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)
Theoretical approximation methods for hybrid differential equations. (English) Zbl 1245.34013
Consider the initial value problem $$\gathered{d\over dt} \Biggl[{x(t)\over f(t,x(t))}\Biggr]= g(t,x(t))\quad\text{for a.e. }t\in (t_0, t_0+ a),\quad a> 0,\\ x(t_0)= x_0\in\bbfR.\endgathered\tag{$*$}$$ The author defines lower and upper solutions to $(*)$ and shows that under some conditions on $f$ and $g$ the existence of lower and upper solutions implies the existence of a solution to $(*)$. His main interest concerns the construction of monotone sequences converging to extremal solutions of $(*)$. In particular, he considers the case $$g(t,x)= g_1(t,x)+ g_2(t,x),$$ where $g_1$ is nonincreasing in $x$ and $g_2$ is nondecreasing in $x$. Using the notation of mixed lower and upper solutions, he constructs sequences which monotoneously converge to extremal mixed solutions.

34A45Theoretical approximation of solutions of ODE
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions