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

In this paper, we thoroughly study [Formula: see text]-Ricci–Bourguignon almost soliton and gradient [Formula: see text]-Ricci–Bourguignon almost soliton in the paracontact geometry, precisely, on [Formula: see text]-paracontact and para-Sasakian manifolds. Here, we prove that if the metric [Formula: see text] of the [Formula: see text]-paracontact manifold endows a [Formula: see text]-Ricci–Bourguignon almost soliton with the nonzero potential vector field [Formula: see text] parallel to [Formula: see text], then the manifold is an Einstein with Einstein constant [Formula: see text]. Next, we show that if a para-Sasakian manifold represents a gradient [Formula: see text]-Ricci–Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature [Formula: see text]. We also discuss [Formula: see text]-Ricci–Bourguignon almost soliton on [Formula: see text]-paracontact manifold.