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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