Existence results for fractional hybrid differential systems in Banach algebras

In this manuscript we investigate the existence of solutions for the following system of fractional hybrid differential equations (FHDEs): {Dp[θ(t)−w(t,θ(t))u(t,θ(t))]=v(t,ϑ(t)),t∈J,Dp[ϑ(t)−w(t,ϑ(t))u(t,ϑ(t))]=v(t,θ(t)),t∈J,0<p<1,θ(0)=0,ϑ(0)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} D^{p} [\frac{\theta(t)-w(t,\theta(t))}{u(t,\theta(t))}] = v(t,\vartheta (t)) ,\quad t \in J,\\ D^{p} [\frac{\vartheta(t)-w(t,\vartheta(t))}{u(t,\vartheta(t))}] = v(t,\theta(t)) ,\quad t \in J, 0 < p < 1,\\ \theta(0) = 0, \quad\quad \vartheta(0) = 0, \end{cases} $$\end{document} where Dr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{r}$\end{document} denotes the Riemann-Liouville fractional derivative of order r, J=[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J=[0,1]$\end{document}, and the functions u:J×R→R∖{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u:J\times \mathbb{R}\rightarrow\mathbb{R}\setminus\{0\}$\end{document}, w:J×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$w:J\times \mathbb{R}\rightarrow\mathbb{R}$\end{document}, w(0,0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$w(0,0)=0$\end{document} and v:J×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v:J\times\mathbb{R} \rightarrow\mathbb{R}$\end{document} satisfy certain conditions.