A nonlinear three-point boundary value problem

AbstractWe consider the differential equation −(py′)′ + qy + λay + μby + f(x, y, y′) = 0, x ϵ (α, γ) subject to the boundary conditions cos(α1) y(α) − sin(α1) y′(α) = 0cos(β1) y(β) − sin(β1) y′(β) = 0 β ϵ (α, γ)cos(γ1) y(γ) − sin(γ1) 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