Existence and Uniqueness of Positive Solutions to a Class of Singular Integral Boundary Value Problems of Fractional Order

In this paper, we are interested in the study of the existence and uniqueness of positive solutions to the nonlinear singular fractional differential equation D0+αu(t)+f(t,u(t),(Hu)(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{\alpha }_{0^+} u(t)+f(t,u(t),(Hu)(t))=0$$\end{document} with 0<t<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<t<1$$\end{document}, where D0+α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{\alpha }_{0^+}$$\end{document} denotes the classical Riemmann Liouville derivative, under the integral boundary conditions u(0)=u′(0)=⋯=u(n-2)(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(0)=u'(0)=\cdots = u^{(n-2)}(0)=0 $$\end{document} and u(1)=λ∫01u(s)ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u(1)=\lambda \int _0^1 u(s)ds $$\end{document}, where λ∈(0,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in (0,\alpha )$$\end{document}, H is an operator defined on C[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}[0,1]$$\end{document} into itself and f:(0,1]×[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}$$f:(0,1]\times [0,\infty )\times [0,\infty )\rightarrow [0,\infty )$$\end{document} is a continuous function which can have a singularity at (0, x, y). To state our results, we use a fixed point theorem recently proved. Finally, we present some examples illustrating the results obtained.