zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
New fixed point theorems for 1-set-contractive operators in Banach spaces. (English) Zbl 1124.47038

This article presents a list of sufficient conditions for the existence of fixed points of semi-closed 1-set-contractive operators in real Banach spaces. All results presented here are evident corollaries of the following analogue for semi-closed 1-set-contractive operators of the Leray-Schauder fixed point principle: If A is a semi-closed 1-set-contractive operator such that

Ax-x 0 k(x-x 0 )forsomex 0 ΩandforallxΩ,1<k<,(*)

then A has at least one fixed point in Ω ¯.

In turn, this result is an evident corollary of the statement that deg(I-A,Ω)=1 provided that A is semi-closed 1-set-contractive operator satisfying the condition Ax-x 0 k(x-x 0 ) for some x 0 Ω and for all xΩ, 1k<. The main part of the article is devoted to inequalities that imply (*); among others, the inequality

Ax-x α Ax α+β -x α ,xΩ,

(α>1, β0) is treated.

MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties