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Existence and multiplicity of positive solutions for singular fractional boundary value problems. (English) Zbl 1247.34006
Summary: We discuss the existence and multiplicity of positive solutions for the singular fractional boundary value problem $$D^{\alpha}_{0+}u(t)+f((t,u(t),D^{\nu}_{0+}u(t),D^{\mu}_{0+}u(t))=0,$$ $$u(0)=u'(0)=u''(0)=u''(1)=0,$$ where $3<\alpha\le 4$, $0<\nu\le 1$, $1<\mu \le 2$, $D^{\alpha}_{0+}$ is the standard Riemann-Liouville fractional derivative, $f$ is a Carathédory function and $f(t,x,y,z)$ is singular at the value $0$ of its arguments $x,y,z$. By means of a fixed point theorem, the existence and multiplicity of positive solutions are obtained.

34A08Fractional differential equations
34B18Positive solutions of nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
45J05Integro-ordinary differential equations
Full Text: DOI
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