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We use the auxiliary principle technique to suggest and analyze an implicit method for solving the equilibrium problems on Hadamard manifolds. The convergence of this new implicit method requires only the speudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered.

Recently, much attention has been given to study the variational inequalities, equilibrium and related optimization problems on the Riemannian manifold and Hadamard manifold. This framework is a useful for the developments of various fields. Several ideas and techniques form the Euclidean space have been extended and generalized to this nonlinear framework. Hadamard manifolds are examples of hyperbolic spaces and geodesics, see [

We now recall some fundamental and basic concepts needed for reading of this paper. These results and concepts can be found in the books on Riemannian geometry [

Let

Let

A Riemannian manifold is complete, if for any

Let

A complete simply-connected Riemannian manifold of nonpositive sectional curvature is called a

We also recall the following well-known results, which are essential for our work.

Let

So from now on, when referring to the geodesic joining two points, we mean the unique minimal normalized one. Lemma

Let

In terms of the distance and the exponential map, the inequality (

Let

From the law of cosines in inequality (

Let

For any

If

Given the sequences

A subset

A real-valued function

The subdifferential of a function

The existence of subgradients for convex functions is guaranteed by the following proposition, see [

Let

For a given bifunction

If

A bifunction

We now use the auxiliary principle technique of Glowinski et al. [

For a given

For a given

For a given

If

For a given

If

For a given

For a given

In a similar way, one can obtain several iterative methods for solving the variational inequalities on the Hadamard manifold.

We now consider the convergence analysis of Algorithm

Let

Let

Let

Let

In this paper, we have suggested and analyzed an implicit iterative method for solving the equilibrium problems on Hadamard manifold. It is shown that the convergence analysis of this methods requires only the speudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also discussed. Results proved in this paper may stimulate research in this area.

The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities. Professor Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279, and NSFC 71161001-G0105.