Multiple positive solutions for some multi-point boundary value problems with -Laplacian

This paper deals with the existence of multiple positive solutions for the quasilinear second-order differential equation ( φ p ( u ′ ( t ) ) ) ′ + a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , subject to one of the following boundary conditions: φ p ( u ′ ( 0 ) ) = ∑ i = 1 m - 2 a i φ p ( u ′ ( ξ i ) ) , u ( 1 ) = ∑ i = 1 m - 2 b i u ( ξ i ) , or u ( 0 ) = ∑ i = 1 m - 2 a i u ( ξ i ) , φ p ( u ′ ( 1 ) ) = ∑ i = 1 m - 2 b i φ p ( u ′ ( ξ i ) ) , where φ p ( s ) = | s | p - 2 s , p > 1 , 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 , and a i , b i satisfy a i , b i ∈ [ 0 , ∞ ) , ( i = 1 , 2 , … , m - 2 ) , 0 < ∑ i = 1 m - 2 a i < 1 , 0 < ∑ i = 1 m - 2 b i < 1 . Using the five functionals fixed point theorem, we provide sufficient conditions for the existence of multiple (at least three) positive solutions for the above boundary value problems.