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The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.
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We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (first-order) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.
The mean value theorem of calculus states that, given a differentiable function $f$ on an interval $[a, b]$, there exists at least one mean value abscissa $c$ such that the slope of the tangent line at $c$ is equal to the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. In this article, we study how the choices of $c$ relate to varying the right endpoint $b$. In particular, we ask: When we can write $c$ as a continuous function of $b$ in some interval? Drawing inspiration from graphed examples, we first investigate this question by proving and using a simplified implicit function theorem. To handle certain edge cases, we then build on this analysis to prove and use a simplified Morses lemma. Finally, further developing the tools proved so far, we conclude that if $f$ is analytic, then it is always possible to choose mean value abscissae so that $c$ is a continuous function of $b$, at least locally.
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.
In this sequel to arXiv1407.4089 by the second author, we extend to multi-dimensional (or infinite-dimensional) settings the Goldie equation arising in the general regular variation of `General regular variation, Popa groups and quantifier weakening, J. Math. Anal. Appl. 483 (2020) 123610, 31 pp. (arXiv1901.05996). The theory focusses on extension of the treatment there of Popa groups, permitting a characterization of Popa homomorphisms (in two complementary theorems, 4A and 4B below). This in turn enables a characterization of the (real-valued) solutions of the multivariate Goldie equation, to be presented in the further sequel arXiv1910.05817. The Popa groups here contribute to a structure theorem describing Banach-algebra value
In this the sequel to arXiv1910.05816, we derive a necessary and sufficient condition characterizing which real-valued continuous solutions of a multivariate Goldie functional equation express homomorphy between the multivariate Popa groups defined and characterized in the earlier work. This enables us to deduce that all (real-valued) continuous solutions are homomorphisms between such groups. We use this result also to characterize as Popa homomorphisms smooth solutions of a related more general equation, also of Levi-Civita type. A key result here (Theorem 2) on purely radial behaviour is generalized in arXiv2105.07794 to a Banach-algebra setting involving radial tilting behaviour.
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