Positive solutions for a four-point boundary value problem with the [formula omitted]-Laplacian

Abstract

In this paper, we study the nonlinear four-point boundary value problem with the p -Laplacian { ( φ p ( u ′ ) ) ′ + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) − α u ′ ( ξ ) = 0 , u ( 1 ) + β u ′ ( η ) = 0 , where φ p ( s ) = | s | p − 2 s , p > 1 , α , β > 0 , 0 < ξ < η < 1 . By applying a fixed point theorem in cones, sufficient conditions are given for the existence of a positive solution and multiple positive solutions. The interesting point is the Sturm–Liouville-like boundary condition, which was rarely treated until now.