Cauchy Functions for Dynamic Equations on a Measure Chain
Aron, Richard M. and Chen, Goong and Krantz, Steven G.
We consider the nth-order linear dynamic equation Px(t) = ∑i = 0npi(t)x(σi(t)) = 0, where pi(t), 0 ≤ i ≤ n, are real-valued functions defined on T. We define the Cauchy function K(t, s) for this dynamic equation, and then we prove a variation of constants formula. One of our main concerns is to see how the Cauchy function for an equation is related to the Cauchy functions for the factored parts of the operator P. Finally we consider the equation Px(t) = ∑i = 0npix(σi(t)) = 0, where each of the pi's is a constant, and obtain a formula for the Cauchy function. For our main results we only consider the time scale T such that every point in T is isolated.
E. Akin, "Cauchy Functions for Dynamic Equations on a Measure Chain," Journal of Mathematical Analysis and Applications, Elsevier, Jan 2002.
The definitive version is available at http://dx.doi.org/10.1006/jmaa.2001.7753
Mathematics and Statistics
Keywords and Phrases
measure chains; time scales; Cauchy functions
Article - Journal
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