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Positive solutions for boundary value problems of singular fractional differential equations. (English) Zbl 1307.34005

Summary: By using a fixed point theorem, we investigate the existence of a positive solution to the singular fractional boundary value problem \[ {^CD^\alpha_{0+}} u+ f(t,u,{^CD^\nu_{0+}u} {^CD^\mu_{0+}u})+ g(t,u{^CD^\nu_{0+}u}, {^CD^\mu_{0+}u})= 0, \]
\[ u(0)= u'(0)= u''(0)= u'''(0)= 0, \] where \(3<\alpha<4\), \(0<\nu< 1\), \(1<\mu<2\), \({^CD^\alpha_{0+}}\) is Caputo fractional derivative, \(f(t,x,y,z)\) is singular at the value \(0\) of arguments \(x\), \(y\), \(z\), and \(g(t,x,y,z)\) satisfies the Lipschitz condition.

MSC:

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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