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Positive solutions for boundary value problem of nonlinear fractional differential equation. (English) Zbl 1079.34048
Summary: We investigate the existence and multiplicity of positive solutions to the boundary value problem $D^\alpha_{0+}u(t)+f \bigl(t,u(t) \bigr)=0,\;0<t<1, \quad u(0)=u(1)=0,$ where $$1<\alpha\leq 2$$ 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 to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 26A33 Fractional derivatives and integrals
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