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Positive solutions to boundary value problems of nonlinear fractional differential equations. (English) Zbl 1210.34009

Summary: We study the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation

${D}_{0+}^{\alpha }u\left(t\right)+\lambda f\left(u\left(t\right)\right)=0,\phantom{\rule{1.em}{0ex}}0
$u\left(0\right)=u\left(1\right)={u}^{\text{'}}\left(0\right)=0,$

where $2<\alpha \le 3$ is a real number, ${D}_{0+}^{\alpha }$ is the Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter, and $f:\left(0,+\infty \right)\to \left(0,+\infty \right)$ is continuous. By the properties of the Green function and the Guo-Krasnosel’skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are determined, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. Some examples are presented to illustrate the main results.

##### MSC:
 34A08 Fractional differential equations 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations 34B09 Boundary eigenvalue problems for ODE