Positive solutions for boundary value problem of nonlinear fractional differential equation.

*(English)*Zbl 1079.34048Summary: We investigate the existence and multiplicity of positive solutions to the boundary value problem

$${D}_{0+}^{\alpha}u\left(t\right)+f\left(t,u\left(t\right)\right)=0,\phantom{\rule{4pt}{0ex}}0<t<1,\phantom{\rule{1.em}{0ex}}u\left(0\right)=u\left(1\right)=0,$$

where $1<\alpha \le 2$ is a real number, ${D}_{0+}^{\alpha}$ is the standard Riemann-Liouville differentiation, and $f:[0,1]\times [0,\infty )\to [0,\infty )$ is continuous. By means of some fixed-point theorems in a cone, existence and multiplicity results positive solutions are obtained. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.

##### MSC:

34K05 | General theory of functional-differential equations |

34B18 | Positive solutions of nonlinear boundary value problems for ODE |

34B15 | Nonlinear boundary value problems for ODE |

26A33 | Fractional derivatives and integrals (real functions) |