Bouncing cosmology with future singularity from modified gravity

We investigate which Jordan frame $F(R)$ gravity can describe a Type IV singular bouncing cosmological evolution, with special emphasis given near the point at which the Type IV singularity occurs. The cosmological bounce is chosen in such a way so that the bouncing point coincides exactly with Type IV singularity point. The stability of the resulting $F(R)$ gravity is examined and in addition, we study the Einstein frame scalar-tensor theory counterpart of the resulting Jordan frame $F(R)$ gravity. Also, by assuming that the Jordan frame metric is chosen in such a way so that, when conformally transformed in the Einstein frame, it yields a quasi de Sitter or de Sitter Friedmann-Robertson-Walker metric, we study the observational indexes which turn out to be consistent with Planck 2015 data in the case of the Einstein frame scalar theory. Finally, we study the behavior of the effective equation of state corresponding to the Type IV singular bounce and after we compare the resulting picture with other bouncing cosmologies, we critically discuss the implications of our analysis.