Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 437391, 9 pages doi:10.1155/2012/437391 Research Article Implicit Methods for Equilibrium Problems on Hadamard Manifolds Muhammad Aslam Noor, 1 Saira Zainab, 2 and Yonghong Yao 3 1 Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan 2 Mathematics Department, HITEC University, Taxila Cantt, Pakistan 3 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China Correspondence should be addressed to Muhammad Aslam Noor, noormaslam@hotmail.com Received 26 March 2012; Accepted 3 April 2012 Academic Editor: Rudong Chen Copyright q 2012 Muhammad Aslam Noor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. Introduction 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 1–8 and the references therein. N´ emeth 9, Tang et al. 7, M. A. Noor and K. I. Noor 5, and Colao et al. 2 have considered the variational inequalities and equilibrium problems on Hadamard manifolds. They have studied the existence of a solution of the equilibrium problems under some suitable conditions. To the best of our knowledge, no one has considered the auxiliary principle technique for solving the equilibrium problems on Hadamard manifolds. In this paper, we use the auxiliary principle technique to suggest and analyze an implicit method for solving the equilibrium problems on Hadamard manifold. As a special case, our result includes the recent result of Noor and Oettli 10 for variational inequalities on Hadamard manifold. This shows that the results obtained in this paper continue to hold for variational