# 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 $\mu$ : $T×T\to R$ with $\mu \left(r,s\right)+\mu \left(s,t\right)=\mu \left(r,t\right)$ for all r,s,t$\in T$ and $\mu \left(r,s\right)>0$ for $r>s$. The measure $\nu$ is induced by $\nu \left(\left[r,s\left[\right)=\mu \left(s,r\right)$. 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:
 39A10 Additive difference equations 06F30 Order topologies (order-theoretic aspects) 54F05 Linearly, generalized, and partial ordered topological spaces
##### References:
 [1] B. Aulbach, Continuous and Discrete Dynamics near Manifolds of Equilibria. Lecture Notes in Mathematics 1058, Springer, Berlin – Heidelberg –New York – Tokyo, 1984. [2] G. Boole, A Treatise on the Calculus of Finite Differences, Dover Publications, New York 1960. [3] L. Brand, Differential and Difference Equations, Wiley & Sons, New York 1966. [4] F.B. Christiansen/ T.M. Fenchel, Theories of Populations in Biological Communities, Springer, Berlin, 1977. [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. [10] S. Hilger, Ein Ma$\beta$kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Dissertation, Univ. Würzburg, 1988. [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. [13] J.P. LaSalle, Stability for Difference Equations. In: Studies in Ordinary Differential Equations, MAA Studies in Mathematics 14, Englewood Cliffs, 1977, 1–31.