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Zbl 1014.47025
Avery, Richard I.; Henderson, Johnny
Two positive fixed points of nonlinear operators on ordered Banach spaces.
(English)
[J] Commun. Appl. Nonlinear Anal. 8, No.1, 27-36 (2001). ISSN 1074-133X

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 $\theta(0) =0$, $\theta(\lambda x)\le\lambda\theta(x)$ $(0\le\lambda\le 1)$, $\gamma\le \theta(x) \le\alpha (x)$ and $\|x\|\le M_\gamma(x)$ 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) $\{x\in P:\alpha(x) <a\}\ne \emptyset$ 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 $f:\bbfR\to [0,\infty)$ is considered.
[Peter Zabreiko (Minsk)]
MSC 2000:
*47H07 Positive operators on ordered topological linear spaces
34B15 Nonlinear boundary value problems of ODE
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: multiple fixed points; nonnegative continuous functionals; completely continuous operator; second order boundary value problem

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