A Three Solutions Theorem for Nonlinear Operator Equations in Ordered Banach Spaces

In this paper we consider the operator equation in a real Banach space E with cone P: where A = KF; here K is a e-positive, e-continuous and completely continuous operator, and F is a strictly increasing and continuous operator which is Fréchet differentiable at θ. Under certain conditions, we show that the operator equation has at least three solutions x₁, x₂, x₃ such that x₁ ∈ P, x₂ ∈ (−P), x₃ ∈ E\(P ∪ (−P)). Now since the third solution x₃ ∈ E\(P ∪ (−P)), we call it a sign-changing solution. As an application of the main results, we investigate the existence of sign-changing solutions for some three-point boundary value problem.