ) + h ( t ) f ( t , x ( t ) , x ′ ( t ) ) = 0 , 0 t 1 , x ′ ( 0 ) − α 1 x ( ξ ) = 0 , x ′ ( 1 ) + α 2 x ( η ) = 0 , where 0 α 1 1 ξ , 0 α 2 1 1 − η , 0 ξ η 1 , α 1 α 2 η − α 1 α 2 ξ + α 1 + α 2 > 0 . By applying fixed-point theorems, we obtain a variety of existence results. In particular, our four-point boundary condition is a new kind Sturm–Liouville-like boundary condition, which has rarely been considered up to now.

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