Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition

By Qilin Song, Zhanbing Bai. Advances in Differential Equations, 2018

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In this article, the following boundary value problem of fractional differential equation with Riemann–Stieltjes integral boundary condition {D0+αu(t)+λf(t,u(t),u(t))=0,0<t<1,n−1<α≤n,u(k)(0)=0,0≤k≤n−2,u(1)=∫01u(s)dA(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} D_{0+}^{\alpha}u(t)+\lambda f(t, u(t),u(t))=0, \quad0< t< 1, n-1< \alpha \leq n, \\ u^{(k)}(0)=0,\quad 0\leq k\leq n-2, \qquad u(1)= \int_{0}^{1}u(s)\,dA(s) \end{cases} $$\end{document} is studied, where n−1<α≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n-1 < \alpha\le n$\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}, D0+α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D_{0+}^{\alpha}$\end{document} is the Riemann–Liouville fractional derivative, A is a function of bounded variation, ∫01u(s)dA(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int_{0}^{1}u(s)\,dA(s)$\end{document} denotes the Riemann–Stieltjes integral of u with respect to A. By the use of fixed point theorem and the properties of mixed monotone operator theory, the existence and uniqueness of positive solutions for the problem are acquired. Some examples are presented to illustrate the main result.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/c0d3c3fd35fd7cb2c573020d495eeabd1f990d57