Surjectivity of the evaluation map for non-autonomous sequential equations

Ascertain whether the evaluation map E: S → C^2, given by E(x) = (x(0), LD^α x(0)), is surjective for non-autonomous order-two L-fractional linear equations with analytic coefficients, i.e., for equations of the form LD^{2α} x(t) + p(t) LD^α x(t) + q(t) x(t) = c(t).

Background

In Section 7, the authors develop power-series solutions for non-autonomous order-two equations but note that general results for first-order non-autonomous systems LD^α z(t) = A(t) z(t) are not yet treated, leaving surjectivity of the evaluation map unsettled.