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:x = Ax, x is an element of E,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 Frechet differentiable at theta. Under certain conditions, we show that the operator equation has at least three solutions x(1), x(2), x(3) such that x(1) is an element of P, x(2) is an element of (- P), x(3) is an element of E\(P boolean OR (- P)). Now since the third solution x(3) is an element of E\(P boolean OR (- 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.