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 theorem of Leggett-Williams type and its application. (English) Zbl 1066.47059

One of the generalizations of Krasnoselskii’s theorem on cone expansion and compression was obtained in [R. W. Leggett, L. R. Williams, J. Math. Anal. Appl., Vol. 76, 91–97 (1980; Zbl 0448.47044)]. In the present paper, the author proves the following Leggett-Williams type theorem: Theorem. Let P be a normal cone in a Banach space E and let γ be the normal constant of P. Assume that Ω 1 and Ω 2 are bounded open sets in E such that 0Ω 1 and Ω ¯ 1 Ω 2 . Let F:P(Ω ¯ 2 Ω 1 )P be a completely continuous operator and u 0 P0. If either γxFx for xP(u 0 )Ω 1 and Fxx for xPΩ 2 , or Fxx for xPΩ 1 and γxFx for xP(u 0 )Ω 2 is satisfied, then F has a fixed point in the set P(Ω ¯ 2 Ω 1 ). The author applies this theorem to prove the existence of positive solutions to the following second order three-point boundary value problem:

x '' (t)+f(t,x(t))=0,
x(0)=0,αx(η)=x(1),

where t[0,1], f is a continuous function, η(0,1), α0, and 1-αη>0.


MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations