A three solutions theorem for nonlinear operator equations in ordered Banach spaces (Q865007)

Consider a Banach space \(E\) partially ordered by a normal solid cone \(P\). The main result of this paper states sufficient conditions under which an operator \(A:E\to E\) has at least four different fixed points: zero, a positive point \(x_1\), a negative point \(x_2\), and a point \(x_3\in E\setminus [P\cup (-P)]\). This abstract result is then applied to show the existence of at least a sign-changing solution (this is the role of the fixed point \(x_3\)) to the following three-point boundary value problem: \[ y''(t)=-f(y(t)), \quad t\in (0,1),\qquad y(0)=0,\quad y(1)=\alpha y(\eta), \] where \(f\) is continuous, \(\eta\in (0,1)\), and \(\alpha\in [0,1)\). The main results obtained for this problem improve those in the previous paper [\textit{X. Xian, J. Sun}, Nonlinear Anal., Theory Methods Appl. 59, No. 4 (A), 491--505 (2004; Zbl 1069.34019)], by weakening a monotonicity assumption on the nonlinearity \(f\).