\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 108, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/108\hfil Continuous solutions]
{Continuous solutions of distributional Cauchy problems}

\author[S. Heikkil\"a\hfil EJDE-2011/108\hfilneg]
{Seppo Heikkil\"a} 

\address{Seppo Heikkil\"a \newline
Department of Mathematical Sciences,
University of Oulu, BOX 3000,
FIN-90014 University of Oulu, Finland}
\email{sheikki@cc.oulu.fi}

\thanks{Submitted February 28, 2011. Published August 25, 2011.}
\subjclass[2000]{26A24, 26A39, 34A12, 34A36, 47B38, 47H07, 58D25}
\keywords{Distribution; primitive integral;
 continuous; Cauchy problem; \hfill\break\indent
fixed point; smallest solution; greatest solution;
minimal solution; maximal solution}

\begin{abstract}
 Existence of the smallest, greatest, minimal, maximal and unique
 continuous solutions to distributional Cauchy problems, as well
 as their dependence on the data, are studied.
 The main tools are a continuous primitive integral and fixed point
 results in function spaces.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\section{Introduction}\label{S1}

New existence results are derived  for
the smallest, greatest, minimal, maximal and unique continuous
solutions of the distributional Cauchy problem
\begin{equation}\label{E1}
 y'=f(y), \quad  y(a)=c.
 \end{equation}
Novel results for dependence of solutions on $f$ and on the initial
value $c\in\mathbb{R}$ are also derived.
The values of $f$ are distributions (generalized functions) on $[a,b]$,
$-\infty< a<b<\infty$.
Definition of such distributions and their main properties needed
in this paper are presented in section 2.

In section 3, existence results are derived for the smallest and
greatest continuous solutions of \eqref{E1}.
Dependence of these solutions both on $f$ and on $c$ are also studied.
 A concrete example is presented.

Existence of minimal and maximal solutions of problem \eqref{E1}
with $c=0$ is studied in section 4.
In sections 5 and 6 we present conditions which ensure that
problem \eqref{E1} has a unique solution that depends continuously
on the initial value $c$. Main tools are  a continuous primitive
integral introduced for distributions in \cite{ET081} and fixed
point results proved  in  \cite{CH11,HeiLak94}.


\section{Preliminaries}\label{S20}

Distributions on a compact real interval $[a,b]$ are (cf. \cite{Tvr94})
continuous linear functionals on the topological vector space
$\mathcal D$ of functions $\varphi:\mathbb{R}\to\mathbb{R}$ possessing
for every $j\in\mathbb N_0$ a derivative $\varphi^{(j)}$ of order $j$
which is continuous on $\mathbb{R}$ and vanishes on $\mathbb{R}\setminus
(a,b)$. The space $\mathcal D$ is endowed with the topology in which
the sequence $(\varphi_k)$ of $\mathcal D$ tends to
$\varphi\in\mathcal D$ if and only if
$\varphi_k^{(j)}\to \varphi_k^{(j)}$ uniformly on $(a,b)$ for all
$j\in\mathbb N_0$ as $k\to\infty$.
As for the theory of distributions; see e.g. \cite{FrJ99,Sch66}.

A distribution $g$ on $[a,b]$ is called  distributionally Denjoy
(shortly $DD$) integrable on $[a,b]$ if $g$ has a continuous primitive;
i.e., if  $g$ is  a distributional derivative $G'$ of a function
$G\in C[a,b]$ (cf. \cite{ET081}).
Thus the value  $\langle g,\varphi\rangle$ of $g$ at
$\varphi\in\mathcal D$ is defined by
$$
\langle g,\varphi\rangle=\langle G',\varphi\rangle =-\langle G,\varphi'\rangle = \int_a^bG(t)\varphi'(t)\,dt.
$$
 The continuous primitive integral function of $g$ is defined  by
\begin{equation}\label{E200}
\sideset{^c}{}{\!\!\!\int_a^t}g = 
\sideset{^c}{}{\!\!\!\int_a^t} G'=G(t)-G(a), \quad t\in[a,b].
\end{equation}
%
It belongs to the function space
$$
\mathcal B_c[a,b]=\{x:[a,b]\to \mathbb{R}:x
\text{ is continuous and vanishes at $a$}\}.
$$
Assuming that   $\mathcal B_c[a,b]$ is ordered pointwise,
it can be shown (cf. \cite{ET081}) that relation $\preceq$,
defined by
\begin{equation}\label{E20}
f\preceq g  \text{ if and only if }
\sideset{^c}{}{\!\!\!\int_a^t} f
\le \sideset{^c}{}{\!\!\!\int_a^t} g \quad \text{for all }  t\in[a,b],
\end{equation}
is a partial ordering on the set of $DD$ integrable distributions
on $[a,b]$.

The following result reduces problem \eqref{E1} to a fixed point
equation on $C[a,b]$.

\begin{lemma}\label{L20}
Assume that $f(x)$ is  $DD$ integrable on $[a,b]$ for every
$x\in C[a,b]$. Then the Cauchy problem \eqref{E1} has a
continuous solution $y$ if and only if $y$ is a solution
of  the fixed point equation
\begin{equation}\label{E2}
    x(t)=F(x)(t):=c+ \sideset{^c}{}{\!\!\!\int_a^t}f(x), \quad t\in[a,b].
\end{equation}
\end{lemma}

\begin{proof}
 Assume first that $y\in C[a,b]$ is a solution of problem \eqref{E1}.
Applying \eqref{E1}, \eqref{E200} and \eqref{E2} we have for
each $t\in[a,b]$,
$$
y(t)=c+y(t)-y(a)=c+ \sideset{^c}{}{\!\!\!\int_a^t}y'
=c+ \sideset{^c}{}{\!\!\!\int_a^t} f(y)=F(y)(t).
$$
Thus $y$ is a solution of \eqref{E2}. Conversely, assume that
$y\in C[a,b]$ satisfies
the fixed point equation \eqref{E2}. Then for every $t\in[a,b]$,
$$
y(t)-c=F(y)(t)-c= \sideset{^c}{}{\!\!\!\int_a^t} f(y).
$$
This result implies also that $y(a)=c$, since 
$ \sideset{^c}{}{\!\!\!\int_a^a}f(y)=0$
by  \eqref{E200}.
On the other hand,
$$
y(t)-c= y(t)-y(a)= \sideset{^c}{}{\!\!\!\int_a^t}y', \quad t\in[a,b].
$$
The above results imply that $ \sideset{^c}{}{\!\!\!\int_a^t}f(y)
= \sideset{^c}{}{\!\!\!\int_a^t}y'$ for
every $t\in [a,b]$, whence $y'=f(y)$
by \eqref{E20}. Thus $y$ is a solution of problem \eqref{E1}.
\end{proof}

\section{Existence of the smallest and greatest solutions}\label{S2}

The application of monotone methods for fixed point problems in
$C[a,b]$ is complicated by the fact that the limit function of
a pointwise convergent monotone sequence of $C[a,b]$ is not
necessarily continuous. For instance, the functions $x_n(t)=t^n$,
$t\in[0,1]$, $n\in\mathbb N$, form such a sequence.
Therefore it is assumed in this section that $f(x)$  is a
distributional derivative of a continuous function  whenever
$x$ belongs to the space $L^1[a,b]$ of Lebesgue integrable
functions on $[a,b]$. This space is equipped with  a.e. pointwise
ordering, and its a.e. equal functions are identified.

The main result of this section reads as follows.

\begin{theorem}\label{T31}
The Cauchy problem \eqref{E1} has the smallest solution $y_*$
and the greatest solution $y^*$ in $C[a,b]$ if the following
hypotheses are valid.
\begin{itemize}
\item[(A0)] $f(x)$ is $DD$ integrable on $[a,b]$ for every
$x\in L^1[a,b]$;
\item[(B0)] $f(x)\preceq f(y)$ whenever $x\le y$ in $L^1[a,b]$;
\item[(C0)] There exist distributions  $f_\pm$ that are  $DD$
integrable on $[a,b]$ such that
$f_-\preceq f(x)\preceq f_+$ for all $x\in L^1[a,b]$.
\end{itemize}
Moreover, both $y_*$ and $y^*$ are increasing with respect to $f$
and to $c$.
\end{theorem}

\begin{proof}
For each $x\in L^1[a,b]$ denote by $F(x)$ the primitive of $f(x)$,
defined in \eqref{E2}, and define $y_\pm\in C[a,b]$ by
$$
 y_\pm(t)=c+ \sideset{^c}{}{\!\!\!\int_a^t} f_\pm, \quad t\in[a,b].
$$
The given hypotheses imply by \eqref{E20} and \eqref{E2} that $F$
is an increasing mapping from $L^1[a,b]$ to its order interval
$[y_-,y_+]=\{x\in L^1[a,b]:y_-\le x\le y_+\}$.
If $(x_n)$ is a monotone sequence in
$L^1[a,b]$, then  $(F(x_n))$ is a monotone sequence in $[y_-,y_+]$.
Thus, by monotone convergence theorem, $(F(x_n))$ converges in
$L^1[a,b]$, and the limit function belongs to $[y_-,y_+]$.
It then follows from \cite[Theorem 1.2.2]{HeiLak94} that $F$ has
the smallest fixed point $y_*$ and the greatest fixed point
$y^*$ in $L^1[a,b]$, and hence also in $C[a,b]$,
 since $F[L^1[a,b]]\subset C[a,b]$ by the hypothesis (A0).
These fixed points are by Lemma \ref{L20} also the smallest
and greatest continuous solutions of \eqref{E1}.
Moreover, according to  \cite[Theorem 1.2.2]{HeiLak94},
\begin{equation}
y_*=\min\{x\in[y_-,y_+]:F(x)\le x\}, \quad
y^*=\max\{x\in[y_-,y_+]:x\le F(x)\}.
\end{equation}
Applying these relations it is easy to show that both $y_*$ and
$y^*$ are increasing with respect to $F$, and hence, by \eqref{E2}
and \eqref{E20}, also with respect to $f$ and to $c$.
\end{proof}

As noticed in \cite{ET081}, the distributional Denjoy integral
contains the wide Denjoy integral, and hence also
integrals called Riemann, Lebesgue, Denjoy and Henstock-Kurzweil.
In the next corollary the Henstock-Kurzweil integral can be replaced
 by any of those integrals listed above.

\begin{corollary}\label{C31}
The results of Theorem \ref{T31} are valid if
$f(x)$ is for every $x\in L^1[a,b]$ the distributional derivative
of a function ($\sideset{^K}{}{\!\!\!\int}$ denotes the 
Henstock-Kurzweil integral)
\begin{equation}\label{E31}
G(x)(t) = \sum_{i=1}^nH_i(t)\,\sideset{^K}{}{\!\!\!\int_a^t} g_i(x)+H(t), \quad t\in[a,b],
\end{equation}
where for each $i=1,\dots,n$, $H_i$ is nonnegative-valued,
$H_i$, $H\in \mathcal B_c[a,b]$, and  $g_i(x):[a,b]\to\mathbb{R}$
satisfies the following hypotheses.
\begin{itemize}
\item[(GI1)] $g_i(x)$ is Henstock-Kurzweil integrable on $[a,b]$
 for all $x\in L^1[a,b]$.
\item[(GI2)] There exist Henstock-Kurzweil integrable functions
$\underline g_i,\overline g_i:[a,b]\to \mathbb{R}$ such that
$$
\sideset{^K}{}{\!\!\!\int_a^t} \underline g_i\le \sideset{^K}{}{\!\!\!\int_a^t} g_i(x)\le \sideset{^K}{}{\!\!\!\int_a^t} g_i(y)
\le \sideset{^K}{}{\!\!\!\int_a^t} \overline g_i,
$$
for $t\in[a,b]$, whenever $x\le y$ in $L^1[a,b]$.
\end{itemize}
\end{corollary}

\begin{proof}
The hypotheses imposed above ensure that \eqref{E31} defines for
every $x\in L^1[a,b]$ a continuous  function
$G(x):[a,b]\to\mathbb{R}$. Moreover, the distributional derivatives
$f(x)$ of $G(x)$ satisfy the hypotheses (A0), (B0) and (C0), where
$$
 \sideset{^c}{}{\!\!\!\int_a^t}f_- = \sum_{i=1}^nH_i(t)\,\sideset{^K}{}{\!\!\!\int_a^t} \underline g_i+H(t),\quad
 \sideset{^c}{}{\!\!\!\int_a^t} f_+ = \sum_{i=1}^nH_i(t)\,\sideset{^K}{}{\!\!\!\int_a^t} \overline g_i+H(t),\quad
t\in[a,b].
$$
Thus the Cauchy problem \eqref{E1} has by Theorem \ref{T31} the
smallest and greatest solutions in $C[a,b]$, and they are increasing
with respect to $f$ and to $c$.
\end{proof}

\begin{remark}\label{R21} \rm
 Under the hypotheses of Theorem \ref{T31}  the smallest fixed point
$y_*$ of $F$ is by \cite[Theorem 1.2.1]{HeiLak94} the maximum of
the chain $C$ of $L^1[a,b]$ that is well-ordered; i.e.,
every nonempty subset of $C$ has the smallest element, and that
satisfies
\begin{itemize}
\item[(I)] $y_- = \min C$,  and if $y_- < x$, then $x\in C$
 if and only if  $x = \sup F[\{y\in C:y < x\}]$.
\end{itemize}
The smallest elements of $C$ are $F^n(y_-)$, $n\in\mathbb N_0$,
as long as $F^n(y_-)=F(F^{n-1}(y_-))$  is defined and
$F^{n-1}(y_-)<F^n(y_-)$, $n\in\mathbb N$. If $F^{n-1}(y_-)=F^n(y_-)$
for some $n\in\mathbb N$, there  is the smallest such an $n$, and
$F^n(y_-)=\sup F[C]=y_*$ is the smallest fixed point of $F$ in
$C[a,b]$.
If $x_\omega=\underset{n\in\mathbb N}{\sup}F^n(y_-)$ is defined
in $L^1[a,b]$ and is a strict upper bound of
$\{F^n(y_-)\}_{n\in\mathbb N}$, then $x_\omega$ is the next
element of $C$. If $x_\omega=F(x_\omega)$, then $y_*=x_\omega$,
otherwise the next elements of $C$ are of the form $F^n(x_\omega)$,
$n\in\mathbb N$, and so on.

The greatest fixed point $y^*$ of $F$ is by
\cite[Proposition 1.2.1]{HeiLak94} the minimum of the chain $D$
of $L^1[a,b]$ that is inversely well-ordered; i.e.,
 every nonempty subset of $D$ has the greatest element, and that
satisfies
\begin{itemize}
\item[(II)] $y_+ = \max D$, and if $y_+ > x$, then $x\in C$
 if and only if  $x = \inf F[\{y\in D: y > x\}]$.
\end{itemize}
The greatest elements of $D$ are $n$-fold iterates $F^n(y_+)$,
as long as they are defined and $F^{n}(y_+)<F^{n-1}(y_+)$, etc.
\end{remark}


\begin{example}\label{Ex31} \rm
Consider the Cauchy problem
\begin{equation}\label{E35}
 y'=f(y), \quad  y(0)=0,
\end{equation}
where $f(x)$  is for each $x\in L^1[0,1]$ the distributional
derivative of the function $G(x) \in \mathcal B_c[0,1]$, defined by
\begin{equation}\label{E33}
 G(x)(t)= H_1(t)\sideset{^K}{}{\!\!\!\int_0^t} g_1(x)+H(t), \quad t\in[0,1],
\end{equation}
where $H\in \mathcal B_c[0,1]$, $H_1$ is the Heaviside step
function on $[0,1]$, and
$$
g_1(x)(t)=\arctan\Big([10^5\int_0^1(x(t)-H(t))\,dt]10^{-4}\Big)
\Big(\frac 1t\cos(\frac 1t)-\sin(\frac 1t)+1\Big),
$$
where $[\cdot]$ denotes the greatest integer function.
Denote
$$
y_\pm(t):=H(t)\pm 4t(1-\sin(\frac 1t)),\quad t\in(0,1], \;
y_\pm(0)=0,
$$
and let $f_\pm$ be distributional derivatives of $y_\pm$.
The validity of the hypotheses (GI1) and (GI2) is easy to verify.
Thus, the Cauchy problem \eqref{E35}
has by Corollary \ref{C31} the smallest and greatest solutions
in $\mathcal B_c[0,1]$.

Calculating the successive approximations
$$
y_{n+1}=G(y_n), \; y_0=y_- \quad \text{and} \quad
z_{n+1}=G(z_n), \; z_0=y_+,
$$
we see,  that $(y_n)_{n=0}^{24}$ is strictly increasing,
that $(z_n)_{n=0}^{24}$ is strictly decreasing, that
$y_{24}=G(y_{24})$, and that $z_{24}=G(z_{24})$.

Thus $y_*=y_{24}$ and $y^*=z_{24}$ are by Remark \ref{R21}
and Lemma \ref{L20} the smallest and greatest solutions
of  \eqref{E35} in $\mathcal B_c[0,1]$ when $f(x)$ is for each
$x\in L^1[a,b]$ the distributional derivative of $G(x)$ defined
 by \eqref{E35}. The exact formulas of $y_*$ and $y^*$ are
\begin{gather*}
y_*(t)=\arctan\left(\frac{8693}{10000}\right)t(\sin(\frac 1t)-1)+H(t),
\; t\in(0,1], \quad y_*(0)=0,\\
y^*(t)= \arctan\left(\frac{869}{1000}\right)t(1-\sin(\frac 1t))+H(t),\;
t\in(0,1], \quad y^*(0)=0.
\end{gather*}
\end{example}

\section{Existence of minimal and maximal solutions}\label{S3}

In this section existence results are derived for local and global
minimal and maximal continuous solutions of the distributional
Cauchy problem
\begin{equation}\label{E36}
 y'=f(y), \quad  y(a)=0.
\end{equation}
The  space $L^1[a,b]$, ordered a.e. pointwise and normed by
$L^1$-norm: $\|x\|_1=\int_a^b|x(s)|\,ds$, is an ordered normed
space $E:=(L^1[a,b],\|\cdot\|_1)$ having the
following properties ($\theta$ denotes the zero-element of $L^1[a,b]$).
\begin{itemize}
\item [(E0)] Bounded and monotone sequences of $E$ converge.
\item [(E1)] $x^+=\sup\{\theta,x\}$ exists, and
$\|x^+\|_1\le \|x\|_1$ for every $x\in E$.
\end{itemize}
Denote
\begin{equation}\label{E30}
B(\theta,R)=\{x\in L^1[a,b]:\|x\|_1\le R\}.
\end{equation}
Because of the properties (E0) and (E1) we obtain the following
result as a consequence of \cite[Theorem 2.44]{CH11}.

\begin{lemma}\label{L31}
Given a subset $P$ of $L^1[a,b]$, assume that $F:P\to P$ is increasing,
and that $F[P]\subseteq B(\theta,R)\subseteq P$ for some $R> 0$.
Then $F$ has
\begin{itemize}
\item[(a)]  minimal and maximal fixed points;
\item[(b)] smallest and greatest fixed  points $y_*$ and
$y^*$ in the order
interval $[\underline y,\overline y]$ of $P$,  where $\underline y$
is the greatest solution of $y=-(-F(y))^+$, and  $\overline y$
is the smallest solution of $y=F(y)^+$.
\end{itemize}
Moreover, $y^*$, $y_*$,  $\underline y$ and  $\overline y$ are all
increasing with respect to $F$.
\end{lemma}

As an application of Lemma \ref{L31} we obtain the following result.

\begin{proposition}\label{P31}
Assume that the hypotheses (A0) and (B0) hold, and that the
primitives $F(x)$ of $f(x)$ in $\mathcal B_c[a,b]$ satisfy the
following hypothesis for some $R > 0$.
\begin{itemize}
\item[(C1)]
 $\|F(x)\|\le R$ for all $x\in L^1[a,b]$, $\|x\|_1\le R$.
\end{itemize}
Then the Cauchy problem \eqref{E36} has
\begin{itemize}
\item[(a)]  minimal and maximal solutions in $B(\theta,R)$;

\item[(b)] smallest and greatest solutions $y_*$ and $y^*$ in the order
interval $[\underline y,\overline y]$ of $B(\theta,R)$,
where $\underline y$
is the greatest solution of $y=-(-F(y))^+$, and  $\overline y$
is the smallest solution of $y=F(y)^+$.
\end{itemize}
Moreover, $y^*$, $y_*$,  $\underline y$ and  $\overline y$ are all
increasing with respect to $f$.
\end{proposition}

\begin{proof}
For each $x\in L^1[a,b]$ the distribution $f(x)$ has by (A0)
the primitive  $F(x)$ in $\mathcal B_c[a,b]\subset L^1[a,b]$.
The hypotheses (B0) and (C1) imply that $F$ satisfies the hypotheses
of Lemma \ref{L31} when
$P=B(\theta,R)$. Thus, by Lemma \ref{L31}(a), $F$ has in
$B(\theta,R)$ minimal and maximal fixed points, which are
by Lemma \ref{L20} also minimal and maximal solutions of \eqref{E36}
 in $B(\theta,R)$. The results of (b) and the last result of
 theorem follow from the corresponding results of Lemma \ref{L31}
and from \eqref{E20}.
\end{proof}

As for the existence of minimal and maximal  solutions of \eqref{E36}
in the whole $\mathcal B_c[a,b]$, we have
the following result.

\begin{theorem}\label{T22}
The distributional Cauchy problem \eqref{E36} has minimal and maximal
solutions in $\mathcal B_c[a,b]$, and they are increasing with respect
to $f$, if the hypotheses (A0) and (B0) hold, and if the
primitives $F(x)$ of $f(x)$ in $\mathcal B_c[a,b]$ satisfy the
following hypothesis.
\begin{itemize}
\item[(C2)]
 $\|F(x)\|_1\le Q(\|x\|_1)$ for all $x\in L^1[a,b]$, where
$Q:\mathbb{R}_+\to\mathbb{R}_+$ is increasing, $R=Q(R)$ for
some $R>0$, and $r\le Q(r)$ implies $r\le R$.
\end{itemize}
\end{theorem}

\begin{proof}
Hypothesis (C2) implies that
$$
\|F(x)\|_1\le Q(\|x\|_1)\le Q(R)=R\quad \text{for every }
 x\in B(\theta,R).
$$
Thus hypothesis (C1) holds, whence
 \eqref{E36} has the by Proposition \ref{P31} minimal and maximal
solutions  in $B(\theta,R)$, and they are increasing with
respect to $f$.

If $y\in B(\theta,r)$ is a solution of \eqref{E1}, then $y$
is a fixed point of $F$ by Lemma \ref{L20}. Hypothesis (C2)
with $r=\|y\|_1$ implies that
$$
\|y\|_1=\|F(y)\|_1\le Q(\|y\|_1)\le Q(R)=R.
$$
Thus all the solutions of \eqref{E36} are in $B(\theta,R)$.
The assertion follows from the above results.
\end{proof}

\section{Existence and uniqueness results}\label{S4}


In this section, conditions are presented for  distributions
$f(x)$, $x\in C[a,b]$, which ensure that \eqref{E1} has for
each $c\in\mathbb{R}$ a unique solution.
Denoting  $\lceil x\rceil =| x(\cdot)|$, $x\in C[a,b]$, we have the
following fixed point result that is basis of our main existence
and uniqueness theorem.

\begin{proposition}[{\cite[Theorem 1.4.9]{HeiLak94}}]\label{P410.1}
 Let $F:C[a,b]\to C[a,b]$ satisfy the hypothesis:
\begin{itemize}
\item[(F0)] There exists a $v\in C_+[a,b]=\{u\in C[a,b]:\theta\le u\}$
and an increasing mapping
$Q:[\theta,v]\to C_+[a,b]$ satisfying $Qv(t) < v(t)$ and
$Q^nv(t)\to 0$  for
each $t\in [a,b]$, such that
\begin{equation}\label{E410.5}
\lceil  F(x) - F(z)\rceil\le Q\lceil  x - z\rceil
\end{equation}
for all $x, z\in C[a,b]$, $\lceil  x- z\rceil\le v$.
\end{itemize}
Then for each $y_0\in C[a,b]$ the sequence $(F^n(y_0))_{n=0}^\infty$
converges uniformly on $[a,b]$ to a unique fixed point of $F$.
\end{proposition}

In our main existence and uniqueness theorem we rewrite
the inequality \eqref{E410.5} in terms of distributions.
The modulus $|g|$ of a distribution $g$ on $(0,b)$ that is $DD$
integrable on $[a,b]$ is defined by
\begin{equation}\label{E0.01}
|g|:=\sup\{g,-g\},
\end{equation}
where the supremum is taken in the partial ordering $\preceq$
defined by \eqref{E20}. $|g|$ exists because $\preceq$ is a
lattice ordering (cf. \cite[Sect. 9]{ET081}).

Now  we are able to prove an existence and uniqueness theorem for
the  solution of the Cauchy problem \eqref{E1}.


\begin{theorem}\label{T411.1}
Assume that distributions $f(x)$, $x\in C[a,b]$, and $h(w)$,
$w\in[\theta,v]$, $v\in C_+[a,b]$, are $DD$ integrable on $[a,b]$,
and that
\begin{equation}\label{E411.00}
|f(x)-f(z)|\preceq h(\lceil x-z \rceil)
\end{equation}
for all $x,\,z\in C[a,b]$ with $\lceil x-z\rceil\le v$, and that
 $Q:[\theta,v]\to C_+[a,b]$, defined by
\begin{equation}\label{E411.1}
Q(w)(t)= \sideset{^c}{}{\!\!\!\int_a^t} h(w), \quad \theta\le w\le v, \ a\le t\le b,
\end{equation}
is increasing, $Q(v)(t))<v(t)$ and $Q^n(v)(t)\to 0$ for each
$t\in [a,b]$.
Then the Cauchy problem \eqref{E1} has a
unique solution $y$ in $C[a,b]$.  Moreover, $y$ is for each choice
of $y_0\in C[a,b]$ the
uniform limit of the sequence $(y_n)_{n=0}^\infty$ of the successive
approximations
\begin{equation}\label{E411.3}
y_{n+1}(t) = c+ \sideset{^c}{}{\!\!\!\int_a^t} f(y_n), \quad t\in [a,b], \;
n\in\mathbb N_0.
\end{equation}
\end{theorem}

\begin{proof}
  It follows from \eqref{E20} and \eqref{E0.01} that  \eqref{E411.00}
holds if and only if
\begin{equation}\label{E411.0}
\big| \sideset{^c}{}{\!\!\!\int_a^t} f(x)-\sideset{^c}{}{\!\!\!\int_a^t} f(z)\big|\le
 \sideset{^c}{}{\!\!\!\int_a^t} h(\lceil x-z \rceil), \quad \text{for all } t\in[a,b].
\end{equation}
Equation \eqref{E2}
defines  a mapping $F:C[a,b]\to C[a,b]$.
The given hypotheses and the equivalence of \eqref{E411.00} and
\eqref{E411.0} imply that
the operators $F$ and $Q$, defined by \eqref{E2} and \eqref{E411.1},
satisfy the hypotheses of Proposition \ref{P410.1}. Thus the iteration
sequence $(F^n(y_0))_{n=0}^\infty$, which equals to the sequence
$(y_n)_{n=0}^\infty$ of successive approximations \eqref{E411.3},
converges for every choice of $y_0\in C[a,b]$ uniformly on $[a,b]$
to a unique fixed point $y$ of $F$.  This result and Lemma \ref{L20}
imply  that $y$ is the uniquely determined continuous
solution of the Cauchy problem \eqref{E1}.
\end{proof}


The following result will be applied to obtain a special case of
 Theorem \ref{T411.1}.

\begin{lemma}[{\cite[Lemma 6.11]{CH11}}]\label{L411.2}
Assume that the function $q:[a,b]\times[0,r]\to\mathbb{R}_+$, $r > 0$,
satisfies the  condition.
\begin{itemize}
\item[(Q0)]  $q(\cdot,x)$
is measurable for all $x\in[0,r]$, $q(\cdot,r)\in L^1([a,b],
\mathbb{R}_+)$, $q(t,\cdot)$ is increasing and right-continuous
for a.e. $t\in [a,b]$, and the zero-function is the only
absolutely continuous
(AC) solution with  $u_0=0$ of the Cauchy problem
\begin{equation}\label{E411.2}
u'(t) =  q(t,u(t))  \text{ a.e. on } [a,b],\quad u(a) = u_0.
\end{equation}
\end{itemize}
Then there exists  an $r_0>0$ such that the Cauchy problem
\eqref{E411.2} has for every $u_0\in [0,r_0]$ the smallest AC
solution $u=u(\cdot,u_0)$, which is increasing with respect to
$u_0$. Moreover, $u(t,u_0)\to 0$ uniformly over $t\in [a,b]$ when
$u_0\to 0$.
\end{lemma}

The next result is an application of Lemma \ref{L411.2} and
Theorem \ref{T411.1}.

\begin{proposition}\label{P411.0}
 The results of Theorem \ref{T411.1} are valid if
the distributions $f(x)$, $x\in C[a,b]$, are $DD$ integrable
on $[a,b]$ and satisfy the following hypothesis.
\begin{itemize}
\item[(F0)] There exists an $r > 0$ such that \eqref{E411.00} holds
for all $x,z\in C[a,b]$ with $\|x-z\|_\infty \le r$ and for
all $t\in [a,b]$, where $h$ is the Nemytskij operator defined by
\begin{equation}\label{E411.02}
h(w) = q(\cdot,w(\cdot)), \quad w\in C_+[a,b], \ \|w\|_\infty\le r,
\end{equation}
and $q:[a,b]\times[0,r]\to\mathbb{R}_+$
satisfies the hypothesis (q) of Lemma \ref{L411.2}.
\end{itemize}
\end{proposition}

\begin{proof}
 According to Lemma \ref{L411.2} the Cauchy
problem \eqref{E411.2} has for some $u_0=r_0> 0$ the smallest AC
solution $v=u(\cdot,r_0)$,  and $r_0\le v(t) \le r$ for each $t\in
[a,b]$. Since $q(s,\cdot)$ is increasing and right-continuous
in $[0,r]$ for a.e. $s\in [a,b]$, and because $q(\cdot, x)$
is measurable for all $x\in[0,r]$ and   $q(\cdot,r)$ is
Lebesgue integrable, it follows from
\cite[Theorem 2.1.1 and Remarks 2.1.1]{CarHei00}
that $q(\cdot,u(\cdot))$ is Lebesgue integrable whenever $u$
belongs to the order interval $[\theta,v]$ of $C_+[a,b]$.
Thus the equation \eqref{E411.1}, where $h$ is the Nemytskij
operator defined by \eqref{E411.02},
defines a  mapping $Q:[\theta,v]\to C_+[a,b]$.
Condition (Q0) ensures that $Q$ is increasing, and the choices
of $r_0$ and $v$ imply that
\begin{equation}\label{E411.03}
r_0 + Q(v) = v.
\end{equation}
Thus $v(t)-Q(v)(t)=r_0>0$ for each $t\in[a,b]$. The sequence
$(Q^n(v))_{n=0}^\infty$ is decreasing because $q(t,\cdot)$ is
increasing for a.e. $t\in [a,b]$. Noticing that the functions
$Q^n(v)$ are also continuous, the reasoning similar to that applied
in the proof of Lemma  \ref{L411.2} shows
that $(Q^n(v))_{n=0}^\infty$ converges uniformly on $[a,b]$
to the zero function.
The above proof shows that the hypotheses of Theorem \ref{T411.1} hold.
\end{proof}


\section{Dependence on the Initial Value}\label{S5}

We shall first prove that under the hypotheses
of Proposition \ref{P411.0} the
difference of   solutions $y$ of \eqref{E1} belonging to initial
values $c$ and $\hat c$, respectively, can be estimated  by the
\emph{smallest solution} of the comparison problem \eqref{E411.2}
with initial value $u_0=|c-\hat c|$. This estimate implies by
Lemma \ref{L411.2} the continuous dependence of $y$ on $c$.

\begin{proposition}\label{P411.1}
 Let the distributions $f(x)$, $x\in C[a,b]$,
satisfy the hypotheses  of Proposition \ref{P411.0}.
If $y = y(\cdot,c)$ denotes
the  solution of the Cauchy problem \eqref{E1} and
$u=u(\cdot,u_0)$ the smallest solution of the Cauchy problem
\eqref{E411.2}, then for all $c,\,\hat c\in \mathbb{R}$, with
$| c-\hat c|$ small enough,
\begin{equation}\label{E411.5}
| y(t,c) - y(t,\hat c))|\le u(t,| c-\hat c|),
 \quad t\in [a,b].
\end{equation}
In particular, $y(\cdot,c)$ depends continuously on $c$ in the
sense that $y(t,\hat c)\to y(t,c)$ uniformly over $t\in [a,b]$ as
$\hat c\to c$.
\end{proposition}

\begin{proof}
 Assume that $c,\hat c\in \mathbb{R}$, and that
$| c-\hat c|\le r_0$, where $r_0$ is chosen as in Lemma
\ref{L411.2}. The solutions $y=y(\cdot,c)$ and
$\hat y = y(\cdot,\hat c)$ exist by Proposition \ref{P411.0},
and they satisfy by Lemma \ref{L20} the fixed point equations
$$
y(t)=F(y)(t) = c + \sideset{^c}{}{\!\!\!\int_a^t}f(y),\ \hbox {and}\
\hat y(t)= \hat F(\hat y)(t) = \hat c +  \sideset{^c}{}{\!\!\!\int_a^t}f(\hat
y), \quad t\in [a,b].
$$
Moreover, $F$ satisfies by the proof of Proposition \ref{P411.0}
the hypotheses of Proposition \ref{P410.1}
with $Q$ defined by \eqref{E411.1}, or equivalently, by
$$
Q(w)(t) = \int_a^t q(s,w(s))\,ds, \quad t\in [a,b],
$$
and $u=u(\cdot,|c-\hat c|)$ is the smallest AC solution of
$$
u=|c-\hat c|+Q(u).
$$
Denote
$$
V = \{y\in C[a,b]: \lceil y - \hat y\rceil\le u\}.
$$
Since $Q$ is increasing, and since
$$
F(\hat y)(t)-\hat y(t)= F(\hat y)(t)-\hat F(\hat y)(t)=c-\hat c
$$
for all $t\in [a,b]$, we have for every $y\in V$,
\begin{align*}
\lceil F(y)-\hat y\rceil
&\le \lceil F(\hat y)-\hat y\rceil+\lceil F(y)-F(\hat y)\rceil\\
&\le \lceil F(\hat y)-\hat y\rceil +Q(\lceil y-\hat y\rceil)\\
&\le |c-\hat c|+Q(u)=u.
\end{align*}
Thus $F[V]\subseteq V$. Since $\hat y\in V$, then $(F^n(\hat y))\in V$
for every $n\in \mathbb{N}_0$. The uniform limit
$y=\lim_nF^n(\hat y)$ exists by Theorem \ref{T411.1} and is
the solution of \eqref{E1}. Because $V$ is closed, then $y\in V$,
so that $\lceil y-\hat y\rceil\le u$. This proves \eqref{E411.5}.
According to Lemma \ref{L411.2}, $u(t,|c-\hat c|)\to 0$ uniformly
over $t\in [a,b]$ as $|c-\hat c|\to 0$. This result and
\eqref{E411.5} imply that the last assertion of the proposition
is true.
 \end{proof}


\begin{remark}\label{R411.1}\rm
If in condition {\rm (Q0)}, $r=\infty$ and
$q(\cdot,z)\le \overline q\in L^1([a,b]$ for each
$z\in\mathbb{R}_+$, then \eqref{E411.5} holds
for all $c,\,\hat c\in \mathbb{R}$.

The hypotheses imposed on $q:[a,b]\times [0,r]\to \mathbb{R}_+$
in (Q0) hold if $q(t,\cdot)$ is increasing for a.e. $t\in [a,b]$,
and if $q$ is an $L^1$-bounded Carath\'eodory function such that
the following local Kamke's condition holds.
$$
u\in C([a,b],[0,r])\text{ and $u(t)\le \int_a^tq(s,u(s))\,ds$
 for all $t\in [a,b]$ imply } \ u(t)\equiv 0.
$$
The hypotheses of (Q0) are valid also for the function
$q:[a,b]\times [0,r]\to \mathbb{R}_+$, defined by
\begin{equation}\label{E411.05}
q(t,s)= p(t)\,\phi(s)\quad t\in[a,b], \ s\in [0,r],
\end{equation}
where $p\in L^1([a,b],\mathbb{R}_+)$, $\phi:[0,r]\to\mathbb{R}_+$
is increasing and right-continuous, and
$\int_0^r\frac{dv}{\phi(v)} =\infty$.

Let $\ln_n$ and $\exp_n$ denote $n$-fold
iterated logarithm and exponential functions, respectively.
The functions $\phi_n$,
$n\in\mathbb{N}$, defined by $\phi_n(0)=0$, and
$$
\phi_n(s) = s\prod_{j=1}^n\ln_j\frac 1s, \quad 0< s\le
\exp_n(1)^{-1},
$$
have properties assumed above for the function $\phi$ when
$r=\exp_n(1)^{-1}$. These properties hold also for the function
$\phi(s)=s$, $s \ge 0$. Thus the following result is a special
case of Theorem \ref{T411.1} and Proposition \ref{P411.1}.
\end{remark}

\begin{corollary}\label{C41}
The Cauchy problem \eqref{E1} has for each $c\in \mathbb{R}$ a
unique  solution $y=y(\cdot,c)$, if $f(x)$ is $DD$ integrable on
$[a,b]$ for all $x\in C[a,b]$, and if there exists a Lebesgue
integrable function $p:[a,b]\to \mathbb{R}_+$ such that
$$
\big|  \sideset{^c}{}{\!\!\!\int_a^t} f(y)- \sideset{^c}{}{\!\!\!\int_a^t}f(z)\big|\le
\int_a^tp(s)|y(s)-z(s)|\,ds
$$
for all $y,z\in C[a,b]$ and for
all $t\in [a,b]$. Moreover,
$$
| y(t,c) - y(t,\hat c))|\le e^{\int_a^tp(s)ds}| c-\hat c|, \quad
 t\in [a,b], \; c,\,\hat c\in\mathbb{R}.
$$
\end{corollary}

In linear case we obtain the following consequence from
Corollary \ref{C41}.

\begin{corollary}\label{C42}
For each $c\in \mathbb{R}$, the linear Cauchy problem
$$
y'=h+py, \quad y(0)=c,
$$
has a unique solution in $C[a,b]$  whenever the distribution
$h$ is  $DD$ integrable on $[a,b]$, and $p:[a,b]\to \mathbb{R}_+$
is Lebesgue integrable.
\end{corollary}


\begin{remark}\label{R61.1} \rm The Cauchy problem
\begin{equation}\label{E611.2}
y'(t) =  g(t,y(t))  \text{ a.e. on } [a,b],\quad y(a) = c,
\end{equation}
is a special case of problem  \eqref{E1} when $f$ is the Nemytskij
operator associated with the function
$g:[a,b]\times\mathbb{R}\to \mathbb{R}$ by
$$
f(x):=g(\cdot,x(\cdot)), \quad x\in L^1[a,b].
$$
\end{remark}

For instance,  Theorem \ref{T31} implies the following result.

\begin{corollary}\label{C600}
The Cauchy problem \eqref{E611.2} has the smallest and greatest
continuous solutions that are increasing with respect to $g$
and $c$, if  the following hypotheses are valid.
\begin{itemize}
\item[(G0)] $g(\cdot,x(\cdot))$ is Henstock-Kurzweil integrable
 on $[a,b]$ for every $x\in L^1[a,b]$.
\item[(G1)] $\sideset{^K}{}{\!\!\!\int_a^t} g(s,x(s))\,ds\le \sideset{^K}{}{\!\!\!\int_a^t} g(s,y(s))\,ds$
 for all $t\in[a,b]$ whenever $x\le y$ in $L^1[a,b]$.
\item[(G2)] There exist Henstock-Kurzweil integrable functions
 $g_\pm:[a,b]\to\mathbb{R}$ such that\\
 $\sideset{^K}{}{\!\!\!\int_a^t} g_-(s)\,ds\le \sideset{^K}{}{\!\!\!\int_a^t} g(s,x(s))\,ds
 \le \sideset{^K}{}{\!\!\!\int_a^t} g_+(s)\,ds$ for all $x\in L^1[a,b]$ and  $t\in[a,b]$.
\end{itemize}
\end{corollary}

\begin{thebibliography}{9}

\bibitem{CarHei00} S. Carl and  S. Heikkil\"a;
Nonlinear Differential Equations in Ordered Spaces.
Chapman \& Hall/CRC, Boca Raton (2000).

\bibitem{CH11} S. Carl, S. Heikkil\"a;
 {Fixed Point Theory in Ordered Spaces and Applications},
Springer, New York, (2011).

\bibitem{FrJ99} F. G. Friedlander and M. Joshi;
 Introduction to the theory of distributions, Cambridge,
Cambridge University Press (1999).

\bibitem{HeiLak94} S. Heikkil\"a and V. Lakshmikantham;
 Monotone Iterative Techniques for Discontinuous Nonlinear Differential
Equations, Marcel Dekker Inc., New York (1994).

\bibitem{Sch66} L. Schwartz;
 Th\'{e}orie des distributions, Paris, Hermann (1966).

\bibitem{ET081} E. Talvila;
 \emph{The distributional Denjoy integral}, Real Anal.
Exchange \textbf{33} (2008), 51--82.

\bibitem{Tvr94} M. Tvrd\'{y};
 \emph{Linear distributional differential equations of the second order},
Mathematica Bohemica \textbf{119},4 (1994), 419--436.

\end{thebibliography}

\end{document}