Jeffrey T. Neugebauer

Conjugate points for fractional differential equations

Let b > 0. Let 1 < α ≤ 2. The theory of u0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D0+αu + λp(t)u = 0, 0 < t < b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation.

Smallest eigenvalues for a fractional difference equation with right focal boundary conditions

We show the existence of and then compare smallest eigenvalues for Atici–Eloe fractional difference equations satisfying a right focal boundary condition. The theory of -positive operators is applied to obtain these results.

Comparison of Green's functions for a family of boundary value problems for fractional difference equations

ABSTRACT In this paper, we obtain sign conditions and comparison theorems for Greens functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Greens function converges to a uniquely defined Greens function of a singular boundary value problem.