Multiple positive solutions for nonlinear singular m-point boundary value problem

Abstract

In this paper, we study the existence of multiple positive solutions for the following nonlinear m-point boundary value problem (BVP) ( φ ( u ′ ) ) ′ + a ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , u ′ ( 0 ) = ∑ i = 1 m - 2 a i u ′ ( ξ i ) , u ( 1 ) = ∑ i = 1 k b i u ( ξ i ) - ∑ i = k + 1 s b i u ( ξ i ) - ∑ i = s + 1 m - 2 b i u ′ ( ξ i ) , where φ : R → R is an increasing homeomorphism and homomorphism and φ ( 0 ) = 0 , 1 ⩽ k ⩽ s ⩽ m - 2 , a i , b i ∈ ( 0 , + ∞ ) with 0 < ∑ i = 1 k b i - ∑ i = k + 1 s b i < 1 , 0 < ∑ i = 1 m - 2 a i < 1 , 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 , a ( t ) ∈ C ( ( 0 , 1 ) , [ 0 , + ∞ ) ) , f ∈ C ( [ 0 , + ∞ ) , [ 0 , + ∞ ) ) . Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery–Henderson and a new fixed-point theorems, respectively. As an application, some examples are included to illustrate the main results. In particular, our results extend and improve some known results.