Existence of positive solutions for a generalized and fractional ordered Thomas-Fermi theory of neutral atoms

The singular boundary value problem we discuss is as follows: D0+αCu(t)=λq(t)f(t,u(t)),0<t<1,α1u(0)+α2u′(0)=a,β1u(1)+β2u′(1)=b,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& {}^{\mathrm{C}}D_{0^{+}}^{\alpha}u(t)=\lambda q(t)f \bigl(t,u(t)\bigr),\quad 0< t< 1, \\& \alpha_{1}u(0)+\alpha_{2}u'(0)=a,\qquad \beta_{1}u(1)+\beta_{2}u'(1)=b, \end{aligned}$$ \end{document} where 1<α≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1<\alpha\leq2$\end{document}, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda>0$\end{document} is a parameter, D0+αC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}^{\mathrm{C}}D_{0^{+}}^{\alpha}$\end{document} is the Caputo fractional derivative. We present the existence of positive solutions for a fractional boundary value problem modeled from the Thomas-Fermi equation subjected to Sturm-Liouville boundary conditions.