\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
$$
w(a)=\frac{u'(a)}{v'(a)}=\frac{u'(b)}{v'(b)}
=\lim_{x\to b, x<b} \frac{u(b)-u(x)}{v(b)-v(x)} \geq w(b).
$$
This implies in view of \eqref{zero} that $w$ is a constant, and therefore
$w'=0$.
\end{proof}

\noindent\textbf{Remark.}
Since $v'(x)\ne 0$ for all $x$, $v$ is a diffeomorphism.
Let $u(v^{-1}(x))=U(x),v(a)=A,v(b)=B$. Then we have
 $U'(x) = u'(v^{-1}(x))/v'(v^{-1}(x))$ so the constraint reads
$U'(A)=U'(B)$. Thus, by Flett's mean value theorem (Theorem 1.1),
$$
\frac{U(C)-U(A)}{C-A} = U'(C),
$$
for some $C\in(A,B)$. But if $C = v^{-1}(c)$,
\[
\frac{U(C)-U(A)}{C-A} = \frac{u(c)-u(a)}{v(c)-v(a)} = U'(C)
= \frac{u'(c)}{v'(c)}.
\]


This remark yields an alternative proof of Lemma 2.1.
The Volterra operator is defined for a function
$f(t)\in L^{2}([0,1])$ (the space of Lebesgue square-integrable function
on $[0,1]$), and a value $t\in [0,1]$, as
$$
V(f)(x)=\int_0^xf(t)dt.
$$
It is a well-known  that $V$ is a bounded linear operator between
Hilbert spaces, with Hermitian adjoint $ V^{*}(f)(x)=\int_x^1f(t)dt$.

This operator has been studied intensively in the last decades because
it is the simplest operator that exhibits a range of phenomena which
can arise when one leaves the normal or finite-dimensional cases.
 Moreover, the Volterra operator is well-known as a quasinilpotent,
but not nilpotent operator with no eigenvalues.

For a continuous real-valued function $\Psi$ and $\phi:[0,1]\to\mathbb{R}$
differentiable with $\phi'(x)\neq 0$ for all $x\in (0,1)$.
Let $V\Psi$ be the function mapping given by $V\Psi(t)=\int_0^t\Psi(x)dx$
and similarly define
$$
V_{\phi}\Psi(t)=\int_0^t\phi(x)\Psi(x)dx.
$$
 Let
$$
 \tilde{C}^1([a, b]):=\{\phi:[a, b]\to\mathbb{R}: \phi\in C^{1}([a, b]);
\phi'(x)\neq 0, x\in [a, b], \phi(a)=0\}.
$$
Denote $C_{\rm null}([a,b])$ the space of continuous functions having
 null integral on the interval $[a, b]$.

Now, we are ready to prove the main results of this note.

\begin{theorem} \label{thm2.2}
Let $f\in C_{\rm null}([a, b])$ and $g\in C^{1}([a, b])$, with
 $g'(x)\neq 0$ for all $x\in [a, b]$. Then there exists
 $c\in (a, b)$ such that
$$
V_{g}f(c)=g(a)\cdot Vf(c).
$$
\end{theorem}

\begin{proof}
 The conclusion asks to prove the existence of $c\in (a, b)$ such that
$$
\int_a^cf(x)g(x)dx=g(a)\int_a^cf(x)dx.
$$
Let us consider the functions $u, v:[a,b]\to\mathbb{R}$ given by
\begin{gather*}
u(t)=\int_a^tf(x)g(x)dx-g(t)\int_a^tf(x)dx, \\
v(t)=g(t), \quad\text{for all } t\in [a, b].
\end{gather*}
Now, it is easy to see that $ u'(t)=-g'(t)\int_a^tf(x)dx$.
By the Lemma 2.1, there exists $c\in (a, b)$ such that
$$
\frac{u(c)-u(a)}{v(c)-v(a)}=\frac{u'(c)}{v'(c)},
$$
which is equivalent to
$$
\frac{\int_a^cf(x)g(x)dx-g(c)\int_a^cf(x)dx}{g(c)-g(a)}
=\frac{-g'(c)\int_a^cf(x)dx}{g'(c)}
$$
which finally reduces to
$$
\int_a^cf(x)g(x)dx-g(c)\int_a^cf(x)dx=-g(c)\int_a^cf(x)dx+g(a)\int_a^cf(x)dx,
$$
and the conclusion follows.
\end{proof}


\noindent\textbf{Remark.}
In the same setting as Theorem 2.2 if we apply Meyers \cite{Meyers}
mean value theorem we obtain the existence of $c\in (a, b)$ such that
$$
(b-c)u'(c)=u(b)-u(c)
$$
which is equivalent to
\begin{gather*}
 -(b-c)g'(c)\int_a^cf(x)dx=\int_a^cf(x)g(x)dx-g(c)\int_a^cf(x)dx,\\
 (g(c)-(b-c)g'(c))\int_a^cf(x)dx=\int_a^cf(x)g(x)dx,
\end{gather*}
and this can be rewritten as
$$ 
(g(c)-g'(c)(b-c))Vf(c)=V_{g}f(c).
$$

\begin{corollary} \label{coro2.3}
If $f\in C_{\rm null}([a, b])$ and $g\in\tilde{C}^1([a, b])$, 
then there is $c\in (a, b)$ such that
$$ 
\int_a^cf(x)g(x)dx=0.
$$
\end{corollary}

The above corollary is evident from Theorem 2.2.

\begin{theorem} \label{thm2.4}
If $f, g$ are continuous real-valued functions on $[0,1]$, then there 
exists $x_0\in (0,1)$ such that 
\begin{align*}
& V_{\phi}f(x_0)\int_0^1g(x)dx-V_{\phi}g(x_0)\int_0^1f(x)dx \\
&=\phi(0)\cdot\Big(Vf(x_0)\int_0^1g(x)dx-Vg(x_0)\int_0^1f(x)dx\Big)
\end{align*}
\end{theorem}

\begin{proof} 
Let us consider the functions $ u, v:[0,1]\to\mathbb{R}$,
\begin{gather*}
 u(t)=(\phi(t)Vf(t)-V_{\phi}f(t))\int_0^1g(x)dx-(\phi(t)Vg(t)
-V_{\phi}g(t))\int_0^1f(x)dx,\\
v(t)=\phi(t).
\end{gather*}
 Clearly, these two functions satisfy the conditions from the Lemma 2.1, 
and thus, there exists $x_0\in (0,1)$ such that
$$
\frac{u(x_0)-u(0)}{v(x_0)-v(0)}=\frac{u'(x_0)}{v'(x_0)},
$$
which is equivalent to
\begin{align*}
&\int_0^{x_0}\phi(x)f(x)dx\int_0^1g(x)dx
 -\int_0^{x_0}\phi(x)g(x)dx\int_0^1f(x)dx \\
&=\phi(0)\Big(\int_0^{x_0}f(x)dx\int_0^1g(x)dx
 -\int_0^{x_0}g(x)\int_0^1f(x)dx\Big),
\end{align*}
which is rewritten as
\begin{align*}
&V_{\phi}f(x_0)\int_0^1g(x)dx-V_{\phi}g(x_0)\int_0^1f(x)dx\\
&=\phi(0)\cdot\Big(Vf(x_0)\int_0^1g(x)dx-Vg(x_0)\int_0^1f(x)dx\Big).
\end{align*}
\end{proof}


\begin{corollary}[\cite{Gologan-Lupu}] \label{coro2.5}
If $\phi(0)=0$, then there exists $x_0\in (0,1)$ such that
$$
\int_0^1f(x)dx\cdot V_{\phi}g(x_0)=\int_0^1g(x)dx\cdot V_{\phi}f(x_0),
$$
\end{corollary}

The above corollary follows immediately from Theorem 2.4.


\begin{corollary}[\cite{Lupu-Lupu1}] \label{coro2.6}
If $f, g:[0,1]\to\mathbb{R}$ are two continuous functions, 
then there exists $x_{1}\in (0,1)$ such that
$$
\int_0^1f(x)dx\int_0^{x_{1}}xg(x)dx=\int_0^1g(x)dx\int_0^{x_{1}}xf(x)dx.
$$
\end{corollary}

The proof of the above corollary follows by applying Corollary 2.5 wiht
$\phi(x)=x$.

\section{Discussion and some examples}

In the proof of Theorem 2.2 we considered the auxiliary function 
$\varphi:[0,1]\to\mathbb{R}$,
$$
\varphi(t)=\int_0^t\phi(x)f(x)dx-\phi(t)\int_0^tf(x)dx.
$$
If we apply Theorem 1.1 for $f\in C_{\rm null}([0,1])$, then there 
exists $c\in (0,1)$ such that $\varphi(c)-\varphi(0)=c\varphi'(c)$ which 
is equivalent to
$$
\int_0^c\phi(x)f(x)dx=(\phi(c)-c\phi'(c))\int_0^cf(x)dx.
$$
One can remark that this result is completely different from Theorem 2.2. 
On the other hand, if we consider $f\in C([0,1])$ such that
$$
\int_0^1\phi(x)f(x)dx=\phi(1)\int_0^1f(x)dx,
$$
then by Rolle's theorem there exists $\tilde{c}\in (0,1)$ such 
that $\varphi'(\tilde{c})=0$,
$$
\int_0^{\tilde{c}}f(x)dx=0.
$$
Now, we present some examples that follow as consequences from Theorem 2.2 
on interval $[0, 1]$.

\begin{example} \label{examp1} \rm
If we replace functions $f, g$ by their squares in Theorem 2.4,
 we have the equality
\begin{align*}
&\int_0^{x_0}\phi(x)f^2(x)dx\int_0^1g^2(x)dx
-\int_0^{x_0}\phi(x)g^2(x)dx \int_0^1g^2(x)dx\\
&=\phi(0)\Big(\int_0^{x_0}f^2(x)dx\int_0^1g^2(x)dx
-\int_0^{x_0}g^2(x)dx\int_0^1f^2(x)dx\Big).
\end{align*}
This equality actually says that given any two continuous functions
 $f, g$ we have
\begin{align*}
&\|f\|_{L^2_{\phi}(0, x_0)}\|g\|_{L^2(0,1)}
-\|g\|_{L^2_{\phi}(0, x_0)}\|f\|_{L^2(0,1)}\\
&=\phi(0)(\|f\|_{L^2(0, x_0)}\|g\|_{L^2(0,1)}
 -\|g\|_{L^2(0, x_0)}\|f\|_{L^2(0,1)}),
\end{align*}
where the quantities are the norms in their respective spaces 
of (weighted) square integrable functions.
 Moreover, if $\phi(0)=0$ we recover \cite[Example 3]{Mingarelli-Pacheco-Plaza}.
\end{example}

\begin{example} \label{examp2}\rm
 For  $i\neq j$ consider $f(x)=P_{i}(x)P_{j}(x)$, where $P_{i}, P_{j}$ 
are orthogonal functions on $[0, 1]$; i.e.,
$$
\int_0^1P_{i}(x)P_{j}(x)dx=0.
$$
Now, by Theorem 2.2, there is a point $c_{ij}\in (0,1)$ such that
$$
\int_0^{c_{ij}}P_{i}(x)P_{j}(x)\phi(x)dx
=\phi(0)\int_0^{c_{ij}}P_{i}(x)P_{j}(x)dx.
$$
If $\phi(0)=0$ we obtain $\int_0^{c_{ij}}P_{i}(x)P_{j}(x)\phi(x)dx=0$
 which is  precisely \cite[Example 4]{Mingarelli-Pacheco-Plaza}.
\end{example}


\subsection*{Acknoledgments} 
The author is indebted to the department of mathematics of
 the Politehnica University of Bucharest for hospitality during
 May-June, 2012, when this work was done.

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\end{document}