Zhao, Yige; Sun, Shurong; Han, Zhenlai; Li, Qiuping The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. (English) Zbl 1221.34068 Commun. Nonlinear Sci. Numer. Simul. 16, No. 4, 2086-2097 (2011). Summary: We study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem \[ \begin{cases} D^\alpha_{0^+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ u(0)=u'(0)=u'(1)=0,\end{cases} \] where \(2<\alpha\leq 3\) is a real number and \(D^\alpha_{0^+}\) is the Riemann-Liouville fractional derivative. Using the properties of the Green’s function, the lower and upper solution method and a fixed-point theorem, some new existence criteria for singular and nonsingular fractional differential equation boundary value problems are established. As applications, examples are presented to illustrate the main results. Cited in 105 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34A08 Fractional ordinary differential equations 34A37 Ordinary differential equations with impulses 47N20 Applications of operator theory to differential and integral equations Keywords:fractional differential equation; boundary value problem; positive solution; fractional Green’s function; fixed-point theorem; lower and upper solution method PDFBibTeX XMLCite \textit{Y. Zhao} et al., Commun. Nonlinear Sci. Numer. Simul. 16, No. 4, 2086--2097 (2011; Zbl 1221.34068) Full Text: DOI References: [1] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equation (1993), John Wiley: John Wiley New York · Zbl 0789.26002 [2] Oldham, K. 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