\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 57, 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 (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/57\hfil Existence of solutions]
{Existence of solutions of an integral equation
of Chandrasekhar type in the theory of radiative transfer}
\author[J. Caballero, A. B. Mingarelli, K. Sadarangani\hfil EJDE-2006/57\hfilneg]
{Josefa Caballero, Angelo B. Mingarelli, Kishin Sadarangani} % in alphabetical order
\address{Josefa Caballero \newline
Departamento de Matem\'aticas, Universidad de
Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las
Palmas de Gran Canaria, Spain}
\email{jmena@ull.es}
\address{Angelo B. Mingarelli \newline
School of Mathematics and Statistics, Carleton University, Ottawa,
Ontario, K1S 5B6, Canada}
\email{amingare@math.carleton.ca}
\address{Kishin Sadarangani \newline
Departamento de Matem\'aticas, Universidad de
Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las
Palmas de Gran Canaria, Spain}
\email{ksadaran@dma.ulpgc.es}
\date{}
\thanks{Submitted March 3, 2006. Published May 1, 2006.}
\subjclass[2000]{45M99, 47H09}
\keywords{Measure of noncompactness; Banach algebra;
integral equation; \hfill\break\indent
functional equation; fixed point; Volterra;
Chandrasekhar H-functions; \hfill\break\indent
Chandrasekhar integral equation; radiative transfer}
\begin{abstract}
We give an existence theorem for some functional-integral
equations which includes many key integral and functional
equations that arise in nonlinear analysis and its applications.
In particular, we extend the class of characteristic functions
appearing in Chandrasekhar's classical integral equation from
astrophysics and retain existence of its solutions. Extensive
use is made of measures of noncompactness and abstract fixed
point theorems such as Darbo's theorem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\section{Introduction}
The study of Chandrasekhar's integral equation \cite{17}
$$
x(t)= 1+ x(t)\int_0^1\frac{t}{t+s}\varphi(s)\,x(s)\,ds
$$
has been a subject of much investigation since its appearance
around fifty years ago. It arose originally in connection with
scattering through a homogeneous semi-infinite plane atmosphere
and has since been used to model diverse forms of scattering via
the H-functions of Chandrasekhar. These in turn are used to write
down specific solutions of the integral equation. The problem of
approximating such solutions is still much in vogue today and many
efficient methods of calculation of these functions have been
found, e.g., see \cite{effi} for details and \cite{jonq} for an
update on the method. Insofar as the theoretical question of the
existence of solutions is concerned, we note that it is known that
in some cases as many as two solutions may exist to one and the
same equation, [\cite{them}, Chapter 2]. We show that an abstract
framework exists in which Chandrasekhar's integral equation above
takes part as a special case. Indeed, we show that for said
equation, the mere continuity of the characteristic function
$\varphi (t)$ along with $\varphi (0)=0$ will guarantee the
existence of at least one solution of (\ref{chandra}). Recall that
normally one assumes that $\varphi (t)$ is an even polynomial, \cite{17}.
Associated with the usual equation (\ref{chandra}) is the modified
integral equation (\ref{chandra33}) first suggested by
Chandrasekhar \cite[Chapter 5, Sect. 38]{17}, namely
\begin{equation}
\label{chandra33} \frac{1}{x(t)}
= (1-2\psi_0)^{1/2}+ \int_0^1\frac{s}{t+s}\psi(s)\,x(s)\,ds,
\end{equation}
where $\psi_0= \int_{0}^{1}\psi(x)\, dx$. Theoretical results,
see (\ref{chandra3}) below, give that $0 < (1-2\psi_0)^{1/2} \leq 1$.
The usefulness of (\ref{chandra33}) lies in part that it is
better suited for numerical approximations than the original one,
(\ref{chandra}). It is known that (\ref{chandra33}) allows for
two solutions each of one distinct sign, see \cite{effi}.
Multiplying (\ref{chandra33}) by $x(t)/(1-2\psi_0)$ and rearranging
terms we find the modified form
\begin{equation}
\label{chandra34}
y(t) = \frac{1}{1-2\psi_0}- \int_0^1\frac{s}{t+s}\psi(s)\,y(t)\,y(s)\,ds,
\end{equation}
where $y(t) = x(t)/(1-2\psi_0)$, provided $\psi_0 \neq 1/2$,
the non-critical case, in which case (\ref{chandra33}) and
(\ref{chandra34}) are equivalent.
On the other hand, one could also start the process with
equation (\ref{chandra34}). In this case, the existence of at
least one real solution of (\ref{chandra34}) is a consequence of
our abstract theorem below, and this for basically any choice of
a characteristic function $\psi(s)$ in the sense that no additional
assumption of the characteristic function at $s=0$ is required in
contrast to the original equation.
Multidimensional (matrix) generalizations of Chandrasekhar's
H-equation can be found in \cite{jmp1} and the references therein.
In this paper we study the existence of solutions of certain
functional integral equations (so, possibly containing delays)
which contain as particular cases many important integral and
functional equations, for example: the nonlinear Volterra
integral equation, and the integral equation of Chandrasekhar
which gives rise to solutions expressible in terms of
Chandrasekhar's H-functions (see \cite{17} for more details). The
main tool used in our research is the fixed-point theorem for the
product of two operators which satisfy the Darbo condition with
respect to a measure of noncompactness in the Banach algebra of
continuous functions on the interval $[0,a]$. Applications to the
theory of radiative transfer are provided at the end of
Section~\ref{three}, while specific applications to other integral
equations such as those mentioned above are given in
Section~\ref{exa}.
\section{Notation and auxiliary facts}
We recall basic results which we will need further on.
Assume that $E$ is a real Banach space with norm $\| \cdot \|$ and
zero element, $0$. Denote by $B(x,r)$ the closed ball centered at $x$ with
radius $r$ and by $B_{r}$ the ball $B(0,r)$. For $X$ a nonempty subset of $E$ we
denote by $\overline{X}$, $Conv X$ the closure and the convex closure of $X$, respectively.
We denote the standard algebraic operations on sets by the symbols $\lambda X$ and $X+Y$ . Finally,
let us denote by ${\mathfrak{M}}_E$ the family of nonempty bounded subsets of $E$
and by ${\mathfrak{N}}_E$ its subfamily consisting of all relatively compact sets.
\begin{definition}[\cite{3}] \label{def1} \rm
A function $\mu : {\mathfrak{M}}_E\to [0,\infty )$ is
said to be a {\em{regular measure of noncompactness}} in the space
$E$ if it satisfies the following conditions:
\begin{enumerate}
\item $\ker \mu = 0 \Longleftrightarrow X \in {\mathfrak{N}}_E$.
\item $X \subset Y \Rightarrow \mu(X) \leq \mu (Y)$.
\item $\mu ({\overline{X}}) = \mu(Conv X)=\mu (X)$.
\item $\mu (\lambda X)= |\lambda| \mu (X)$, for $\lambda \in \mathbb{R}$.
\item $\mu ( X + Y)\leq \mu (X)+\mu(Y)$.
\item $\mu (X\cup Y)=\max \{\mu(X),\mu(Y)\}$.
\item If $\{X_n\}_n$ is a sequence of nonempty, bounded, closed subsets of $E$ such that
$X_{n+1}\subset X_n$ for $n=1,2,\ldots$ and $\lim_{n\to \infty}
\mu (X_n)=0$ then the set $X_{\infty}=\bigcap_{n=1}^{\infty}X_n$
is nonempty.
\end{enumerate}
\end{definition}
Further facts concerning measures of noncompactness and their
properties may be found in \cite{3}. Now, let us assume that
$\Omega$ is a nonempty subset of a Banach space $E$ and $T:
\Omega \to E$ is a continuous operator mapping bounded
subsets of $\Omega$ to bounded ones. Moreover, let $\mu$ be a
regular measure of noncompactness in $E$.
\begin{definition}[\cite{3}] \rm
We say that $T$ satisfies the Darbo condition with a constant $Q$
with respect to a measure of noncompactness $\mu$ provided
\[
\mu(TX)\leq Q\cdot \mu (X)
\]
for each $X\in \mathfrak{M}_E$ such that $X\subset \Omega$.
\end{definition}
If $Q< 1$, then $T$ is called a {\it contraction} with respect
to the measure $\mu$ (always assumed to be a measure of noncompactness
in the sequel).
For our purposes we will need the following fixed point theorem \cite{3}.
\begin{lemma}\label{teorema1}
Let $N$ be a nonempty, bounded, closed, convex subset of the
Banach space $E$ and let $T: N\to N$ be a contraction
with respect to a measure $\mu$. Then $T$ has a fixed point in the
set $N$.
\end{lemma}
In what follows we will work in the classical Banach space
$C[0,a]$ consisting of all real functions defined and continuous
on the interval $[0,a]$.This space is equipped with the standard
(uniform) norm
\[
\|x\|=\max \{|x(t)|:\ t\in [0,a]\}.
\]
Obviously, the space $C[0,a]$ also has the structure of a Banach
algebra. Now we present the definition of a special measure of
noncompactness in $C[0,a]$ which will be used in the sequel, a
measure that was introduced and studied in \cite{3}.
To do this let us fix a nonempty bounded subset $X$ of $C[0,a]$. For
$\varepsilon >0$ and $x \in X$ denote by $w(x,\varepsilon)$ the
{\it modulus of continuity} of $x$ defined by
\[
w(x,\varepsilon)= \sup \{| x(t)-x(s)| : t,s \in [0,a],\; | t-s|
\leq \varepsilon\}
\]
Further, let us put
\begin{gather*}
w(X,\varepsilon)=\sup\{w(x,\varepsilon) : x\in X\}\\
w_0 (X)= \lim_{\varepsilon \to 0} w(X,\varepsilon),
\end{gather*}
It can be shown (see \cite{11}) that the function $\mu(X)=w_0 (X)$
is a regular measure of noncompactness in the space $C[0,a]$.
Moreover, the following theorem (\cite{11}) holds, a result which
is essential in the proof of our main result.
\begin{lemma}\label{producto} \label{lem2}
Assume that $\Omega$ is a nonempty, bounded, convex, closed subset of $C[0,a]$ and the operators $F$ and $G$ transform continuously the
set $\Omega$ into $C[0,a]$ in such a way that $F(\Omega)$ and $G(\Omega)$ are bounded. Moreover, assume that the operator $T=F\cdot G$
transforms $\Omega$ into itself. If the operators $F$ and $G$ each satisfy the Darbo condition on the set $\Omega$ (with respect to the
measure of noncompactness $w_0$) with constant $Q_1$ and $Q_2$, respectively, then the operator $T$ satisfies the Darbo
condition on $\Omega$ with the constant
\[
\|F(\Omega)\|Q_2 + \|G(\Omega)\|Q_1.
\]
In particular, if $ \|F(\Omega)\|Q_2 + \|G(\Omega)\|Q_1< 1$ then $T$
is a contraction with respect to $w_0$ and so has at least
one fixed point in $\Omega$.
\end{lemma}
\section{Main Result}
\label{three}
In this section, we will study the solvability of the
functional-integral equation
\begin{equation}\label{ecuacionprincipal}
x(t)=f\Big( t,\int_0^t v(t,s,x(s))\, ds,\, x(\alpha(t))\Big)\cdot
g\Big( t,\int_0^a x(t)\,u(t,s,x(s))\, ds,\, x(\beta(t))\Big),
\end{equation}
for $x\in C[0,a]$. The methods used will be shown to be sufficiently
general to allow applications to complex functional integral equations
that include Chandrasekhar's H-functions as solutions
(see \cite[Chapter 5]{17}).
In what follows we will assume that the functions involved in
(\ref{ecuacionprincipal}) verify the following
conditions:
\begin{itemize}
\item[(i)] $f, g:[0,a]\times \mathbb{R}\times \mathbb{R} \to \mathbb{R}$
are continuous and there exist nonnegative
constants
$c_1,c_2,d_1,d_2$ such that
\begin{gather*}
|f(t,0,x)|\leq c_1 +d_1 |x|\\
|g(t,0,x)|\leq c_2 +d_2 |x|
\end{gather*}
\item[(ii)] The functions $f(t,y,x), g(t,y,x)$ satisfy a Lipschitz condition with respect to the va\-ria\-bles $y$ and $x$ with
constants $k, k'\geq 0$ respectively, i.e.,
\begin{gather*}
|f(t,y_1,x)-f(t,y_2,x)|\leq k|y_1-y_2|\\
|g(t,y_1,x)-g(t,y_2,x)|\leq k|y_1-y_2|,
\end{gather*}
for all $ t\in [0,a]$, and $ y_1,y_2,x \in \mathbb{R}$, and
\begin{gather*}
|f(t,y,x_1)-f(t,y,x_2)|\leq k'|x_1-x_2|\\
|g(t,y,x_1)-g(t,y,x_2)|\leq k'|x_1-x_2|,
\end{gather*}
for all $ t\in [0,a]$ and $ x_1,x_2,y \in \mathbb{R}$.
\item[(iii)] $u,v:[0,a]\times [0,a]\times \mathbb{R}\to \mathbb{R}$
are continuous.
\item[(iv)] $\alpha , \beta : [0,a]\to [0,a]$ are continuous and satisfy,
\begin{gather*}
|\alpha(t_1)-\alpha(t_2)|\leq |t_1 -t_2|,\\
|\beta(t_1)-\beta(t_2)|\leq |t_1 -t_2|,
\end{gather*}
for all $t_1,t_2 \in [0,a]$.
\item[(v)] (Sublinear nonlinearity) There exist nonnegative constants
$\alpha_1, \beta_1$ ,$\alpha_2$ and $\beta_2$ such that
\[
|v(t,s,x)|\leq \alpha_1 +\beta_1 |x| ,\quad
|u(t,s,x)| \leq \alpha_2 + \beta_2 |x|.
\]
for all $t,s \in [0,a]$ and $x\in \mathbb{R}$.
\item[(vi)] The inequality
$$
\big[k{(\tilde{\alpha}+\tilde{\beta} r})\cdot a + (c+dr)\big]
\big[k{(\tilde{\alpha}+\tilde{\beta} r)\cdot r }\cdot a + (c+dr)\big]\leq r
$$
has a positive solution $r_0$, where
$\tilde{\alpha}=\max \{\alpha_1, \alpha_2\}, \tilde{\beta}
= \max\{\beta_1, \beta_2\}, c= \max \{c_1,c_2\}$
and $d= \max\{ d_1,d_2\}$.
\item[(vii)]
$$
k'\big[ k (\tilde{\alpha}+ \tilde{\beta}r_0)\cdot a \cdot(1+ r_0)
+2(c+dr_0)\big]< 1.
$$
\end{itemize}
On this basis we have the following result.
\begin{theorem}\label{teorema4}
Under the tacit assumptions (i)-(vii) above, the functional-integral
equation
\begin{equation}\label{ecuacionprincipal2}
x(t)=f\Big( t,\int_0^t v(t,s,x(s))\, ds, x(\alpha(t))\Big)\cdot g
\Big( t,\int_0^a x(t)\,u(t,s,x(s))\, ds,\, x(\beta(t))\Big),
\end{equation}
has at least one solution $x\in C[0,a]$.
\end{theorem}
\noindent{{\bf Remark:}} \rm
Assumption (v) is essentially a {\it sublinear nonlinearity}
assumption on the kernels $u, v$ appearing in (\ref{ecuacionprincipal}). I
n order to handle a quadratic type of nonlinearity as can occur in, say,
the integral equation of Chandrasekhar (see \cite{17})
\begin{equation}
\label{chandra}
x(t)= 1+ x(t)\int_0^1\frac{t}{t+s}\varphi(s)\,x(s)\,ds
\end{equation}
we need to show that our technique can be used so as to include
this class of important integral equations.
We note that usually the existence of solutions of (\ref{chandra})
is derived under the additional assumption that the so-called
{\it characteristic} function $\varphi$ appearing in (\ref{chandra})
is an even polynomial in $s$, (cf., \cite[Chapter 5]{17}).
For such characteristic functions it is known that the resulting
solutions can be expressed in terms of Chandrasekhar's H-functions
\cite[Chapters 4 \& 5]{17}.
In our case, we derive the existence of solutions of this equation
(\ref{chandra}) under the much weaker assumption of continuity of
$\varphi$ along with $\varphi(0) =0$. The condition $\varphi(0)
=0$ is actually physically meaningful in some cases of radiative
transfer (see [\cite{17}, p.102, eq.(74)]). In this context,
there does remain an interesting question, that is, in the case of
a general characteristic function, can the solutions we obtain be
expressed as an infinite linear combination of classical
H-functions?
\begin{proof}
To prove this result using Lemma~\ref{producto} as our main tool,
we need to define operators $F$ and $G$ on the space $C[0,a]$ in
the following way:
\begin{gather*}
(Fx)(t)= f\Big(t,\, \int_0^t v(t,s,x(s))\, ds,\, x(\alpha (t))
\Big),\\
(Gx)(t)= g\Big(t,\, \int_0^a x(t)\, u(t,s,x(s))\, ds,\, x(\beta (t))\Big).
\end{gather*}
Next, we prove that the operators $F$ and $G$ transform the space
$C[0,a]$ into itself. To this end we are going to prove that $F, G$
are compositions of continuous functions defined on $[0,a]$;
that is, the operator $F$ can be expressed as the composition of
the following functions:
\[
\begin{array}{rccll}
[0,a]&\stackrel{Id\times \int v\times (x\circ \alpha)}{\to} & [0,a]\times \mathbb{R}\times \mathbb{R}&
\stackrel{f}{\to}& \mathbb{R}\\
t&\longmapsto & \Big(t,\int_0^t v(t,s,x(s))\,ds,\,x(\alpha(t))\Big)&\longmapsto & Fx(t)
\end{array}
\]
Now, taking into account assumptions (i), (iii) and (iv) it follows
that above functions are continuous, and therefore $F$ transforms
the Banach algebra $C[0,a]$ into itself. Similarly, one can prove
that the operator $G$ transforms $C[0,a]$ into itself.
The required operator $T$ on $C[0,a]$ is defined by setting
\[
Tx = (Fx)\cdot (Gx).
\]
Obviously, $T$ transforms $C[0,a]$ into itself. Also using assumptions (ii), (iv) and (v) we get that for every $t\in [0,a]$,
\begin{equation}
\begin{aligned}
|Fx(t)|&= \Big|f\Big(t,\int_0^t v(t,s,x(s))\,ds,\,x(\alpha (t))
\Big)\Big| \\
&\leq \Big|f\Big(t,\int_0^t v(t,s,x(s))\,ds,\,x(\alpha (t))\Big)
-f(t,0,x(\alpha (t)))\Big| + |f(t,0,x(\alpha (t)))| \\
&\leq k \Big|\int_0^t v(t,s,x(s))\,ds\Big|+ c_1 +d_1|x(\alpha(t))| \\
&\leq k(\alpha_1 +\beta_1 \|x\|)\cdot a + (c_1 +d_1\|x\|)\label{ec1}.
\end{aligned}
\end{equation}
On the other hand, by (ii), (iv), and (v) again, we have
\begin{equation}
\begin{aligned}
|Gx(t)|&= \Big|g\Big(t,\int_0^a x(t)\, u(t,s,x(s))\,ds,\,x(\beta (t))
\Big)\Big| \\
&\leq \Big|g\Big(t,\int_0^a x(t)\,u(t,s,x(s))\,ds,\,x(\beta (t))\Big)
-g(t,0,x(\beta (t)))\Big| \\
&\quad + |g(t,0,x(\beta (t)))| \\
&\leq k \Big|\int_0^a x(t)\,u(t,s,x(s))\, ds\Big|+ c_2
+d_2|x(\beta(t))| \\
&\leq k\|x\|(\alpha_2 +\beta_2 \|x\|)\cdot a + (c_2 +d_2\|x\|)\label{ec2}.
\end{aligned}
\end{equation}
Linking (\ref{ec1}) and (\ref{ec2}) we obtain
\begin{align*}
&|Tx(t)|\\
&= |Fx(t)|\cdot |Gx(t)| \\
&\leq \big[k\,(\alpha_1 +\beta_1 \|x\|)\cdot a + (c_1 +d_1\|x\|)\big]
\big[k\, \|x\|\, (\alpha_2 +\beta_2 \|x\|)\cdot a
+ (c_2 +d_2\|x\|)\big].
\end{align*}
Hence,
\[
\|Tx\|\leq \left[k{(\tilde{\alpha}+\tilde{\beta} \|x\|})\, a + (c+d\|x\|)\right]\left[k\, \|x\| \,{(\tilde{\alpha}+\tilde{\beta} \|x\|)}\,a + (c+d\|x\|)\right]
\]
Taking into account assumption (vi) we deduce that the operator $T$
maps the ball $B_{r_0}~\subset ~C[0,a]$ into itself.
Next, we show that the operator F is continuous on $B_{r_0}$. To do this fix $\varepsilon >0$ and take $x,y\in B_{r_0}$ such that
$\|x-y\|\leq \varepsilon$. Then, for $t\in [0,a]$ we get
\begin{align*}
&|Fx(t)-Fy(t)|\\
&= \Big|f\Big(t,\int_0^t v(t,s,x(s))\, ds,\, x(\alpha (t))\Big)
-f\Big(t,\int_0^t v(t,s,y(s))\, ds, y(\alpha (t))\Big)\Big| \\
&\leq \Big|f\Big(t,\int_0^t v(t,s,x(s))\, ds,\, x(\alpha (t))\Big)
-f\Big(t,\int_0^t v(t,s,y(s))\, ds, x(\alpha (t))\Big)\Big| \\
&\quad + \Big|f\Big(t,\int_0^t v(t,s,y(s))\, ds,\, x(\alpha (t))\Big)
-f\Big(t,\int_0^t v(t,s,y(s))\, ds, y(\alpha (t))\Big)\Big| \\
&\leq k\int_0^t |v(t,s,x(s))-v(t,s,y(s))|ds+ k' |x(\alpha (t))
-y(\alpha (t))|\\
&\leq k\cdot w(v,\varepsilon)\cdot a+ k'\|x-y\|\\
&\leq k\cdot w(v,\varepsilon)\cdot a+k'\varepsilon,
\end{align*}
where
\[
w(v,\varepsilon)= \sup \{|v(t,s,x_1)-v(t,s,x_2)|:
t,s\in [0,a], x_1, x_2 \in [-r_0,r_0], |x_1 -x_2|\leq \varepsilon\}.
\]
Using the fact that the function $v$ is uniformly continuous on
the bounded subset $[0,a]\times[0,a]\times[-r_0 , r_0]$, we
infer that $w(v,\varepsilon)\to 0$ as $\varepsilon \to 0$. Thus,
the above estimate shows that the operator $F$ is continuous on
$B_{r_0}$. Similarly, one can infer that the operator $G$ is
continuous on $B_{r_0}$ and consequently deduce $T$ is a
continuous operator on $B_{r_0}$.
Now, we prove that the operators $F$ and $G$ satisfy the Darbo
condition with respect to the measure $w_0$, defined in Section 2,
in the ball $B_{r_0}$. Take a nonempty subset $X$ of $B_{r_0}$
and $x\in X$. Then, for a fixed $\varepsilon >0$ and $t_1,t_2\in
[0,a]$ such that $t_1\leq t_2$ and $t_2-t_1\leq \varepsilon$, we
obtain
\begin{equation}
\begin{aligned}
&|Fx(t_2)-Fx(t_1)|\\
&= \Big|f\Big(t_2,\int_0^{t_2}v(t_2,s,x(s))\, ds,\, x(\alpha
(t_2))\Big)-
f\Big(t_1,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_1))\Big)\Big| \\
&\leq \Big|f\Big(t_2,\int_0^{t_2}v(t_2,s,x(s))\, ds,\, x(\alpha (t_2))
\Big)- f\Big(t_2,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))
\Big)\Big| \\
&\quad + \Big|f\Big(t_2,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))
\Big)- f\Big(t_1,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_1))\Big)
\Big| \\
&\leq k \big| \int_0^{t_2}v(t_2,s,x(s))\, ds
-\int_0^{t_1}v(t_1,s,\, x(s))\, ds\big| \\
&\quad + \Big|f\Big(t_2,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))\Big)
- f\Big(t_1,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))\Big)\Big| \\
&\quad + \Big|f\Big(t_1,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))
\Big)- f\Big(t_1,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_1))\Big)
\Big| \\
&\leq k\Big[\int_0^{t_1} |v(t_2,s,x(s))-v(t_1,s,x(s))\, ds|
+ \int_{t_1}^{t_2}|v(t_2,s,x(s))|ds\Big] \\
&\quad + \Big|f\Big(t_2,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))\Big)
- f\Big(t_1,\int_0^{t_1}v(t_1,s,x(s))\, ds,\, x(\alpha (t_2))\Big)\Big|\\
&\quad + k' |x(\alpha(t_2))-x(\alpha(t_1))|
\end{aligned}\label{ec3}
\end{equation}
At this point we introduce the notation:
\begin{gather*}
\begin{aligned}
w_v(\varepsilon,\cdot,\cdot)
=\sup \big\{&|v(t,s,x)-v(t',s,x)|:
t,t',s\in [0,a], |t-t'|\leq \varepsilon \\
&\textrm{ and } x\in [-r_0,r_0]\big\},
\end{aligned}\\
L= \sup \big\{|v(t,s,x)|: \ t,s \in [0,a], x\in [-r_0, r_0]\big\}, \\
\begin{aligned}
w_f(\varepsilon,\cdot,\cdot)
= \sup \big\{&|f(t,x,y)-f(t',x,y)|: t,t'\in [0,a],
|t-t'|\leq \varepsilon,\\
&x\in [-L\,r_0\, a, L\,r_0\, a], y \in [-r_0,r_0]\big\}.
\end{aligned}
\end{gather*}
Then, using (\ref{ec3}) we obtain the estimate
\[
|Fx(t_2)-Fx(t_1)|\leq k\cdot[ w_v(\varepsilon,\cdot, \cdot)\cdot a + L\cdot \varepsilon]+ w_f(\varepsilon,\cdot,\cdot)+ k'
|x(\alpha(t_2))-x(\alpha(t_1))|.
\]
Now, assumption (iv) allows us to deduce
\[
w(Fx,\varepsilon)\leq k\cdot[ w_v(\varepsilon,\cdot, \cdot)\cdot a + L\cdot \varepsilon]+ w_f(\varepsilon,\cdot,\cdot)+ k'
w(x,\varepsilon).
\]
Thus, taking the supremum in $X$, then the limit as $\varepsilon
\to 0 $, and taking into account the uniform continuity of the
functions $f$ and $v$ in bounded sets, we can deduce that
\begin{equation}\label{ec4}
w_0 (FX)\leq k' w_0 (X).
\end{equation}
Similarly, one can prove that
\begin{equation}\label{ec5}
w_0 (GX)\leq k' w_0 (X).
\end{equation}
Finally, linking (\ref{ec1})-(\ref{ec2}),
(\ref{ec4})-(\ref{ec5}) and keeping in mind Lemma~\ref{producto},
we infer that the operator $T$
satisfies the Darbo condition on $B_{r_0}$ with respect to the measure
$w_0$ with constant
$$
Q = k'\Big[ k (\tilde{\alpha}+ \tilde{\beta}r_0)\cdot a
\cdot(1+ r_0) +2(c+dr_0)\Big],
$$
(see assumption (vii)). Moreover, from assumption (vii) we deduce
that the operator $T$ is a contraction on $B_{r_0}$.
Therefore, applying Darbo's theorem we get that $T$ has at least
one fixed point in $B_{r_0}$. Consequently, the functional-integral
equation (\ref{ecuacionprincipal}) has at least one solution
in $B_{r_0}$. This completes the proof.
\end{proof}
\noindent{\bf Remark:} Moreover, in going through the estimates
leading to a solution of (\ref{chd}) we note that, in actuality,
for this specific choice of $f$ and $g$ condition (vi) can be
relaxed to $$k\|\varphi\|ar^2 + c \leq r,$$ which, of course,
shows that Chandrasekhar's equation
has a real continuous solution in this setting where $k=1$ and $c=1$
provided the characteristic function $\varphi$
and the interval $[0,a]$ are related by the inequality
\begin{equation}
\label{chandra2}
4\|\varphi\| a < 1.
\end{equation}
In [\cite{17}, Section 38, Corollary 1] Chandrasekhar proves that a necessary condition for a solution of
equation (\ref{chandra}) to be real is that, in the case $a=1$, we have
\begin{equation}
\label{chandra3}
\int_{0}^{1}\varphi (s)\, ds \leq \frac{1}{2}.
\end{equation}
But we have shown above that if (\ref{chandra2}) holds then the solution of
(\ref{chandra}) that we found must be {\it real} and continuous. Under assumption (\ref{chandra2}), however,
it is easy to see that, when $a=1$,
$$
\int_{0}^{1} \varphi (s)\, ds \leq \|\varphi\| < \frac{1}{4}.
$$
This result is consistent with the stated one in (\ref{chandra3})
for the existence of a real solution. Indeed, when $a=1$ it is
easy to see that $r_0 = 1/\sqrt{\|\varphi\|}$ is a solution of the
inequality $\|\varphi\|r^2 + 1 \leq r$. Since the fixed point of
our operator T (i.e., our solution $x(t)$) must lie in the ball
with radius $r_0$, it follows that our solution(s) lie in this
ball, and so there holds the {\it a priori} estimate
$$
\|x\| \leq \frac{1}{\sqrt{\|\varphi\|}}.
$$
Such a result, in the case of a general characteristic function,
does not appear in the literature (nor in \cite{17}), and so is
new. We therefore state this as a separate result.
\begin{theorem} \label{thm2}
Any solution $x(t)$ of Chandrasekhar's integral
equation \eqref{chd} on $[0,a]$ necessarily satisfies
\begin{equation}
\label{boud}
\|x\| \leq \frac{1}{\sqrt{\|\varphi\|}}
\end{equation}
for any choice of the characteristic function $\varphi(t)$ subject
to it only being continuous on $[0,a]$ and $\varphi(0)=0$.
\end{theorem}
The bound on the right of (\ref{boud}) can likely be improved for
specific classes of characteristic functions. Finally, we present
an additional concrete example of a functional-integral equation
where all the functions involved in the equation satisfy our
conditions.
\section{Applications}
\label{exa}
In this section we present some examples of classical integral and
functional equations considered in nonlinear analysis which are
particular cases of equation (\ref{ecuacionprincipal}) and
consequently, the existence of their solutions can be established
using Theorem \ref{teorema4}.
\begin{example} \label{ex1} \rm
First we note that equation (1) concerns the well-known functional
equation of the first order with a possible delay of the form
\[
x(t)=f_1\left(t,\, x(\alpha(t))\right),
\]
see \cite{16}. To obtain this example it is sufficient to put
$f(t,y,x)=f_1(t,x)$ and $g(t,y,x)=1$.
\end{example}
\begin{example} \label{ex2} \rm
Next, setting $g(t,y,x)\equiv 1$ and $f(t,y,x)= a(t)+y$, equation
(\ref{ecuacionprincipal}) reduces to the well-known nonlinear
Volterra integral equation
\[
x(t)= a(t)+ \int_0^t v(t,s,x(s))\, ds.
\]
\end{example}
\begin{example} \label{cch} \rm
On the other hand, if we choose $f(t,y,x)\equiv 1$, $g(t,y,x)=
1+y$, $u(t,s,y)= \frac{t}{t+s}\varphi (s) y$, and $\beta(t)=t$ in
Theorem \ref{teorema4}, equation (\ref{ecuacionprincipal2}) now
takes the form
\begin{equation}\label{chd}
x(t)= 1+ x(t)\int_0^a\frac{t}{t+s}\varphi(s)x(s)\, ds,
\end{equation}
and this is the famous quadratic integral equation of Chandrasekhar
discussed above and considered in many papers
and monographs (e.g., \cite{2,17}).
\end{example}
\noindent{\bf Remark.} Applying our technique to the specific
equation (\ref{chd}) we see that in order for all the assumptions
(i)-(vii) to be satisfied in Theorem~\ref{teorema4} we only need
to impose the additional condition that the characteristic
function $\varphi$ defined in (\ref{ecuacionprincipal2}) is
continuous and satisfies $\varphi(0) =0$. This previous condition
will ensure that the kernel $u(t,s,x)$ defined by
$$
u(t,s,x) = \begin{cases}
0, & s=0, t \geq 0, x \in \mathbb{R} \\
\frac{t}{t+s}\varphi(s)x,& s\neq 0, t \geq 0, x \in \mathbb{R}
\end{cases}
$$
is continuous on $[0,a]\times[0,a]\times \mathbb{R}$ in accordance
with assumption (iii).
To see this let $\varphi(0) =0$ along with $u(0,0,x)=0$.
Since $\varphi$ is continuous at $s=0$, given $\varepsilon > 0$
we can choose $\delta_1 > 0$ so small that
$|\varphi(s)| < \sqrt{\varepsilon}$
whenever $|s| <\delta_1$. Next, let $(t,s,x)$ be such that
$\sqrt{t^2 + s^2 +x^2} < \delta_1$.
Then $|u(t,s,x) | \leq |\varphi(s)||x| < \sqrt{\varepsilon}\delta_1
< \varepsilon$ provided we choose $\delta_1 < \sqrt{\varepsilon}$.
Thus $u(t,s,x)$ is continuous at $(0,0,0)$, and clearly at every
other point in $[0,a]\times[0,a]\times \mathbb{R}$.
\begin{example} \label{ex4} \rm
In addition, setting $f(t,x,y)\equiv 1$ and $g(t,y,x)= b(t)+y + x$,
we obtain existence results for the functional integral equation
\begin{equation}\label{chd1}
x(t) = b(t)+k\,x(ct)+ \int_0^a x(t)\,u(t,s,x(s))\, ds,
\end{equation}
where $k\in \mathbb{R}$ and $0 \leq c \leq 1$ are constants. This is an equation that includes the modified equation of radiative transfer (\ref{chandra33}) since we can fix the function $b(t) = 1/(1-2\psi_0)$, to be a constant function and $k=0$. In this case, $u(t,s,y)= - s \psi (s) y/(t+s)$, and this function is automatically continuous at $s=0$. Thus, it is not necessary to assume anything about the value of $\psi(x)$ at $x=0$, in contrast with Example~\ref{cch} and the arguments in the Remark above concerning equation (\ref{chandra}).
\end{example}
\begin{example} \label{ex5} \rm
Let us take $f,g:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$
defined by $f(t,y,x)= \frac{1}{9}y+ \frac{1}{10}\sin x$ and
$g(t,y,x)\equiv 1$. It is easy to prove that these functions are
continuous and satisfy hypothesis (i) with $c_1=\frac{1}{10}$,
$d_1=0$, $c_2 = 1$ and $d_2=0$. In this case
$c=\max \{c_1,c_2\}=1$ and $d=\max \{d_1,d_2\}= 0$.
Also the functions $f$ and $g$ verify the Lipschitz condition with
respect to the variables $y$ and $x$ with constants $k=\frac{1}{9}$
and $k'= \frac{1}{10}$, respectively. On the other hand, we define
the continuous functions $v(t,s,x)= t\cdot s \arctan x$ and
$u(t,s,x)\equiv 0$. It is clear that
\[
|v(t,s,x)|\leq |\arctan x|\leq |x|
\]
then $v$ satisfies assumption (v) with $\alpha _1 = 0$ and $\beta_1=1$.
Moreover, it is obvious that $u$ satisfies the same hypothesis with
$\alpha_2=0$ and $\beta_2 =0$. Consequently,
$\tilde {\alpha}=\max \{\alpha_1 ,\alpha _2\}=0$ and
$\tilde{\beta}=\max\{\beta_1,\beta_2\}=1$.
Next, we take $\alpha (t)=1/(1+t)$ and $\beta (t)=t$, each of which
satisfies assumption (v). Taking into account the above estimates
we obtain that the inequality of hypothesis (vi) has the form
\[
\big(\frac{r}{9} + 1\big)\big( \frac{r^2}{9} + 1\big)\leq r.
\]
However, it is easy to see that there is a root $r_0$ of this
inequality with $r_0 \in (0,3)$. For this value of $r_0$,
we have that assumption (vii) is satisfied. Now taking into account
all the functions defined previously, the functional-integral
equation is
\[
x(t)=\frac{t}{9}\int_0^t s \arctan x(s)\, ds + \frac{1}{10}
\sin \big(\frac{1}{1+t}\big)\,.
\]
Applying the result obtained in Theorem \ref{teorema4}, we deduce
that this equation has at least one solution in $B_{r_0}\subset C[0,a]$.
\end{example}
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\end{document}