  In this paper, is introduced a new proposal of resolvent for equilibrium
problems in terms of the Busemann's function. A great advantage of this new
proposal is that, in addition to be a natural extension of the proposal in the
linear setting by Combettes and Hirstoaga in [20], the new term that performs
regularization is a convex function in general Hadamard manifolds, being a
first step to fully answer to the problem posed by Cruz Neto et al. in [21,
Section 5]. During our study, some elements of convex analysis are explored in
the context of Hadamard manifolds, which are interesting on their own. In
particular, we introduce a new definition of convex combination (now
commutative) of any finite collection of points and present the realization of
an associated Jensen-type inequality.
