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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 437391, 9 pages
http://dx.doi.org/10.1155/2012/437391
Research Article

Implicit Methods for Equilibrium Problems on Hadamard Manifolds

1Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
2Mathematics Department, HITEC University, Taxila Cantt, Pakistan
3Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 26 March 2012; Accepted 3 April 2012

Academic Editor: Rudong Chen

Copyright © 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.

Abstract

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 [18] and the references therein. Németh [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 inequalities on Hadamard manifold, which are due to M. A. Noor and K. I. Noor [5], Tang et al. [7], and Németh [9]. We hope that the technique and idea of this paper may stimulate further research in this area.

2. Preliminaries

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 [13, 6].

Let 𝑀 be a simply connected 𝑚-dimensional manifold. Given 𝑥𝑀, the tangent space of 𝑀 at 𝑥 is denoted by 𝑇𝑥𝑀 and the tangent bundle of 𝑀 by 𝑇𝑀=𝑥𝑀𝑇𝑥𝑀, which is naturally a manifold. A vector field 𝐴 on 𝑀 is a mapping of 𝑀 into 𝑇𝑀 which associates to each point 𝑥𝑀, a vector 𝐴(𝑥)𝑇𝑥𝑀. We always assume that 𝑀 can be endowed with a Riemannian metric to become a Riemannian manifold. We denote by ,, the scalar product on 𝑇𝑥𝑀 with the associated norm 𝑥, where the subscript 𝑥 will be omitted. Given a piecewise smooth curve 𝛾[𝑎,𝑏]𝑀 joining 𝑥 to 𝑦 (i.e., 𝛾(𝑎)=𝑥 and 𝛾(𝑏)=𝑦,) by using the metric, we can define the length of 𝛾 as 𝐿(𝛾)=𝑏𝑎𝛾(𝑡)𝑑𝑡. Then, for any 𝑥,𝑦𝑀, the Riemannian distance 𝑑(𝑥,𝑦), which includes the original topology on 𝑀, is defined by minimizing this length over the set of all such curves joining 𝑥 to 𝑦.

Let Δ be the Levi-Civita connection with (𝑀,,). Let 𝛾 be a smooth curve in 𝑀. A vector field 𝐴 is said to be parallel along 𝛾 if Δ𝛾𝐴=0. If 𝛾 itself is parallel along 𝛾, we say that 𝛾 is a geodesic and in this case 𝛾 is constant. When 𝛾=1,𝛾 is said to be normalized. A geodesic joining 𝑥 to 𝑦 in 𝑀 is said to be minimal if its length equals 𝑑(𝑥,𝑦).

A Riemannian manifold is complete, if for any 𝑥𝑀 all geodesics emanating from 𝑥 are defined for all 𝑡𝑅. By the Hopf-Rinow Theorem, we know that if 𝑀 is complete, then any pair of points in 𝑀 can be joined by a minimal geodesic. Moreover, (𝑀,𝑑) is a complete metric space and bounded closed subsets are compact.

Let 𝑀 be complete. Then the exponential map exp𝑥𝑇𝑥𝑀𝑀 at 𝑥 is defined by exp𝑥𝑣=𝛾𝑣(1,𝑥) for each 𝑣𝑇𝑥𝑀, where 𝛾(.)=𝛾𝑣(.,𝑥) is the geodesic starting at 𝑥 with velocity 𝑣(i.e.,𝛾(0)=𝑥 and 𝛾(0)=𝑣). Then exp𝑥𝑡𝑣=𝛾𝑣(𝑡,𝑥) for each real number 𝑡.

A complete simply-connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. Throughout the remainder of this paper, we always assume that 𝑀 is an 𝑚-manifold Hadamard manifold.

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

Lemma 2.1 (See [6]). Let 𝑥𝑀. Then exp𝑥𝑇𝑥𝑀𝑀 is a diffeomorphism, and, for any two points 𝑥,𝑦𝑀, there exists a unique normalized geodesic joining 𝑥 to 𝑦,𝛾𝑥,𝑦, which is minimal.

So from now on, when referring to the geodesic joining two points, we mean the unique minimal normalized one. Lemma 2.1 says that 𝑀 is diffeomorphic to the Euclidean space 𝑅𝑚. Thus 𝑀 has the same topology and differential structure as 𝑅𝑚. It is also known that Hadamard manifolds and euclidean spaces have similar geometrical properties. Recall that a geodesic triangle (𝑥1,𝑥2,𝑥3) of a Riemannian manifold is a set consisting of three points 𝑥1, 𝑥2, 𝑥3 and three minimal geodesics joining these points.

Lemma 2.2 (See [2, 3, 6] (comparison theorem for triangles)). Let (𝑥1,𝑥2,𝑥3) be a geodesic triangle. Denote, for each 𝑖=1,2,3(mod3), by 𝛾𝑖[0,𝑙𝑖]𝑀, the geodesic joining 𝑥𝑖 to 𝑥𝑖+1, and 𝛼𝑖;=𝐿(𝛾𝑖(0),𝛾𝑙(𝑖1)(𝑙𝑖1)), the angle between the vectors 𝛾𝑖(0) and 𝛾𝑖1(𝑙𝑖1), and 𝑙𝑖;=𝐿(𝛾𝑖). Then, 𝛼1+𝛼2+𝛼3𝑙𝜋,(2.1)2𝑙+𝑙2𝑖+12𝐿𝑖𝑙𝑖+1cos𝛼𝑖+1𝑙2𝑖1.(2.2)

In terms of the distance and the exponential map, the inequality (2.2) can be rewritten as𝑑2𝑥𝑖,𝑥𝑖+1+𝑑2𝑥𝑖+1,𝑥𝑖+22exp𝑥1𝑖+1𝑥𝑖,exp𝑥1𝑖+1𝑥𝑖+2𝑑2𝑥𝑖1,𝑥𝑖,(2.3) sinceexp𝑥1𝑖+1𝑥𝑖,exp𝑥1𝑖+1𝑥𝑖+2𝑥=𝑑𝑖,𝑥𝑖+1𝑑𝑥𝑖+1,𝑥𝑖+2cos𝛼𝑖+1.(2.4)

Lemma 2.3 (See [6]). Let (𝑥,𝑦,𝑧) be a geodesic triangle in a Hadamard manifold 𝑀. Then, there exist 𝑥,𝑦,𝑧𝑅2 such that 𝑑𝑥(𝑥,𝑦)=𝑦𝑦,𝑑(𝑦,𝑧)=𝑧𝑧,𝑑(𝑧,𝑥)=𝑥.(2.5) The triangle (𝑥,𝑦,𝑧) is called the comparison triangle of the geodesic triangle (𝑥,𝑦,𝑧), which is unique up to isometry of 𝑀.

From the law of cosines in inequality (2.3), we have the following inequality, which is a general characteristic of the spaces with nonpositive curvature [6]:exp𝑥1𝑦,exp𝑥1𝑧+exp𝑦1𝑥,exp𝑦1𝑧𝑑2(𝑥,𝑦).(2.6) From the properties of the exponential map, we have the following known result.

Lemma 2.4 (See [6]). Let 𝑥0𝑀 and {𝑥𝑛}𝑀 such that 𝑥𝑛𝑥0. Then the following assertions hold.(i)For any 𝑦𝑀, exp𝑥1𝑛𝑦exp𝑥10𝑦,exp𝑦1𝑥𝑛exp𝑦1𝑥0.(2.7)(ii)If {𝑣𝑛} is a sequence such that 𝑣𝑛𝑇𝑥𝑛𝑀 and 𝑣𝑛𝑣0, then 𝑣0𝑇𝑥0𝑀.(iii)Given the sequences {𝑢𝑛} and {𝑣𝑛} satisfying 𝑢𝑛,𝑣𝑛𝑇𝑥𝑛𝑀, if 𝑢𝑛𝑢0 and 𝑣𝑛𝑣0, with 𝑢0,𝑣0𝑇𝑥0𝑀, then 𝑢𝑛,𝑣𝑛𝑢0,𝑣0.(2.8)

A subset 𝐾𝑀 is said to be convex if for any two points 𝑥,𝑦𝐾, the geodesic joining 𝑥 and 𝑦 is contained in 𝐾,𝐾 that is, if  𝛾[𝑎,𝑏]𝑀 is a geodesic such that 𝑥=𝛾(𝑎) and 𝑦=𝛾(𝑏), then 𝛾((1𝑡)𝑎+𝑡𝑏)𝐾,for all 𝑡[0,1]. From now on 𝐾𝑀 will denote a nonempty, closed, and convex set, unless explicitly stated otherwise.

A real-valued function 𝑓 defined on 𝐾 is said to be convex, if, for any geodesic 𝛾 of 𝑀, the composition function 𝑓𝛾𝑅𝑅 is convex, that is,[](𝑓𝛾)(𝑡𝑎+(1𝑡)𝑏)𝑡(𝑓𝛾)(𝑎)+(1𝑡)(𝑓𝛾)(𝑏),𝑎,𝑏𝑅,𝑡0,1.(2.9)

The subdifferential of a function 𝑓𝑀𝑅 is the set-valued mapping 𝜕𝑓𝑀2𝑇𝑀 defined as𝜕𝑓(𝑥)=𝑢𝑇𝑥𝑀𝑢,exp𝑥1𝑦𝑓(𝑦)𝑓(𝑥),𝑦𝑀,𝑥𝑀,(2.10) and its elements are called subgradients. The subdifferential 𝜕𝑓(𝑥) at a point 𝑥𝑀 is a closed and convex (possibly empty) set. Let 𝐷(𝜕𝑓) denote the domain of 𝜕𝑓 defined by𝐷(𝜕𝑓)={𝑥𝑀𝜕𝑓(𝑥)}.(2.11)

The existence of subgradients for convex functions is guaranteed by the following proposition, see [8].

Lemma 2.5 (See [6, 8]). Let 𝑀 be a Hadamard manifold and let 𝑓𝑀𝑅 be convex. Then, for any 𝑥𝑀, the subdifferential 𝜕𝑓(𝑥) of 𝑓 at 𝑥 is nonempty. That is, 𝐷(𝜕𝑓)=𝑀.

For a given bifunction 𝐹(,)𝐾×𝐾𝑅, we consider the problem of finding 𝑢𝐾 such that𝐹(𝑢,𝑣)0,𝑣𝐾,(2.12) which is called the equilibrium problem on Hadamard manifolds. This problem was considered by Colao et al. [2]. They proved the existence of a solution of the problem (2.12) using the KKM maps. Colao et al. [2] have given an example of the equilibrium problem defined in a Euclidean space whose set 𝐾 is not a convex set, so it cannot be solved using the technique of Blum and Oettli [11]. However, if one can reformulate the equilibrium problem on a Riemannian manifold, then it can be solved. This shows the importance of considering these problems on Hadamard manifolds. For the applications, formulation, and other aspects of the equilibrium problems in the linear setting, see [4, 922].

If 𝐹(𝑢,𝑣)=𝑇𝑢,exp𝑢1𝑣, where 𝑇 is a single valued vector filed 𝑇𝐾𝑇𝑀, then problem (2.12) is equivalent to finding 𝑢𝐾 such that𝑇𝑢,exp𝑥1𝑣0,𝑣𝐾,(2.13) which is called the variational inequality on Hadamard manifolds. Németh [9], Colao et al. [2], Noor and Oettli [10], and M. A. Noor and K. I. Noor [5] studied variational inequalities on Hadamard manifold from different point of views. In the linear setting, variational inequalities have been studied extensively, see [5, 10, 11, 1328] and the references therein.

Definition 2.6. A bifunction 𝐹(,) is said to be speudomonotone, if and only if 𝐹(𝑢,𝑣)0,𝐹(𝑣,𝑢)0,𝑢,𝑣𝐾.(2.14)

3. Main Results

We now use the auxiliary principle technique of Glowinski et al. [23] to suggest and analyze an implicit iterative method for solving the equilibrium problems (2.12).

For a given 𝑢𝐾 satisfying (2.12), consider the problem of finding 𝑤𝐾 such that𝜌𝐹(𝑤,𝑣)+exp𝑢1𝑤,exp𝑤1𝑣0,𝑣𝐾,(3.1) which is called the auxiliary equilibrium problem on Hadamard manifolds. We note that, if 𝑤=𝑢, then 𝑤 is a solution of (2.12). This observation enables to suggest and analyze the following implicit method for solving the equilibrium problems (2.12). This is the main motivation of this paper.

Algorithm 3.1. For a given 𝑢0, compute the approximate solution by the iterative scheme 𝑢𝜌𝐹𝑛+1+,𝑣exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣0,𝑣𝐾.(3.2) Algorithm 3.1 is called the implicit (proximal point) iterative method for solving the equilibrium problem on the Hadamard manifold. Algorithm 3.1 can be written in the following equivalent form.

Algorithm 3.2. For a given 𝑢0𝐾, find the approximate solution 𝑢𝑛+1 by the iterative scheme: 𝑢𝜌𝐹𝑛+,𝑣exp𝑢1𝑛𝑦𝑛,exp𝑦1𝑛𝑣𝑦0,𝑣𝐾,𝜌𝐹𝑛+,𝑣exp𝑦1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣0,𝑣𝐾.(3.3) Algorithm 3.2 is a two-step iterative method for solving the equilibrium problems on Hadamard manifolds. This method can be viewed as the extragradient method for solving the equilibrium problems.

If 𝐾 is a convex set in 𝑅𝑛, then Algorithm 3.1 collapses to the following.

Algorithm 3.3. For a given 𝑢0𝐾, find the approximate solution 𝑢𝑛+1 by the iterative scheme: 𝑢𝜌𝐹𝑛+1+𝑢,𝑣𝑛+1𝑢𝑛,𝑣𝑢𝑛+10,𝑣𝐾,(3.4) which is known as the implicit method for solving the equilibrium problem. For the convergence analysis of Algorithm 3.2, see [16, 19, 20].

If 𝐹(𝑢,𝑣)=𝑇𝑢,exp𝑢1𝑣, where 𝑇 is a single valued vector filed 𝑇𝐾𝑇𝑀, then Algorithm 3.1 reduces to the following implicit method for solving the variational inequalities.

Algorithm 3.4. For a given 𝑢0𝐾, compute the approximate solution 𝑢𝑛+1 by the iterative scheme 𝜌𝑇𝑢𝑛+1+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣0,𝑣𝐾.(3.5) Algorithm 3.4 is due according to Tang et al. [7] and M. A. Noor and K. I. Noor [5]. We can also rewrite Algorithm 3.4 in the following equivalent form.

Algorithm 3.5. For a given 𝑢0𝐾, compute the approximate solution 𝑢𝑛+1 by the iterative scheme 𝜌𝑇𝑢𝑛+exp𝑢1𝑛𝑦𝑛,exp𝑦1𝑛𝑣0,𝑣𝐾,𝜌𝑇𝑦𝑛+exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑣0,𝑣𝐾,(3.6) which is the extragradient method for solving the variational inequalities on Hadamard manifolds and appears to be a new one.

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 3.1 and this is the motivation of our next result.

Theorem 3.6. Let 𝐹(,) be a speudomonotone bifunction. Let 𝑢𝑛 be the approximate solution of the equilibrium problem (2.12) obtained from Algorithm 3.1, then 𝑑2𝑢𝑛+1,𝑢+𝑑2𝑢𝑛+1,𝑢𝑛𝑑2𝑢𝑛,𝑢,(3.7) where 𝑢𝐾 is a solution of the equilibrium problem (2.12).

Proof. Let 𝑢𝐾 be a solution of the equilibrium problem (2.12). Then, by using the speudomonotonicity of the bifunction 𝐹(,), we have 𝐹(𝑣,𝑢)0,𝑣𝐾.(3.8) Taking 𝑣=𝑢𝑛+1 in (3.9), we have 𝐹𝑢𝑛+1,𝑢0.(3.9) Taking 𝑣=𝑢 in (3.2), we have 𝑢𝜌𝐹𝑛+1+,𝑢exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑢0.(3.10) From (3.10) and (3.9), we have exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑢0.(3.11) For the geodesic triangle (𝑢𝑛,𝑢𝑛+1,𝑢), the inequality (2.3) can be written as 𝑑2𝑢𝑛+1,𝑢+𝑑2𝑢𝑛+1,𝑢𝑛exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑢𝑑2𝑢𝑛,𝑢.(3.12) Thus, from (3.12) and (3.11), we obtained the inequality (3.8), the required result.

Theorem 3.7. Let 𝑢𝐾 be solution of (2.12) and let 𝑢𝑛+1 be the approximate solution obtained from Algorithm 3.1, then lim𝑛𝑢𝑛+1=𝑢.

Proof. Let ̂𝑢 be a solution of (2.12). Then, from (3.8), it follows that the sequence {𝑢𝑛} is bounded and 𝑛=0𝑑2𝑢𝑛+1,𝑢𝑛𝑑2𝑢0,𝑢,(3.13) then it follows that lim𝑛𝑑𝑢𝑛+1,𝑢𝑛=0.(3.14) Let ̂𝑢 be a cluster point of {𝑢𝑛}. Then there exists a subsequence {𝑢𝑛𝑖} such that {𝑢𝑢𝑖} converges to ̂𝑢. Replacing 𝑢𝑛+1 by 𝑢𝑛𝑖 in (3.2), taking the limit, and using (3.14), we have 𝐹(̂𝑢,𝑣)0,𝑣𝐾.(3.15) This shows that ̂𝑢𝐾 solves (2.12) and 𝑑2𝑢𝑛+1,̂𝑢𝑑2𝑢𝑛,̂𝑢(3.16) which implies that the sequence {𝑢𝑛} has unique cluster point and lim𝑛𝑢𝑛=̂𝑢 is a solution of (2.12), the required result.

4. Conclusion

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.

Acknowledgments

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.

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