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Positive solutions of a singular fractional boundary value problem with a fractional boundary condition

Lyons J. W., Neugebauer J. T.

Opuscula Mathematica

2017

Vol. 37, no. 3

421--434

For α ∈ (1,2] the singular fractional boundary value problem [formula] satisfying the boundary conditions [formula] where β ∈ (0,α - 1], μ ∈ (0,α - 1], and [formula] are Riemann-Liouville derivatives of order α, β, and μ respectively, is considered. Here ƒ satisfies a local Carathéodory condition, and ƒ (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.

Right focal boundary value problems for difference equations

Henderson J., Liu X., Lyons J. W., Neugebauer J. T.

Opuscula Mathematica

2010

Vol. 30, no. 4

447-456

An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.