Positive solutions for mixed problems of singular fractional differential equations |
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| Authors: | Ravi P Agarwal Donal O'Regan Svatoslav Staněk |
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| Institution: | 1. Department of Mathematics, Texas A&M University–Kingsville, 700 University Blvd., Kingsville, TX 78363‐8202, USA;2. Department of Mathematics, National University of Ireland, Galway, Ireland;3. Department of Mathematical Analysis, Faculty of Science, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic |
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| Abstract: | We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a Lq‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones. |
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| Keywords: | Fractional differential equation Caputo fractional derivative singular mixed problem positive solution regularization fixed point theorem on cones MSC (2010) 26A33 34B16 |
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