Triple positive solutions for a class of boundary value problems for second-order neutral functional differential equations

Abstract

We obtain sufficient conditions for the existence of at least three positive solutions for the second-order neutral functional differential equation x ″ ( t ) + q ( t ) f ( t , x ( t ) , x ( t − τ ) , x ′ ( t ) , x ′ ( t − τ ) ) = 0 , 0 < t < 1 , τ > 0 , subject to one of the following two pairs of boundary conditions: { x ( t ) = ξ ( t ) , − τ ≤ t ≤ 0 , x ( 1 ) = 0 { x ( t ) = ξ ( t ) , − τ ≤ t ≤ 0 , x ′ ( 1 ) = 0 and generalize and correct some conditions of Theorem 3.2 in [X.B. Shu, Y.T. Xu, Triple positive solutions for a class of boundary value problems of second-order functional differential equations, Nonlinear Anal. 61 (8) (2005) 1401–1411]. This is an application of a new fixed-point theorem introduced by Avery and Peeterson [R.I. Avery, A.C. Peeterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42 (2001) 313–322].