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)
A fixed point theorem of Krasnoselskii-Schaefer type. (English) Zbl 0896.47042
The authors focus on three fixed point theorems and an integral equation. Schaefer’s fixed point theorem yields a $T$-periodic solution of $$x(t)= a(t)+ \int^t_{t- h}D(t, s)g(s, x(s))ds\tag 1$$ if $D$ and $g$ satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder’s theorem (known as Krasnoselskii’s theorem) yields a $T$-periodic solution of $$x(t)= f(t, x(t))+ \int^t_{t- h}D(t,s)g(s, x(s))ds\tag 2$$ if $f$ defines a contraction and if $D$ and $g$ are small enough. The authors prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer’s theorem which yields a $T$-periodic solution of (2) when $f$ defines a contraction mapping, while $D$ and $g$ satisfy the aforementioned sign conditions.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
45M15Periodic solutions of integral equations
Full Text: DOI