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The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. (English) Zbl 1221.34068

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

$\left\{\begin{array}{c}{D}_{{0}^{+}}^{\alpha }u\left(t\right)+f\left(t,u\left(t\right)\right)=0,\phantom{\rule{1.em}{0ex}}0

where $2<\alpha \le 3$ is a real number and ${D}_{{0}^{+}}^{\alpha }$ 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:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34A08 Fractional differential equations 34A37 Differential equations with impulses 47N20 Applications of operator theory to differential and integral equations