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We study mimetic $F(R)$ gravity with potential and Lagrange multiplierconstraint. In the context of these theories, we introduce a reconstructiontechnique which enables us to realize arbitrary cosmologies, given the Hubblerate and an arbitrarily chosen $F(R)$ gravity. We exemplify our method byrealizing cosmologies that are in concordance with current observations (Planckdata) and also well known bouncing cosmologies. The attribute of our method isthat the $F(R)$ gravity can be arbitrarily chosen, so we can have the appealingfeatures of the mimetic approach combined with the known features of some$F(R)$ gravities, which unify early-time with late-time acceleration. Moreover,we study the existence and the stability of de Sitter points in the context ofmimetic $F(R)$ gravity. In the case of unstable de Sitter points, it isdemonstrated that graceful exit from inflation occurs. We also study theEinstein frame counterpart theory of the Jordan frame mimetic $F(R)$ gravity,we discuss the general properties of the theory and exemplify our analysis bystudying a quite interesting from a phenomenological point of view, model withtwo scalar fields. We also calculate the observational indices of the twoscalar field model, by using the two scalar field formalism. Furthermore, weextensively study the dynamical system that corresponds to the mimetic $F(R)$gravity, by finding the fixed points and studying their stability. Finally, wemodify our reconstruction method to function in the inverse way and thusyielding which $F(R)$ gravity can realize a specific cosmological evolution,given the mimetic potential and the Lagrange multiplier.

Accelerating Cosmology and Phase Structure of F(R) Gravity with Lagrange Multiplier Constraint: Mimetic Approach