\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 37, 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{9mm}} \begin{document} \title[\hfilneg EJDE-2013/37\hfil Uniqueness and asymptotic behavior] {Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary-value problem} \author[X. Zhou, W. Wang \hfil EJDE-2013/37\hfilneg] {Xiangbing Zhou, Wenquan Wu} \address{Xiangbing Zhou \newline Department of Computer Science, Aba Teachers College, WenChuan 623002, Sichuan, China} \email{studydear@gmail.com} \address{Wenquan Wu \newline Department of Computer Science, Aba Teachers College, WenChuan 623002, Sichuan, China} \email{wenquanwu@163.com} \thanks{Submitted December 11, 2012. Published February 1, 2013.} \subjclass[2000]{26A33, 34B10} \keywords{Upper and lower solution; fractional differential equation; \hfill\break\indent Schauder's fixed point theorem; positive solution} \begin{abstract} In this note, we extend the results by Jia et al \cite{Jia} to a more general case. By refining the conditions imposed on $f$ and finding more suitable upper and lower solution, we remove some key conditions used in \cite{Jia}, and still establish their results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives, see \cite{Yuan, SZhang, Jia, BAhmad, MFeng, Zhang2, Zhang3, Zhang4,ZBai}. Yuan \cite{Yuan} studied the $(n-1,1)$-type conjugate boundary-value problem \label{1.1} \begin{gathered} \mathscr{D}_t^{\alpha}u(t)+f(t,u(t))=0,\quad 00$of a function$x:(0,+\infty)\to \mathbb{R}$is given by $$I^{\alpha}x(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}x(s)ds$$ provided that the right-hand side is pointwise defined on$(0,+\infty)$. \end{definition} \begin{definition}[\cite{KMiller,IPodlubny}] \label{def2.2} \rm The Riemann-Liouville fractional derivative of order$\alpha >0$of a function$x:(0,+\infty)\to \mathbb{R}$is given by $$\mathscr{D}_t^{\alpha}x(t)=\frac{1}{\Gamma(n-\alpha)} (\frac{d}{dt})^{n}\int^{t}_{0}(t-s)^{n-\alpha-1}x(s)ds,$$ where$n=[\alpha]+1$,$[\alpha]$denotes the integer part of number$\alpha$, provided that the right-hand side is pointwise defined on$(0,+\infty)$. \end{definition} \begin{proposition}[\cite{KMiller,IPodlubny}] \label{prop2.1} (1) If$x\in L^1(0, 1), \nu>\sigma> 0$, then $$I^{\nu}I^{\sigma}x(t)=I^{\nu+\sigma}x(t), \quad \mathscr{D}_t^{\sigma}I^{\nu} x(t)=I^{\nu-\sigma} x(t),\quad \mathscr{D}_t^{\sigma}I^{\sigma} x(t)=x(t).$$ (2) If$\alpha>0, \sigma>0$, then $$\mathscr{D}_t^{\alpha} t^{\sigma-1} =\frac{\Gamma(\sigma)}{\Gamma(\sigma-\alpha)}t^{\sigma-\alpha-1}.$$ \end{proposition} \begin{proposition}[\cite{KMiller,IPodlubny}] \label{prop2.2} Let$\alpha > 0$, and$f(x)$be integrable, then $$I^{\alpha}\mathscr{D}_t^{\alpha}f(x) =f(x)+c_{1}x^{\alpha-1}+c_{2}x^{\alpha-2}+\cdot\cdot\cdot+c_{n}x^{\alpha-n},$$ where$c_{i}\in \mathbb{R}(i=1,2,\dots,n)$,$n$is the smallest integer greater than or equal to$\alpha$. \end{proposition} Let $$x(t)=I^{\mu_{n-2}}y(t),\quad y(t)\in C[0,1],$$ by Propositions \ref{prop2.1}-\ref{prop2.2} and a discussion similar to \cite{{Jia}}, we easily reduce the order of \eqref{1.4} to the equivalent problem $$\begin{gathered} -\mathscr{D}_t^{\mu-\mu_{n-2}} y(t) = f(t,I^{\mu_{n-2}}y(t),I^{\mu_{n-2}-\mu_1}y(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}y(t), y(t)), \\ y(0)=0, \quad y(1)=\int^1_0 y(s)dA(s). \end{gathered}\label{2.1}$$ \begin{lemma}[\cite{ZBai}] \label{lem2.1} Given$h\in L^1(0, 1)$, then the problem \begin{gathered} \mathscr{D}_t^{\mu-\mu_{n-2}}y(t)+h(t)=0 ,\quad 0 0$, for all $t,s\in (0,1)$. (2) $$\frac{t^{\mu-\mu_{n-2}-1}}{1-\mathcal{C}}\mathcal{G}_A(s) \le K(t,s)\le\mathcal{H}(s)t^{\mu-\mu_{n-2}-1}, \label{2.7}$$ where $$\mathcal{H}(s)=\frac{(1-s)^{\mu-\mu_{n-2}-1}}{\Gamma(\mu-\mu_{n-2})} + \frac{\mathcal{G}_A(s)}{1-\mathcal{C}}.$$ \end{lemma} \begin{definition} \label{def2.3} A continuous function $\Psi(t)$ is called a lower solution of \eqref{2.1}, if it satisfies \begin{gather*} -\mathscr{D}_t^{\mu-\mu_{n-2}} \Psi(t)(t) \le f(t,I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t), \dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)), \\ \Psi(0)\ge0, \quad \Psi(1)\ge\int^1_0 \Psi(s)dA(s). \end{gather*} \end{definition} \begin{definition} \label{def2.4} A continuous function $\Phi(t)$ is called a upper solution of \eqref{2.1}, if it satisfies \begin{gather*} -\mathscr{D}_t^{\mu-\mu_{n-2}} \Phi(t)(t) \ge f(t,I^{\mu_{n-2}}\Phi(t),I^{\mu_{n-2}-\mu_1}\Phi(t), \dots,I^{\mu_{n-2}-\mu_{n-3}}\Phi(t), \Phi(t)), \\ \Phi(0)\le0, \quad \Phi(1)\le\int^1_0 \Phi(s)dA(s). \end{gather*} \end{definition} \section{Main results} Let $E= C[0,1]$. Define the following continuous functions on $E$: \begin{gather*} \kappa_0(t)=I^{\mu_{n-2}}s^{\mu-\mu_{n-2}-1} =\int_0^t\frac{(t-s)^{\mu_{n-2}-1}s^{\mu-\mu_{n-2}-1}}{\Gamma(\mu_{n-2})}ds =\frac{\Gamma(\mu-\mu_{n-2})}{\Gamma(\mu)}s^{\mu-1}, \\ \begin{aligned} \kappa_1(t)=I^{\mu_{n-2}-\mu_1}s^{\mu-\mu_{n-2}-1} &=\int_0^t\frac{(t-s)^{\mu_{n-2}-\mu_1-1}s^{\mu-\mu_{n-2}-1}} {\Gamma(\mu_{n-2}-\mu_1)}ds\\ &=\frac{\Gamma( {\mu-\mu_{n-2}})}{\Gamma(\mu-\mu_1)}t^{\mu-1-\mu_{1}}, \end{aligned}\\ \dots \\ \begin{aligned} \kappa_{n-3}(t)=I^{\mu_{n-2}-\mu_{n-3}}s^{\mu-\mu_{n-2}-1} &=\int_0^t\frac{(t-s)^{\mu_{n-2}-\mu_{n-3}-1}s^{\mu-\mu_{n-2}-1}} {\Gamma(\mu_{n-2}-\mu_{n-3})}ds\\ &=\frac{\Gamma( {\mu-\mu_{n-2}})}{\Gamma(\mu-\mu_{n-3})}t^{\mu-1-\mu_{n-3}}, \end{aligned}\\ \kappa_{n-2}(t)=t^{\mu-1-\mu_{n-2}}. \end{gather*} Set \begin{aligned} P&=\Big\{y\in E:\text{ there exist positive numbers } 01 \text{ such that }\\ &\quad l_y \kappa_{n-2}(t)\le y(t)\le L_y \kappa_{n-2}(t), \; t\in[0,1\Big\}. \end{aligned}\label {3.1} Clearly, $P$ is nonempty since $\kappa_{n-2}(t)\in P$. For any $y\in P$, define an operator $T$ by $$(T y)(t)= \int_0^1 K(t,s)f(s,I^{\mu_{n-2}}y(s),I^{\mu_{n-2} -\mu_1}y(s),\dots,I^{\mu_{n-2}-\mu_{n-3}}y(s), y(s))ds. \label{3.2}$$ In this note, we will use the following conditions: \begin{itemize} \item[(H1)] $f\in C((0,1)\times(0,\infty)^{n-1},[0,+\infty))$, and $f(t,x_0,x_1,x_2,\dots,x_{n-2})$ is nonincreasing in $x_i>0$ for $i=0,1,2,\dots,n-2$; \item[(H2)] For any $\lambda_i>0$, $$0<\int_0^1\mathcal{H}(s)f(s,\lambda_0 \kappa_0(s), \lambda_1 \kappa_1(s), \lambda_2 \kappa_2(s),\dots, \lambda_{n-2} \kappa_{n-2}(s))ds<+\infty.$$ \end{itemize} \begin{lemma} \label{lem3.1} Suppose {\rm (H0)--(H2)} hold. Then $T$ is well defined, $T (P)\subset P$, and $T$ is nonincreasing relative to $y$. \end{lemma} \begin{proof} For any $y\in P$, by the definition of $P$, there exist two positive numbers $01$ such that $$l_y \kappa_{n-2}(s)\le y(t)\le L_y \kappa_{n-2}(s)\label{3.3}$$ for any $s\in [0,1]$. It follows from \eqref{2.7} and (H1)--(H2) that \begin{aligned} &(Ty)(t)\\ &= \int_0^1 K(t,s)f(s,I^{\mu_{n-2}}y(s),I^{\mu_{n-2} -\mu_1}y(s),\dots,I^{\mu_{n-2}-\mu_{n-3}}y(s), y(s))ds\\ &\le \kappa_{n-2}(s)\int_0^1 \mathcal{H}(s)f(s,l_y\kappa_0(s), l_y\kappa_1(s),\dots,l_y\kappa_{n-3}(s), l_y\kappa_{n-2}(s))ds\\ &<+\infty. \end{aligned}\label{3.4} By \eqref{2.7}, \eqref{3.3} and \eqref{3.4}, we have \begin{aligned} & (Ty)(t)\\ &=\int_0^1 K(t,s)f(s,I^{\mu_{n-2}}y(s),I^{\mu_{n-2} -\mu_1}y(s),\dots,I^{\mu_{n-2}-\mu_{n-3}}y(s), y(s))ds\\ &\ge \frac{t^{\mu-\mu_{n-2}-1}}{1-\mathcal{C}} \int_0^1 \mathcal{G}_A(s)f(s,L_y\kappa_0(s),L_y\kappa_1(s),\dots, L_y\kappa_{n-3}(s), L_y\kappa_{n-2}(s))ds. \end{aligned}\label{3.5} Take $$\begin{gathered} l'_y=\min\Big\{1,\, \frac1{1-\mathcal{C}}\int_0^1 \mathcal{G}_A(s) f(s,L_y\kappa_0(s),L_y\kappa_1(s),\dots, L_y\kappa_{n-3}(s), L_y\kappa_{n-2}(s))ds\Big\},\\ L'_y=\max\Big\{1,\, \int_0^1 \mathcal{H}(s) f(s,l_y\kappa_0(s),l_y\kappa_1(s),\dots, l_y\kappa_{n-2}(s))ds \Big\}. \end{gathered} \label{3.6}$$ It follows from \eqref{3.3}-\eqref{3.6} that $T$ is well defined and $T (P)\subset P$. Moreover, by (H1), $T$ is nonincreasing relative to $y$. \end{proof} \begin{theorem}[Existence] \label{thm3.1} Suppose {rm(H0)--(H2)} hold. Then \eqref{1.4} has at least one positive solution $x(t)$. \end{theorem} \begin{proof} From \eqref{3.2} and simple computation, we have $$\begin{gathered} -\mathscr{D}_t^{\mu-\mu_{n-2}} (Ty)(t) = f(t,I^{\mu_{n-2}}y(t),I^{\mu_{n-2}-\mu_1}y(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}y(t), y(t)), \\ (Ty)(0)=0, \quad (Ty)(1)=\int^1_0 (Ty)(s)dA(s). \end{gathered}\label{3.7}$$ Let $$\alpha(t) =\min\{\kappa_{n-2}(t), (T\kappa_{n-2})(t)\},\quad \beta(t)=\max\{\kappa_{n-2}(t), (T\kappa_{n-2})(t)\},\label{3.8}$$ then, if $\kappa_{n-2}(t)=(T\kappa_{n-2})(t)$, the conclusion of Theorem \ref{thm3.1} holds. If $\kappa_{n-2}(t)\not=(T\kappa_{n-2})(t)$, clearly, $\alpha(t), \beta(t)\in P$ and $$\alpha(t)\le \kappa_{n-2}(t)\le \beta(t).\label{3.9}$$ Set $$\Phi(t)=(T\beta)(t), \Psi(t)=(T\alpha)(t),$$ then by \eqref{3.8}-\eqref{3.9} and Lemma \ref{lem3.1}, one has $$\begin{gathered} \Phi(t)=(T\beta)(t)\le (T\kappa_{n-2})(t)\le T(\alpha)(t)=\Psi(t),\\ \Phi(t)\le (T\kappa_{n-2})(t)\le \beta(t), \quad \Psi(t)\ge (T\kappa_{n-2})(t)\ge \alpha(t), \end{gathered}\label{3.10}$$ and $\Phi(t), \Psi(t)\in P$. On the other hand, by \eqref{3.7}, \eqref{3.10} and Lemma \ref{lem3.1}, we have \begin{aligned} &\mathscr{D}_t^{\mu-\mu_{n-2}}\Phi(t)+ f(t, I^{\mu_{n-2}}\Phi(t), I^{\mu_{n-2}-\mu_1}\Phi(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\Phi(t), \Phi(t))\\ &\ge\mathscr{D}_t^{\mu-\mu_{n-2}}(T\beta)(t)+ f(t,I^{\mu_{n-2}}\beta(t), I^{\mu_{n-2}-\mu_1}\beta(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\beta(t), \beta(t))\\ &=-f(t,I^{\mu_{n-2}}\beta(t),I^{\mu_{n-2}-\mu_1}\beta(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}\beta(t), \beta(t))\\ &\quad + f(t,I^{\mu_{n-2}}\beta(t),I^{\mu_{n-2}-\mu_1}\beta(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}\beta(t), \beta(t))=0,\\ & (T\Phi)(0)=0, \quad (T\Phi)(1)=\int^1_0 (T\Phi)(s)dA(s). \end{aligned} \label{3.11} and \begin{aligned} &\mathscr{D}_t^{\mu-\mu_{n-2}}\Psi(t) + f(t, I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t), \dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t))\\ &\le\mathscr{D}_t^{\mu-\mu_{n-2}}(T\alpha)(t) + f(t,I^{\mu_{n-2}}\alpha(t),I^{\mu_{n-2}-\mu_1}\alpha(t), \dots,I^{\mu_{n-2}-\mu_{n-3}}\alpha(t), \alpha(t))\\ &=- f(t,I^{\mu_{n-2}}\alpha(t),I^{\mu_{n-2}-\mu_1}\alpha(t), \dots,I^{\mu_{n-2}-\mu_{n-3}}\alpha(t)\\ &\quad + f(t,I^{\mu_{n-2}}\alpha(t),I^{\mu_{n-2} -\mu_1}\alpha(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\alpha(t)=0,\\ & (T\Psi)(0)=0, \quad (T\Psi)(1)=\int^1_0 (T\Psi)(s)dA(s). \end{aligned} \label{3.12} Inequalities \eqref{3.10}-\eqref{3.12} imply that $\Phi(t),\Psi(t)$ are lower and upper solution of \eqref{2.1}, respectively. Define the function $F$ and the operator $A$ in $E$ by \begin{aligned} &F(t,y)\\ &=\begin{cases} f(t, I^{\mu_{n-2}}\Phi(t),I^{\mu_{n-2}-\mu_1}\Phi(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}\Phi(t), \Phi(t)), & y<\Phi(t),\\ f(t,I^{\mu_{n-2}}y(t),I^{\mu_{n-2}-\mu_1}y(t),\dots,I^{\mu_{n-2} -\mu_{n-3}}y(t), y(t)), &\Phi(t)\le y\le \Psi(t),\\ f(t, I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)),& y>\Psi(t), \end{cases} \end{aligned}\label{3.13} and $$(\mathfrak{A}y)(t)= \int_0^1 K(t,s)F(s,y(s))ds, \quad \forall y\in E.$$ Clearly, $F:[0,1]\times[0,+\infty)\to [0,+\infty)$ is continuous by \eqref{3.13}. Consider the following boundary value problem \begin{gathered} -\mathscr{D}_t^{\mu-\mu_{n-2}} y(t)=F(t,y),\quad 0\Psi(t)$. According to the definition of$F, we have \begin{aligned} &-\mathscr{D}_t^{\mu-\mu_{n-2}} y(t)=F(t,y(t))\\ &=f(t, I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t), \dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)). \end{aligned} \label{3.16} On the other hand, it follows from\psi$is an upper solution to \eqref{2.1} that $$-\mathscr{D}_t^{\mu-\mu_{n-2}}\Psi(t)\ge f(t, I^{\mu_{n-2}}\Psi(t), I^{\mu_{n-2}-\mu_1}\Psi(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)). \label{3.17}$$ Let$z(t)=\Psi(t)-y(t)$, \eqref{3.15}-\eqref{3.17} imply that $$\mathscr{D}_t^{\mu-\mu_{n-2}}z(t)=\mathscr{D}_t^{\mu-\mu_{n-2}}\Psi(t) -\mathscr{D}_t^{\mu-\mu_{n-2}}y(t)\le 0,$$ and $$z(0)=0, z(1)=\int^1_0 z(s)dA(s).$$ It follows from \eqref{2.6} that $$z(t)\ge 0,$$ i.e.,$y(t) \le \Psi(t)$on$[0, 1]$, which contradicts$y(t)>\Psi(t)$. Hence,$y(t)>\Psi(t)$is impossible. By the same way, we also have$y(t) \ge \Phi(t)$on$[0, 1]$. So $$\Phi(t)\le y(t)\le \Psi(t), \quad t\in [0,1]. \label{3.18}$$ Consequently,$F(t,y(t))=f(t,I^{n-2}y(t),I^{n-3}y(t),\dots,I^{1}y(t),y(t))$,$t\in [0,1]$. Then$y(t)$is a positive solution of the problem \eqref{2.1}. It follows from \eqref{2.1} that$x(t)=I^{\mu_{n-2}}y(t)$is positive solution of \eqref{1.4}. \end{proof} \begin{remark} \label{rmk3.1}\rm In this work, we not only extend the main result of \cite{Jia} to more general form with fractional derivatives in nonlinearity and boundary condition, but also by finding more suitable upper and lower solution, we omit the following key conditions of \cite{Jia}: \begin{itemize} \item[(i)] For any$\lambda_i>0$,$ f(t,\lambda_0t^{n-2},\lambda_1t^{n-3},\dots,\lambda_{n-3}t,\lambda_{n-2}) \not\equiv0$,$t\in(0,1)$. \item[(ii)] $$\int_0^1\mathcal{G}_A(s)f\Big(s,\frac Ll\kappa_0(s),\frac Ll\kappa_1(s), \dots, \frac Ll\kappa_{n-3}(s),\frac Ll\kappa_{n-2}(s)\Big)ds\ge{1-\mathcal{C}}.$$ \end{itemize} This implies our result essentially improves those of \cite{Jia}. \end{remark} \begin{theorem}[Asymptotic Behavior] \label{thm3.2} Suppose Suppose {\rm (H0)--(H2)} hold. Then there exist two constants$\mathcal{B}_1, \mathcal{B}_2$such that the positive solution$x(t)$of \eqref{1.4} satisfies $$\mathcal{B}_1\kappa_{n-2}(t)\le x(t)\le \mathcal{B}_2\kappa_{n-2}(t).\label{3.19}$$ \end{theorem} \begin{proof} By \eqref{3.18}, and$\Phi,\Psi\in P$, we know that there exist two positive constants$01$such that $$l_\Phi\kappa_{n-2}(t)\le \Phi(t)\le y(t)\le \Psi(t)\le L_\Psi \kappa_{n-2}(t).$$ Notice that$ x(t)=I^{\mu_{n-2}}y(t), we have \begin{align*} \frac{l_\Phi\Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)}t^{\mu-1} &=l_\Phi I^{\mu_{n-2}}\kappa_{n-2}(s)\le x(t)\\ &\le L_\Psi I^{\mu_{n-2}}\kappa_{n-2}(s) =\frac{L_\Psi \Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)}t^{\mu-1}. \end{align*} Let $$\mathcal{B}_1=\frac{l_\Phi\Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)},\quad \mathcal{B}_2=\frac{L_\Psi \Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)},$$ then \eqref{3.19} holds. \end{proof} If\mu>1$is a integer, then we also have the following uniqueness result similar to \cite{Jia}. \begin{theorem}[Uniqueness] \label{thm3.3} Suppose Suppose {\rm (H0)--(H2)} hold, and$\mu=n>1$. Then the positive solution$x(t)of \eqref{1.4} is unique. \end{theorem} \begin{proof} Notice that \begin{align*} & f(t,I^{\mu_{n-2}}w_2(t),I^{\mu_{n-2}-\mu_1}w_2(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}w_2(t),w_2(t))\\ & \le f(t,I^{\mu_{n-2}}w_1(t),I^{\mu_{n-2}-\mu_1}w_1(t),\dots, I^{\mu_{n-2}-\mu_{n-3}}w_1(t),w_2(t)), \end{align*} for $$w_2(t)\ge w_1(t),\quad t\in [a,b].$$ Thus similar to \cite{Jia}, the proof is completed. \end{proof} \begin{example} \label{examp3.1} \rm Consider the existence of positive solutions for the nonlinear fractional differential equation \begin{gathered} -\mathscr{D}_t^{11/3} x(t)=t^{2/3} \big[x^{-1/2}+(\mathscr{D}_t^{2/3}x )^{-1/3} +(\mathscr{D}_t^{7/3}x )^{-2/3}\big], \quad 00,\ i=0,1,2, we have \begin{align*} 0&<\int_0^1\mathcal{H}(s)f(s,\lambda_0\kappa_0(s),\lambda_1\kappa_1(s), \lambda_2\kappa_2(s))ds\\ &=\int_0^1 \big[\frac{(1-s)^{1/3}}{\Gamma(4/3)} + \frac{\mathcal{G}_A(s)}{0.2637}\big]s^{2/3} \Big[(\frac{\Gamma(4/3)}{\Gamma(11/3)}\lambda_0)^{-1/2} s^{-\frac4{3}}\\ &+(\frac{\Gamma(4/3)}{\Gamma(3)}\lambda_1)^{-1/3}s^{-2/3} +\lambda_2^{-2/3}s^{-\frac29}\Big] ds \\ &<+\infty. \end{align*} Thus (H2) is satisfied. 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