The open inflation model recently proposed by Hawking and Turok is investigated in scalar-tensor gravity context. If the dilaton-like field has no potential, the instanton of our model is singular but has a finite action. The GibbonsHawking surface term vanishes and hence, can not be used to make Ω 0 nonzero.To obtain a successful open inflation one should introduce other matter fields or a potential for the dilaton-like fields.PACS numbers: 98.80.Cq 1

The oscillating inflation model recently proposed by Damour and Mukhanov is investigated with a non-minimal coupling. Numerical study confirms an inflationary behavior and the density perturbation is obtained. A successful inflation requires the gravity-dilaton coupling to be small. PACS numbers: 98.80.Cq 1