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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 29, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/29\hfil Existence of positive solutions]
{Existence of positive solutions for a nonlinear fractional 
 differential equation}

\author[H. Maagli \hfil EJDE-2013/29\hfilneg]
{Habib Maagli} 

\address{Habib Maagli \newline
King Abdulaziz University, College of Sciences
ant Arts, Rabigh Campus, Department of Mathematics P.O. Box 344,
Rabigh 21911, Kingdom of Saudi Arabia}
\email{habib.maagli@fst.rnu.tn}

\thanks{Submitted November 25, 2012. Published January 28, 2013.}
\subjclass[2000]{34A08}
\keywords{Riemann-Liouville fractional derivative; fixed point theorem}

\begin{abstract}
 Using the Schauder fixed point theorem, we prove an existence of
 positive solutions for the  fractional differential problem
 in the half line $\mathbb{R}^+=(0,\infty)$:
 $$
 D^{\alpha}u=f(x,u),\quad \lim_{x \to 0^+}u(x)=0,
 $$
 where $\alpha \in (1,2]$ and $f$ is a Borel measurable function
 in $\mathbb{R}^+\times \mathbb{R}^+$ satisfying some appropriate
 conditions.
\end{abstract}

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

\section{Introduction}

Recently, fractional differential equations have been studied extensively. 
The motivation for these studies stems from the fact that fractional 
differential equations serve as an excellent tool to describe many phenomena
 in various fields of science
and engineering such as control, porous media, electrochemistry, viscoelasticity,
electromagnetic, etc (see\cite{DF,Koe, KST, L1, Pod}).
Therefore, the theory of fractional differential equations has been
developed very quickly and the investigation for the existence of solutions
of fractional differential equations has  attracted a considerable
attention from  researches
 (see \cite{AG1, AG2, BL, BashA, BN1, CHS, DM, K, LWX, QB, ZSHL}
and the references therein).

To the best of our knowledge, most of the related results focus
on developing the  existence and uniqueness of  solutions
on the finite interval $[0,1]$.
In this note, we consider the following fractional differential problem 
in the half line ${\mathbb{R}}^+=(0,\infty)$:
\begin{equation} \label{P1}
\begin{gathered}
D^{\alpha}u=f(x,u),\\
u>0 \quad \text{in }  {\mathbb{R}}^+,\\
\lim_{x \to 0^+}u(x)=0,
\end{gathered}
\end{equation}
where $1<\alpha \leq 2$, $f(x,y)$ is a Borel measurable function in 
${\mathbb{R}}^+\times {\mathbb{R}}^+$
satisfying the following hypotheses:
\begin{itemize}
\item[(H1)] $f$ is continuous with respect to the second variable.
\item[(H2)] There exists a nonnegative measurable function $q$ 
defined on ${\mathbb{R}}^+ \times {\mathbb{R}}^+$
such that
\begin{itemize}
\item[(i)] $|f(x,y)|\leq y q(x,y)$  for all $ x, y\in {\mathbb{R}}^+$.
\item[(ii)] The function $y \to q(x,y)$ is nondecreasing and 
$\lim_{y\to 0^+}q(x,y)=0$.
\item[(iii)] The integral $\int_0^{\infty}t^{\alpha-1}q(t,t^{\alpha-1})\,dt$ 
converges.
\end{itemize}
\end{itemize}
We recall that for a measurable function $v$, the Riemann-Liouville 
fractional integral $I_{\beta}v$ and the
Riemann-Liouville derivative $D^{\beta}v$ of order $\beta>0$ are 
respectively defined by
\[
I_{\beta}v(x)=\frac{1}{\Gamma(\beta)}\int_0^x(x-t)^{\beta-1} v(t)\,dt\,
\]
and
\[
D^{\beta} v(x)
= \frac{1}{\Gamma(n-\beta)}\big(\frac{d}{dx}\big)^n
\int_0^x(x-t)^{n-\beta-1} v(t)\,dt
= \big(\frac{d}{dx}\big)^n I_{n-\beta} v(x)\,,
\]
provided that the right hand sides are pointwise defined for $x>0$. 
Here $n=[\beta]+1$ and $[\beta]$ means the
integral part of the number $\beta$ and $\Gamma$ is the Euler 
Gamma function.

Our main result is the following.

\begin{theorem}\label{thm1}
Assume {\rm (H1)--(H2)}, then problem \eqref{P1}  has infinitely many solutions.
More precisely, there exists a number $b>0$ such that for each $c \in (0,b]$, 
problem \eqref{P1} has a continuous solution $u$ satisfying
\begin{itemize}
\item[(i)] $u(x)= c x^{\alpha-1}+\int_0^{\infty}\left(x^{\alpha-1}-
\left((x-t)^+\right)^{\alpha-1}\right)f(t,u(t))\,dt.$
\item[(ii)] $\lim_{x \to \infty} x^{1-\alpha}u(x)=c$.
\item[(iii)] $\frac{c}{2} x^{\alpha-1}\leq u(x)\leq \frac{3}{2} x^{\alpha-1}$, 
for $x>0$.
\end{itemize}
\end{theorem}
Note that Theorem \ref{thm1} generalizes a result established by 
Zhao \cite{ZZ} in the case $\alpha=2$ (see also \cite{MM}).
As a special but important case of the above
general setting is the problem
\begin{equation} \label{P2}
\begin{gathered}
D^{\alpha}u=k(x)u^{p},\\
u>0 \quad \text{in }  {\mathbb{R}}^+,\\
\lim_{x \to 0^+}u(x)=0,
\end{gathered}
\end{equation}
where $p>1$ and $k$ is a Borel measurable function in ${\mathbb{R}}^+$ 
satisfying
\begin{equation}\label{condintegr2}
 \int_0^{\infty}t^{p(\alpha-1)}\,|k(t)|\,dt<\infty.
\end{equation}
An immediate consequence of Theorem \ref{thm1} is the following.

\begin{corollary}
Let $k$ be a Borel measurable function satisfying \eqref{condintegr2}, then 
the conclusion of Theorem \ref{thm1} holds for problem \eqref{P2}.
\end{corollary}

In the sequel, we denote by $C([0,\infty])$ the set of continuous functions $v$
on ${\mathbb{R}}^+$ such that $\lim_{x \to 0^+}v(x)$  
and $\lim_{x \to \infty}v(x)$ exist. 
It is easy to see that $C([0,\infty])$ is a Banach space with the norm
${\|v\|}_{\infty}=\sup_{x>0}|v(x)|$.
Finally, for $\lambda \in {\mathbb{R}}$,
we put $\lambda^+=\max(\lambda,0)$.

\section{Proof of Theorem \ref{thm1}}

Let ${\mathcal{F}}=\{v \in C([0,\infty]): {\|v\|}_{\infty}\leq 1\}$.
To prove Theorem \ref{thm1}, we need the following Lemma.

\begin{lemma}\label{lem1}
Assume {\rm (H1)--(H2)}, then the family of functions
$$
\big\{x\to \int_0^x(1-\frac{t}{x})f(t,t^{\alpha-1} v(t))\,dt:
 v\in {\mathcal{F}}\big\}
$$
is relatively compact in $C([0,\infty])$.
\end{lemma}

\begin{proof}
For $v \in {\mathcal{F}}$ and $x>0$, put 
$Sv(x)=\int_0^x(1-\frac{t}{x}) f(t,t^{\alpha-1} v(t))\,dt$.
By (H2), we have for all $v\in {\mathcal{F}}$ and $x>0$,
\begin{align*}
|Sv(x)|&\leq \int_0^{\infty}|f(t,t^{\alpha-1} v(t))|\,dt\\
&\leq \int_0^{\infty}t^{\alpha-1}q(t,t^{\alpha-1})\,dt<\infty.
\end{align*}
Thus the family $S({\mathcal{F}})$ is uniformly bounded.

 Now,we prove the equicontinuity of $S({\mathcal{F}})$ in $[0,\infty]$. 
Let $x,x'\in {\mathbb{R}}^+$ and
$v \in {\mathcal{F}}$, then we have
\begin{gather*}
|Sv(x)-Sv(x')|\leq \int_0^{\infty}|((1-\frac{t}{x})^+)^{\alpha-1}-((1-\frac{t}{x'})^+)^{\alpha-1}|
t^{\alpha-1}q(t,t^{\alpha-1})\,dt\,, 
\\
|Sv(x)| \leq \int_0^xt^{\alpha-1}q(t,t^{\alpha-1})\,dt\,,
\\
\Big|Sv(x)-\int_0^{\infty} f(t,t^{\alpha-1} v(t))\,dt\Big|
\leq \int_0^{\infty}\Big(1-((1-\frac{t}{x})^+
)^{\alpha-1}\Big)t^{\alpha-1} q(t,t^{\alpha-1})\,dt.
\end{gather*}
Using the Lebesgue's theorem, we deduce from the above inequalities that 
$S({\mathcal{F}})$ is equicontinuous in $[0,\infty]$.
 Hence, by Ascoli's theorem, we conclude that $S({\mathcal{F}})$
is relatively compact in $C([0,\infty])$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
By (H2) and Lebesgue's theorem, it follows that 
$$
\lim_{\beta \to 0} \int_0^{\infty}t^{\alpha-1} q(t,\beta t^{\alpha-1})\,dt=0.
$$
Hence we can fix a number
$\beta>0$ such that 
$$
\int_0^{\infty}t^{\alpha-1} q(t,\beta t^{\alpha-1})\,dt\leq \frac{1}{3}.
$$
Let $b=\frac{2}{3}\beta$ and $c \in (0,b]$. In order to apply a fixed point 
argument , set
$$ 
\Lambda= \{ v \in C([0,\infty]): \frac{c}{2}\leq v(x)\leq \frac{3}{2}c\,,\;
 \text{ for all } x>0\}\,.
$$
Then $\Lambda$ is a nonempty closed bounded and convex set in $C([0,\infty])$.
Define the operator $T$
on $\Lambda$ by
$$
Tv(x)=c+\int_0^{\infty}\Big(1-((1-\frac{t}{x})^+
)^{\alpha-1}\Big)f(t,t^{\alpha-1} v(t))\,dt\,,\quad  \text{for }
 x \in {\mathbb{R}}^+.
$$
First, we shall prove that the operator $T$ maps $\Lambda$ into itself. 
Let $v \in \Lambda$, then for any $x \in {\mathbb{R}}^+$, we have
\begin{align*}
|Tv(x)-c|&\leq \int_0^{\infty}t^{\alpha-1} v(t) q(t,t^{\alpha-1}v(t))\,dt\\
&\leq  \frac{3}{2}c \int_0^{\infty}t^{\alpha-1} q(t,\beta t^{\alpha-1})\,dt
\leq  \frac{c}{2}.
\end{align*}
It follows that $\frac{c}{2}\leq Tv\leq \frac{3}{2}c$ and since by 
Lemma \ref{lem1}, $T(\Lambda) \subset C([0,\infty])$, we deduce 
that $T(\Lambda)\subset \Lambda$.\\ Next, we shall prove
the continuity of $T$ in the supremum norm. Let $(v_k)_k$ 
be a sequence in $\Lambda$ which converges
uniformly to $v$ in $\Lambda$. It follows by (H1), (H2) and Lebesgue's
theorem that $Tv_k(x)\to Tv(x)$ as $k\to \infty$, for $x \in {\mathbb{R}}^+$.
 Since $T(\Lambda)$ is relatively compact in $C([0,\infty])$,  the pointwise 
convergence implies the uniform convergence.
Thus we have proved that $T$ is a compact mapping from $\Lambda$ to itself.

 Now, the Schauder fixed point theorem implies the existence of 
$\omega \in \Lambda$ such that $T\omega=\omega$. That is
$$
\omega(x)=c+\frac{1}{x^{\alpha-1}}\int_0^{\infty}\left(x^{\alpha-1}-((x-t)^+
)^{\alpha-1}\right)f(t,t^{\alpha-1}\,\omega(t))\,dt\,,\quad \text{for } x >0.
$$
Put $u(x)=x^{\alpha-1}\omega(x)$, for $x>0$. Then we have
$$
u(x)=cx^{\alpha-1}+\int_0^{\infty}\left(x^{\alpha-1}-((x-t)^+
)^{\alpha-1}\right)f(t,u(t))\,dt.
$$
Moreover,  for $x>0$, we have
\begin{gather*}
\frac{c}{2}x^{\alpha-1}\leq u(x)\leq \frac{3}{2}x^{\alpha-1},\\
\lim_{x\to \infty}x^{1-\alpha}u(x)=c.
\end{gather*}
It remains to show that  $u$ is a solution of problem \ref{P1}. 
 Indeed, for $x>0$, $u$ satisfies 
$$
u(x)=\Big(c+\int_0^{\infty}f(t,u(t))\,dt\Big)x^{\alpha-1}
-I_{\alpha}(f(.,u))(x).
$$
So $D^{\alpha}u(x)=-f(x,u(x))$, for $x>0$.
\end{proof}

\begin{example} \rm
Let $\beta>0$, $\gamma \in {\mathbb{R}}$ and $p>\gamma$. 
Let $k$ be a Borel measurable
function in ${\mathbb{R}}^+$ such that 
$\int_0^{\infty}t^{(p-\gamma+1)(\alpha-1)}\,|k(t)|\,dt<\infty$.
Then there exists $b>0$ such that for each $c \in (0,b]$, the  problem
\begin{equation*}
\begin{gathered}
D^{\alpha}u=\frac{k(x)u^{p+1}}{x^{\beta}+u^{\gamma}},\\
u>0 \quad \text{ in }  {\mathbb{R}}^+,\\
\lim_{x \to 0^+}u(x)=0, \quad \lim_{x \to \infty}x^{1-\alpha}\,u(x)=c
\end{gathered}
\end{equation*}
has a continuous solution $u$ in ${\mathbb{R}}^+$.
\end{example}

\begin{thebibliography}{00}

\bibitem{AG1}  Agarwal, Ravi P; O'Regan, Donal;  Stan\u{e}k, Svatoslav; 
\emph{Positive solutions for Dirichlet problems
of singular nonlinear fractional differential equations}.  
J. Math. Anal. Appl. {\textbf{371}} (2010), no.1, 57-68.

\bibitem{AG2} Agarwal, Ravi P; Benchohra, Mouffak; Hamani, Samira; Pinelas, Sandra.
\emph{Boundary value problems for differential equations involving 
Riemann-Liouville fractional derivative on the half line}.
Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal. {\textbf{18}} (2011),
no.2, 235-244.

\bibitem{BL} Bai, Zhanbing; L\"{u}, Haishen; 
\emph{Positive solutions for boundary value problem of nonlinear fractional 
differential equation}, J. Math. Anal. Appl. {\textbf{311}} (2005), no.2,  495-505.

\bibitem{BashA} Ahmad, Bashir; 
\emph{Existence of solutions for irregular boundary value problems of nonlinear 
fractional differential equations},
Appl. Math. Lett. {\textbf{23}} (2010), no.4,  390-394.

\bibitem{BN1} Ahmad, Bashir ;  Nieto, Juan J.;
 \emph{Riemann-Liouville fractional integro-differential
equations with fractional nonlocal integral boundary conditions}, 
Bound. Value Probl. (2011), 2011:36, 9 pp.

\bibitem{CHS} Caballero, J; Harjani, J; Sadarangani, K.;
\emph{Positive solutions for a class of singular
fractional boundary value problems}. Comput. Math. Appl. {\textbf{62}} (2011), no.3, 1325-1332.

\bibitem{DF} K, Diethelm; A. D,  Freed;
 \emph{On the solution of nonlinear fractional order differential equations used
in the modelling of viscoplasticity}, in "Scientifice Computing in  Chemical Engi-neering II-Computational Fluid Dynamics,
Reaction Engineering and Molecular Properties", (F. Keil, W. Mackens, H. Voss, and J.Werther, Eds),
pp 217-224, Springer-Verlag, Heidelberg, 1999.

\bibitem{DM}Deng, Jiqin; Ma, Lifeng;
\emph{Existence and uniqueness of solutions of initial value problems for
nonlinear fractional differential equations}, Appl. Math. Lett. {\textbf{23}} (2010), no. 6,  676-680.

\bibitem{Koe} Koeller, R. C.;
 \emph{Application of fractional calculus to the theory of viscoelasticity}. 
Trans. ASME J. Appl. Mech.
 {\textbf{51}} (1984), no. 2,   299-307.

\bibitem{K} Kosmatov, Nickolai;
 \emph{A singular boundary value problem for nonlinear differential
equations of fractional order}. 
J. Appl. Math. Comput. {\textbf{29}} (2009), no. 1-2, 125-135.

\bibitem{KST} Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J.;
\emph{Theory and Applications of Fractional Differential Equations}. 
North-Holland Mathematics Studies, 204.
Elsevier Science B.V., Amsterdam, 2006.

\bibitem{L1} Lin, Wei;
 \emph{Global existence theory and chaos control of fractional differential 
equations}. J. Math. Anal. Appl. {\textbf{332}} (2007), no. 1,  709-726.

\bibitem{LWX} Liu, Yang; Zhang, weiguo; Liu, Xiping;
\emph{A sufficient condition for the existence of a
positive solution for a nonlinear fractional differential equation
 with the Riemann–Liouville derivative}.
Appl. Math. Lett. {\textbf{25}} (2012), no. 11,  1986-1992.

\bibitem{MM} Maagli, Habib;  Masmoudi, Syrine;
\emph{Existence theorem of nonlinear singular
boundary value problem}.  Nonlinear Anal. {\textbf{46}} (2001), no. 4,  465-473.

\bibitem{Pod} Podlubny, Igor;
\emph{Geometric and physical interpretation of fractional integration and 
fractional differentiation}, Fract. Calc. Appl. Anal {\textbf{5}} (2002), 
no. 4,  367-386.

\bibitem{QB} Qiu, Tingting; Bai, Zhanbing;
\emph{Existence of positive solutions for singular fractional differential 
equations}. Electron. J. Differential Equations  (2008) , no. 146, pp. 1–9.

\bibitem{ZSHL} Zhao, Yige; Sun, Shurong; Han, Zhenlai; Li, Qiuping;
\emph{Positive Solutions to Boundary Value Problems of
Nonlinear Fractional Differential Equations}. Abstr. Appl. Anal.
2011, Article ID 390543, 16 pp.

\bibitem{ZZ} Z. Zhao;
 \emph{Positive solutions of nonlinear second order ordinary differential 
equations}, Proc. Amer. Math. Soc.  {\textbf{121}} (1994) , no. 2,  p.465-469.

\end{thebibliography}

\end{document}