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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2019 (2019), No. 11, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2019 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2019/11\hfil Causal terminal value problems]
{Quasilinearization method for causal terminal value problems
involving Riemann-Liouville fractional derivatives}

\author[C. Yakar, M. Arslan \hfil EJDE-2019/11\hfilneg]
{Co\c{s}kun Yakar, Mehmet Arslan}

\address{Co\c{s}kun Yakar \newline
Gebze Technical University,
Faculty of Fundamental Sciences,
Department of Mathematics, Applied Mathematics,
Gebze, Kocaeli, Turkey}
\email{cyakar@gtu.edu.tr}

\address{Mehmet Arslan \newline
Malatya Science and Art Center,
Ye\c{s}ilyurt, Malatya, Turkey}
\email{marslanmat@gmail.com}


\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted December 20, 2018. Published January 23, 2019.}
\subjclass[2010]{34A08, 34A34, 34A45, 34A99}
\keywords{Causal operator; fractional causal terminal value problem; 
\hfill\break\indent Riemann-Liouville derivative;
quasilinearization method; quadratic convergence}

\begin{abstract}
 In this work, we  construct new definitions for a causal terminal
 value problem involving Riemann-Liouville fractional derivatives,
 and study the unique solution by combining techniques from generalized
 quasilinearization.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

It has  been shown that causal differential equations 
\cite{2, 7, 8, 14,21, 22, 30, 31,32} 
provide excellent models for real world problems \cite{7} and 
in a variety of disciplines. This is the main
advantage of causal differential equations in comparison with the
traditional models \cite{12}. There has also been a growing interest 
to study causal dynamic systems \cite{7, 14}. The theory of 
terminal value problems \cite{1, 3, 10, 23, 25, 27, 31, 32} for ordinary
differential equations is more complicated than that of  initial value
problems of ordinary differential equations, and it is such an interesting
theory to study. The study of a terminal value problem for ordinary
differential equations using the method of lower and upper solutions can be
found in \cite{12}. The information is given at the end point of the interval
and one has to work backwards to find the initial value at which the
solution must start in order to reach the prescribed value at the end point
of the interval. This problem becomes more interesting in the case of a
fractional differential equation where it closely resembles a boundary value
problem, in the sense that the initial value is inherently involved in the
definition of the differential operator, and the terminal value provides the
condition at the right end point of the interval.

The study of differential equations with causal operators has rapidly
developed in recent years; see for example \cite{7, 14}. The term for
causal operators was adopted from the engineering literature, 
and the theory these operators have is the powerful quality of 
unifying the fractional order differential equations \cite{4, 33}, 
ordinary differential equations \cite{7},
integro-differential equations \cite{26}, differential equations with finite or
infinite delay \cite{9}, Volterra integral equations \cite{26}, and neutral functional
equations \cite{7, 14, 22}. Especially, they are very common equations for
modeling problems in mechanical engineering, physical engineering, electric
and electronics engineering \cite{11, 19, 20, 24}. Moreover, causality is a
basic concept in physical sciences to describe the process of cause and
effect in a particular situation.

The most important application of the quasilinearization method 
\cite{3, 5, 6, 13, 14,15, 17, 23, 28, 29, 30, 31,32,33, 34}
 in fractional causal differential equations
has been to obtain a sequence of lower and upper bounds which are the solutions of
linear fractional causal differential equations, that converge quadratically.
As a result, the method has been popular in applied areas. However, the
convexity assumption that is demanded by the method of quasilinearization
has been a stumbling block for further development of the theory. Recently,
this method has been generalized, refined and extended in several directions
so as to be applicable to a much larger class of nonlinear problems by not
demanding convexity and concavity property. Moreover, other possibilities
that have been explored make the method of generalized quasilinearization
universally useful in applications \cite{16}.

In a fractional causal terminal value problem \eqref{e2.2}  is used
to obtain upper and lower sequences in terms of the solutions of a linear
fractional causal terminal value problem, and bound the solutions of a given
nonlinear fractional causal terminal value problem. Moreover, we have also
shown that these sequences converge to the unique solution of the
nonlinear equation uniformly and quadratically.

\section{Preliminaries}

In this section, we state some fundamental definitions and useful theorems
used for  proving the main result. Let $E=C[ J,X]$ where $J$ is an
 appropriate time interval, $X$ represents either finite
or infinite dimensional space, depending on the requirement of the context,
so that $E$ is a function space.

An operator $Q:E\to E$ is said to be a causal operator if, for each
couple of elements $x,y$ in $E$ such that $x(s)=y(s)$ for 
$0\leq t_0\leq s\leq t$, the equality $(Qx)(s)=(Qy)(s)$ holds for
 $0\leq t_0\leq s\leq t$, $t<T$, $T$ is a given number.

If $E$ is a space of measurable functions on $[ t_0,T) $ for 
$t_0\geq 0$, then the definition needs a slight modification, requiring the
property to be valid almost everywhere on $[t_0,T]$. One can point out
that for causal operators, a notation identical with what is encountered for
a general equation with a memory can be stated as follows. A representation
of the form
\[
x(t)=(Qx)(t)
\]
where for each $t\in [ t_0,T)$. The functional $(Qx)(t)$ on $E$ which takes
values in $X$, for each $t$, while the whole family of
functionals, $t\in [ t_0,T) $, define the operator from $E=C([ t_0,T) , X)$
to itself.

For illustration, let us take $E=C[ [ t_0,T) ,\mathbb{R}^n] $ 
as the underlying space. Let $\{Q_n\}$ be a sequence of
causal operators on $E$ such that
\begin{equation}
\lim_{n\to \infty }(Q_nx)(t)=(Qx)(t)  \label{e2.1}
\end{equation}
for each $(t,x)\in [ t_0,T) \times E$. The question is whether
we can infer that the limit $Q:E\to E$ is also a causal operator.
The answer is yes because the causality of $\{Q_n\}$ implies 
\[
(Q_nx)(s)=(Q_ny)(s),\quad s\in E[ t_0,T) .
\]

If we let $n\to \infty $ on both sides, in the above relation and
use \eqref{e2.1} for each fixed $s\in [ t_0,T)$, we obtain the
causality of $Q$.

The Riemann-Liouville Fractional Causal Terminal Value Problem (FCTVP) 
is defined as follows,
\begin{equation}
D^qu(t) =(Qu) (t) ,\quad u(T)=u^T=u(t) (T-t) ^{1-q}\big|_{t=T}  \label{e2.2}
\end{equation}
where $0<q<1$ and the terminal value $T$ and the solution $u(T,t_0,u_0) =u^T$.
The corresponding Volterra fractional integral equation is given by
\begin{equation}
u(t) =u^T(t) +\frac{1}{\Gamma (q) }
\int _t^T(t-\tau ) ^{q-1}(Qu) (\tau ) d\tau  \label{e2.3}
\end{equation}
where $u^T(t) =\frac{u^T(T-t) ^{q-1}}{\Gamma(q) }$ and $\Gamma (q) $
is the standard Gamma function.

Let $p=1-q$ and 
\[
C_p([ t_0,T] ,\mathbb{R}) =\{ u:u\in C([t_0,T],\mathbb{R}) \text{ and }
(T-t) ^pu(t) \in C([t_0,T] ,\mathbb{R})\} 
\]
consider the fractional terminal value problem (FTVP)
\begin{equation}
D^q u(t) =f\  (t,u(t) ) ,\quad 
 u(T) =u^{_{T}}=u(t) (T-t) ^{1-q}\big| _{t=T} \label{e2.4}
\end{equation}
where $f\in C[ [ t_0,T] \times\mathbb{R},\mathbb{R}] $ and 
$u^T(t) =\frac{u^T(T-t) ^{q-1}}{\Gamma (q) }$.
In fact, the terminal condition $u(T)=u^{_{T}}$ and $u(t) $ is a solution of
\eqref{e2.4}.

\begin{definition} \label{def2.1}\rm
 A function $f:(t_0,T] \to \mathbb{R}$ is H\"{o}lder continuous if there 
are nonnegative real constants $C,\alpha $ such that $| f(x) -f(y)
| \leq C| x-y| ^{\alpha }$ for all $x,y\in (t_0,T] $.
\end{definition}

\begin{lemma} \label{lem2.1}
Let $m\in C_p[ [ t_0,T] ,\mathbb{R}] $ be locally H\"{o}lder continuous with 
exponent $\lambda >q$, and for any $t_1\in (t_0,T]$ we have that on $(t_1,T]$:
$m(t_1) =0$, $m(t) \leq 0$ and $m(t) (T-t) ^{1-q}\big|_{t=T}\leq 0$  for 
$t_0\leq t\leq t_1$.
Then
\begin{equation}
D^qm(t_1) \leq 0.  \label{e2.5}
\end{equation}
\end{lemma}

\begin{proof} 
By definition of the Riemann-Liouville fractional derivative is
\[
D^qm(t) =\frac{1}{\Gamma (p) }\frac{d}{dt}\int
_t^T(s-t) ^{p-1}m(s) ds.
\]
Let $H(t) =\int _t^T(s-t) ^{p-1}m(s) ds$. For small $h>0$, consider
\begin{align*}
&H(t_1+h) -H(t_1) \\
&=\int_{t_1+h}^T(s-t_1-h) ^{p-1}m(s) ds
 -\int _{t_1}^T(s-t_1) ^{p-1}m(s) ds \\
&=\int _{t_1+h}^T[ (s-t_1-h) ^{p-1}-(s-t_1) ^{p-1}] m(s) ds
 -\int_{t_1}^{t_1+h}(s-t_1) ^{p-1}m(s) ds \\
&=I_1-I_2
\end{align*}
Since $[(s-t_1-h)^{p-1}-(s-t_1)^{p-1}]>0$ for $t_1\leq s\leq T$ and
$m(s)\leq 0$ by hypothesis one has $I_1\leq 0$. This leads to
\[
H(t_1+h) -H(t_1) =-\int_{t_1}^{t_1+h}(s-t_1) ^{p-1}m(s)ds=-I_2.
\]
Since $m(t)$ is locally H\"{o}lder continuous there exists a $k(t_1)>0$
such that for $t_1-h\leq s\leq t_1+h$,
\[
-k(t_1)(s-t_1)^{\lambda }\leq m(s)-m(t_1)\leq k(t_1)(s-t_1)^{\lambda }
\]
where $0<\lambda <1$ is such that $\lambda >q$. By H\"{o}lder continuity and
from the fact that $m(t_1) =0$ we obtain
\begin{align*}
\int _{t_1}^{t_1+h}(s-t_1) ^{p-1}m(s)ds 
&\geq \int _{t_1}^{t_1+h}(s-t_1) ^{p-1}[m(t_1)-k(t_1)(s-t_1) ^{\lambda }] ds \\
&=-k(t_1)\int _{t_1}^{t_1+h}(s-t_1)^{p-1+\lambda }ds.
\end{align*}
Thus
\[
-I_2=\int _{t_1}^{t_1+h}(s-t_1) ^{p-1}m(s) ds
\leq k(t_1)\int _{t_1}^{t_1+h}(s-t_1)^{p-1+\lambda }ds=k(t_1)
\frac{h^{p+\lambda }}{p+\lambda }.
\]
Hence
\[
H(t_1+h) -H(t_1) -k(t_1)\frac{h^{p+\lambda }}{p+\lambda }\leq 0
\]
for sufficiently small $h>0$.
Letting $h\to 0$, we obtain $\frac{d}{dt}H(t_1) \leq 0$, which implies 
that $D^qm(t_1) \leq 0$ and the proof is complete.
\end{proof}

\begin{lemma} \label{lem2.2} 
Let $\{ u_{\epsilon }(t)\}$
be a family of continuous functions on $[ t_0,T] $, for 
$\epsilon >0$, such that
\begin{gather*}
D^qu_{\epsilon }(t) =f(t,u_{\epsilon }(t)), \\
u_{\epsilon }^T=u_{\epsilon }(t) (T-t) ^{1-q}\big|_{t=T},\quad
| f(t,u_{\epsilon }(t)) | \leq M\quad \text{for }t_0\leq t\leq T.
\end{gather*}
Then the family of functions $\{ u_{\epsilon }(t) \} $ is equicontinuous
 on $[ t_0,T] $.
\end{lemma}

The proof of the above lemma can be found in \cite{18}.

\begin{definition} \label{def2.2} \rm 
Function $v,w\in C_p[ [ t_0,T] ,\mathbb{R}] $ are said to be 
lower and the upper solutions of \eqref{e2.2} if $v$ and $w$
satisfy the differential inequalities, respectively, 
\begin{gather*}
D^qv(t) \geq (Qv) (t) ,\quad v(T) \leq u^T \\
D^qw(t) \leq (Qw) (t) ,\quad w(T) \geq u^T
\end{gather*}
where the causal operator $Q\in E=C(\mathbb{R}_{+},\mathbb{R})$, $Q:E\to E$
is continuous.
\end{definition}

\begin{definition} \label{def2.3} \rm
The causal operator $Q:E\to E$ is said to
be semi nondecreasing in $t$ for each $x$ if
\[
(Qx) (t_1) =(Qy) (t_1)\quad  \text{and}\quad 
(Qx) (t)  \leq (Qy) (t),\; 0\leq t<t_1<T,\; T\in\mathbb{R}_{+}
\]
for
\[
x(t_1) =y(t_1) ,\quad x(t)<y(t),\quad 0\leq t<t_1<T,\;T\ \in\mathbb{R}_{+}.
\]
\end{definition}

\begin{definition} \label{def2.4}\rm
Let the causal operator $Q\in C(\mathbb{R}_{+},\mathbb{R})$. At $x\in E$,
\[
(Q(x+h) )(t)=(Qx)(t)+L(x,h) (t) +\| h\| \eta (x,h) (t)
\]
where $\lim_{\| h\| \to 0}\| \eta (x,h)(t)\| =0$ and $L(x,\cdot )(t)$ 
is a linear operator. $L(x,h)(t) $ is said to be Fr\'{e}chet derivative 
of $Q$ at $x$ with the increment $h$ for the remainder $\eta (x,h)(t)$.
\end{definition}

\begin{theorem} \label{thm2.1}
Assume that $(Qu) (t) \in C[\mathbb{R}_{+}\times\mathbb{R},\mathbb{R}]$, 
where the causal operator $Q\in E=C(\mathbb{R}_{+},\mathbb{R})$, 
$Q:E\to E\ $is continuous. In addition to $v,w\in C_p[[ t_0,T] ,\mathbb{R}] $ 
be with continuous exponent $\lambda >q$, such that
\begin{itemize}
\item[(i)] $ D^qv(t) \geq (Qv) (t)$;

\item[(ii)] $D^qw(t) \leq (Qw) (t)$;

\item[(iii)] $ (Qu) (t)$ is nondecreasing in $u$ for each $t$, 
 $t_0\leq t\leq T$ with one of the inequalities $(i)$ or $(ii)$ being
strict.
\end{itemize}
Then $v(T) <w(T)$, where $v(T)=v^T=v(t) (T-t) ^{1-q}\big|_{t=T}\leq v(t_0)$ and 
$w(T) =w^T=w(t) (T-t) ^{1-q}\big|_{t=T}\geq w(t_0) $, implies 
$v(t) <w(t)$, $t\in [ t_0,T] $.
\end{theorem}

\begin{proof} 
 Assume that one of the inequalities is strict; let (i) be
strict and then set $m(t)=v(t)-w(t)$. If the conclusion of the theorem is
not true, then there exists $t_1\in (t_0,T] $\ such that 
$m(t_1)=0$, $m(t)\leq 0$ for $t_0\leq t\leq t_1$.

Consider the case when $t_1\in (t_0,T] $, then $m(t_1)=0$,
$m(t)\leq 0$ on $(t_0,t_1) $. By using Lemma \ref{lem2.1}, we obtain
to be $D^qm(t_1) \leq 0$. Thus
\begin{gather*}
(Qv) (t_1) <D^qv(t_1) \leq D^qw(t_1) \leq (Qw) (t_1), \\
(Qv) (t_1) <(Qw) (t_1)
\end{gather*}
which is a contradiction. Therefore $v(t)<w(t)$.

We set, for the nonstrict inequality
\[
\widetilde{v}(t) =v(t) -\epsilon [ (T-t) ^{q-1}E_{q,q}[ -2L(t-t_0) ^q] ]
\]
for $\epsilon, L>0$, where $E_{q,q}$ is the Mittag-Leffler function that
define as $E_{q,q}(z)=\sum_{k=0}^{\infty } \frac{z^{k}}{\Gamma ((k+1) q)}$, $q>0$.
This implies that
\[
\widetilde{v}(t) (T-t) ^{1-q}\big|_{t=T}=\widetilde{v
}^T=v(t) (T-t) ^{1-q}\big|_{t=T}-\epsilon g(t) (T-t) ^{1-q}\big|_{t=T}.
\]
So that  $\widetilde{v}^T=v^T-\epsilon g^T$. 
Then $\widetilde{v}(t) <v(t) $ for $t\in [ t_0,T]$ and $\widetilde{v}(T) <v(T) $.
Thus, it follows from (i) and the fact that $(Qu) (t) $ is nondecreasing, that
\begin{align*}
D^q\widetilde{v}(t) 
&= D^qv(t) -\epsilon D^qg(t) \geq (Qv) (t) +2\epsilon Lg(t) \\
&\geq (Q\widetilde{v}) (t) +2\epsilon Lg(t) >(Q\widetilde{v}) (t) .
\end{align*}
It  follows by the earlier argument that $\widetilde{v}(t)<w(t)$. 
Finally,  letting $\varepsilon \to 0$, we have $v(t) \leq w(t) $.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm2.2}
Assume that $v,w\in C_p[ [ t_0,T] ,\mathbb{R}] $ such that 
$v(t) \leq w(t)$, $t\in [t_0,T] $ and $Q:\Omega \to\mathbb{R}$ is 
the continuous causal operator where $\Omega =[ (t,u):v(t) \leq u\leq w(t) ] $.
Suppose further that
\begin{itemize}
\item[(i)] $ D^qv(t) \geq (Qv) (t)$;

\item[(ii)] $D^qw(t) \leq (Qw) (t)$;

\item[(iii)] $(Qu) (t) \leq \lambda (t)$ 
on $\Omega$ such that $\lambda \in L^{1}[\mathbb{R}_{+},\mathbb{R}] $.
\end{itemize}
Then  \eqref{e2.2} has a solution which satisfies 
$v(t) \leq u(t) \leq w(t)$ on $[ t_0,T]$ provided that $v(T) \leq u(T) \leq w(T) $
for some$\ t_0\geq 0$.
\end{theorem}

\begin{proof} 
Consider $P:[t_0,T]\times\mathbb{R}\to\mathbb{R}$ defined by
\begin{equation}
(Pu) (t) =\max \{ v(t) ,  \min \{ u, w(t) \} \} .  \label{e2.6}
\end{equation}
Then $Q$ is a continuous causal operators and by the assumption (iii), we
have $(Qu) (t) \leq \lambda (t) $. So
that $Q(t,(Pu) (t) ) $ defines a
continuous extension of $Q$ to $[t_0,T]\times \mathbb{R}$
which is also bounded. Therefore, the FCTVP
\begin{equation}
D^qu=Q(t,(Pu) (t) ) ,\quad u(T) =u^T  \label{e2.7}
\end{equation}
has a solution $u(t) $ on $[t_0,T]$. We show $v(t)\leq u(t) \leq w(t) $ 
for $t\in [ t_0,T]$ and therefore $u(t) $ is a solution of \eqref{e2.2}.

For $\epsilon , L>0$, consider 
\begin{equation} \label{e2.8}
\begin{gathered}
\widetilde{v}(t) (T-t) ^{1-q} \big|_{t=T}=v(t) (T-t) ^{1-q}\big|_{t=T}-\epsilon g(t)
(T-t) ^{1-q}\big|_{t=T}   \\
\widetilde{w}(t) (T-t) ^{1-q} \big|_{t=T}
=w(t) (T-t) ^{1-q}\big|_{t=T}+\epsilon g(t)(T-t) ^{1-q}\big|_{t=T}  
\end{gathered}
\end{equation}
where $g(t) =(T-t) ^{q-1}E_{q,q}[-2L(t-t_0) ^q] $.

Then $\widetilde{w}(t) >w(t)$, $\widetilde{v}(t) <v(t) $ and 
$\widetilde{v}(T) <u(T) <\widetilde{w}(T)$.
We claim that $\widetilde{v}(t) <u(t) <\widetilde{w}(t) $ on 
$[t_0,T]$. Suppose that it is not true and thus there exists 
$t_1\in [t_0,T]$ such that $u(t_1) =\widetilde{w}(t_1) $ and 
$\widetilde{v}(t) <u(t) <\widetilde{w}(t)$, $t_0\leq t\leq t_1$.

Then $u(t_1) >w(t_1)$ and hence $(Pu) (t_1) =w(t_1)$. Also 
$v(t_1)\leq (Pu) (t_1) \leq w(t_1)$. 
Setting $m(t)=u(t) -\widetilde{w}(t) $, we have 
$m(t_1)=0$ and $m(t)\leq 0,\ t_0\leq t\leq t_1$. Hence by Lemma \ref{lem2.1}, 
we obtain $D^qm(t_1) \leq 0$ that yields
\begin{align*}
Q(t_1,(Pu) (t_1) ) 
&= D^qu(t_1) \leq D^q\widetilde{w}(t_1) =D^qw( t_1) -2\epsilon Lg(t_1) \\
&\leq Q(t_1,w_{t_1}) -2\epsilon Lg(t_1)
=Q(t_1,(Pu) (t_1) ) -2\epsilon Lg(t_1) \\
&< Q(t_1,(Pu) (t_1) )
\end{align*}
which is a contradiction. Then, we have 
$u(t) <\widetilde{w} (t)$ on $[ t_0,T]$ provided that 
$u(T) \leq w(T) $ for some $t_0\geq 0$. Similarly, the
other case $\widetilde{v}(t) <u(t) $ for $t_0\leq t\leq T$ can be proved.

Consequently,  combining the proved results, we have 
$\widetilde{v}(t) <u(t) <\widetilde{w}(t)$ on 
$t\in [t_0,T]$. Letting $\epsilon \to 0$, we obtain $v(t)\leq u(t)\leq w(t)$,
on $[0,T]$. The proof is complete.
\end{proof}

\section{Quasilinearization Method}

In this section, we extend the generalized quasilinearization method
for nonlinear terminal value problems in \cite{3}. We prove the main theorem that
gives several  conditions to apply the method of quasilinearization
to the nonlinear causal terminal value problem involving Riemann-Liouville
fractional derivatives.

\begin{theorem} \label{thm3.1}
Assume that $Q,\ \Phi :C[\mathbb{R}_{+},\mathbb{R}] 
\to C[\mathbb{R}_{+},\mathbb{R}] $
are continuous causal operator such that $(Qu) (t)$,  
$(\Phi u) (t) \in C[\mathbb{R}_{+}\times\mathbb{R},\mathbb{R}] $ and
\begin{itemize}
\item[(M1)] $| (Qu) (t)| \leq \lambda (t) | u(t)| $ on 
$\Omega =[ (t,u) \in [ t_0,T]\times C^q[ [ t_0,T] ,\mathbb{R}] :v(t)
 \leq u\leq w(t) ] $, where $\lambda \in L^{1}[ 0,\infty )$;

\item[(M2)] $v,w\in C^q[ [ t_0,T] ,\mathbb{R}] $ are the lower and upper 
solutions  of \eqref{e2.2} such that $v(t) \leq w(t)$, $t\in [ t_0,T]$;

\item[(M3)] $v_0,w_0\in C^q[ [ t_0,T] ,\mathbb{R}] $ with $v_0(t) \leq w_0(t) $ on
 $[t_0,T] $, $v_0(T)$, $w_0(T) $ exist and
\begin{itemize}
\item[(a)] $D^qv_0(t) \geq (Qv_0) (t)$, $v_0(T) \leq u^T$ for $t\in [ t_0,T]$;

\item[(b)] $D^qw_0(t) \leq (Qw_0) (t) $, $w_0(T) \geq u^T$ for
$t\in [ t_0,T]$;
\end{itemize}

\item[(M4)] $Q,  \Phi \in C^q[\mathbb{R}_{+},\mathbb{R}] $ and for 
$(t,u) \in \Omega $ the Fr\'{e}chet derivatives $(Q_{u}u) (t)$,
$(\Phi_{u}u) (t)$, $(Q_{uu}u) (t) $ and
$(\Phi _{uu}u) (t)$ exists and are continuous on 
$[ 0,\infty ) $ such that $(Q_{u}u) (t) \leq B$, 
$(Q_{uu}u) (t) +$ $(\Phi_{uu}u) (t) \leq 0$ for some function $\Phi $ with 
$| (\Phi u) (t) | \leq \lambda_1(t) | u(t) |$,  
$|(\Phi _{u}u) (t) | \leq F$ and 
$(Q_{uu}u) (t) \geq 0$, $(\Phi _{uu}u) (t) \leq 0$ on 
$\mathbb{R}_{+}\times\mathbb{R}$, where 
$B, F, \lambda _1\in L^{1}[ 0,\infty ) $.
\end{itemize}
Then there exist the monotone sequences $\{ v_n\}$ and 
$\{ w_n\} $ which converge uniformly to the unique solution 
$u(t) =u^T(t) +\frac{1}{\Gamma (q) }\int _t^T(t-\tau ) ^{q-1}(Qu) (
\tau ) d\tau $ that satisfy 
$u(T,t_0,u_0) =u^T$ of \eqref{e2.2} on $[ t_0,T] $. 
Moreover, the convergence is quadratic.
\end{theorem}

\begin{proof} 
 Let us initially define a continuous causal operator 
$\Psi :C[\mathbb{R}_{+},\mathbb{R}] \to C[\mathbb{R}_{+},\mathbb{R}] $ 
and $(\Psi u) (t) \in C[\mathbb{R}_{+}\times\mathbb{R},\mathbb{R}] $, 
such that
\begin{equation}
(\Psi u) (t) =(Qu) (t)+(\Phi u) (t).  \label{e3.1}
\end{equation}

In view of (M4), we see that $(\Psi _{uu}u)(t) \leq 0$, and 
$| (\Psi u) (t) | \leq (\lambda (t) +\lambda_1(t) ) | u(t) |=P| u(t) | $, 
where $P=(\lambda(t) +\lambda _1(t) ) \in L^{1}[0,\infty ) $. Also,
 $| (\Psi _{u}u) (t) | \leq B+F=P_1\in L^{1}[ 0,\infty ) $.
Using the generalized mean value theorem and \eqref{e3.1}, we have
\[
(Qu) (t) \leq (\Psi \alpha ) (t) +(\Psi _{u}\alpha ) (t) (u-\alpha
) -(\Phi u) (t)
\]
where $u,\alpha \in C^q[ [ t_0,T] ,\mathbb{R}] $ such that 
$\alpha (t) \leq u(t)$, $t\in[ t_0,T] $. Get
\begin{equation}
(Gu\alpha ) (t) =(\Psi \alpha ) (t) +(\Psi _{u}\alpha ) (t) (u-\alpha
) -(\Phi u) (t)  \label{e3.2}
\end{equation}
and observe that
\begin{equation} \label{e3.3}
\begin{gathered}
(Gu\alpha ) (t) \geq (Qu) (t) \\
(Guu) (t) =(Qu) (t) .
\end{gathered}
\end{equation}
Further, in view of the nonincreasing property of $(\Phi _{u}u)(t) $, we obtain
\[
(G_{u}u\alpha ) (t) =(\Psi _{u}\alpha )(t) -(\Phi _{u}u) (t)
\geq (\Psi_{u}\alpha ) (t) -(\Phi _{u}\alpha ) (t)
\geq (Q_{u}\alpha ) (t) \geq 0.
\]
Thus, $(Gu\alpha ) (t) $ is nondecreasing in $u$ for
each fixed $(t,\alpha ) \in [ t_0,T] \times C^q[ [ t_0,T] ,\mathbb{R}] $.
Further,
\[
(Gu\alpha ) (t) =(\Psi \alpha ) (t) +(\Psi _{u}u) (t) (u-\alpha )-(\Phi u) (t) ,
\]
which, together with (M1), (M4) and \eqref{e3.2} implies that
\begin{equation}
(Gu\alpha ) (t)
=P| \alpha |+B(| u| +| \alpha | )+\lambda _1| u|
=P_2(t) |\alpha | +P_3(t) | u| =(H| u| ) (t) ,  \label{e3.4}
\end{equation}
where $P_2=P+B,\ P_3=\lambda _1+B\in L^{1}[ 0,\infty ) $.
Now, using the mean value theorem and the nonincreasing nature of
$(\Psi _{u}u) (t) $, we obtain
\begin{equation} \label{e3.5}
\begin{aligned}
(Gu\alpha _1) (t) -(Gu\alpha _2)(t) 
&\leq (\Psi _{u}\mu _1) (t)(\alpha _1-\alpha _2) +(\Psi _{u}\alpha _2)
(t) (\alpha _2-\alpha _1)   \\
&=(\Psi _{uu}\mu _2) (t) (\mu _1-\alpha_2) (\alpha _1-\alpha _2)
\leq 0
\end{aligned}
\end{equation}
where $\alpha _2\leq \mu _2\leq \mu _1\leq \alpha _1$. Expression
\eqref{e3.5} implies that $(Gu\alpha ) (t) $ is
nonincreasing in $\alpha $ for each fixed
$(t,u) \in [t_0,T] \times C^q[ [ t_0,T] ,\mathbb{R}] $.
Set $v=\beta _0$ and consider the  FCTVP
\begin{equation}
D^qu(t) =(Gu\beta _0) (t) ,\quad u(T) =\gamma ^T  \label{e3.6}
\end{equation}
Because of expression \eqref{e3.4}, the problem \eqref{e3.6} has a unique
solution $\beta_1(t) $ on $[ a,\infty )$, $a>0$ satisfying
$u_1(T) =u^T$. Also, in view of $(M2) $ and \eqref{e3.3}, we have
\begin{gather*}
D^q\beta _0\geq (Q\beta _0) (t)
=(G\beta_0\beta _0) (t) ,\quad  \beta _0(T) \leq \gamma ^T, \\
D^qw(t) \leq (Qw) (t) \leq (Gw\beta _0) (t) ,\quad  w(T) \geq \gamma ^T
\end{gather*}
which imply
\[
v(t) \leq u_1(t) \leq w(t) \quad \text{for some }a\geq 0.
\]
Next, we consider the FCTVP
\begin{equation}
D^qu(t) =(Gu\beta _1) (t) ,\quad  u(T) =\gamma ^T  \label{e3.7}
\end{equation}
As above, we can show that  \eqref{e3.7} has a unique solution
$\beta _2(t) $ satisfying $\beta _2(T) =\gamma ^T$.
Using \eqref{e3.3} and the nonincreasing property of
$(Gu\alpha )(t) $ in $\alpha $, we have
\[
D^q\beta _1(t) =(G\beta _1\beta _0) (t) \geq (G\beta _1\beta _1) (t) ,\quad
 \beta_1(T) =\gamma ^T
\]
which implies that $\beta _1(t) $ is a lower solution of \eqref{e3.7}
and
\[
D^qw(t) \leq (Qw) (t) \leq (Gw\beta _1) (t) ,\quad
 \beta (T) \geq \gamma ^T
\]
implies that $w(t) $ is an upper solution of \eqref{e3.7}. Further,
$\beta _1(T) \leq \beta _2(T) \leq w(T) $. Again, by  Theorem \ref{thm2.2},
we obtain
\[
\beta _1(t) \leq \beta _2(t) \leq w(
t) ,\quad  t\in [ a,T)
\]
 for some $a\geq 0$.
Continuing this process successively, we obtain
\[
v\leq \beta _1\leq \beta _2\leq \beta _3\leq \dots\leq \beta _{n-1}\leq
\beta _n\leq w\  \text{on }[ t_0,T]
\]
where the elements of the monotone sequence $\{ \beta _n\} $
are the solutions of the  problem
\[
D^qu(t) =(Gu\beta _{(n-1) }) (t) ,\ u(T) =\gamma ^T.
\]
Since the sequence $\{ \beta _n\} $ is monotone, it follows
that it has a pointwise limit $\beta (t) $. To show that $\beta
(t) $ is in fact a solution of \eqref{e2.2}, we observe that $\beta
_n $ is a solution of the  linear FCTVP
\begin{equation}
D^qu(t) =(G\beta _n\beta _{(n-1) })(t) =F_n(t) ,\quad  \beta _n(T)
=\gamma ^T  \label{e3.8}
\end{equation}
where
\[
F_n(t) =(\Psi \beta _{(n-1) }) (
t) +(\Psi _{u}\beta _{(n-1) }) (t)
(\beta _n-\beta _{n-1}) -(\Phi \beta _n) (
t) .
\]
Since $G$ is continuous on $\mathbb{R}_{+}$, therefore, in view of \eqref{e3.4},
it follows that for each $n\in\mathbb{N}$, the sequence $\{ F_n(t) \} $ is a
sequence of continuous  functions and is bounded by
$(H\beta _n) (t) \in L^{1}[ 0,\infty ) $. Consequently,
$\int_t^{\infty }F_n(s) ds<\infty $. Now, taking the
limits both side as $n\to \infty $, we have
\[
\lim_{n\to \infty }F_n(t) =\lim_{n\to \infty }(G\beta _n\beta _{(n-1) }) (t)
=(Q\beta ) (t) .
\]
Now, by using the Lebesque dominated convergence theorem, we obtain
\[
\lim_{n\to \infty }\int _t^{\infty }F_n(s)ds
=\int _t^{\infty }(Qu) (s) ds
\]
which implies
$ \int _t^{\infty }(Qu) (s)ds<\infty$.
Now, the solution of \eqref{e3.8} is
\[
\beta _n(t) =\gamma ^T-\int _t^{\infty}F_n(s) ds
\]
which, by taking the limit $n\to \infty $, yields
\[
\beta (t) =\gamma ^T-\int _t^{\infty }(Qu) (s) ds.
\]
This shows that $\beta (t) $ is solution of the \eqref{e2.2}.

To prove the quadratic convergence of $\{ \alpha _n\}$ and
$\{ \beta _n\} $ to the unique solution, we consider 
\[
\sigma _n(t) =\beta (t) -\beta _n(t) ,\quad n=1,2,3,\dots
\]
Observe that $\sigma _n(t) \geq 0$ and $\sigma _n(\infty ) =0$. 
Here we  use the mean value theorem and assumption (M4), to obtain
\begin{equation} \label{e3.9}
\begin{aligned}
&D^q\sigma _{n+1}(t) \\
&=D^q\beta (t) -D^q\beta_{n+1}(t)  \\
&=(Q\beta ) (t) -[ (\Psi \beta_n) (t)
 +(\Psi _{u}\beta _n) (t) (\beta _{n+1}-\beta _n)
 -(\Phi \beta _{(n+1) }) (t) ]   \\
&=(\Psi _{u}\beta _n) (t) (\beta -\beta_n) +(\Psi _{uu}\xi ) (t)
 \frac{(\beta-\beta _n) ^2}{2!}\\
&\quad - (\Psi _{u}\beta _n) (t) (\beta_{n+1}-\beta _n)
 -((\Phi \beta ) (t)-(\Phi \beta _{(n+1) }) (t) ) \\
&= (\Psi _{u}\beta _n) (t) (\beta -\beta _{n+1})
 +(\Psi _{uu}\xi ) (t) \frac{(\beta -\beta _n) ^2}{2!}
 -(\Psi _{u}\xi _1) (t) (\beta -\beta _{n+1})   \\
&\geq (Q_{u}\beta _n) (t) \sigma _{n+1}(t)
 +(\Psi _{uu}\zeta _1) (t) \frac{(\sigma _n(t) ) ^2}{2!}   \\
&\geq -B(t) \sigma _{n+1}(t) -\frac{D^qP(t) }{2}
 (\sigma _n(t) ) ^2,
\end{aligned}
\end{equation}
$\sigma_{n+1}(\infty ) =0$,
where $\beta _n\leq \zeta \leq \beta$.
From \eqref{e3.9} and using the definition of lower solution and
Theorem \ref{thm2.2}, we have
$\sigma _{n+1}(t) \leq r(t)$  for some $t\geq a>0$,
where
\[
r(t) =\exp \Big(\int _t^{\infty }B(s)ds\Big)
\Big[ \int _t^{\infty }\frac{D^qP(s) }{2}(\sigma _n(s) ) ^2
\exp \Big(-\int _t^{\infty }B(l) dl\Big) ds\Big] ,
\]
which is a unique solution of the  nonhomogeneous linear problem
\[
D^qr(t) =-B(t) r(t) -\frac{D^qP(t) }{2}(\sigma _n(t) ) ^2,\quad
\beta (\infty ) =0.
\]
Thus,
\[
\sigma _{n+1}(t) \leq \exp \Big(\int _t^{\infty
}B(s) ds\Big) \Big[ \int _t^{\infty }\frac{
D^qP(s) }{2}(\sigma _n(s) ) ^2\exp
(-\int _t^{\infty }B(l) dl) ds\Big] .
\]
Hence,
\begin{align*}
| \sigma _{n+1}(t) |
&\leq |\exp (\int _t^{\infty }B(s) ds) |
\Big| \int _t^{\infty }\frac{D^qP(s) }{2}
(\sigma _n(s) ) ^2\exp (-\int
_t^{\infty }B(l) dl) ds\Big| \\
&\leq K| \sigma _n(s) | ^2T=A| \sigma _n(s) | ^2,
\end{align*}
where $| \exp (\int _t^{\infty }B(s) ds) | \leq K$,
\[
\big| \int _t^{\infty }\frac{
D^qP(s) }{2}(\sigma _n(s) ) ^2\exp
(-\int _t^{\infty }B(l) dl) ds\big|\leq 2T
\]
and $A=KT$.
This establishes the quadratic convergence and therefore completes the proof.
\end{proof}


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\end{document}