On generalizations of Flett's Theorem

Abstract

In 1958 T. M. Flett proved a theorem which is a variant of the Lagrange mean value theorem; namely, let $f:[ a,b] \to\mathbb{R}$ be a differentiable function in $[a,b]$ and $f^{\prime}( a)=f^{\prime}(b)$. Then there exists a number $\eta\in(a,b)$ such that $f(\eta)-f(a)=(\eta-a)\cdot f^{\prime}(\eta)$. Manav Das, Thomas Riedel and Prasanna K. Sahoo have given generalizations of Flett's theorem for approximately differentiable functions. Here we provide generalizations of these theorems for some local $\mathcal{S}$-systems.