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)
Positive solutions of a second-order Neumann boundary value problem with a parameter. (English) Zbl 1256.34019
The authors are concerned with the following second-order Neumann boundary value problem: $$ \cases -(p(t)x(t))'+q(t)x(t)=\lambda g(t)f(x(t)), &0<t<1,\\ x'(0)=x'(1)=0,&\\ \endcases $$ where $\lambda>0$ is a parameter, $p\in C^1[0,1]$ ($p(t)>0)$, $q\in C[0,1]$ ($q(t)\ge0)$, $f, g: [0,\infty)\to[0,\infty)$ are continuous functions with $f\not\equiv0$ and $\int_0^1g(s)ds>0$. Under some assumptions regarding the limits $\lim\limits_{x\to+\infty,0^+}\sup\frac{f(x)}{x}$, the authors first prove the existence of single and twin positive solutions. They use Krasnosel’skii’s fixed point theorem of cone compression and expansion both with a fixed point theorem for strongly completely continuous mappings; a discussion upon the parameter $\lambda$ is given and a nonexistence result presented. When the nonlinearity $f$ further satisfies a monotonicity condition, a uniqueness result is obtained together with the continuous dependence of the solutions on $\lambda$.

34B18Positive solutions of nonlinear boundary value problems for ODE
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34B08Parameter dependent boundary value problems for ODE
Full Text: DOI