  We study the Hawking-Turok (HT) instanton solutions which have been employed
to describe the creation of an open inflationary universe, in the context of
higher derivative theories. We consider the effects of adding quadratic and
cubic terms of the forms $\alpha R^{2}$ and $\beta R^{3}$ to the gravitational
action. Using a conformal transformation to convert the higher derivative
theories into theories of self interacting scalar fields minimally coupled to
Einstein gravity, we argue that the cubic term represents a generic
perturbation of the polynomial type to the action and obtain the conditions on
the parameters of these theories for the existence of singular and non-singular
instanton solutions. We find that, relative to the quadratic case, there are
significant changes in the nature of the constraints on the parameters for the
existence of these instantons, once cubic (and higher order perturbations) are
added to the action.
