## Positive solutions of a boundary value problem for a nonlinear fractional differential equation.(English)Zbl 1183.34007

Summary: We give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem
\begin{aligned} & D^{\alpha}u + a(t) f(u) = 0, \quad 0<t<1,\quad 1<\alpha\leq 2,\\ & u(0) = 0,\;u'(1)= 0,\end{aligned}
where $$D^{\alpha}$$ is the Riemann-Liouville differential operator of order $$\alpha , f: [0,\infty)\to [0,\infty)$$ is a given continuous function and $$a$$ is a positive and continuous function on $$[0,1]$$.

### MSC:

 34A08 Fractional ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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