A nonlinear three-point boundary value problem

Abstract

We consider the differential equation −( py′)′ + qy + λay + μby + f( x, y, y′) = 0, x ϵ ( α, γ) subject to the boundary conditions cos( α 1) y( α) − sin( α 1) y′( α) = 0cos( β 1) y( β) − sin( β 1) y′( β) = 0 β ϵ ( α, γ)cos( γ 1) y( γ) − sin( γ 1) y′( γ) = 0. The functions p, g, a, b, and f are well-behaved functions of x; f is smooth and of “higher order” in y and y′; the scalars λ and μ are eigenparameters. With mild restrictions on a and b it is known that the linearized problem, f ≡ 0, has eigensolutions, ( λ ∗, μ ∗, ψ ∗ ). In this paper we use an Implicit Function Theorem argument to establish the existence of a local branch of solutions, bifurcating from ( λ ∗, μ ∗, 0 ), to the above nonlinear two-parameter eigenvalue problem.