Volume 2012, Issue 1 308791

Research Article

Open Access

Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

Prasit Cholamjiak,

Prasit Cholamjiak

School of Science, University of Phayao, Phayao 56000, Thailand up.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand cmu.ac.th

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Suthep Suantai,

Suthep Suantai

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand cmu.ac.th

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand cmu.ac.th

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Yeol Je Cho,

Corresponding Author

Yeol Je Cho

[email protected]

Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea gnu.ac.kr

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Prasit Cholamjiak,

Prasit Cholamjiak

School of Science, University of Phayao, Phayao 56000, Thailand up.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand cmu.ac.th

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Suthep Suantai,

Suthep Suantai

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand cmu.ac.th

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand cmu.ac.th

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Yeol Je Cho,

Corresponding Author

Yeol Je Cho

[email protected]

Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea gnu.ac.kr

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First published: 16 January 2012

https://doi.org/10.1155/2012/308791

Citations: 2

Academic Editor: Yonghong Yao

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Abstract

We introduce a Halpern-type 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 E be a real Banach space, C a nonempty, closed, and convex subset of E, and E^* the dual space of E. Let T : C → C be a nonlinear mapping. The fixed points set of T is denoted by F(T), that is, F(T) = {x ∈ C : x = Tx}.

One classical way often used to approximate a fixed point of a nonlinear self-mapping T on C was firstly introduced by Halpern [1] which is defined by x₁ = x ∈ C and

()

where {α_n} is a real sequence in [0,1]. He proved, in a real Hilbert space, a strong convergence theorem for a nonexpansive mapping T when α_n = n^−a for any a ∈ (0,1).

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 f : C × C → ℝ be a bifunction, A : C → E^* a mapping, and φ : C → ℝ a real-valued function. The generalized mixed equilibrium problem is to find

such that

()

The solutions set of (1.2) is denoted by GMEP (f, A, φ) (see Peng and Yao [10]).

If A ≡ 0, 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 MEP (f, φ) (see Ceng and Yao [11]).

If f ≡ 0, 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 VI (C, A, φ) (see Noor [12]).

If φ ≡ 0, 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 GEP (f, A) (see Moudafi [13]).

If φ ≡ 0, 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 EP (f) (see Combettes and Hirstoaga [14]).

If f ≡ 0, 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 CMP(φ).

If φ ≡ 0, 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 VI (C, A) (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)
f(x, x) = 0 for all x ∈ C;
(A2)
f is monotone, that is, f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C;
(A3)
for all x, y, z ∈ C, limsup _t↓0f(tz + (1 − t)x, y) ≤ f(x, y);
(A4)
for all x ∈ C, f(x, ·) is convex and lower semicontinuous.

The purpose of this paper is to investigate strong convergence of Halpern-type 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 E be a real Banach space and let U = {x ∈ E : ∥x∥ = 1} be the unit sphere of E. A Banach space E is said to be strictly convex if, for any x, y ∈ U,

()

It is also said to be uniformly convex if, for any ε ∈ (0,2], there exists δ > 0 such that, for any x, y ∈ U,

()

It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function δ : [0,2]→[0,1] called the modulus of convexity of E as follows:

()

Then E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0,2]. A Banach space E is said to be smooth if the limit

()

exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for x, y ∈ U. The normalized duality mapping

is defined by

()

for all x ∈ E. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E (see [35]).

Let E be a smooth Banach space. The function ϕ : E × E → ℝ is defined by

()

Remark 2.1. We know the following: for any x, y, z ∈ E,

(1)
(∥x∥−∥y∥) ² ≤ ϕ(x, y)≤(∥x∥+∥y∥) ²;
(2)
ϕ(x, y) = ϕ(x, z) + ϕ(z, y) + 2〈x − z, Jz − Jy〉;
(3)
ϕ(x, y) = ∥x − y∥² in a real Hilbert space.

Lemma 2.2 (see [36].)Let E be a uniformly convex and smooth Banach space and let {x_n} and {y_n} be sequences of E such that {x_n} or {y_n} is bounded and lim _n→∞ϕ(x_n, y_n) = 0. Then lim _n→∞∥x_n − y_n∥ = 0.

Let E be a reflexive, strictly convex, and smooth Banach space and let C be a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber [37], is a mapping Π_C : E → C, that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(y, x), that is,

, where

is the solution to the minimization problem:

()

In fact, we have the following result.

Lemma 2.3 (see [37].)Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space E and let x ∈ E. Then there exists a unique element x₀ ∈ C such that ϕ(x₀, x) = min {ϕ(z, x) : z ∈ C}.

Lemma 2.4 (see [36], [37].)Let C be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space E, x ∈ E, and z ∈ C. Then z = Π_Cx if and only if

()

Lemma 2.5 (see [36], [37].)Let C be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space E and let x ∈ E. Then

()

Lemma 2.6 (see [38].)Let E be a uniformly convex and uniformly smooth Banach space and C a nonempty, closed, and convex subset of E. Then Π_C is uniformly norm-to-norm continuous on every bounded set.

We make use of the following mapping V studied in Alber [37]:

()

for all x ∈ E and x^* ∈ E^*, that is, V(x, x^*) = ϕ(x, J⁻¹(x^*)).

Lemma 2.7 (see [39].)Let E be a reflexive, strictly convex, smooth Banach space. Then

()

for all x ∈ E and x^*, y^* ∈ E^*.

Lemma 2.8 (see [25].)Let C be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × C to ℝ which satisfies conditions (A1)–(A4), and let r > 0 and x ∈ E. Then there exists z ∈ C such that

()

Following [25, 40], we know the following lemma.

Lemma 2.9 (see [41].)Let C be a nonempty closed and convex subset of a smooth, strictly convex, and reflexive Banach space E. Let A : C → E^* be a continuous and monotone mapping, let f be a bifunction from C × C to ℝ satisfying (A1)–(A4), and let φ be a lower semicontinuous and convex function from C to ℝ. For all r > 0 and x ∈ E, there exists z ∈ C such that

()

Define the mapping T_r : E → 2^C as follows:

()

Then, the followings hold:

(1)
T_r is single-valued;
(2)
T_r is firmly nonexpansive-type mapping [42], that is, for all x, y ∈ E,
()
(3)
F(T_r) = GMEP (f, A, φ);
(4)
GMEP (f, A, φ) is closed and convex.

Remark 2.10. It is known that T is of firmly nonexpansive type if and only if

()

for all x, y∈ dom T (see [42]).

The following lemmas give us some nice properties of real sequences.

Lemma 2.11 (see [43].)Assume that {a_n} is a sequence of nonnegative real numbers such that

()

where {α_n} is a sequence in (0,1) and {b_n} is a sequence such that

(a)
;
(b)
limsup _n→∞b_n/α_n ≤ 0 or .

Then lim _n→∞a_n = 0.

Lemma 2.12 (see [44].)Let {γ_n} be a sequence of real numbers such that there exists a subsequence of {γ_n} such that for all j ≥ 1. Then there exists a nondecreasing sequence {m_k} of ℕ such that lim _k→∞m_k = ∞ and the following properties are satisfied by all (sufficiently large) numbers k ≥ 1:

()

In fact, m_k is the largest number n in the set {1,2, …, k} such that the condition γ_n < γ_n+1 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 C be a nonempty closed and convex subset of a reflexive, strictly convex, and uniformly smooth Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4), A : C → E^* a continuous and monotone mapping, and φ a lower semicontinuous and convex function from C to ℝ such that GMEP (f, A, φ) ≠ ∅. Let {r_n}⊂(0, ∞) be such that lim inf _n→∞r_n > 0. For each n ≥ 1, let be defined as in Lemma 2.9. Suppose that x ∈ C and {x_n} is a bounded sequence in C such that . Then

()

where p = Π_{GMEP (f,A,φ)}x and Π_{GMEP (f,A,φ)} is the generalized projection of C onto GMEP (f, A, φ).

Proof. Let x ∈ C and put p = Π_{GMEP (f,A,φ)}x. Since E is reflexive and {x_n} is bounded, there exists a subsequence of {x_n} such that and

()

Put

. Since

, we have

. On the other hand, since E is uniformly smooth, J is uniformly norm-to-norm continuous on bounded subsets of E. So we have

()

Since

()

By the definition of

, for any y ∈ C, we see that

()

By (A2), for each y ∈ C, we obtain

()

For any t ∈ (0,1) and y ∈ C, we define y_t = ty + (1 − t)v. Then y_t ∈ C. It follows by the monotonicity of A that

()

By (A4), (3.4), and the weakly lower semicontinuity of φ, letting k → ∞, we obtain

()

By (A1), (A4), and the convexity of φ, we have

()

It follows that

()

By (A3), the weakly lower semicontinuity of φ, and the continuity of A, letting t → 0, we obtain

()

This shows that v ∈ GMEP (f, A, φ). By Lemma 2.4, we have

()

This completes the proof.

Theorem 3.2. Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4), A : C → E^* a continuous and monotone mapping, and φ a lower semicontinuous and convex function from C to ℝ such that GMEP (f, A, φ) ≠ ∅. Define the sequence {x_n} as follows: x₁ = x ∈ C and

()

where {α_n}⊂(0,1) and {r_n}⊂(0, ∞) satisfy the following conditions:

(a)
lim _n→∞α_n = 0;
(b)
;
(c)
lim inf _n→∞r_n > 0.

Then {x_n} converges strongly to Π_{GMEP (f,A,φ)}x, where Π_{GMEP (f,A,φ)} is the generalized projection of C onto GMEP (f, A, φ).

Proof. From Lemma 2.9(4), we know that GMEP (f, A, φ) is closed and convex. Let p = Π_{GMEP (f,A,φ)}x. Put and z_n = J⁻¹(α_nJx + (1 − α_n)Jy_n) for all n ∈ ℕ. So, by Lemma 2.5, we have

()

By induction, we can show that ϕ(p, x_n) ≤ ϕ(p, x) for each n ∈ ℕ. Hence {ϕ(p, x_n)} is bounded and thus {x_n} is also bounded.

We next show that if there exists a subsequence of {x_n} such that

()

then

()

Since

()

Since J is uniformly norm-to-norm continuous on bounded subsets of E, so is J⁻¹. It follows that

()

Since E is uniformly smooth and uniformly convex, by Lemma 2.6, Π_C is uniformly norm-to-norm continuous on bounded sets. So we obtain

()

and hence

()

Furthermore,

. Indeed, by the definition of ϕ, 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. ϕ(p, x_n+1) ≤ ϕ(p, x_n) for all sufficiently large n. Hence the sequence {ϕ(p, x_n)} is bounded and nonincreasing. So lim _n→∞ϕ(p, x_n) exists. This shows that lim _n→∞(ϕ(p, x_n+1) − ϕ(p, x_n)) = 0 and hence

()

Since

is of firmly nonexpansive type, by Remark 2.10, we have

()

which implies

()

Hence

()

as n → ∞. By Lemma 2.2, we obtain

()

Proposition 3.1 yields that

()

It also follows that

()

Finally, we show that x_n → p. Using Lemma 2.7, we see that

()

Set a_n = ϕ(p, x_n) and b_n = α_n(〈Jx − Jp, z_n − y_n〉+〈Jx − Jp, y_n − p〉). We see that limsup _n→∞b_n/α_n ≤ 0. By Lemma 2.11, since

, we conclude that lim _n→∞ϕ(p, x_n) = 0. Hence x_n → p as n → ∞.

Case 2. There exists a subsequence of {ϕ(p, x_n)} such that for all j ∈ ℕ. By Lemma 2.12, there exists a strictly increasing sequence {m_k} of positive integers such that the following properties are satisfied by all numbers k ∈ ℕ:

()

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 C be nonempty closed and convex subset of a uniformly smooth and uniformly convex Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4) and φ a lower semicontinuous and convex function from C to ℝ such that MEP (f, φ) ≠ ∅. Define the sequence {x_n} as follows: x₁ = x ∈ C and

()

where {α_n}⊂(0,1) and {r_n}⊂(0, ∞) satisfy the following conditions:

(a)
lim _n→∞α_n = 0;
(b)
;
(c)
lim inf _n→∞r_n > 0.

Then {x_n} converges strongly to Π_MEP (f,φ)x, where Π_MEP (f,φ) is the generalized projection of C onto MEP (f, φ).

Corollary 3.4. Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4), and A : C → E^* a continuous and monotone mapping such that GEP (f, A) ≠ ∅. Define the sequence {x_n} as follows: x₁ = x ∈ C and

()

where {α_n}⊂(0,1) and {r_n}⊂(0, ∞) satisfy the following conditions:

(a)
lim _n→∞α_n = 0;
(b)
;
(c)
lim inf _n→∞r_n > 0.

Then {x_n} converges strongly to Π_GEP (f,A)x, where Π_GEP (f,A) is the generalized projection of C onto GEP (f, A).

Corollary 3.5. Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let A : C → E^* be a continuous and monotone mapping, and φ a lower semicontinuous and convex function from C to ℝ such that VI (C, A, φ) ≠ ∅. Define the sequence {x_n} as follows: x₁ = x ∈ C and

()

where {α_n}⊂(0,1) and {r_n}⊂(0, ∞) satisfy the following conditions:

(a)
lim _n→∞α_n = 0;
(b)
;
(c)
lim inf _n→∞r_n > 0.

Then {x_n} converges strongly to Π_{VI (C,A,φ)}x, where Π_{VI (C,A,φ)} is the generalized projection of C onto VI (C, A, φ).

4. Examples and Numerical Results

In this section, we give examples and numerical results for our main theorem.

Example 4.1. Let E = ℝ and C = [−1,1]. Let f(x, y) = −9x² + xy + 8y², φ(x) = 3x², and Ax = 2x. Find such that

()

Solution 4. It is easy to check that f, φ, and A satisfy all conditions in Theorem 3.2. For each r > 0 and x ∈ [−1,1], Lemma 2.9 ensures that there exists z ∈ [−1,1] such that, for any y ∈ [−1,1],

()

Put G(y) = 11ry² + (3rz + z − x)y − (14rz² + z² − xz). Then G is a quadratic function of y with coefficient a = 11r, b = (3rz + z − x), and c = −(14rz² + z² − xz). We next compute the discriminant Δ of G as follows:

()

We know that G(y) ≥ 0 for all y ∈ [−1,1] if it has at most one solution in [−1,1]. So Δ ≤ 0 and hence x = 25rz + z. Now we have z = T_rx = x/(25r + 1).

Let be the sequence generated by x₁ = x ∈ [−1,1] and

()

and, equivalently,

()

We next give two numerical results for algorithm (4.5).

Algorithm 4.2. Let α_n = 1/80n and r_n = n/(n + 1). Choose x₁ = x = 1. Then algorithm (4.5) becomes

()

Numerical Result I See Table 1.

n	x_n
1	1.0000
2	0.0856
3	0.0111
4	0.0047
5	0.0033
⋮	⋮
261	0.0001
262	0.0000

Algorithm 4.3. Let α_n = 1/100n and r_n = (n + 1)/2n. Choose x₁ = x = −1. Then algorithm (4.5) becomes

()

Numerical Result II See Table 2.

n	x_n
1	−1.0000
2	−0.0481
3	−0.0074
4	−0.0038
5	−0.0027
⋮	⋮
217	−0.0001
218	0.0000

5. Conclusion

Tables 1 and 2 show that the sequence {x_n} converges to 0 which solves the generalized mixed equilibrium problem. On the other hand, using Lemma 2.9(3), we can check that GMEP (f, A, φ) = F(T_r) = {0}.

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. 2011-0021821).

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Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

Abstract

1. Introduction

2. Preliminaries and Lemmas

3. Main Results

4. Examples and Numerical Results

5. Conclusion

Acknowledgments

References

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Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

Abstract

1. Introduction

2. Preliminaries and Lemmas

3. Main Results

4. Examples and Numerical Results

5. Conclusion

Acknowledgments

References

Citing Literature

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