zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $\theta(0) =0$, $\theta(\lambda x)\le\lambda\theta(x)$ $(0\le\lambda\le 1)$, $\gamma\le \theta(x) \le\alpha (x)$ 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.

47H07Monotone and positive operators on ordered topological linear spaces
34B15Nonlinear boundary value problems for ODE
47H10Fixed-point theorems for nonlinear operators on topological linear spaces