Two positive fixed points of nonlinear operators on ordered Banach spaces. (English) Zbl 1014.47025

This article deals with the following theorem: Let \(P\) be a cone in a real Banach space \(E\), \(\alpha\) and \(\gamma\) increasing nonnegative continuous and \(\theta\) nonnegative continuous functionals on \(P\) with \(\theta(0) =0\), \(\theta(\lambda x)\leq\lambda\theta(x)\) \((0\leq\lambda\leq 1)\), \(\gamma\leq \theta(x) \leq\alpha (x)\) and \(\|x\|\leq M_\gamma(x)\) for all \(x\in\overline {\{x\in P:\gamma (x)<c\}}\), and let a completely continuous operator \(A:\overline {\{x\in P:\gamma (x)<c\}}\to P\) satisfy, for some \(a,b,a <b<c\), the following conditions: (i) \(\gamma(Ax)>c\) for all \(x\in\partial \{x\in P:\gamma(x) <c\}\); (ii) \(\theta(Ax) <b\) for all \(x\in\partial \{x\in P:\theta (x)<b\}\); (iii) \(\{x\in P:\alpha(x) <a\}\neq \emptyset\) and \(\alpha(Ax) >a\) for all \(x\in\partial \{x\in P:\alpha(x) <a\}\). Then \(A\) has at least two fixed points \(x_1\) and \(x_2\) such that \(a<\alpha (x_1)\), \(\theta(x_1)<b\) and \(b<\theta (x_2)\), \(\gamma(x_2)<c\). As an application, the second order boundary value problem \(y''+f(y)=0\), \(y(0)=y(1)=0\) with a continuous function \(f:\mathbb{R}\to [0,\infty)\) is considered.


47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems