\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 291, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/291\hfil Asymptotic behavior of solutions] {Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems} \author[M. D. Kassim, K. M. Furati, N.-E. Tatar \hfil EJDE-2016/291\hfilneg] {Mohammed D. Kassim, Khaled M. Furati, Nasser-eddine Tatar} \address{Mohammed D. Kassim \newline King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261, Saudi Arabia} \email{dahan@kfupm.edu.sa} \address{Khaled M. Furati \newline King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261, Saudi Arabia} \email{kmfurati@kfupm.edu.sa} \address{Nasser-eddine Tatar \newline King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261, Saudi Arabia} \email{tatarn@kfupm.edu.sa} \thanks{Submitted July 7, 2016. Published October 31, 2016.} \subjclass[2010]{34C11, 42B20, 34E10} \keywords{Regularization technique; Mittag-Leffler function; power type decay; \hfill\break\indent weighted space} \begin{abstract} We study the boundedness and asymptotic behavior of solutions for a class of nonlinear fractional differential equations. These equations involve two Riemann-Liouville fractional derivatives of different orders. We determine fairly large classes of nonlinearities and appropriate underlying spaces where solutions are bounded, exist globally and decay to zero as a power type function. Our results are obtained by using generalized versions of Gronwall-Bellman inequality, appropriate regularization techniques and several properties of fractional derivatives. Three examples are given to illustrate our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{property}[theorem]{Property} \allowdisplaybreaks \section{Introduction} The field of fractional calculus is concerned with the generalization of the integer order differentiation and integration to an arbitrary real (or complex) order \cite{Miller+Ross,Oldham+Spanier,Podl-(1999)}. Many events in diverse fields of engineering can be portrayed better and more accurately by differential equations of non-integer order \cite{G,Heymans+podl}.\newline In this article, we consider the fractional differential problem \begin{equation} \begin{gathered} D_0^{\alpha }y(t) =f(t,y(t) ,D_0^{\beta}y(t) ) ,\quad 0\leq \beta <\alpha <1,\; t>0, \\ I_0^{1-\alpha }y(t) \big|_{t=0}=b,\quad b\in \mathbb{R}, \end{gathered} \label{sr1} \end{equation} in an appropriate space of continuous functions, where $f$ is a continuous nonlinear function with respect to all of its arguments, $I_0^{\sigma }$ and $D_0^{\sigma }$ are the Riemann-Liouville fractional integral and fractional derivative defined by \begin{gather} I_{a}^{\alpha }f(t) =\frac{1}{\Gamma (\alpha )}\int_{a}^{t}\frac{f(s) }{(t-s)^{1-\alpha }}ds,\quad t>a, \label{ri} \\ D_{a}^{\alpha }f(t) =\frac{d}{dt}I_{a}^{1-\alpha }f(t),\quad t>a,\; 0<\alpha <1, \label{rd} \end{gather} respectively. Here $\Gamma (\alpha )$ is the Gamma function. When $\alpha =0 $, we define $I_{a}^{0}f$ $=f$. In particular, when $\alpha =1$ we have $D_{a}^{\alpha }f$ $=Df$, $D=\frac{d}{dt}$ and when $\alpha =0$, $D_{a}^{0}f=f$. When $\beta =0$, Problem \eqref{sr1} reduces to \begin{gather*} D_0^{\alpha }y(t) =f(t,y(t) ) ,\quad 0<\alpha <1,\quad t>0, \\ I_0^{1-\alpha }y(t) \big|_{t=0}=b. \end{gather*} The existence and uniqueness of solutions in a weighted space of continuous functions for problem \eqref{sr1} have been established in \cite[p 168, Theorem 3.14]{Kilbas} when $f$ is a real-valued continuous function and satisfies the Lipschitz condition. The study of the long time behavior of solutions of differential problems is in general extremely useful in applications. It has attracted many researchers, see \cite{G1,M1,M2,S} and many other references in \cite{Kilbas-4}. The question of asymptotic behavior of solutions of general differential problems consists often of determining sufficient conditions ensuring a certain specific (or just exploring the) behavior for large values of time. This task may be simple for simple problems. Things become even more complicated when dealing with nonlinear fractional differential equations. Therefore, observing the behavior through the explicit solution is not always possible. The behavior of solutions of various classes of FDEs (fractional differential equations) has been considered in many papers in the literature, see for example \cite{Barrett,Campos,Delb,Diethelm,Fujiwara,F1,F2,F3,F4,F5,F6,F7,K1, K2,Kilbas-0,Kilbas-1,Kilbas-2,Kilbas-4,L+Tatar,Tazali,Zhang}, and the references therein, to cite but a few. In particular, the behavior of solutions of the nonlinear equation \begin{equation} \begin{gathered} D_0^{\alpha }y(t) =f(t,y) ,\quad 0<\alpha <1,\; t>0, \\ t^{1-\alpha }y(t) \big|_{t=0}=b \end{gathered} \label{fu1} \end{equation} has been considered by Furati and Tatar in \cite{F4}. They proved that solutions decay polynomially on their interval of existence provided that $f(t,y) $ satisfies the condition \begin{equation} | f(t,y) | \leq t^{\mu }e^{-\sigma t}\varphi (t) | y| ^{m},\quad \mu \geq 0, \; m>1,\;\sigma >0, \label{cond. decay} \end{equation} where $\varphi (t) $ is a continuous function on $\mathbb{R}_{+}:=[0,\infty )$. In 2012, Furati, Kassim and Tatar \cite{F1} studied the nonlinear fractional differential problem \begin{equation} \begin{gathered} D_0^{\alpha ,\beta }y(t) =f(t,y) ,\quad 0<\alpha <1,\; t>0, \\ t^{(1-\alpha ) (1-\beta ) }y(t)\big|_{t=0}=b, \end{gathered} \label{fu2} \end{equation} where \[ D_0^{\alpha ,\beta }=I_0^{\beta (1-\alpha ) }DI_0^{(1-\beta ) (1-\alpha ) }, \] is the Hilfer fractional derivative of order $0<\alpha <1$ and type $0\leq \beta \leq 1$. They showed that solutions of \eqref{fu2} decay as a power function under the same condition \eqref{cond. decay} on the function $f(t,y) $. Notice when $\beta =0$ in \eqref{fu2} we obtain the same derivative in \eqref{fu1}. In 2013, P{\l}ociniczak \cite{pt} considered the linear fractional differential equation \begin{equation} ^{c}\!D^{\alpha } y(t) = \lambda \, q(t) y(t), \qquad 0<\alpha <1, \quad t>0, \label{pt} \end{equation} where $^{c}\!D^{\alpha }$ is the Caputo derivative of order $\alpha $ and $q(t) \sim C_{q} t^{\mu }>0$, $\mu >0$. He proved that the solution of \eqref{pt} for $\lambda >0$ obeys the asymptotic property $$ y(t) \sim C_1\exp (\lambda ^{1/\alpha }\int q^{1/\nu}(t) dt) \quad\text{as } t\to \infty, \quad\text{for some }C_1, $$ while for $\lambda <0$, \begin{equation*} y(t) \sim \frac{y(0) }{1-\lambda \Gamma (1-\alpha) q(t) t^{\alpha }} \quad \text{as }t\to \infty . \end{equation*} In this article, we study the behavior of solutions of \eqref{sr1}. We determine sufficient conditions on the nonlinear term which guarantee that solutions of \eqref{sr1} decay for all time in a weighted space of continuous functions. In particular, we prove that solutions decay like the power function $t^{\alpha -1}$. We mention here that the right hand side of \eqref{sr1} may contain several ``fractional derivatives'' but for simplicity we restrict ourselves to only one derivative. The presence of singular kernels in these derivatives is one of the main challenges we have to face. This article is organized as follows. In the next section, we introduce some material needed in our study. In Section 3, we establish some inequalities involving some special classes of functions. Sections 4 and 5 are devoted to our results. In Section 6, we illustrate our findings by three examples. \section{Fractional calculus and preliminaries} In this section we present some definitions, lemmas, properties and notation related to our results. For more details, we refer the reader to \cite{Kilbas,Podl-(1999),Samko}. For a finite interval $[a,b]$, let $C[a,b] $ and $C^{n}[a,b] $ denote the spaces of continuous and $n$-times continuously differentiable functions on $[a,b]$, respectively. \begin{definition}\label{def:sapace} \rm We consider the weighted spaces of continuous functions \begin{gather*} C_{\gamma }[a,b] =\{f:(a,b]\to \mathbb{R}: (t-a) ^{\gamma }f(t) \in C[a,b] \},\quad 0<\gamma <1,\\ C[a,b]=C_0[a,b], \end{gather*} and \begin{gather*} C_{\gamma }^{n}[a,b] =\{f\in C^{n-1}[a,b]:f^{(n)}\in C_{\gamma }[a,b] \},\quad n\in\mathbb{N}, \\ C_{\gamma }[a,b]= C_{\gamma }^{0}[a,b]. \end{gather*} \end{definition} \begin{remark}\label{rem:C^n} \rm Note that $C_{\gamma }^{n}[a,b] \subset AC^{n}[a,b] $ for $n\geq 1$, where \begin{equation*} AC^{n}[a,b] =\{ f:[a,b]\to \mathbb{R} \text{ and } f^{(n-1) }\in AC[a,b]\}, \end{equation*} and $AC[a,b] $ is the space of absolutely continuous functions on $[a,b]$. \end{remark} \begin{lemma}[\cite{Kou}] \label{lem:mapping} Let $\alpha >0$ and $0\leq \gamma <1$. Then, $I_{a}^{\alpha }$ is bounded from $C_{\gamma }[a,b] $ into $ C_{\gamma }[a,b] $. \end{lemma} \begin{lemma}[\cite{K3}] \label{lem:g''} Let $g$ be a continuous function on $(a,b]$. Then, $g^{(n) }\in C_{\gamma }[a,b] $ if and only if $g\in C_{\gamma }^{n}[a,b]$, $0\leq \gamma <1$. \end{lemma} For power functions we have the following property. \begin{property}[\cite{Kilbas}]\label{pro:power} \rm If $\alpha \geq 0$ and $\beta >0$, then \begin{equation*} D_{a}^{\alpha }(t-a) ^{\beta -1} =\frac{\Gamma (\beta) }{\Gamma (\beta -\alpha ) }(t-a) ^{\beta -\alpha -1},\quad t>a. \end{equation*} \end{property} A composition property between the fractional differentiation operator and the fractional integration operator is given next. \begin{property}[\cite{Kilbas}] \label{pro:comp} \rm Let $0<\beta <\alpha $ and $0\leq \gamma <1$. If $f\in C_{\gamma }[a,b]$, then the relation \begin{equation*} D_{a}^{\beta }I_{a}^{\alpha }f(t) =I_{a}^{\alpha -\beta }f(t) \end{equation*} holds at any point $t\in (a,b]$. When $f\in C[a,b] $ this relation is valid at any point $t\in \lbrack a,b]$. \end{property} The following result provides another composition of the fractional integration operator $I_{a}^{\alpha }\ $with the fractional differentiation operator $D_{a}^{\alpha }$. \begin{lemma}[\cite{Kilbas}] \label{lem:ID} Let $0<\alpha <1$, $0\leq \gamma <1$. If $f\in C_{\gamma }[a,b] $ and $I_{a}^{1-\alpha }f\in C_{\gamma }^{1}[a,b] $, then the equality \begin{equation} I_{a}^{\alpha }D_{a}^{\alpha }f(t) =f(t) -\frac{(I_{a}^{1-\alpha }f) (a) }{\Gamma (\alpha ) }(t-a) ^{\alpha -1}, \label{lemma 10 2} \end{equation} holds at any point $t\in (a,b]$. \end{lemma} \begin{lemma}[\cite{K3}] \label{lem:lower} Let $0<\alpha <1$ and $0\leq \gamma <1$. If $f\in C_{\gamma }[a,b] $ and $I_{a}^{1-\alpha }f\in C_{\gamma }^{1}[a,b]$, then for $0<\beta \leq \alpha <1$ we have \begin{equation*} D_{a}^{\beta }f(t) =I_{a}^{\alpha -\beta }D_{a}^{\alpha }f(t) +\frac{I_{a}^{1-\alpha }f(a) }{\Gamma (\alpha -\beta ) }(t-a) ^{\alpha -\beta -1},\quad t>a. \end{equation*} \end{lemma} \section{Useful inequalities} In this section we establish some inequalities involving special classes of functions. These inequalities are used in a crucial manner to prove our main results. \begin{remark} \label{rem:equiv} \rm In the rest of the paper we use the following equivalency. If $p,q>1$ and $\frac{1}{p}+\frac{1}{q}=1$, then \begin{equation} p(\alpha -1)+1>0\Longleftrightarrow q\alpha >1,\quad \alpha >0. \label{rem1} \end{equation} \end{remark} \begin{lemma}[\cite{Michalski}] \label{lem:estmofexp} If $\lambda ,\nu ,\omega >0$, then for any $t>0$, we have \begin{equation*} \int_0^{t}(t-s) ^{\nu -1}s^{\lambda -1}e^{-\omega s}ds\leq Ct^{\nu -1}, \end{equation*} where \begin{equation*} C=\max \{ 1,2^{1-\nu }\} \Gamma (\lambda ) ( 1+\lambda (\lambda +1) /\nu ) \omega ^{-\lambda }>0. \end{equation*} \end{lemma} Based on this result we prove here the following result. \begin{lemma} \label{lem:res3} Let $w>0$ and $\upsilon$, $\lambda >1/q$, for some $q>1$. Then, for any $t>0$ and any nonnegative continuous function $h$ defined on $\mathbb{R}_{+}$, we have \begin{equation} \int_0^{t}(t-s) ^{\upsilon -1}s^{\lambda -1}e^{-ws}h(s) ds \leq C t^{\upsilon -1}\Big(\int_0^{t}h^q(s)ds\Big) ^{1/q}, \label{sr7} \end{equation} where \begin{equation*} C=\Big[\max\{ 1,2^{1-\lambda _1}\} \Gamma (\lambda _2) \Big(1+\frac{(\lambda _2) (\lambda _2+1) }{\lambda _1}\Big) (pw) ^{-\lambda _2}\Big]^{1/p}, \end{equation*} $1/p+1/q=1$, $\lambda _1=p(\upsilon -1) +1$, and $\lambda _2=p(\lambda -1) +1$. \end{lemma} \begin{proof} Applying H\"{o}lder inequality to the left hand side of \eqref{sr7}, for $t>0$, we obtain \begin{equation} \begin{aligned} &\int_0^{t}(t-s) ^{\upsilon -1}s^{\lambda -1}e^{-ws}h(s) ds \\ &\leq \Big(\int_0^{t}(t-s) ^{p(\upsilon -1) }s^{p(\lambda -1) }e^{-pws}\Big) ^{1/p} \Big(\int_0^{t}h^q(s) ds\Big) ^{1/q}. \end{aligned} \label{sr7.5} \end{equation} From the hypotheses stated in the lemma, we have \begin{equation*} \lambda _1=p(\upsilon -1) +1>0\text{ and }\lambda _2=p(\lambda -1) +1>0. \end{equation*} Applying Lemma \ref{lem:estmofexp} to \eqref{sr7.5} (with $\upsilon $ replaced by $\lambda _1$, $\lambda $ replaced by $\lambda _2$ and $w$ replaced by $pw$), gives the result. \end{proof} \begin{lemma}[\cite{Anastassiou,Kuczma}] \label{lem:power} Let $a>0$ and $b>0$. Then \begin{gather*} a^{r}+b^{r}\leq (a+b) ^{r}\leq 2^{r-1}(a^{r}+b^{r}) ,\quad r\geq 1, \\ 2^{r-1}(a^{r}+b^{r}) \leq (a+b) ^{r}\leq a^{r}+b^{r}, \quad 0\leq r\leq 1. \end{gather*} \end{lemma} We recall now the Bihari inequality. \begin{theorem}[\cite{Pachpatte}] \label{thm:Bih.ineq} Let $u$ and $f$ be nonnegative continuous functions defined on $\mathbb{R}_{+}$. Let $w(u)$ be a continuous nondecreasing function defined on $\mathbb{R}_{+}$ and $w(u)>0$ on $(0,\infty ) $. If \begin{equation*} u(t) \leq k+\int_0^{t}f(s) w(u(s) ) ds, \end{equation*} for $t\in\mathbb{R}_{+}$, where $k$ is a nonnegative constant, then for $0\leq t\leq t_1$, \begin{equation*} u(t) \leq G^{-1}\Big(G(k) +\int_0^{t}f(s) ds\Big) , \end{equation*} where \begin{equation*} G(r) =\int_{r_0}^{r}\frac{ds}{w(s) },\quad r>0,\;r_0>0, \end{equation*} $G^{-1}$ is the inverse function of $G$, and $t_1\in\mathbb{R}_{+}$ is chosen so that \begin{equation*} G(k) +\int_0^{t}f(s) ds\in \operatorname{Dom}(G^{-1}) , \end{equation*} for $0\leq t\leq t_1$. \end{theorem} From Theorem \ref{thm:Bih.ineq} we have the following corollaries. \begin{corollary} \label{lem:res1} Let $z$ and $h$ be nonnegative continuous functions defined on $\mathbb{R}_{+}$. Let $w(z)$ be a continuous nondecreasing function defined on $\mathbb{R}_{+}$ and $w(z)>0$ on $(0,\infty )$. If \begin{equation} z(t) \leq K_1+K_2(\int_0^{t}h(s) w(z(s) ) ds) ^{1/q},\quad q>1,\; t>0, \label{sr14} \end{equation} where $K_i$, $i=1,2$, are nonnegative constants, then for $0\leq t\leq t_1$ \begin{equation*} z(t) \leq \Big[G^{-1}\Big(G(2^{q-1}K_1) +2^{q-1}K_2\int_0^{t}h(s) ds\Big) \Big] ^{1/q}, \end{equation*} where \begin{equation*} G(x) =\int_{x_0}^{x}\frac{ds}{w(s^{1/q}) },\quad x>x_0>0, \end{equation*} and $G^{-1}$ is the inverse function of $G$, and $t_1\in\mathbb{R}_{+}$ is chosen so that \begin{equation*} G(2^{q-1}K_1) +2^{q-1}K_2\int_0^{t}h(s) ds\in \operatorname{Dom}(G^{-1}) , \end{equation*} for $0\leq t\leq t_1$. \end{corollary} \begin{proof} Raising both sides of \eqref{sr14} to the power $q$ and using Lemma \ref{lem:power}, we have \begin{equation} z^q(t) \leq B_1+B_2\int_0^{t}h(s) w(z(s) ) ds,\quad t>0, \label{sr16} \end{equation} where $B_i=2^{q-1}K_i$, $i=1,2$. Now, let $u(t) =z^q(t) $, then \eqref{sr16} can be written as \begin{equation} u(t) \leq B_1+B_2\int_0^{t}h(s) g(u(s) ) ds,\quad t>0, \label{sr17} \end{equation} where \begin{equation} g(r) =w(r^{1/q}) . \label{sr18} \end{equation} Since $w$ is a continuous and nondecreasing functions, then $g$ is a continuous and nondecreasing function. Applying Bihari's inequality (Theorem\ref{thm:Bih.ineq}) to \eqref{sr17}, we obtain the result. \end{proof} \begin{corollary} \label{lem:res2} Let $z$, $h_i$, $w_i$, $i=1,2$, and $q$ be as in Corollary \ref{lem:res1}. If \begin{equation} z(t) \leq K_1+K_2\Big[\Big(\int_0^{t}h_1(s) w_1(z(s) ) ds\Big) ^{1/q} +\Big(\int_0^{t}h_2(s) w_2(z(s) ) ds\Big) ^{1/q}\Big] , \label{sr44} \end{equation} for $t>0$, then, for $0\leq t\leq t_1$, \begin{equation} z(t) \leq \Big[G^{-1}\Big(G(2^{q-1}K_1^q\quad ) +2^{2(q-1) }K_2^q\int_0^{t}[h_1( s) +h_2(s) ] ds\Big) \Big] ^{1/q}, \label{sr51} \end{equation} where \begin{equation*} G(x) =\int_{x_0}^{x}\frac{ds}{g(s) } =\int_{x_0}^{x}\frac{ds}{w_1(s^{1/q}) +w_2( s^{1/q}) },\quad x>x_0>0, \end{equation*} and $G^{-1}$ is the inverse function of $G$, and $t_1\in \mathbb{R}_{+}$ is chosen so that \begin{equation*} G(2^{q-1}K_1^q) +2^{2(q-1) }K_2^q\int_0^{t}[h_1(s) +h_2(s) ] ds\in \operatorname{Dom}(G^{-1}) , \end{equation*} for $0\leq t\leq t_1$. \end{corollary} \begin{proof} Raising both sides of \eqref{sr44} to the power $q$, we have \begin{equation} z^q(t) \leq B_1+B_2[\int_0^{t}h_1(s) w_1(z(s) ) ds+\int_0^{t}h_2(s) w_2(z(s) ) ds] , \quad t>0, \label{sr45} \end{equation} where \begin{equation*} B_1=2^{q-1}K_1^q, \quad B_2=2^{2(q-1) }K_2^q. \end{equation*} Furthermore, we have \begin{equation} h_1(s) w_1(z(s) ) +h_2(s) w_2(z(s) ) \leq [h_1(s) +h_2(s) ] [w_1(z(s) ) +w_2(z(s) ) ] . \label{sr46} \end{equation} Now, let $u(t) =z^q(t) $, then by \eqref{sr45} and \eqref{sr46} we can write \begin{equation} u(t) \leq B_1+B_2\int_0^{t}[h_1(s) +h_2(s) ] g(u(s) ) ds,\quad t>0, \label{sr50} \end{equation} where \begin{equation} g(r) =w_1(r^{1/q}) +w_2(r^{1/q}) . \label{sr49} \end{equation} Since $w_i$, $i=1,2$, are continuous and nondecreasing functions, then $g$ is a continuous, nondecreasing function. Applying Bihari's inequality to \eqref{sr50}, we obtain the result. \end{proof} \section{Preliminaries} In this section we prove some lemmas which will be used to prove the main results. In the sequel, we consider the following assumptions: \begin{itemize} \item[(A1)] The function $f:(0,\infty ) \times\mathbb{R}^{2}\to\mathbb{R}$ is such that $f(\cdot,u(\cdot) ,v(\cdot) ) \in C_{1-\alpha }[0,\infty )$ for any $u$, $v\in C_{1-\alpha }[0,\infty )$. \item[(A2)] There exist continuous functions $h$, $\varphi _1$, $\varphi _2:\mathbb{R}_{+}\to \mathbb{R}_{+}$, such that \begin{equation} | f(t,u,v) | \leq t^{\gamma }e^{-\delta t}h(t) \varphi _1(t^{1-\alpha }| u| ) \varphi _2(t^{1-(\alpha -\beta ) }|v| ) , \label{sr2} \end{equation} where $h\in L_{q}(0,\infty ) $ for some $q>\frac{1}{\alpha-\beta }$, $\gamma >\frac{1}{q}-1$, $\delta >0$, and $\varphi _i$, $i=1,2$, are nondecreasing functions. \item[(A3)] There exist continuous functions $h_i$, $\varphi _i: \mathbb{R}_{+}\to \mathbb{R}_{+}$, such that \begin{equation} | f(t,u,v) | \leq t^{\gamma _1}e^{-\delta _1t}h_1(t) \varphi _1(t^{1-\alpha }| u| ) +t^{\gamma _2}e^{-\delta _2t}h_2(t) \varphi _2(t^{1-(\alpha -\beta ) }|v| ) , \label{sr35} \end{equation} where $h_i\in L_{q}(0,\infty ) $ for some $q>\frac{1}{\alpha -\beta }$, $\gamma _i>\frac{1}{q}-1$, $\delta _i>0$, and $\varphi _i$, $i=1,2$, are nondecreasing functions. \end{itemize} The following results provide useful estimates for the solutions of \eqref{sr1}. \begin{lemma} \label{lem:res4} Assume that $y\in C_{1-\alpha }[0,\infty )$ is a global solution of \eqref{sr1} and $f$ satisfies {\rm (A1)} and {\rm (A2)}. Then \begin{equation*} \max \big\{ t^{1-\alpha }| y(t) | ,t^{1-(\alpha -\beta ) }| D^{\beta }y(t) | \big\} \leq z(t) ,\quad t>0, \end{equation*} where \begin{gather} z(t) =K_1+K_2\Big(\int_0^{t}h^q(s) \varphi _1^q(s^{1-\alpha }| y(s) | ) \varphi _2^q(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds\Big) ^{1/q}, \label{sr11} \\ K_1=| b| \max \big\{ \frac{1}{\Gamma (\alpha ) },\frac{1}{\Gamma (\alpha -\beta ) }\big\} , \quad K_2=\max \big\{ C_1,C_2\big\} , \nonumber\\ C_1=\frac{1}{\Gamma (\alpha ) }\Big(\max\{1,2^{p(1-\alpha ) }\} \Gamma (p\gamma +1)\Big(1+\frac{(p\gamma +1) (p\gamma +2) }{p( \alpha -1) +1}\Big) (p\delta ) ^{-(p\gamma+1) }\Big) ^{1/p}, \nonumber \\ \begin{aligned} C_2&=\frac{1}{\Gamma (\alpha -\beta ) }\Big(\max \{ 1,2^{p(1-(\alpha -\beta ) ) }\} \Gamma (p\gamma +1) \\ &\quad\times \Big(1+\frac{(p\gamma +1) (p\gamma +2) }{p(\alpha -\beta -1) +1}\Big) (p\delta ) ^{-(p\gamma +1) }\Big) ^{1/p}. \end{aligned} \nonumber \end{gather} \end{lemma} \begin{proof} Since $f\in C_{1-\alpha }[0,\infty )$, Equation \eqref{sr1} implies that $D_0^{\alpha }y=DI_0^{1-\alpha }y\in C_{1-\alpha }[0,\infty )$. By Lemma \ref{lem:g''}, we have $I_0^{1-\alpha }y\in C_{1-\alpha }^{1}[0,\infty )$. Applying $I_0^{\alpha }$ to \eqref{sr1} and using Lemma \ref{lem:ID}, having in mind that $y\in C_{1-\alpha }[0,\infty )$ and $I_0^{1-\alpha }y\in C_{1-\alpha }^{1}[0,\infty )$, we obtain \begin{equation} y(t) =\frac{b}{\Gamma (\alpha ) }t^{\alpha -1}+\frac{1}{\Gamma (\alpha ) } \int_0^{t}(t-s) ^{\alpha-1}f(s,y(s) ,D_0^{\beta }y(s) ) ds, \quad t>0. \label{sr4} \end{equation} Next, multiplying both sides of \eqref{sr4} by $t^{1-\alpha }$ and using the assumption \eqref{sr2}, we obtain \begin{equation} \begin{aligned} t^{1-\alpha }| y(t) | &\leq \frac{|b| }{\Gamma (\alpha ) } +\frac{t^{1-\alpha }}{\Gamma (\alpha ) }\int_0^{t}( t-s) ^{\alpha -1}s^{\gamma }e^{-\delta s}h(s) \varphi _1(s^{1-\alpha }| y(s) | ) \\ &\quad \times \varphi _2(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds,\quad t>0. \end{aligned} \label{sr5} \end{equation} In view of Lemma \ref{lem:res3}, we find \begin{equation} \begin{aligned} t^{1-\alpha }| y(t) | & \leq \frac{|b| }{\Gamma (\alpha ) } +C_1\Big(\int_0^{t}h^q(s) \varphi _1^q(s^{1-\alpha}| y(s) | ) \\ &\quad\times \varphi _2^q(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds\Big) ^{1/q},\quad t>0. \end{aligned} \label{sr8} \end{equation} Since $0<\beta <\alpha <1$, by Lemma \ref{lem:lower}, we see that \begin{equation} \begin{aligned} D_0^{\beta }y(t) &=\frac{bt^{\alpha -\beta -1}}{\Gamma ( \alpha -\beta ) }+\frac{1}{\Gamma (\alpha -\beta ) } \int_0^{t}(t-s) ^{\alpha -\beta -1}D_0^{\alpha }y(s) ds \\ &=\frac{bt^{\alpha -\beta -1}}{\Gamma (\alpha -\beta) } +\frac{1}{\Gamma (\alpha -\beta ) }\int_0^{t}(t-s) ^{\alpha -\beta -1} f(s,y(s) ,D_0^{\beta}y(s) ) ds, \end{aligned} \label{sr9} \end{equation} for $t>0$. Multiplying both sides of \eqref{sr9} by $t^{1-(\alpha -\beta )} $ and using the assumption \eqref{sr2}, for $t>0$, we deduce \begin{align*} t^{1-(\alpha -\beta ) }| D_0^{\beta }y(t)| &\leq \frac{| b| }{\Gamma (\alpha -\beta) } +\frac{t^{1-(\alpha -\beta ) }}{\Gamma (\alpha -\beta ) } \int_0^{t}(t-s) ^{\alpha -\beta -1}s^{\gamma }e^{-\delta s}h(s) \\ &\quad\times \varphi _1(s^{1-\alpha}| y(s) | ) \varphi _2(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s)| ) ds. \end{align*} Again, by Lemma \ref{lem:res3}, we conclude that \begin{equation} \begin{aligned} t^{1-(\alpha -\beta ) }| D_0^{\beta }y(t)| &\leq \frac{| b| }{\Gamma (\alpha -\beta) } +C_2\Big(\int_0^{t}h^q(s) \varphi _1^q(s^{1-\alpha }| y(s) |) \\ &\quad\times \varphi _2^q(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds\Big) ^{1/q}, \quad t>0. \end{aligned} \label{sr10} \end{equation} Therefore, the result follows from \eqref{sr11}, \eqref{sr8} and \eqref{sr10}. \end{proof} \begin{lemma} \label{lem:res6} Assume that $y\in C_{1-\alpha }[0,\infty )$ is a global solution of \eqref{sr1} and $f$ satisfies {\rm (A1)} and {\rm (A3)}. Then \begin{equation*} \max \big\{ t^{1-\alpha }| y(t) |,t^{1-(\alpha -\beta ) }| D^{\beta }y(t) | \big\} \leq z(t) ,\quad t>0, \end{equation*} where, for $t>0$, \begin{gather} \begin{aligned} z(t)&=K_1 +K_2\Big[\Big(\int_0^{t}h_1^q(s) \varphi _1^q(s^{1-\alpha }| y(s) | ) ds\Big) ^{1/q} \\ &\quad +\Big(\int_0^{t}h_2^q(s) \varphi_2^q(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds\Big) ^{1/q}\Big] , \end{aligned} \label{sr41} \\ K_1=| b| \max \big\{ \frac{1}{\Gamma (\alpha) },\frac{1}{\Gamma (\alpha -\beta ) } \big\}, \quad K_2=\max \{ C_3,C_3'\} , \nonumber \\ C_3=\max\{ C_1,C_2\} ,\quad C_3'=\max \{C_1',C_2'\} ,\nonumber \\ C_i=\frac{1}{\Gamma (\alpha ) }\Big(\max \{1,2^{p(1-\alpha ) }\} \Gamma (p\gamma _i+1) \Big(1+\frac{p\gamma _i+1}{p(\alpha -1) +1}\Big) (p\delta _i) ^{-(p\gamma _i+1) }\Big)^{1/p}, \nonumber \\ C_i'=\Big(\frac{\max \{ 1,2^{p(1-(\alpha-\beta ) ) }\} \Gamma (p\gamma _i+1) }{\Gamma ^{p}(\alpha -\beta ) } \Big(1+\frac{p\gamma _i+1}{ p(\alpha -\beta -1) +1}\Big) (p\delta _i)^{-(p\gamma _i+1) }\Big) ^{1/p}, \nonumber \end{gather} for $i=1,2$. \end{lemma} \begin{proof} Multiplying both sides of \eqref{sr4} by $t^{1-\alpha }$ and using the assumption \eqref{sr35}, we obtain \begin{align*} & t^{1-\alpha }| y(t) | \\ & \leq \frac{|b| }{\Gamma (\alpha ) } +\frac{t^{1-\alpha }}{\Gamma(\alpha ) } \int_0^{t}(t-s) ^{\alpha -1}s^{\gamma_1}e^{-\delta _1s}h_1(s) \varphi _1(s^{1-\alpha}| y(s) | ) ds \\ &\quad +\frac{t^{1-\alpha }}{\Gamma (\alpha ) }\int_0^{t}(t-s) ^{\alpha -1}s^{\gamma _2}e^{-\delta _2s}h_2(s) \varphi _2(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds, \quad t>0. \end{align*} Since $q>\frac{1}{\alpha -\beta }$, $\gamma _i>\frac{1}{q}-1$, $\delta_i>0$, then $p(\alpha -1) +1>0$, $p\gamma _i+1>0$ and $p\delta _i>0$, $i=1$, $2$, so we can apply Lemma \ref{lem:res3}, for $t>0$, we obtain \begin{equation} \begin{aligned} t^{1-\alpha }| y(t) | &\leq \frac{|b| }{\Gamma (\alpha ) } +C_3\Big[\Big(\int_0^{t}h_1^q(s) \varphi _1^q(s^{1-\alpha }| y(s) | )ds\Big) ^{1/q} \\ &\quad +\Big(\int_0^{t}h_2^q(s) \varphi _2^q(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | ) ds\Big) ^{1/q}\Big] . \end{aligned} \label{sr38} \end{equation} Multiplying both sides of \eqref{sr9} by $t^{1-(\alpha -\beta )} $ and using \eqref{sr35}, we obtain \begin{align*} &t^{1-(\alpha -\beta ) }| D_0^{\beta }y(t)| \\ &\leq \frac{| b| }{\Gamma (\alpha -\beta ) } +\frac{t^{1-(\alpha -\beta ) }}{\Gamma (\alpha -\beta ) } \int_0^{t}(t-s) ^{\alpha-\beta -1}s^{\gamma _1}e^{-\delta _1s}h_1(s) \varphi _1(s^{1-\alpha }| y(s) | ) ds \\ &\quad +\frac{t^{1-(\alpha -\beta ) }}{\Gamma (\alpha -\beta ) }\int_0^{t}(t-s) ^{\alpha -\beta -1}s^{\gamma _2}e^{-\delta _2s}h_2(s) \varphi _2\big( s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | \big) ds,\quad t>0. \end{align*} By Lemma \ref{lem:res3}, for $t>0$, we can write \begin{equation} \begin{aligned} &t^{1-(\alpha -\beta ) }| D_0^{\beta }y(t)| \\ &\leq \frac{| b| }{\Gamma (\alpha -\beta) } +C_3'\Big[\Big(\int_0^{t}h_1^q(s) \varphi _1^q(s^{1-\alpha }| y(s) | ) ds\Big) ^{1/q} \\ &\quad +\Big(\int_0^{t}h_2^q(s) \varphi _2^q\big(s^{1-(\alpha -\beta ) }| D_0^{\beta }y(s) | \big) ds\Big) ^{1/q}\Big] . \end{aligned}\label{sr40} \end{equation} Therefore, the result follows from \eqref{sr41}, \eqref{sr38} and \eqref{sr40}. \end{proof} \section{Power-type decay} We consider the space \begin{equation} C_{1-\alpha }^{\alpha }[0,\infty ) =\{ y\in C_{1-\alpha }[0,\infty):D_0^{\alpha }y\in C_{1-\alpha }[0,\infty )\} . \label{space} \end{equation} \begin{theorem} \label{thm:res1} Suppose that $f$ satisfies {\rm (A1)} and {\rm (A2)}, then, for any global solution $y\in C_{1-\alpha }[0,\infty )$ of \eqref{sr1}, there exists a positive constant $C$ such that \begin{equation*} | y(t) | \leq Ct^{\alpha -1}\quad\text{and}\quad | D_0^{\beta }y(t) | 0, \end{equation*} provided that \begin{equation*} \int_{x_0}^{\infty }\frac{ds}{\varphi _1^q(s^{1/q}) \varphi _2^q(s^{1/q}) }=\infty ,\quad x_0>0. \end{equation*} \end{theorem} \begin{proof} From Lemma \ref{lem:res4} we conclude \begin{equation} \varphi _1(t^{1-\alpha }| y(t) |) \leq \varphi _1(z(t) ),\quad \varphi_2(t^{1-(\alpha -\beta ) }| D^{\beta }y(t) | ) \leq \varphi _2(z(t) ) ,\quad t>0, \label{in1} \end{equation} where $z(t) $\ is as in \eqref{sr11}. Using the inequalities in \eqref{in1}, from \eqref{sr11} it follows that \begin{equation*} z(t) \leq K_1+K_2\big(\int_0^{t}h^q(s) \varphi _1^q(z(s) ) \varphi _2^q( z(s) ) ds\Big) ^{1/q},\quad t>0. \end{equation*} Therefore, by Corollary \ref{lem:res1} with $w(t) =\varphi_1^q(t) \varphi _2^q(t) $, we deduce that \begin{equation*} z(t) \leq \Big[G^{-1}\Big(G(2^{q-1}K_1) +2^{q-1}K_2\int_0^{t}h^q(s) ds\Big) \Big] ^{1/q},\quad t>0. \end{equation*} Since $h\in L_{q}(0,\infty ) $, we have \begin{equation*} z(t) \leq C=\Big[G^{-1}\Big(G(2^{q-1}K_1) +2^{q-1}K_2\int_0^{\infty }h^q(s) ds\Big) \Big]^{1/q}<\infty . \end{equation*} Again, by Lemma \ref{lem:res4}, \begin{equation*} | y(t) | \leq Ct^{\alpha -1}\quad\text{and}\quad | D_0^{\beta }y(t) | 0. \end{equation*} \end{proof} \begin{theorem} \label{thm:res3} Suppose that $f$ satisfies {\rm (A1)} and {\rm (A3)}. Then, for each global solution $y\in C_{1-\alpha}[0,\infty )$ of \eqref{sr1}, there exists a positive constant $C$ such that \begin{equation*} | y(t) | \leq Ct^{\alpha -1}\quad \text{and} \quad | D_0^{\beta }y(t) | 0, \end{equation*} provided that \begin{equation*} \int_{x_0}^{\infty }\frac{ds}{\varphi _1^q(s^{1/q}) +\varphi _2^q(s^{1/q}) }=\infty ,\quad x_0>0. \end{equation*} \end{theorem} \begin{proof} By Lemma \ref{lem:res6} we see that \begin{equation} \varphi _1(t^{1-\alpha }| y(t) |) \leq \varphi _1(z(t) ), \quad \varphi_2(\big(t^{1-(\alpha -\beta ) }| D^{\beta }y(t) | \big) \leq \varphi _2(z(t)) ,\quad t>0. \label{in3} \end{equation} Taking into account \eqref{sr41} and \eqref{in3}, we have \begin{align*} z(t) &\leq K_1+K_2\Big[\Big(\int_0^{t}h_1^q(s) \varphi _1^q(z(s) ) ds\Big)^{1/q} +\Big(\int_0^{t}h_2^q(s) \varphi _2^q( z(s) ) ds\Big) ^{1/q}] ,\quad t>0. \end{align*} Therefore, by Corollary \ref{lem:res2} with $w_i(t) =\varphi _i^q(t) $, $i=1,2$, we find \begin{equation*} z(t) \leq \Big[G^{-1}\Big(G(2^{q-1}K_1^q) +2^{2(q-1) }K_2^q\int_0^{t}[h_1^q(s) +h_2^q(s) ] ds\Big) \Big] ^{1/q},\quad t>0. \end{equation*} Since $h_i\in L_{q}(0,\infty ) $, \begin{equation*} z(t) \leq C=\Big[G^{-1}\Big(G(2^{q-1}K_1) +2^{q-1}K_2\int_0^{\infty }h^q(s) ds\Big) \Big]^{1/q}<\infty . \end{equation*} Thus \begin{equation*} | y(t) | \leq Ct^{\alpha -1}\quad \text{and}\quad | D_0^{\beta }y(t) | 0. \end{equation*} \end{proof} \section{Examples} In this section, we provide three examples, where we apply Theorems \ref{thm:res1} and \ref{thm:res3} to show that all global solutions decay like $t^{\alpha -1}$. Unlike the first example, the solutions of the second and third examples may not be available explicitly. \subsection*{Example 1} Consider the problem \begin{equation} \begin{gathered} D_0^{\alpha }y(t) =-\lambda t^{\frac{\alpha -1}{q}}[ E_{\alpha ,\alpha }(-\lambda t^{\alpha }) ] ^{1/q}( y(t) ) ^{1-1/q}\quad t>0, \\ I_0^{1-\alpha }y(t) |_{t=0}=1,\quad 0<\alpha <1,\; q>1,\; \lambda >0, \end{gathered} \label{exact-exp} \end{equation} where $E_{\alpha ,\beta }(z) $ is the Mittag-Leffler function \begin{equation*} E_{\alpha ,\beta }(z) =\sum_{k=0}^{\infty } \frac{z^{k}}{\Gamma (k\alpha +\beta ) },\quad \alpha, \beta >0. \end{equation*} We can rewrite the right-hand side of \eqref{exact-exp} as \begin{align*} &| -\lambda t^{\frac{\alpha -1}{q}}[E_{\alpha ,\alpha }( -\lambda t^{\alpha }) ] ^{1/q}(y(t) )^{1-1/q}| \\ &=\lambda t^{\frac{\alpha -1}{q}-(1-\alpha ) (1-\frac{1}{q}) }[E_{\alpha ,\alpha }(-\lambda t^{\alpha }) ] ^{1/q}(t^{1-\alpha }y(t)) ^{1-\frac{1}{q}} \\ &=\lambda t^{\alpha -1}[E_{\alpha ,\alpha }(-\lambda t^{\alpha }) ] ^{1/q}(t^{1-\alpha }y(t) )^{1-1/q}\\ &=h(t) \varphi (t^{1-\alpha }y(t)) , \end{align*} where $h(t) =\lambda t^{\alpha -1}[E_{\alpha ,\alpha }(-\lambda t^{\alpha }) ] ^{1/q}$ and $\varphi (t) =t^{1-1/q}$. Notice that $h\in L_{q}(0,\infty ) $ since \cite[p 44, equation (1.8.33)]{mlf} \begin{equation} \int_0^{\infty }t^{s-1}E_{\alpha ,\beta }(-wt) dt=\frac{1}{ w^{s}}\frac{\Gamma (s) \Gamma (1-s) }{\Gamma ( \beta -\alpha s) },\quad \alpha , \beta , s, w>0, \label{cor:mlf} \end{equation} and $\varphi $ is a positive, continuous and nondecreasing function. Clearly, all conditions of Theorem \ref{thm:res1} are satisfied. Therefore, any global solution $y\in C_{1-\alpha }[0,\infty )$ of \eqref{exact-exp} satisfies \begin{equation*} | y(t) | \leq Ct^{\alpha -1},\quad t>0. \end{equation*} In fact the function \begin{equation*} y(t) =t^{\alpha -1}[E_{\alpha ,\alpha }(-\lambda t^{\alpha }) ] ,\quad t>0, \end{equation*} is a global solution in $C_{1-\alpha }[0,\infty )$ of \eqref{exact-exp} and clearly \begin{equation*} | y(t) | =| t^{\alpha -1}[ E_{\alpha ,\alpha }(-\lambda t^{\alpha }) ] | \leq Ct^{\alpha -1},\quad t>0. \end{equation*} \subsection*{Example 2} Consider the problem \begin{equation} \begin{gathered} D_0^{1/2}y(t) =t^{2}e^{-2t}(\cos y^{2}) (y(t) ) ^{1/5}(D_0^{1/3}y(t) )^{1/3}, \quad t>0, \\ I_0^{1/2}y(t) \big|_{t=0}=b. \end{gathered} \label{exa1} \end{equation} Then the right-hand side satisfies \begin{align*} | f(t,y(t) ,D_0^{1/3}y(t) )| &=| t^{2}e^{-2t}(\cos y^{2}) (y(t) ) ^{1/5}\big(D_0^{1/3}y(t) \big)^{1/3}| \\ &\leq t^{\gamma }e^{-t}h(t) \varphi _1(t^{1-1/2}y(t) ) \varphi_2\big(t^{1-(1/2-1/3) }D_0^{1/3}y(t) \big) , \end{align*} where $\gamma =73/45$, $h(t) =e^{-t}$, $\varphi _1(t) =t^{1/5}$ and $\varphi _2(t) =t^{1/3}$. All the conditions of Theorem \ref{thm:res1} are satisfied. Therefore, any global solution $y\in C_{1/2}[0,\infty )$ of \eqref{exa1} satisfies \begin{equation*} | y(t) | \leq Ct^{\alpha -1}\quad\text{and}\quad | D_0^{\beta }y(t) | 0, \\ I_0^{1/2}y(t) \big|_{t=0}=b. \end{gathered} \label{exa2} \end{equation} Note that this example is different from the previous ones. In fact, the right hand side of \eqref{exa2} is sum of two terms, similar to that of the assumption (A3). We can rewrite the right hand side of \eqref{exa2} as follows \begin{align*} &| t^{2}e^{-2t}(\cos y) (y(t) )^{1/3}+t^{3}e^{-4t}(\sin t^{3}) \big(D_0^{1/4}y(t) \big) ^{1/3}| \\ &\leq t^{\gamma_1}e^{-t}h_1(t) \varphi _1\big(t^{1-1/2}y(t)\big) +t^{\gamma _2}e^{-2t}h_2(t) \varphi _2\big( t^{1-(1/2-1/4) }D_0^{1/4}y(t) \big) , \end{align*} where $\gamma _1=11/6$, $\gamma _2=33/12$, $h_1(t)=e^{-t}$, $h_2(t) =e^{-2t}$ and $\varphi _1(t)=\varphi _2(t) =t^{1/3}$. All the conditions of Theorem \ref{thm:res3} are satisfied and therefore any global solution $y\in C_{1/2}[0,\infty )$ of \eqref{exa2} satisfies \begin{equation*} | y(t) | \leq Ct^{\alpha -1}, \quad | D_0^{\beta }y(t) | 0. \end{equation*} \subsection*{Acknowledgements} The authors acknowledge the support provided by King Fahd University of Petroleum and Minerals (KFUPM) through project number IN151035. \begin{thebibliography}{99} \bibitem{Anastassiou} G. A. Anastassiou; \emph{Opial type Inequalities involving Riemann-Liouville fractional derivatives of two functions with applications}, Math. Comput Modelling, 48 (2008), 344--374. \bibitem{Barrett} J. H. Barrett; \emph{Differential equations of non-integer order}, Canad. J. Math. 6 (4) (1954), 529--541. \bibitem{Campos} L. M. B. C. Campos; \emph{On the solution of some simple fractional differential equations}, Internat. J. Math. Sci. 13 (3) (1990), 481--496. \bibitem{Delb} D. Delbosco, L. Rodino; \emph{Existence and uniqueness for a nonlinear fractional differential equation}, J. Math. Anal. Appl. 204 (2) (1996), 609--625. \bibitem{Diethelm} K. Diethelm, N. J. Ford; \emph{Analysis of fractional differential equations}, J. Math. Anal. Appl. 265 (2002), 229--248. \bibitem{Fujiwara} M. Fujiwara; \emph{On the integration and differentiation of an arbitrary order}, Tohoku Math. J. 37 (1933), 110--121. \bibitem{F1} K. M. Furati, M. D. Kassim, N.-E. Tatar; \emph{Existence and uniqueness for a problem involving Hilfer fractional derivative}, Comput. Math. Appl., vol 64 (2012), 1616-1626. \bibitem{F2} K. M. Furati, M. D. Kassim, N.-E. Tatar; \emph{Non-existence of global solutions for a differential equation involving Hilfer fractional derivative}, Electron. J. Diff. Equ., vol 2013, no. 235, (2013), 1-10. \bibitem{F3} K. M. Furati, N-e. Tatar; \emph{An existence result for a nonlocal fractional differential problem}, J. Fract. Calc. Vol. 26 (2004), 43-51. \bibitem{F4} K. M. Furati, N-e. Tatar; \emph{Power type estimates for a nonlinear fractional differential equation}, Nonlinear Analysis T.M.A. 62 (2005), 1025-1036. \bibitem{F5} K. M. Furati, N-e. Tatar; \emph{Behavior of solutions for a weighted Cauchy-type fractional differential problem}, J. Fract. Calc. 28 (2005), 23-42. \bibitem{F6} K. M. Furati, N-e. Tatar; \emph{Long time behaviour for a nonlinear fractional model}, J. Math. Anal. Appl., Vol. 332, Issue 1 (2007), 441-454. \bibitem{F7} K. M. Furati, N.-E. Tatar; \emph{Inequalities for fractional differential equations}, Math. Inequal. Appl., vol. 12, no. 2 (2009), 279--293. \bibitem{G1} W. G. Gl\"{o}ckle, T. F. Nonnenmacher; \emph{A fractional calculus approach to self-similar protein dynamics}, Biophys. J., 68 (1995), 46-53. \bibitem{G} R. Gorenflo, F. Mainardi; \emph{Some recent advances in theory and simulation of fractional diffusion processes}, J. Comput. Appl. Math., 229 (2009), 400-415. \bibitem{Heymans+podl} N. Heymans, I. Podlubny; \emph{Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives}. Rheol Acta, 45 (2006), 765--771. \bibitem{K3} M. Kassim, K. Furati, N.-E. Tatar; \emph{Asymptotic behavior of solutions to nonlinear fractional differential equations}, Math. Model Anal., 21:5 (2016), 610-629. \bibitem{K1} M. D. Kassim, K. M. Furati, N.-E. Tatar; \emph{On a differential equation involving Hilfer-Hadamard fractional derivative}, Abstr. Appl. Anal., vol. 2012, Article ID 391062, 17 pages, 2012. \bibitem{K2} M. D. Kassim, N.-E. Tatar; \emph{Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative}, Abstr. Appl. Anal., vol. 2013, Article ID 605029, 12 pages, 2013. \bibitem{Kilbas-0} A. A. Kilbas, B. Bonilla, J. J. Trujillo; \emph{Existence and uniqueness theorems for nonlinear fractional differential equations}, Demonstratio Math. 33 (3) (2000), 538--602. \bibitem{Kilbas-1} A. A. Kilbas, B. Bonilla, J. J. Trujillo; \emph{Fractional integrals and derivatives and differential equations of fractional order in weighted spaces of continuous functions}, Dokl. Nats. Akad. Nauk Belarusi. 44 (6) (2000). \bibitem{Kilbas-2} A. A. Kilbas, M. Saigo; \emph{On solution in closed from of nonlinear integral and differential equations of fractional order}, Surikaisekiken-Kyusho Kokyuroku, vol. 963 (1996), 39--50. \bibitem{Kilbas} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; \emph{Theory and Applications of Fractional Differential Equations}, North-Holland Mathematics Studies 204, Editor: Jan van Mill, Elsevier, Amsterdam, The Netherlands 2006. \bibitem{Kilbas-4} A. A. Kilbas, J. J. Trujillo; \emph{Differential equations of fractional order: methods, results and problem}, Appl. Anal. 78 (1--2) (2001), 153--192. \bibitem{Kou} C. Kou, J. Liu, Y. Ye; \emph{Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equation}s, Discrete Dyn. Nat. Soc., Vol. 2010, Article ID 142175, (2010), 1-15. \bibitem{Kuczma} M. Kuczma; \emph{An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality}, Birkh\"{a}user, 2009. \bibitem{L+Tatar} Y. Laskri, N.-E. Tatar; \emph{The critical exponent for an ordinary fractional differential problem}, Comput. Math. Appl., 59, (2010), 1266-1270. \bibitem{M1} R. L. Magin, O. Abdullah, D. B\u{a}leanu. J. Z. Xiaohong; \emph{Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation}, J. Magn. Reson., 190 (2) (2008), 255--270. \bibitem{M2} R. Metzler, W. Schick, H. G. Kilian, T. F. Nonennmacher; \emph{Relaxation in filled polymers: A fractional calculus approach}, J. Chem. Phys., 103 (1995), 7180-7186. \bibitem{Michalski} M. W. Michalski; \emph{Derivatives of non integer order and their applications}, Warszawa, Poland: Polska Akademia Nauk, Institut Matematyczny, 1993. \bibitem{Miller+Ross} K. S. Miller, B. Ross; \emph{An Introduction to the Fractional Calculus and Fractional Differential Equations}, John Wiley, New York, 1993. \bibitem{Oldham+Spanier} K. B. Oldham, J. Spanier; \emph{The Fractional Calculus}, Academic Press, New York, London, 1974. \bibitem{Pachpatte} B. G. Pachpatte; \emph{Inequalities for Differential and Integral Equations}, vol 197 of Mathematics in Science and Engineering, Acad. Press, San Diego-London. Edited by W. F. Ames, 1998. \bibitem{Podl-(1999)} I. Podlubny; \emph{Fractional Differential Equations}, Mathematics in Sciences and Engineering. 198, Academic Press, San-Diego, 1999. \bibitem{pt} {\L}. P{\l}ociniczak; \emph{On Asymptotics of Some Fractional Differential Equations}, Math. Model Anal., 18:3 (2013), 358-373. \bibitem{S} J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado; \emph{Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering}, Springer-Verlag, Dordrecht, 2007. \bibitem{Samko} S. G. Samko, A. A. Kilbas, O. I. Marichev; \emph{Fractional Integrals and Derivatives: Theory and Applications}, Gordon and Breach, Switzerland, 1993. \bibitem{mlf} A. K. Shukla, J. C. Prajapati; \emph{On a generalization of Mittag-Leffler function and its properties}, J. Math. Anal. Appl. 336 (2007), 797--811. \bibitem{Tazali} A. Zain, A. M. Tazali; \emph{Local existence theorems for ordinary differential equations of fractional order}, Ordinary and Partial Differential Equations, vol. 964 (1982), 652--665. \bibitem{Zhang} S. Zhang; \emph{The existence of a positive solution for a nonlinear fractional differential equation}, J. Math. Anal. Appl. 252 (2000), 804--812. \end{thebibliography} \end{document}