Convergence of diagonally stationary sequences in convex optimization

LetX be a real normed linear space,f, fⁿ, n , be extended real-valued proper closed convex functions onX. A sequence {x_n} inX is called diagonally stationary for {fⁿ} if for alln there existsx*_n fⁿ (x_n) such that x*_{n *} 0. Such sequences arise in approximation methods for the problem of minimizingf. Some general convergence results based upon variational convergence theory and appropriate equi-well-posedness are presented.