Positive Solutions of a Singular Fractional Boundary Value Problem with a Fractional Boundary Condition
For α ∈ (1,2], the singular fractional boundary value problem
D0α+x + f t,x,D0µ+x = 0, 0 < t < 1,
satisfying the boundary conditionsα β µ x(0) = D0β+x(1) = 0, where β ∈ (0,α−1], µ ∈ (0,α−1], and D0+, D0+ and D0+ are Riemann-Liouville derivatives of order α, β and µ respectively, is considered. Here f satisfies a local Carathéodory condition, and f(t,x,y) may be singular at the value 0 in its space variable x. Using regularization and sequential techniques and Krasnosel’skii’s fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.