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Zbl 1221.34068
Zhao, Yige; Sun, Shurong; Han, Zhenlai; Li, Qiuping
The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations.
(English)
[J] Commun. Nonlinear Sci. Numer. Simul. 16, No. 4, 2086-2097 (2011). ISSN 1007-5704

Summary: We study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem $$\cases D^\alpha_{0^+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ u(0)=u'(0)=u'(1)=0,\endcases$$ where $2<\alpha\le 3$ 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.
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34A08
34A37 Differential equations with impulses
47N20 Appl. 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

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