I assume that when you say "equilibrium" you mean the equilibrium price ...

In perfect competition, the firm and the industry have the same equilibrium price.  This is because all firms in perfect competition are price takers and must, in the long run, charge the same price.

The AFC curve refers to the Average Fixed Cost of producing each quantity of goods.  Fixed costs refers to costs, like rent on a building, that remain the same no matter how many units a firm produces.  In that case, the AFC (TFC/quantity produced) must by definition decline as the quantity produced rises.