Monotonicity of functions and sign changes of their Caputo derivatives

By Kai Diethelm. 2015

57 citations recorded by the source corpus.

Abstract It is well known that a continuously differentiable function is monotone in an interval [a, b] if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders α ∈ (0, 1) with starting point a of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it is not sufficient if the Caputo derivatives have a constant sign for a few values of α ∈ (0, 1) only.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b1c247a140c7d536ad1b4a432f0a6a507a92439b

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Positivity, monotonicity, and convexity for convolution operators (2020, background): “It is easy to verify that a ( t ) satisﬁes all the conditions of the corollary but, however, the condition ( g 1 − α ∗ u (cid:48) )( t ) ≥ 0 alone does not imply that u is monotone, see [12, Theorem 2.2].”