Some multipoint boundary value problems of Neumann–Dirichlet type involving a multipoint p-Laplace like operator

• Published in: Journal of mathematical analysis and applications 2007-09, Vol.333 (1), p.247-264
• Main Authors: García-Huidobro, M., Gupta, Chaitan P., Manásevich, R.
• Format: Article
• Language: English
• Subjects: Continuation lemmas; Exact sciences and technology; Global analysis, analysis on manifolds; Leray–Schauder; Manifolds and cell complexes; Mathematical analysis; Mathematics; Multipoint; Nonlinear BVP; Numerical analysis; Numerical analysis. Scientific computation; p-Laplace like; Partial differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2006.09.054

Let ϕ and θ be two increasing homeomorphisms from R onto R with ϕ ( 0 ) = 0 , θ ( 0 ) = 0 . Let f : [ 0 , 1 ] × R × R ↦ R be a function satisfying Carathéodory's conditions, and for each i, i = 1 , 2 , … , m − 2 , let a i : R ↦ R , be a continuous function, with ∑ i = 1 m − 2 a i ( 0 ) = 1 , ξ i ∈ ( 0 , 1 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 . In this paper we first prove a suitable continuation lemma of Leray–Schauder type which we use to obtain several existence results for the m-point boundary value problem: ( ϕ ( u ′ ) ) ′ = f ( t , u , u ′ ) , t ∈ ( 0 , 1 ) , u ′ ( 0 ) = 0 , θ ( u ( 1 ) ) = ∑ i = 1 m − 2 θ ( u ( ξ i ) ) a i ( u ′ ( ξ i ) ) . We note that this problem is at resonance, in the sense that the associated m-point boundary value problem ( ϕ ( u ′ ( t ) ) ) ′ = 0 , t ∈ ( 0 , 1 ) , u ′ ( 0 ) = 0 , θ ( u ( 1 ) ) = ∑ i = 1 m − 2 θ ( u ( ξ i ) ) a i ( u ′ ( ξ i ) ) has the non-trivial solution u ( t ) = ρ , where ρ ∈ R is an arbitrary constant vector, in view of the assumption ∑ i = 1 m − 2 a i ( 0 ) = 1 .