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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 235, pp. 1--10.\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{8mm}}

\begin{document}
\title[\hfilneg EJDE-2013/235\hfil Non-existence of global solutions]
{Non-existence of global solutions for a differential equation involving
 Hilfer fractional derivative}

\author[K. M. Furati, M. D. Kassim, N.-e. Tatar \hfil EJDE-2013/235\hfilneg]
{Khaled M. Furati, Mohammed D. Kassim, Nasser-eddine Tatar}  % in alphabetical order

\address{Khaled M. Furati \newline
King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia}
\email{kmfurati@kfupm.edu.sa}

\address{Mohammed D. Kassim \newline
King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia}
\email{dahan@kfupm.edu.sa}

\address{Nasser-eddine Tatar \newline
King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia}
\email{tatarn@kfupm.edu.sa}

\thanks{Submitted September 3, 2013. Published October 22, 2013.}
\subjclass[2000]{26D10, 42B20, 26A33, 35J05, 35J25}
\keywords{Cauchy problem; critical exponent;
 fractional differential inequality; \hfill\break\indent
 Hilfer fractional derivative;  test function method}

\begin{abstract}
We consider a basic fractional differential inequality with a fractional
 derivative named after Hilfer and a polynomial source. A non-existence of
 global solutions result is proved in an appropriate space and the critical
 exponent is shown to be optimal.
\end{abstract}

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

\section{Introduction}

We study the Cauchy problem of fractional order with a polynomial
nonlinearity
\begin{equation}
\begin{gathered}
( D_{0^{+}}^{\alpha ,\beta }u) ( t) \geq t^{\delta}| u( t) | ^{m},\quad 
t>0,\;m>1,\;\delta \in \mathbb{R} \\
( D_{0^{+}}^{\gamma -1}u) ( 0) =b>0,
\end{gathered} \label{1}
\end{equation}
where
\begin{equation}
( D_{0^{+}}^{\alpha ,\beta }y) ( x) =\big(
I_{0^{+}}^{\beta ( 1-\alpha ) }\frac{d}{dx}I_{0^{+}}^{(
1-\beta ) ( 1-\alpha ) }f\big) ( x)  \label{2}
\end{equation}
is the Hilfer fractional derivative (HFD) of order $0<\alpha <1$
and type $0\leq \beta \leq 1$, $\gamma =\alpha +\beta -\alpha \beta $ and 
$I_{0^{+}}^{\sigma }$, $\sigma >0$, is the usual Riemann-Liouville fractional
integral of order $\sigma$. This type of derivatives were introduced by
Hilfer in \cite{h1,h2}.
These references provide information about the applications of this 
derivative and how it arises. It is easy to see that
this derivative interpolates the Riemann-Liouville fractional derivative 
($\beta =0$) and the Caputo fractional derivative ($\beta =1$) 
(see \cite{k5,p1}).
The special case $\beta =0$ has been discussed in \cite{l1}.

In this article we find the range of values of $m$ for which solutions do not
exist globally and establish an optimal exponent (in some sense) by showing
that solutions do exist beyond this bound in a certain space. The existence
and uniqueness for the general problem
\begin{gather*}
( D_{a^{+}}^{\alpha ,\beta }u) (t) =f(t,u),\quad
0<\alpha <1,\; 0<\beta <1,\; t>a, \\
( D_{a^{+}}^{\gamma -1}u) ( a+) =c>0,
\end{gather*}
has been established in \cite{f1} in the space
\[
C_{1-\gamma }^{\alpha ,\beta }[ a,b] =\{ y\in C_{1-\gamma
}[ a,b] ,\;D_{a+}^{\alpha ,\beta }y\in C_{1-\gamma }[a,b] \}
\]
where $C_{1-\gamma }[ a,b] $ is the weighted space of continuous
functions on $(a,b]$
\[
C_{1-\gamma }[ a,b] =\{ g:(a,b]\to \mathbb{R}:( x-a) ^{1-\gamma }g( x) 
\in C[ a,b]\} .
\]
The special cases $\beta =0$ and $\beta =1$ may be found in \cite{k1,k2,k3,k4,k5}.
These cases correspond to the Riemann-Liouville derivative and the Caputo
derivative cases, respectively. Problems with such derivatives have been
treated in many papers, we cite a few of them 
\cite{b4,d1,d2,e1,e2,e3,e4,f3,f4,f5,f6,k7,k8,l1,z1},
 and refer the reader to the books \cite{k5,p1,s1} 
for many other properties of such derivatives. 
The applications of these types of derivatives are numerous.
Some of them may be found in \cite{b1,b2,b3,g2,k6,m1,m2,p1,p2,s1}.
However, we cannot find much on Hilfer type derivatives.

The next section contains some definitions, notation and some lemmas which
will be useful later in our proof. In Section 3 we state and prove our
non-existence result. Finally, in Section 4 we give an example showing the
existence of solutions in case the exponent is higher than the critical one
found in the previous section.

\section{Preliminaries}

In this section we present some definitions, lemmas, properties and notation
which will be used in our results later.

\begin{definition} \label{def1} \rm
Let $\Omega =[ a,b] $ be a finite
interval and $0\leq \gamma <1$, we introduce the weighted space 
$C_{\gamma }[ a,b] $ of continuous functions $f$  on $(a,b]$
\[
C_{\gamma }[ a,b] =\{ f:(a,b]\to \mathbb{R}:(x-a) ^{\gamma }f( x) \in C[ a,b] \} .
\]
In the space $C_{\gamma }[ a,b]$, we define the norm
\[
\| f\| _{C_{\gamma }}=\| ( x-a) ^{\gamma }f(x) \| _{C},\quad 
C_0[ a,b] =C[ a,b] .
\]
\end{definition}

\begin{definition} \label{def2} \rm
The Riemann-Liouville left-sided fractional integral 
$I_{a+}^{\alpha }f$ of order $\alpha >0$ is defined by 
\[
( I_{a^{+}}^{\alpha }f) (x):=\frac{1}{\Gamma (\alpha )}
\int_a^x\frac{f(t)}{(x-t)^{1-\alpha }}dt,\quad (a<x\leq b,\;\alpha >0)
\]
provided that the integral exists. Here 
$\Gamma (\alpha )$ is the Gamma function. 
When  $\alpha =0$, we define
$I_{a^{+}}^{0}f=f$.
In fact, one can prove that $I_{a^{+}}^{\alpha }f$ converges to $f$ when 
$\alpha \to 0$. 
\end{definition}

\begin{definition} \label{def3} \rm
The Riemann-Liouville right-sided fractional integral 
$I_{b^{\_}}^{\alpha }f$  of order $\alpha >0$  is defined by
\[
( I_{b^{-}}^{\alpha }f) (x):=\frac{1}{\Gamma (\alpha )}
\int_{x}^{b}\frac{f(t)}{(t-x)^{1-\alpha }}dt,\quad ( a\leq x<b,\;\alpha >0)
\]
provided that the integral exists. When $\alpha =0$, we
define
$I_{b^{-}}^{0}f=f$.
\end{definition}

\begin{definition} \label{def4}\rm
The Riemann-Liouville left-sided fractional derivative 
$D_{a+}^{\alpha }f$ of order $\alpha $ ($0\leq \alpha <1$)
is defined by
\[
( D_{a+}^{\alpha }f) ( x) =\frac{d}{dx}(I_{a+}^{1-\alpha }f) (x);
\]
that is,
\[
( D_{a+}^{\alpha }f) =\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}
\int_a^x\frac{f(t)}{(x-t)^{\alpha }}dt\quad ( x>a,\;0<\alpha <1) ,
\]
when $\alpha =1$ we have $D_{a+}^{\alpha }f=Df$. In particular, when
$\alpha =0$, $D_{a+}^{0}f=f$.
\end{definition}

\begin{definition} \label{def5} \rm
The Riemann-Liouville right-sided fractional derivative 
$ D_{b^{-}}^{\alpha }f$ of order $\alpha $ 
($0\leq \alpha <1$) is defined by 
\[
( D_{b^{-}}^{\alpha }f) ( x) =-\frac{d}{dx}(I_{a+}^{1-\alpha }f) (x);
\]
that is,
\[
( D_{b^{-}}^{\alpha }f) =-\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}
\int_{x}^{b}\frac{f(t)}{(t-x)^{\alpha }}dt\quad
( a\leq x<b,\;0<\alpha <1) .
\]
In particular, when $\alpha =0$, 
$D_{b^{-}}^{0}f=f$.
\end{definition}

\begin{definition} \label{def6} \rm
We define the space
\[
C_{1-\gamma }^{\gamma }[ a,b] =\{ y\in C_{1-\gamma }[a,b] ,
D_{a+}^{\gamma }y\in C_{1-\gamma }[ a,b] \} .
\]
\end{definition}

\begin{lemma}[\cite{k5,s1}]\label{lem1}
Let $0<\alpha <1$  and $0\leq \gamma <1$.
If $f\in C_{\gamma }^{1}$, 
the space of continuous functions on $[a,b]$ such that their derivatives 
are in $C_\gamma$,
then the fractional derivatives 
$D_{a^{+}}^{\alpha }$ and $D_{b^{-}}^{\alpha }$ exist on 
$(a,b]$ and $[a,b)$ respectively, and can be represented in the
forms
\[
( D_{a^{+}}^{\alpha }f) ( x) =\frac{1}{\Gamma (
1-\alpha ) }\Big[ \frac{f( a) }{( x-a) ^{\alpha
}}+\int_a^x\frac{f'( t) dt}{(x-t) ^{\alpha }}\Big]
\]
and
\[
( D_{b^{-}}^{\alpha }f) ( x) =\frac{1}{\Gamma (
1-\alpha ) }\Big[ \frac{f( b) }{( b-x) ^{\alpha
}}-\int_{x}^{b}\frac{f'( t) dt}{(t-x) ^{\alpha }}\Big] .
\]
\end{lemma}

Next, we have the Semigroup property of the fractional integration
operator $I_{a+}^{\alpha }$.

\begin{lemma}[\cite{k5,s1}]  \label{lem2}
 Let $\alpha >0$, $\beta >0$ and $0\leq \gamma <1$. If 
$f\in L_{p}(a,b)$, $1\leq p\leq \infty $ then the equation
\[
I_{a+}^{\alpha }I_{a+}^{\beta }f=I_{a+}^{\alpha +\beta }f
\]
holds at almost every point $x\in [a,b)]$.
When $\alpha +\beta >1$,  this relation is valid at any point $x\in [a,b]$.
\end{lemma}

Next is the fractional integration by parts.

\begin{lemma}[\cite{k5,s1}] \label{lem3}
Let $\alpha>0$, $p\geq 1$, $q\geq 1$ and 
$\frac{1}{p}+ \frac{1}{q}\leq 1+\alpha $ 
($p\neq 1$ and $q\neq 1$ in the case when 
$\frac{1}{p}+\frac{1}{q}=1+\alpha $). If
$\varphi \in L_{p}( a,b) $ and $\psi \in L_{q}(a,b)$,  then
\[
\int_a^{b}\varphi ( x) ( I_{a+}^{\alpha }\psi) ( x) dx
=\int_a^{b}\psi ( x) (I_{b-}^{\alpha }\varphi ) ( x) dx.
\]
\end{lemma}

\begin{definition} \label{def7} \rm
The fractional derivative $^{c}D_{a+}^{\alpha }f$ of order
$\alpha \in \mathbb{R}$ $( 0<\alpha <1) $ on 
$[ a,b] $ defined by
\[
^{c}D_{a+}^{\alpha }f=I_{a+}^{1-\alpha }Df,
\]
where $D=\frac{d}{dx}$, is called the Caputo fractional
derivative of $f$ of order $\alpha \in \mathbb{R}$.
\end{definition}

\begin{theorem}[Young's inequality] \label{thm1}
If $a$ and $b$ are
nonnegative real numbers and $p$ and $q$ are positive
real numbers such that $1/p+1/q=1$ then we have
\[
ab\leq \frac{a^{p}}{p}+\frac{b^{q}}{q}.
\]
\end{theorem}

\section{Non-existence result}

In this section we establish sufficient conditions ensuring non-existence of
global solutions. In particular we find a range of values for the exponent 
$m $ for which solutions cannot be continued for all time. The proof is based
mainly on the test function method developed by Mitidieri and Pohozaev 
\cite{m3} and some adequate manipulations of the fractional derivatives and integrals.
In addition to the results stated in the Preliminaries Section we need the
following lemma.

\begin{lemma} \label{lem4}
If $\alpha >0$ and $f\in C[ a,b] $,
then
\[
( I_{a+}^{\alpha }f) ( a) =\lim_{t\to a}(I_{a+}^{\alpha }f) ( t) =0
\]
and
\[
( I_{b^{-}}^{\alpha }f) ( b) =\lim_{t\to b}( I_{b^{-}}^{\alpha }f) ( t) =0.
\]
\end{lemma}

\begin{proof}
 Since $f\in C[ a,b] $,  on $[a,b]$, we have
 $ | f( t) | <M$ for some positive constant $M$.
Therefore
\begin{align*}
| ( I_{a+}^{\alpha }f) ( t) | 
&\leq \frac{1}{ \Gamma ( \alpha ) }\int_a^{t}( t-s)^{\alpha -1}| f( s) | ds\\
&\leq \frac{M}{\Gamma (\alpha ) }\int_a^{t}( t-s) ^{\alpha -1}ds \\
&\leq \frac{M}{\alpha \Gamma ( \alpha ) }[ -( t-s)
 ^{\alpha }] _{s=a}^{t}=\frac{M}{\Gamma ( \alpha +1) }(t-a) ^{\alpha }.
\end{align*}
As $\alpha >0$ we see that
\[
( I_{a+}^{\alpha }f) ( a) =\lim_{t\to a}(I_{a+}^{\alpha }f) ( t) =0.
\]
The second part is proved similarly.
\end{proof}

\begin{theorem} \label{thm2}
Assume that $\delta >-\alpha $  and $1<m\leq \frac{\delta
+1}{1-\alpha }$.  Then, Problem \eqref{1} does not admit global nontrivial
solutions in $C_{1-\gamma }^{\gamma }$, when $b>0$.
\end{theorem}

\begin{proof} Assume, on the contrary, that a nontrivial solution $u$
exists for all time $t>0$. Let $\varphi \in C^{1}( [0,\infty )) $
be a test function satisfying: $\varphi ( t) \geq 0$ and $\varphi $
is non-increasing such that
\[
\varphi (t) := \begin{cases}
1, & t\in [ 0,T/2], \\
0, & t\in [T,\infty ),
\end{cases}
\]
for some $T>0$. Multiplying the inequality in \eqref{1} by $\varphi (t) $ 
and integrating we obtain
\begin{equation}
\int_0^T( D_{0^{+}}^{\alpha ,\beta }u) (
t) \varphi ( t) dt\geq \int_0^Tt^{\delta
}| u( t) | ^{m}\varphi ( t) dt  \label{3}
\end{equation}
and from the definition of $( D_{0^{+}}^{\alpha ,\beta }u) (t) $ 
(see \eqref{2}) we can write
\begin{equation}
\int_0^TI_{0^{+}}^{\beta ( 1-\alpha ) }\frac{d}{dt}
( I_{0^{+}}^{1-\gamma }u) ( t) \varphi ( t)
dt\geq \int_0^Tt^{\delta }| u( t) |
^{m}\varphi ( t) dt.  \label{4}
\end{equation}
By Lemma \ref{lem3}, we may deduce from \eqref{4} that
\begin{equation}
\int_0^T\frac{d}{dt}\big( I_{0^{+}}^{1-\gamma }u\big)(t)
 \big(I_{T-}^{\beta ( 1-\alpha ) }\varphi \big) ( t)
dt\geq \int_0^Tt^{\delta }| u( t) |
^{m}\varphi ( t) dt.  \label{5}
\end{equation}
An integration by parts  yields
\begin{align*}
&[ ( I_{0^{+}}^{1-\gamma }u) ( t) (
I_{T-}^{\beta ( 1-\alpha ) }\varphi ) ( t)
] _{t=0}^T-\int_0^T( I_{0^{+}}^{1-\gamma }u)
( t) \frac{d}{dt}( I_{T-}^{\beta ( 1-\alpha )
}\varphi ) ( t) dt \\
&\geq \int_0^T t^{\delta }| u( t) |
^{m}\varphi ( t) dt.
\end{align*}
Using Lemma \ref{lem4} we see that $( I_{T-}^{\beta ( 1-\alpha )
}\varphi ) ( T) =0$ and $( I_{0^{+}}^{1-\gamma
}u) ( 0) =( D_{0^{+}}^{\gamma -1}u) (0) =b$, so
\[
-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (
0) -\int_0^T( I_{0^{+}}^{1-\gamma }u) (
t) \frac{d}{dt}( I_{T-}^{\beta ( 1-\alpha ) }\varphi
) ( t) dt\geq \int_0^Tt^{\delta }| u( t) | ^{m}\varphi ( t) dt.
\]
From Definition \ref{def5}, it follows that
\[
-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (
0) +\int_0^T( I_{0^{+}}^{1-\gamma }u) (
t) ( D_{T-}^{1-\beta ( 1-\alpha ) }\varphi )
( t) dt\geq \int_0^Tt^{\delta }| u(t) | ^{m}\varphi ( t) dt
\]
and from Lemma \ref{lem1} we see that
\begin{equation}
\begin{aligned}
&-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (0) \\
&+\int_0^T( I_{0^{+}}^{1-\gamma }u) (
t) \Big[ \frac{1}{\Gamma [ \beta ( 1-\alpha ) ]
}\Big( \frac{\varphi ( T) }{( T-t) ^{1-\beta (1-\alpha ) }}
 -\int_{t}^T\frac{\varphi '(s) ds}{( s-t) ^{1-\beta ( 1-\alpha ) }}\Big) \Big] \\
&\geq \int_0^Tt^{\delta }| u( t) | ^{m}\varphi ( t) dt.
\end{aligned}
\label{6}
\end{equation}
Since $\varphi ( T) =0$, relation (6) becomes
\[
-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (
0) -\int_0^T( I_{0^{+}}^{1-\gamma }u) (
t) ( I_{T^{-}}^{\beta ( 1-\alpha ) }\varphi ') ( t) dt\geq \int_0^Tt^{\delta }|
u( t) | ^{m}\varphi ( t) dt.
\]
Lemma \ref{lem3} allows us to write
\[
-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (
0) -\int_0^T\varphi '( t) (
I_{0^{+}}^{\beta ( 1-\alpha ) }I_{0^{+}}^{1-\gamma }u)
( t) dt\geq \int_0^Tt^{\delta }| u(
t) | ^{m}\varphi ( t) dt,
\]
and by Lemma \ref{lem2}
\begin{equation}
-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (
0) -\int_0^T\varphi '( t) (
I_{0^{+}}^{1-\alpha }u) ( t) dt\geq
\int_0^Tt^{\delta }| u( t) | ^{m}\varphi
( t) dt.  \label{7}
\end{equation}
Notice that
\begin{align*}
&-\int_0^T\varphi '( t) (
I_{0^{+}}^{1-\alpha }u) ( t) dt=\frac{-1}{\Gamma (
1-\alpha ) }\int_0^T\varphi '( t)
\int_0^{t}\frac{u( s) }{( t-s) ^{\alpha }}
ds\,dt \\
&\leq \frac{1}{\Gamma ( 1-\alpha ) }\int_0^T|
\varphi '( t) | \int_0^{t}\frac{
| u( s) | }{( t-s) ^{\alpha }}dsdt.
\end{align*}
Since $\varphi ( t) $ is nonincreasing,
 $\varphi ( s) \geq \varphi ( t) $ for all $t\geq s$, and
\[
\frac{1}{\varphi ( s) ^{1/m}}\leq \frac{1}{\varphi (
t) ^{1/m}},\quad 0\leq s\leq t<T,\;m>1.
\]
Also we have
\[
\varphi '( t) =0,\quad t\in [ 0,T/2] .
\]
Therefore,
\begin{align*}
-\int_0^T\varphi '( t) (
I_{0^{+}}^{1-\alpha }u) ( t) dt 
& \leq  \frac{1}{\Gamma
( 1-\alpha ) }\int_0^T| \varphi '( t) | \int_0^{t}\frac{| u( s)
| }{( t-s) ^{\alpha }}\frac{\varphi ( s) ^{1/m}}{
\varphi ( s) ^{1/m}}dsdt \\
& \leq  \frac{1}{\Gamma ( 1-\alpha ) }\int_0^T\frac{
| \varphi '( t) | }{\varphi (
t) ^{1/m}}\int_0^{t}\frac{| u( s) | }{
( t-s) ^{\alpha }}\varphi ( s) ^{1/m}dsdt \\
& \leq  \frac{1}{\Gamma ( 1-\alpha ) }\int_{T/2}^T
\frac{| \varphi '( t) | }{\varphi (
t) ^{1/m}}\int_0^{t}\frac{| u( s) | }{
( t-s) ^{\alpha }}\varphi ( s) ^{1/m}dsdt.
\end{align*}
Hence,
\[
-\int_0^T\varphi '( t) ( I_{0^{+}}^{1-\alpha }u) ( t) dt
\leq \int_{T/2}^T \frac{| \varphi '( t) | }{\varphi (
t) ^{1/m}}( I_{0^{+}}^{1-\alpha }\varphi ^{1/m}| u|
) ( t) dt\,.
\]
By Lemma \ref{lem3}, 
\begin{equation}
-\int_0^T\varphi '( t) (I_{0^{+}}^{1-\alpha }u) ( t) dt
\leq \int_{T/2}^T\Big( I_{T-}^{1-\alpha }\frac{| \varphi
'| }{\varphi ^{1/m}}\Big) ( t) \varphi (
t) ^{1/m}| u( t) | dt.  \label{8}
\end{equation}
(Note that we may assume that $| \varphi '( t)
| \varphi ( t) ^{-1/m}$ is summable even though $\varphi
( t) \to 0$ as $t\to T$, for otherwise we
consider $\varphi ^{\lambda }( t) $ with sufficiently large
exponent $\lambda $). Next, we multiply by $t^{\delta /m}.t^{-\delta /m}$
inside the integral in the right hand side of (8)
\[
-\int_0^T\varphi '( t) (I_{0^{+}}^{1-\alpha }u) ( t) dt
\leq \int_{T/2}^T \Big( I_{T-}^{1-\alpha }\frac{| \varphi
'| }{\varphi ^{1/m}}\Big) ( t) \varphi (
t) ^{1/m}\frac{t^{\delta /m}}{t^{\delta /m}}| u( t)
| dt.
\]
For $-\alpha <\delta <0$ we have $t^{-\delta /m}<T^{-\delta /m}$ 
(because $ t<T$) and for $\delta >0$ we obtain 
$t^{-\delta /m}<2^{\delta /m}T^{-\delta /m}$
(because $T/2<t$), that is
\[
t^{-\delta /m}<\max \{ 1,2^{\delta /m}\} T^{-\delta /m}.
\]
Therefore,
\begin{equation}
\begin{aligned}
&-\int_0^T\varphi '( t) (
I_{0^{+}}^{1-\alpha }u) ( t) dt \\
&\leq \max \{ 1,2^{\delta /m}\} T^{-\delta
/m}\int_{T/2}^T( I_{T-}^{1-\alpha }\frac{| \varphi'| }{\varphi ^{1/m}}) 
( t) t^{\delta /m}\varphi ( t) ^{1/m}| u( t) | dt.
\end{aligned}
\label{9}
\end{equation}
A simple application of the Young inequality (Theorem \ref{thm1}) with $m$ and 
$m'$ such that $\frac{1}{m}+\frac{1}{m'}=1$ gives
\begin{align*}
&-\int_0^T\varphi '( t) (I_{0^{+}}^{1-\alpha }u) ( t) dt\\
&\leq \frac{1}{m}\int_{T/2}^Tt^{\delta }\varphi ( t)
| u( t) | ^{m}dt+\frac{( \max \{ 1,2^{\delta
/m}\} ) ^{m'}}{m'}T^{-\frac{\delta m'}{m}}
\int_{T/2}^T\Big( I_{T-}^{1-\alpha }\frac{
| \varphi '| }{\varphi ^{1/m}}\Big) ^{m'}( t) dt \\
&\leq \frac{1}{m}\int_0^Tt^{\delta }\varphi ( t)
| u( t) | ^{m}dt+\frac{( \max \{ 1,2^{\delta
/m}\} ) ^{m'}}{m'}T^{-\frac{\delta
m'}{m}}\int_{T/2}^T\Big( I_{T-}^{1-\alpha }\frac{
| \varphi '| }{\varphi ^{1/m}}\Big) ^{m'}( t) dt.
\end{align*}
or
\begin{equation}
\begin{aligned}
&\int_0^T\varphi '( t) (I_{0^{+}}^{1-\alpha }u) ( t) dt \\
&\geq -\frac{1}{m}\int_0^Tt^{\delta }\varphi ( t)| u( t) | ^{m}dt
-\frac{( \max \{ 1,2^{\delta /m}\} ) ^{m'}}{m'}
T^{-\frac{\delta m'}{m}}\int_{T/2}^T( I_{T-}^{1-\alpha }\frac{
| \varphi '| }{\varphi ^{1/m}}) ^{m'}( t) dt.
\end{aligned}
\label{10}
\end{equation}
Clearly from \eqref{7} and \eqref{10}, we see that
\begin{align*}
&-b( I_{T-}^{\beta ( 1-\alpha ) }\varphi ) (
0) +\frac{( \max \{ 1,2^{\delta /m}\} )
^{m'}}{m'}T^{-\frac{\delta m'}{m}
}\int_{T/2}^T( I_{T-}^{1-\alpha }\frac{| \varphi
'| }{\varphi ^{1/m}}) ^{m'}(t) dt \\
&\geq ( 1-\frac{1}{m}) \int_0^Tt^{\delta }|
u( t) | ^{m}\varphi ( t) dt,
\end{align*}
or since $b>0$,
\[
\frac{1}{m'}\int_0^Tt^{\delta }| u(
t) | ^{m}\varphi ( t) dt\leq \frac{( \max
\{ 1,2^{\delta /m}\} ) ^{m'}}{m'}
T^{-\frac{\delta m'}{m}}\int_{T/2}^T\Big(
I_{T-}^{1-\alpha }\frac{| \varphi '| }{\varphi ^{1/m}}
\Big) ^{m'}( t) dt.
\]
Therefore, by Definition \ref{def3} we have
\begin{align*}
&\int_0^Tt^{\delta }| u( t) | ^{m}\varphi
( t) dt \\
&\leq ( \max \{ 1,2^{\delta /m}\} ) ^{m'}T^{-\frac{\delta m'}{m}}
\int_{T/2}^T\Big( \frac{1
}{\Gamma ( 1-\alpha ) }\int_{t}^T( s-t)
^{-\alpha }\frac{| \varphi '( s) | }{
\varphi ( s) ^{1/m}}ds\Big) ^{m'}dt.
\end{align*}
The change of variable $\sigma T=t$ yields
\begin{align*}
&\int_0^Tt^{\delta }| u( t) | ^{m}\varphi
( t) dt \\
&\leq ( \max \{ 1,2^{\delta /m}\} ) ^{m'}T^{-\frac{\delta m'}{m}}\int_{1/2}^{1}
( \frac{1}{\Gamma ( 1-\alpha ) }
\int_{\sigma T}^T\Big(s-\sigma T\Big) ^{-\alpha }
\frac{| \varphi '(s) | }{\varphi ( s) ^{1/m}}ds) ^{m'}Td\sigma .
\end{align*}
Another change of variable $s=rT$ gives
\begin{align*}
&\int_0^Tt^{\delta }| u( t) | ^{m}\varphi ( t) dt \\
&\leq ( \max \{ 1,2^{\delta /m}\} ) ^{m'}T^{-\frac{\delta m'}{m}}
 \int_{1/2}^{1}\Big( \frac{1 }{\Gamma ( 1-\alpha ) }\int_{\sigma }^{1}(
rT-\sigma T) ^{-\alpha }\frac{| \varphi '(
r) | }{\varphi ( r) ^{1/m}}dr\Big) ^{m'}Td\sigma ,
\end{align*}
or
\begin{equation}
\begin{aligned}
&\int_0^Tt^{\delta }| u( t) | ^{m}\varphi ( t) dt \\
&\leq \frac{( \max \{ 1,2^{\delta /m}\} ) ^{m'}}{\Gamma ^{m'}
( 1-\alpha ) }T^{1-\alpha m'-\delta m'/m}\int_{1/2}^{1}
\Big(\int_{\sigma }^{1}( r-\sigma ) ^{-\alpha }\frac{|
\varphi '( r) | }{\varphi ( r) ^{1/m}
}dr\Big) ^{m'}d\sigma .
\end{aligned}\label{11}
\end{equation}
It is clear that we may assume that the integral term in the 
right-hand side of \eqref{11} is bounded; that is,
\[
\int_{1/2}^{1}\Big( \int_{\sigma }^{1}( r-\sigma
) ^{-\alpha }\frac{| \varphi '( r) |
}{\varphi ( r) ^{1/m}}dr\Big) ^{m'}d\sigma \leq
K_{1},
\]
for some positive constant $K_{1}$,  otherwise we consider 
$\varphi^{\lambda }( r) $ with some sufficiently large $\lambda $.
Therefore,
\begin{equation}
\int_0^Tt^{\delta }| u( t) | ^{m}\varphi
( t) dt\leq K_{2}T^{1-\alpha m'-\delta m'/m},  \label{12}
\end{equation}
with
\[
K_{2}:=\frac{( \max \{ 1,2^{\delta /m}\} )
^{m'}}{\Gamma ^{m'}( 1-\alpha ) }K_{1}.
\]
If $m<\frac{\delta +1}{1-\alpha }$ we see that $1-\alpha m'-\delta m'/m<0$ 
and consequently $T^{1-\alpha m'-\delta m'/m}\to 0$ as $T\to \infty $. Then
from \eqref{12} we obtain
\[
\lim_{T\to \infty }\int_0^Tt^{\delta }| u(t) | ^{m}\varphi ( t) dt=0.
\]
This is a contradiction since the solution is supposed to be nontrivial.

In the case $m=\frac{\delta +1}{1-\alpha }$ we have
 $1-\alpha m'-\delta m'/m=0$ and the relation \eqref{12} ensures that
\begin{equation}
\lim_{T\to \infty }\int_0^Tt^{\delta }| u(t) | ^{m}\varphi ( t) dt\leq K_{2}. 
 \label{13}
\end{equation}
Moreover, it is clear that
\begin{align*}
&\int_{T/2}^T( I_{T-}^{1-\alpha }\frac{| \varphi
'| }{\varphi ^{1/m}}) ( t) t^{\delta
/m}\varphi ( t) ^{1/m}| u( t) | dt \\
&\leq \Big[ \int_{T/2}^T\Big( I_{T-}^{1-\alpha }\frac{|
\varphi '| }{\varphi ^{1/m}}\Big) ^{m'}( t) dt\Big] ^{1/m'}\Big[
\int_{T/2}^Tt^{\delta }\varphi ( t) | u(
t) | ^{m}dt\big] ^{1/m}.
\end{align*}
This relation, together with \eqref{7} and \eqref{9}, implies that
\[
\int_0^Tt^{\delta }\varphi ( t) | u(
t) | ^{m}dt\leq K_{3}\Big[ \int_{T/2}^Tt^{\delta
}\varphi ( t) | u( t) | ^{m}dt\Big] ^{1/m}
\]
for some positive constant $K_{3}$, with
\[
\lim_{T\to \infty }\int_{T/2}^Tt^{\delta }\varphi (t) | u( t) | ^{m}dt=0
\]
due to the convergence of the integral in \eqref{13}. This leads again to a
contradiction. The proof is complete. 
\end{proof}

\section{Sharpness of the bound}

In this section we want to prove that the exponent 
$\frac{\delta +1}{ 1-\alpha }$ is sharp in some sense.
 We will show that solutions exist for
exponents strictly bigger than $\frac{\delta +1}{1-\alpha }$. For that we
need the following lemma

\begin{lemma} \label{lem5}
The following identity holds
\[
( D_{a^{+}}^{\alpha ,\beta }[ ( s-a) ^{\sigma
-1}] ) ( t) =\frac{\Gamma ( \sigma ) }{
\Gamma ( \sigma -\alpha ) }( t-a) ^{\sigma -\alpha
-1},\quad t>a,\;\sigma >0,
\]
where $0<\alpha <1$ and $0\leq \beta \leq 1$.
\end{lemma}

\begin{example} \label{examp1} \rm
Consider the following differential equation with Hilfer
fractional derivative of order $0<\alpha <1$ and $0\leq \beta \leq 1$,
\begin{equation}
( D_{a^{+}}^{\alpha ,\beta }y) ( t) =\lambda (
t-a) ^{\delta }[ y( t) ] ^{m},\quad t>a,\;m>1
\label{14}
\end{equation}
with $\lambda $, $\delta \in \mathbb{R}$ ($\lambda \neq 0$).

Look for a solution of the form 
$y(t) =c(t-a) ^{\nu }$ for some $\nu \in \mathbb{R}$. 
Let us find the values of $c$ and $\nu $. By using Lemma \ref{lem5} we have
\[
( D_{a^{+}}^{\alpha ,\beta }[ c( s-a) ^{\nu }]
) ( t) =\frac{c\Gamma ( \nu +1) }{\Gamma (
\nu -\alpha +1) }( t-a) ^{\nu -\alpha },\quad \nu>-1,\; t>a.
\]
Plugging this expression in \eqref{14} yields
\[
\frac{c\Gamma ( \nu +1) }{\Gamma ( \nu -\alpha +1) }
( t-a) ^{\nu -\alpha }=\lambda ( t-a) ^{\delta }[
c( t-a) ^{\nu }] ^{m}.
\]
We obtain $\nu =\frac{\alpha +\delta }{1-m}$ and 
$c=[ \frac{\Gamma ( \frac{\alpha +\delta }{1-m}+1) }{\lambda \Gamma ( \frac{
m\alpha +\delta }{1-m}+1) }] ^{1/( m-1) }$. That is,
\[
y( t) =\Big[ \frac{\Gamma ( \frac{\alpha +\delta }{1-m}
+1) }{\lambda \Gamma ( \frac{m\alpha +\delta }{1-m}+1) }
\Big] ^{1/( m-1) }( t-a) ^{( \alpha +\delta) /( 1-m) }
\]
is a solution of \eqref{14}. 
One can easily check that $y\in C_{1-\gamma }$ with 
$ m=1+\frac{\alpha +\delta }{1-\gamma }$ which is clearly bigger than the
critical exponent $\frac{\delta +1}{1-\alpha }$ if $\delta >-\alpha $.
Moreover, the condition $( D_{a^{+}}^{\gamma -1}u) (0) =b$ 
is satisfied with
\[
b=\Big[ \frac{\Gamma ( \frac{\alpha +\delta }{1-m}+1) }{\lambda
\Gamma ( \frac{m\alpha +\delta }{1-m}+1) }\Big] ^{1/(m-1) }.
\]
\end{example}

\subsection*{Acknowledgments} 
The authors are very grateful for the financial
support and the facilities provided by King Fahd University of Petroleum and
Minerals through Project No. 101003.


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\end{document}