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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 \(3<\alpha\leq 4\), \(0<\nu\leq 1\), \(1<\mu \leq 2\), \(D^{\alpha}_{0+}\) 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.

MSC:

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
45J05 Integro-ordinary differential equations
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