\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 53, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/53\hfil Mean value problems]
{Mean value problems of Flett type for a Volterra operator}
\author[C. Lupu\hfil EJDE-2013/53\hfilneg]
{Cezar Lupu} % in alphabetical order
\dedicatory{Dedicated to the memory of Professor Ion Cucurezeanu}
\address{Cezar Lupu \newline
University of Pittsburgh, Department of Mathematics,
Pittsburgh, PA 15260, USA.\newline
University of Craiova, Department of Mathematics,
Str. Alexandru Ioan Cuza 13, RO -- 200585, Craiova, Romania}
\email{cel47@pitt.edu, lupucezar@gmail.com}
\thanks{Submitted November 10, 2012. Published February 18, 2013.}
\subjclass[2000]{26A24, 26A36, 28A15}
\keywords{Flett's mean value theorem; Volterra operator;
integral equation; \hfill\break\indent differentiable function}
\begin{abstract}
In this note we give a generalization of a mean value problem which
can be viewed as a problem related to Volterra operators.
This problem can be seen as a generalization of a result concerning
the zeroes of a Volterra operator in the Banach space of continuous
functions with null integral on a compact interval.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
Mean value theorems have always been an important tool in mathematical analysis.
It is worth mentioning the pioneering contributions of Fermat, Rolle,
Lagrange, Cauchy, Darboux and others.
A variation of Lagrange's mean value theorem with a Rolle type condition
was given by Flett \cite{Flett} in 1958 and it was later
extended in \cite{Trahan} and generalized in \cite{Molnarova, Pawlikowska}.
\begin{theorem} \label{thm1.1}
Let $f:[a, b]\to\mathbb{R}$ be a continuous function on $[a, b]$,
differentiable on $(a, b)$ and $f'(a)=f'(b)$. Then there exists
$c\in (a, b)$ such that
$$
f'(c)=\frac{f(c)-f(a)}{c-a}.
$$
\end{theorem}
For instance, Trahan \cite{Trahan} extended Theorem 1.1 by replacing
the condition $ f'(a)=f'(b)$ with $ (f'(a)-m)(f'(b)-m)>0$, where
$ m=\frac{f(b)-f(a)}{b-a}$. Moreover, Meyers \cite{Meyers} proved in
the same condition, $ f'(a)=f'(b)$ that there exists $c\in (a, b)$ such that
$$
f'(c)=\frac{f(b)-f(c)}{b-c}.
$$
Riedel and Sahoo \cite{Riedel-Sahoo} removed the boundary assumption
on the derivatives and proved the following
\begin{theorem} \label{thm1.2}
Let $f:[a, b]\to\mathbb{R}$ be a differentiable on $[a, b]$.
Then, there exists a point $c\in (a, b)$ such that
$$
f(c)-f(a)=(c-a)f'(c)-\frac{1}{2}\cdot\frac{f'(b)-f'(a)}{b-a}(c-a)^2.
$$
\end{theorem}
Moreover, Theorems 1.1 and 1.2 were used in \cite{Das-Riedel-Sahoo}
and \cite{Lee-Xu-Ye} in proving Hyers-Ulam stability results
for Flett's and Sahoo-Riedel's points. Another results of Flett-type
appears also in \cite{Radulescu-Radulescu},
\begin{theorem} \label{thm1.3}
If $f:[a, b]\to\mathbb{R}$ is a twice differentiable function such
that $f''(a)=f''(b)$, then there exists $c\in (a, b)$ such that
$$
f(c)-f(a)=(c-a)f'(c)-\frac{(c-a)^2}{2}f''(c).
$$
\end{theorem}
Theorems 1.1 and 1.3 have been generalized by Pawlikowska \cite{Pawlikowska}.
Moreover, Molnarova \cite{Molnarova} gave a new proof for the generalized
Flett mean value theorem of Pawlikowska using only Theorem 1.1.
Furthermore, Molnarova establishes a Trahan-type condition in the general case.
For more details, see \cite{Molnarova}. We also point out other contributions
in this direction in
\cite{Abel-Ivan-Riedel, Cakmak-Tiryaki, Das-Riedel-Sahoo, Lee-Xu-Ye, Lupu-Lupu2}.
In \cite{Gologan-Lupu} it is proved that a Volterra operator has
at least one zero in the space of functions having the integral equal
to zero without assuming differentiability of the function.
The same problem is also discussed in \cite{Mingarelli-Pacheco-Plaza}
in the setting of $C^1$ positive functions with nonnegative derivative.
More exactly, in \cite{Mingarelli-Pacheco-Plaza} it is showed that for
$ f, g$ real-valued continuous functions on $[0,1]$, the problem
$$
\int_0^1f(x)dx\int_0^c\phi(x)g(x)dx=\int_0^1g(x)dx\int_0^c\phi(x)f(x)dx,
$$
has a solution $c\in (0, 1)$ for a class of weight functions.
As stated above, the same problem has been studied in \cite{Gologan-Lupu}
for a ``smaller'' class of weight functions. Curiously, both ideas of
proofs from \cite{Gologan-Lupu} and \cite{Mingarelli-Pacheco-Plaza}
can be found in \cite[page 6.]{Lupu-Lupu2},
In this paper, assuming differentiability, we prove a more general
result from the one provided in \cite{Gologan-Lupu} and
\cite{Mingarelli-Pacheco-Plaza} by employing an extension of Flett's
theorem (Theorem 1.1). The main result and some consequences
are given in the section that follows.
\section{Main results}
The following lemma is an extension of Theorem 1.1 from the previous section.
\begin{lemma} \label{lem}
Let $u, v:[a, b]\to\mathbb{R}$ be two differentiable functions on
$[a, b]$ with $v'(x)\neq 0$ for all $x\in [a, b]$ and
$$
\frac{u'(a)}{v'(a)}=\frac{u'(b)}{v'(b)}.
$$
Then there exists $c\in (a, b)$ such that
$$
\frac{u(c)-u(a)}{v(c)-v(a)}=\frac{u'(c)}{v'(c)}.
$$
\end{lemma}
\begin{proof}
Define $w:[a, b]\to\mathbb{R}$ by
$$
w(x)=\begin{cases}
\frac{u(x)-u(a)}{v(x)-v(a)}, &\text{if } x\neq a\\
\frac{u'(a)}{v'(a)}, &\text{if } x=a
\end{cases}
$$
Clearly, $w$ is continuous on $[a, b]$, and by Weierstrass theorem
$w$ attains its bounds. If $w$ does not attain its bounds simultaneously
in $a$ and $b$, then it follows that there exists $x_0\in (a, b)$
extremum point. By Fermat's theorem, we have $w'(x_0)=0$; i.e.,
$$
\frac{u(x_0)-u(a)}{v(x_0)-v(a)}=\frac{u'(x_0)}{v'(x_0)}.
$$
If $w$ attains its bounds in $a$ and $b$, then we have the following
situations:
\begin{equation}\label{zero}
w(a)\leq w(x)\leq w(b)
\end{equation}
or
\begin{equation}\label{zero2}
w(b)\leq w(x)\leq w(a)
\end{equation}
for all $x\in [a, b]$. We can assume without loss of generality
that \eqref{zero} holds. Moreover, possibly replacing $u$ by $-u$
and $v$ by $-v$, we can also admit that $v'(x)>0$, $x\in [a, b]$.
This enables us to establish the following inequality:
$$
u(x)\leq u(a)+w(b)(v(x)-v(a)),
$$
for all $x\in [a, b]$. Now, for all $x\in [a, b]$ we obtain
$$
\frac{u(b)-u(x)}{v(b)-v(x)}\geq\frac{u(b)-u(a)-w(b)(v(x)-v(a))}{v(b)-v(x)}
=\frac{u(b)-u(a)}{v(b)-v(a)}=w(b).
$$
Passing to the limit as $x$ to $b$, we obtain
$$
w(a)=\frac{u'(a)}{v'(a)}=\frac{u'(b)}{v'(b)}
=\lim_{x\to b, x