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)


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