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)
Leray-Schauder continuation theorems in the absence of a priori bounds. (English) Zbl 0912.47040

The Leray-Schauder continuation principle uses the following basic idea: Let X be a Banach space, ΩX×[0,1] a bounded open set, F:Ω ¯X a compact continuous operator, and Σ the (possibly empty) set of solutions of the equation x=F(x,λ) in Ω ¯. If the a priori estimate ΣΩ= holds and the map I-F(·,0) has nonzero degree, then Σ contains a continuum along which λ assumes all values between 0 and 1. Loosely speaking, this principle is useful whenever it is difficult to find invariant balls for the operator F(·,1), and thus classical fixed point principles do not apply.

In this interesting survey article, the author discusses several variants of continuation principles where a priori bounds are not required explicitly. This essentially enlarges the class of nonlinear problems for which existence may be established by means of topological arguments. Some of the existence theorems given here are taken from previous papers by A. Capietto [Boll. Unione Mat. Ital., VII. Ser. B 8, No. 1, 135-150 (1994; Zbl 0812.47062)]and A. Capietto, F. Zanolin and the author [J. Differ. Equations 88, No. 2, 347-395 (1990; Zbl 0718.34053)], M. Henrard [Differ. Equations Dyn. Syst. 3, No. 1, 43-60 (1995; Zbl 0874.34056)], R. Precup [this journal 5, No. 2, 385-396 (1995; Zbl 0847.34028)], A. Capietto, M. Henrard, F. Zanolin and the author [this journal 3, No. 1, 81-100 (1994; Zbl 0808.34028) and 6, No. 1, 175-188 (1995; Zbl 0849.34018)], and others. The abstract results are illustrated by various applications to ordinary and functional differential equations.

MSC:
47J05Equations involving nonlinear operators (general)
34B15Nonlinear boundary value problems for ODE
34C25Periodic solutions of ODE