Existence of three positive solutions for m -point boundary-value problems with one-dimensional p -Laplacian

• Published in: Nonlinear analysis Vol. 68; no. 7; pp. 2017 - 2026
• Main Authors: Feng, Hanying, Ge, Weigao
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.04.2008; Elsevier
• Subjects: Avery–Peterson’s fixed point theorem; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Multipoint boundary value problem; Numerical analysis; Numerical analysis. Scientific computation; One-dimensional [formula omitted]-Laplacian; Partial differential equations, boundary value problems; Positive solution; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Avery–Peterson’s fixed point theorem; One-dimensional p -Laplacian; Multipoint boundary value problem; Positive solution; P laplacian; Boundary value problem; Fixed point theorem; Existence of solution; Avery-Peterson's fixed point theorem; Nonlinear analysis; One-dimensional p-Laplacian; Mathematical model
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2007.01.029

In this paper, we consider the multipoint boundary value problem for the one-dimensional p -Laplacian ( ϕ p ( u ′ ) ) ′ + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , subject to the boundary conditions: u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 a i u ( ξ i ) , where ϕ p ( s ) = | s | p − 2 s , p > 1 , ξ i ∈ ( 0 , 1 ) with 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 and a i ∈ [ 0 , 1 ) , 0 ≤ ∑ i = 1 m − 2 a i < 1 . Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem. The interesting point is that the nonlinear term f explicitly involves a first-order derivative.