Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations (Q986570)

Consider the existence of a positive solution for the singular fractional boundary value problem \[ D^\alpha u(t)+ f(t,u(t),D^\mu u(t))=0,\,u(0)=u(1)=0, \] where \(1<\alpha<2\), \(\mu>0\) with \(\alpha-\mu\geq 1,\) \(D^\alpha\) is the standard Riemann-Liouville fractional derivative, the function \(f\) is positive, satisfies the Carathéodory conditions on \( [0,1]\times (0,\infty)\times {\mathbb R}\) and \(f(t,x,y)\) is singular at \(x=0\). The proofs are based on regularization and sequential techniques and the results are obtained by means of fixed point theorem of cone compression type due to [\textit{M. A. Krasnosel'skij}, Positive solutions of operator equations. Groningen: The Netherlands: P.Noordhoff Ltd. (1964; Zbl 0121.10604)].