Existence and Asymptotic Behavior of Positive Solutions for a Coupled Fractional Differential System

In this paper, we take up the existence and the asymptotic behavior of positive and continuous solutions to the following coupled fractional differential system Dαu=a(x)upvrin(0,1),Dβv=b(x)usvqin(0,1),u(0)=u(1)=Dα-3u(0)=u′(1)=0,v(0)=v(1)=Dβ-3v(0)=v′(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle D^{\alpha } u=a(x)\displaystyle u^{p }\displaystyle v^{r}\quad \text { in }(0,1) , \\ \displaystyle D^{\beta } v=b(x)\displaystyle u^{s }\displaystyle v^{q}\quad \text { in }(0,1) , \\ u(0)= u(1)= D^{\alpha -3}u(0)= u^{\prime }(1)=0,\\ v(0)= v(1)= D^{\beta -3}v(0)= v^{\prime }(1)=0, \end{array} \right. \end{aligned}$$\end{document}where α,β∈(3,4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha , \beta \in (3,4]$$\end{document}, p,q∈(-1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p, q\in (-1,1)$$\end{document}, r,s∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r, s\in \mathbb {R}$$\end{document} such that (1-|p|)(1-|q|)-|rs|>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-|p|)(1-|q|)-|rs|> 0$$\end{document}, D is the standard Riemann–Liouville differentiation and a, b are nonnegative and continuous functions in (0, 1) allowed to be singular at x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0$$\end{document} and x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document} and they are required to satisfy some appropriate conditions related to Karamata regular variation theory.