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)
First- and second-order dynamic equations with impulse. (English) Zbl 1100.39018
The authors present existence results for discontinuous first- and continuous second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. First, they consider first-order dynamic equation $$u^\nabla (t)=g(t,u(t)),\quad t\in J\backslash \{t_1\},$$ $$u(t_1^+)=I(u(t_1)),$$ $$B(u(0), u(T))=0,$$ where $J=[0,T]\cap \mathbb{T}$, $\mathbb{T}$ is a time scale with $0, t_1, T\in \mathbb{T}$ and $0<t_1<T$, $I: \mathbb{R}\to \mathbb{R}$ is continuous and nondecreasing, and $B: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous and for each $x\in [\alpha(0), \beta(0)], B(x, \cdot)$ is non-increasing, where $\alpha, \beta$ are lower and upper solutions of the above equation with $\alpha(t)\leq \beta(t)$ for all $t\in J$. For the discontinuous case with $g$($g$ is a Carathéodory-type function), they prove that the impulsive equation has a solution $u$ such that $u\in [\alpha, \beta]$ by using upper and lower solutions and Schauder’s fixed point theorem. Secondly, the authors are concerned with second-order dynamic equations $$y^{\Delta\Delta}(t)=f(t,y^\sigma(t)),\quad t\in \mathbb{T}^k\equiv [a,b]\setminus \{t_1\},$$ $$L_1(y(a),y^\Delta(a),y(\sigma^2(b)),y^\Delta(\sigma(b)))=0,$$ $$L_2(y(a),y(\sigma^2(b)))=0,$$ $$y(t_1^+)-y(t_1^-)=r_1,$$ $$y^\Delta (t_1^+)-y^\Delta (t_1^-)=I(y(t_1), y^\Delta (t_1^-)),$$ where $t_1\in \mathbb{T}$ with $a<t_1<b$ and $t_1$ right dense, $r_1\in \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is continuous. Assuming the existence of a lower solution $\alpha$ and an upper solution $\beta$ with $\alpha\leq \beta$ on $\mathbb{T}$, they prove that the above impulsive boundary value problem has a solution $u$ such that $u\in [\alpha, \beta]$ under $L_1, L_2$ and $I$ satisfying some appropriate monotone conditions.

MSC:
39A12Discrete version of topics in analysis
WorldCat.org
Full Text: DOI EuDML