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)
Analysis on measure chains - a unified approach to continuous and discrete calculus. (English) Zbl 0722.39001
A chain is a linearly ordered set with the order topology. A measure chain is a chain T, where any nonvoid subset, which is bounded above, has a l.u.b., and where exists a continuous mapping $\mu$ : $T\times T\to R$ with $\mu (r,s)+\mu (s,t)=\mu (r,t)$ for all r,s,t$\in T$ and $\mu (r,s)>0$ for $r>s$. The measure $\nu$ is induced by $\nu ([r,s[)=\mu (s,r)$. On this background a calculus is basically described, which contains the usual differentiation and the difference calculus as special cases. The integration is introduced as the inverse of the differentiation. The theory is used to study the solutions of dynamical equations. The solutions of linear equations are represented as generalized exponential functions. Remark: The statement on p. 20 that the infimum there exists always is correct only after the introduction of Axiom 2 concerning the existence of a l.u.b.
Reviewer: L.Berg (Rostock)

MSC:
39A10Additive difference equations
06F30Order topologies (order-theoretic aspects)
54F05Linearly, generalized, and partial ordered topological spaces
WorldCat.org
Full Text: DOI
References:
[1] B. Aulbach, Continuous and Discrete Dynamics near Manifolds of Equilibria. Lecture Notes in Mathematics 1058, Springer, Berlin -- Heidelberg --New York -- Tokyo, 1984. · Zbl 0535.34002
[2] G. Boole, A Treatise on the Calculus of Finite Differences, Dover Publications, New York 1960. · Zbl 0090.29701
[3] L. Brand, Differential and Difference Equations, Wiley & Sons, New York 1966. · Zbl 0223.34001
[4] F.B. Christiansen/ T.M. Fenchel, Theories of Populations in Biological Communities, Springer, Berlin, 1977. · Zbl 0354.92025
[5] J. Dieudonné, Grundzüge der modernen Analysis Bd. I, Vieweg, Braunschweig 1985.
[6] J. Dieudonné, Grundzüge der modernen Analysis Bd. II, Vieweg, Braunschweig 1987.
[7] M. Erné, Einführung in die Ordnungstheorie. BI-Wissenschaftsverlag, Mannheim, 1982.
[8] S. Goldberg, Introduction to Difference Equations, Wiley & Sons, New York 1958.
[9] P. Hartman, Difference Equations: Disconjugacy, Principal Solutions, Green’s Functions, Complete Monotonicity. Trans. AMS 246 (1978), 1--30. · Zbl 0409.39001
[10] S. Hilger, Ein Ma{$\beta$}kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Dissertation, Univ. Würzburg, 1988. · Zbl 0695.34001
[11] R.E. Kalman/ J.E. Bertram, Control System Analysis and Design Via the ”Second Method of Lyapunov”, I. Continuous Time Systems, II. Discrete Time Systems. Trans. ASME Ser. D, J. of Basic Engineering, 1960, 371 393, 394--400. · doi:10.1115/1.3662605
[12] V. Lakshmikantham/ D. Trigiante, Theory of Difference Equations, Academic Press, Inc., San Diego 1988. · Zbl 0683.39001
[13] J.P. LaSalle, Stability for Difference Equations. In: Studies in Ordinary Differential Equations, MAA Studies in Mathematics 14, Englewood Cliffs, 1977, 1--31. · Zbl 0397.39009