Ricci-Yamabe solitons in $f(R)$-gravity

Abstract

The main objective of this paper is to describe the perfect fluid spacetimes fulfilling $f(R)$-gravity, when Ricci-Yamabe, gradient Ricci-Yamabe and $\eta$-Ricci-Yamabe solitons are its metrics. We acquire conditions for which the Ricci-Yamabe and the gradient Ricci-Yamabe solitons are expanding, steady or shrinking. Furthermore, we investigate $\eta$-Ricci-Yamabe solitons and deduce a Poisson equation and with the help of this equation, we acquire some significant results.