Certain paracontact metrics satisfying gradient ρ-Ricci-Bourguignon almost solitons

In this paper, we thoroughly study rho-Ricci-Bourguignon almost soliton and gradient rho-Ricci-Bourguignon almost soliton in the paracontact geometry, precisely, on K-paracontact and para-Sasakian manifolds. Here, we prove that if the metric g of the K-paracontact manifold endows a rho-Ricci-Bourguignon almost soliton with the nonzero potential vector field V parallel to xi, then the manifold is an Einstein with Einstein constant -2n. Next, we show that if a para-Sasakian manifold represents a gradient rho-Ricci-Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature -2n(2n + 1). We also discuss rho-Ricci-Bourguignon almost soliton on (kappa,mu)-paracontact manifold.