Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions

• Main Author: Santos, Ana I.
• Publication Date: 2006
• Other Authors: Cabada, Alberto, Minhós, Feliz M.
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10174/2715
• topic: third order functional problems; Nagumo-type condition; Lower and upper solutions; degree theory

We prove an existence and location result for the third order functional nonlinear boundary value problem u′′′(t) = f(t,u,u′(t),u′′(t)), for t∈[a,b], 0 = L₀(u,u′,u(t₀)), 0 = L₁(u,u′,u′(a),u′′(a)), 0 = L₂(u,u′,u′(b),u′′(b)), with t₀∈[a,b] given, f:I×C(I)×R²→R is a L¹- Carathéodory function allowing some discontinuities on t and L₀,L₁, L₂ are continuous functions depending functionally on u and u′. The arguments make use of an a priori estimate on u′′, lower and upper solutions method and degree theory. Applications to a multipoint problem and to a beam equation will be presented.