Positive Solutions of a Singular Fractional Boundary Value Problem with a Fractional Boundary Condition

Jeffrey W. Lyons, Eastern Kentucky University
Jeffrey T. Neugebauer, Eastern Kentucky University

Abstract

For _α ∈ (1_,2], the singular fractional boundary value problem

D₀^α+x + f t,x,D₀^µ+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.