## 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.

### MSC:

 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