Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Cited in: Zbl 1228.34100 Zbl 1220.34083 Zbl 1206.34116 Zbl 1194.34176 Zbl 1145.34004 Zbl 1117.39007 Zbl 1115.34019 Zbl 1093.39003 Zbl 1082.39002 Zbl 1053.34014 Zbl 1070.34036 Zbl 1047.34075 Zbl 1003.39019 Zbl 1001.39025

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]