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)
Fixed point theorems of 1-set-contractive operators in Banach spaces. (English) Zbl 1113.47045
The article deals with degree theory and fixed point theorems for $1$-set-contractive operators in Banach spaces. The authors present 14 theorems about conditions under which the degree of vector fields with $1$-set-contractive operators is equal to $1$ or $0$ and corresponding fixed point results; all these theorems are modifications for vector fields with $1$-set-contractive operators of well-known ones by E. Rothe, J. Leray and J. Schauder, M. Krasnosel’skij, M. Altmann, W. V. Petryshin and other mathematicians concerning vector fields with completely continuous or condensing operators. Nevertheless, the authors’ results appear doubtful. First of all, some known results are cited in the article in incorrect form; for example, the authors write (as if due to E. Rothe) that a condensing operator $A: \overline{D} \to E$ ($D$ is a bounded open subset in a Banach space $E$), without fixed points on $\partial D$ and satisfying the condition $\Vert Ax\Vert \le \Vert x\Vert $ for each $x \in \partial D$, has at least one fixed point. This statement is false even for one-dimensional $E$; nonetheless, later the authors prove it in the case when $A$ is a semi-closed $1$-set-contractive $A$. Second, the authors do not formulate exact definitions of $1$-set-contractive operators and this does not allow to check and estimate the authors’ arguments.

MSC:
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H11Degree theory (nonlinear operators)
47H09Mappings defined by “shrinking” properties
WorldCat.org
Full Text: DOI
References:
[1] Petryshyn, W. V.: Remark on condensing and k-set-contractive mappings. J. math. Anal. appl. 39, 717-741 (1972) · Zbl 0238.47041
[2] Nussbaum, R. D.: Degree theory for local condensing maps. J. math. Anal. appl. 37, 741-766 (1972) · Zbl 0232.47062
[3] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach space. SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044
[4] Nussbaum, R. D.: The fixed point index and asymptotic fixed point theorems for k-set-contractions. Bull. amer. Math. soc. 75, 490-495 (1969) · Zbl 0174.45402
[5] Li, G. Z.: The fixed point index and the fixed point theorems of 1-set-contraction mappings. Proc. amer. Math. soc. 104, 1163-1170 (1988) · Zbl 0694.47041
[6] Guo, D. J.; Sun, J. X.: The topological degree computing and application. J. math. Res. exposition 8, 469-480 (1988) · Zbl 0687.47047
[7] Kirk, W. A.; Morales, C.: Condensing mappings and the Leray--Schauder boundary condition. Nonlinear anal. 3, No. 4, 533-538 (1979) · Zbl 0408.47041
[8] Istratescu, Vasile I.: Fixed point theory. (1981) · Zbl 0465.47035
[9] Liu, L. S.: Approximation theorems and fixed point theorems for various class of 1-set-contractive mappings in Banach spaces. Acta math. Sinica 17, 103-112 (2001) · Zbl 1013.47021