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In discrete systems, Shannon entropy is well known to be characterized by the Shannon-Khinchin axioms. Recently, this set of axioms was generalized for Tsallis entropy, one-parameter generalization of Shannon entropy. In continuous systems, Shannon differential entropy has been introduced as a natural extension of the above Shannon entropy without using an axiomatic approach. We derive the generalized...
.We find the continuous solutions of some functional equations arising from an axiomatization of weighted utility. The axiomatization itself will be presented in a subsequent paper. The results have potential applications to the “utility of gambling” problem.
Summary. Let X be a Banach space. We prove the stability of the functional equation $$ L\left( {\sum\limits_{j = 1}^3 {k_j f(p_j )} } \right) = \sum\limits_{j = 1}^3 {k_j g(p_j )} $$ for where $f:[0,1] \to \mathbb{R}_ + ,g:[0,1] \to X$ and $L:\mathbb{R}_ + \to X$ are unknown...
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