## 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:

 39A10 Additive difference equations 06F30 Ordered topological structures 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
Full Text:

### 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. · Zbl 0504.06001 [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. [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.