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In this paper we investigate how to describe in a unified way a constant-rollinflationary era with a dark energy era, by using the theoretical framework of$F(R)$ gravity. To this end, we introduce some classes of appropriately chosen$F(R)$ gravity models, and we examine in detail how the unification of earlyand late-time acceleration eras can be achieved. We study in detail theinflationary era, and as we demonstrate it is possible to achieve a viableinflationary era, for which the spectral index of primordial curvatureperturbations and the scalar-to-tensor ratio can be compatible with the latestobservational data. In addition, the graceful exit issue is briefly discussedfor a class of models. Finally, we discuss the dark energy oscillations issue,and we investigate which model from one of the classes we introduced, canproduce oscillations with the smallest amplitude.

Unification of Constant-roll Inflation and Dark Energy with Logarithmic $R^2$-corrected and Exponential $F(R)$ Gravity