The existence of symmetric positive solutions for Sturm–Liouville-like four-point boundary value problem with a p-Laplacian operator

Abstract

In this paper, the Sturm–Liouville-like four-point boundary value problem with a p-Laplacian operator ( ϕ p ( u ′ ) ) ′ ( t ) + λ a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) - α u ′ ( ξ ) = 0 , u ( 1 ) + α u ′ ( η ) = 0 is studied, where ϕ p ( s ) = | s | p - 2 s , p > 1 . By the use of fixed point index theory, Leray–Schauder degree and upper and lower solution method, some existence, nonexistence and multiplicity results of symmetric positive solutions are acquired.