Research Article  Open Access
Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications
Abstract
We introduce a Halperntype iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.
1. Introduction
Let be a real Banach space, a nonempty, closed, and convex subset of , and the dual space of . Let be a nonlinear mapping. The fixed points set of is denoted by , that is, .
One classical way often used to approximate a fixed point of a nonlinear selfmapping on was firstly introduced by Halpern [1] which is defined by and where is a real sequence in . He proved, in a real Hilbert space, a strong convergence theorem for a nonexpansive mapping when for any .
Subsequently, motivated by Halpern [1], many mathematicians devoted time to study algorithm (1.1) in different styles. Several strong convergence results for nonlinear mappings were also continuously established in some certain Banach spaces (see also [2–9]).
Let be a bifunction, a mapping, and a realvalued function. The generalized mixed equilibrium problem is to find such that The solutions set of (1.2) is denoted by (see Peng and Yao [10]).
If , then the generalized mixed equilibrium problem (1.2) reduces to the following mixed equilibrium problem: finding such that The solutions set of (1.3) is denoted by (see Ceng and Yao [11]).
If , then the generalized mixed equilibrium problem (1.2) reduces to the following mixed variational inequality problem: finding such that The solutions set of (1.4) is denoted by (see Noor [12]).
If , then the generalized mixed equilibrium problem (1.2) reduces to the following generalized equilibrium problem: finding such that The solutions set of (1.5) is denoted by (see Moudafi [13]).
If , then the mixed equilibrium problem (1.3) reduces to the following equilibrium problem: finding such that The solutions set of (1.6) is denoted by (see Combettes and Hirstoaga [14]).
If , then the mixed equilibrium problem (1.3) reduces to the following convex minimization problem: finding such that The solutions set of (1.7) is denoted by .
If , then the mixed variational inequality problem (1.4) reduces to the following variational inequality problem: finding such that The solutions set of (1.8) is denoted by (see Stampacchia [7]).
The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others. For more details on these topics, see, for instance, [14–34].
For solving the generalized mixed equilibrium problem, let us assume the following [25]:(A1) for all ;(A2) is monotone, that is, for all ;(A3) for all , ;(A4) for all is convex and lower semicontinuous.
The purpose of this paper is to investigate strong convergence of Halperntype iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. As applications, our main result can be deduced to mixed equilibrium, generalized equilibrium, mixed variational inequality problems, and so on. Examples and numerical results are also given in the last section.
2. Preliminaries and Lemmas
In this section, we need the following preliminaries and lemmas which will be used in our main theorem.
Let be a real Banach space and let be the unit sphere of . A Banach space is said to be strictly convex if, for any , It is also said to be uniformly convex if, for any , there exists such that, for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function called the modulus of convexity of as follows: Then is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . The normalized duality mapping is defined by for all . It is also known that if is uniformly smooth, then is uniformly normtonorm continuous on each bounded subset of (see [35]).
Let be a smooth Banach space. The function is defined by
Remark 2.1. We know the following: for any ,(1);(2);(3) in a real Hilbert space.
Lemma 2.2 (see [36]). Let be a uniformly convex and smooth Banach space and let and be sequences of such that or is bounded and . Then .
Let be a reflexive, strictly convex, and smooth Banach space and let be a nonempty closed and convex subset of . The generalized projection mapping, introduced by Alber [37], is a mapping , that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem:
In fact, we have the following result.
Lemma 2.3 (see [37]). Let be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space and let . Then there exists a unique element such that .
Lemma 2.4 (see [36, 37]). Let be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space , , and . Then if and only if
Lemma 2.5 (see [36, 37]). Let be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space and let . Then
Lemma 2.6 (see [38]). Let be a uniformly convex and uniformly smooth Banach space and a nonempty, closed, and convex subset of E. Then is uniformly normtonorm continuous on every bounded set.
We make use of the following mapping studied in Alber [37]: for all and , that is, .
Lemma 2.7 (see [39]). Let be a reflexive, strictly convex, smooth Banach space. Then for all and .
Lemma 2.8 (see [25]). Let be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to which satisfies conditions (A1)–(A4), and let and . Then there exists such that
Following [25, 40], we know the following lemma.
Lemma 2.9 (see [41]). Let be a nonempty closed and convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a continuous and monotone mapping, let be a bifunction from to satisfying (A1)–(A4), and let be a lower semicontinuous and convex function from to . For all and , there exists such that Define the mapping as follows: Then, the followings hold:(1) is singlevalued;(2) is firmly nonexpansivetype mapping [42], that is, for all , (3);(4) is closed and convex.
Remark 2.10. It is known that is of firmly nonexpansive type if and only if for all dom (see [42]).
The following lemmas give us some nice properties of real sequences.
Lemma 2.11 (see [43]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(a);(b) or .Then .
Lemma 2.12 (see [44]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence of such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, is the largest number in the set such that the condition holds.
3. Main Results
In this section, we prove our main theorem in this paper. To this end, we need the following proposition.
Proposition 3.1. Let be a nonempty closed and convex subset of a reflexive, strictly convex, and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4), a continuous and monotone mapping, and a lower semicontinuous and convex function from to such that . Let be such that . For each , let be defined as in Lemma 2.9. Suppose that and is a bounded sequence in such that . Then where and is the generalized projection of onto .
Proof. Let and put . Since is reflexive and is bounded, there exists a subsequence of such that and Put . Since , we have . On the other hand, since is uniformly smooth, is uniformly normtonorm continuous on bounded subsets of . So we have Since , By the definition of , for any , we see that By (A2), for each , we obtain For any and , we define . Then . It follows by the monotonicity of that By (A4), (3.4), and the weakly lower semicontinuity of , letting , we obtain By (A1), (A4), and the convexity of , we have It follows that By (A3), the weakly lower semicontinuity of , and the continuity of , letting , we obtain This shows that . By Lemma 2.4, we have This completes the proof.
Theorem 3.2. Let be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let be a bifunction from to satisfying (A1)–(A4), a continuous and monotone mapping, and a lower semicontinuous and convex function from to such that . Define the sequence as follows: and where and satisfy the following conditions:(a);(b);(c).Then converges strongly to , where is the generalized projection of onto .
Proof. From Lemma 2.9(4), we know that is closed and convex. Let . Put and for all . So, by Lemma 2.5, we have
By induction, we can show that for each . Hence is bounded and thus is also bounded.
We next show that if there exists a subsequence of such that
then
Since ,
Since is uniformly normtonorm continuous on bounded subsets of , so is . It follows that
Since is uniformly smooth and uniformly convex, by Lemma 2.6, is uniformly normtonorm continuous on bounded sets. So we obtain
and hence
Furthermore, . Indeed, by the definition of , we observe that
It follows from (3.19) and (3.20) that . On the other hand, from Remark 2.1(2), we have
It follows from (3.20) and (3.21) that
We next consider the following two cases.
Case 1. for all sufficiently large . Hence the sequence is bounded and nonincreasing. So exists. This shows that and hence
Since is of firmly nonexpansive type, by Remark 2.10, we have
which implies
Hence
as . By Lemma 2.2, we obtain
Proposition 3.1 yields that
It also follows that
Finally, we show that . Using Lemma 2.7, we see that
Set and . We see that . By Lemma 2.11, since , we conclude that . Hence as .Case 2. There exists a subsequence of such that for all . By Lemma 2.12, there exists a strictly increasing sequence of positive integers such that the following properties are satisfied by all numbers :
So we have
This shows that
Following the proof line in Case 1, we can show that
This implies
Hence . Using this and (3.33) together, we conclude that
This completes the proof.
As a direct consequence of Theorem 3.2, we obtain the following results.
Corollary 3.3. Let be nonempty closed and convex subset of a uniformly smooth and uniformly convex Banach space . Let be a bifunction from to satisfying (A1)–(A4) and a lower semicontinuous and convex function from to such that . Define the sequence as follows: and where and satisfy the following conditions:(a);(b);(c).Then converges strongly to , where is the generalized projection of onto .
Corollary 3.4. Let be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let be a bifunction from to satisfying (A1)–(A4), and a continuous and monotone mapping such that . Define the sequence as follows: and where and satisfy the following conditions:(a);(b);(c).Then converges strongly to , where is the generalized projection of onto .
Corollary 3.5. Let be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let be a continuous and monotone mapping, and a lower semicontinuous and convex function from to such that . Define the sequence as follows: and where and satisfy the following conditions:(a);(b);(c).Then converges strongly to , where is the generalized projection of onto .
4. Examples and Numerical Results
In this section, we give examples and numerical results for our main theorem.
Example 4.1. Let and . Let , , and . Find such that
Solution 4. It is easy to check that , , and satisfy all conditions in Theorem 3.2. For each and , Lemma 2.9 ensures that there exists such that, for any ,
Put . Then is a quadratic function of with coefficient , , and . We next compute the discriminant of as follows:
We know that for all if it has at most one solution in . So and hence . Now we have .
Let be the sequence generated by and
and, equivalently,
We next give two numerical results for algorithm (4.5).
Algorithm 4.2. Let and . Choose . Then algorithm (4.5) becomes
Numerical Result I
See Table 1.

Algorithm 4.3. Let and . Choose . Then algorithm (4.5) becomes
Numerical Result II
See Table 2.

5. Conclusion
Tables 1 and 2 show that the sequence converges to 0 which solves the generalized mixed equilibrium problem. On the other hand, using Lemma 2.9(3), we can check that .
Remark 5.1. In the view of computation, our algorithm is simple in order to get strong convergence for generalized mixed equilibrium problems.
Acknowledgments
The first and the second authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 20110021821).
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