Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions

Open Access
Research

Abstract

In this article, we introduce a new mapping generated by an infinite family of κ i - strict pseudo-contractions and a sequence of positive real numbers. By using this mapping, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Strong convergence theorem of the purposed iteration is established in the framework of Hilbert spaces.

Keywords

nonexpansive mappings strongly positive operator generalized equilibrium problem strict pseudo-contraction fixed point 

1 Introduction

Let C be a closed convex subset of a real Hilbert space H, and let G : C × C → ℝ be a bifunction. We know that the equilibrium problem for a bifunction G is to find xC such that
The set of solutions of (1.1) is denoted by EP(G). Given a mapping T : CH, let G(x, y) = 〈Tx, y - x〉 for all x, y ∈. Then, zEP(G) if and only if 〈Tz, y - z〉 ≥ 0 for all yC, i.e., z is a solution of the variational inequality. Let A : CH be a nonlinear mapping. The variational inequality problem is to find a uC such that
for all vC. The set of solutions of the variational inequality is denoted by V I(C, A). Now, we consider the following generalized equilibrium problem:
The set of such zC is denoted by EP(G, A), i.e.,

In the case of A ≡ 0, EP(G, A) is denoted by EP(G). In the case of G ≡ 0, EP(G, A) is also denoted by V I(C, A). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics reduce to find a solution of (1.3) (see, for instance, [1]-[3]).

A mapping A of C into H is called inverse-strongly monotone (see [4]), if there exists a positive real number α such that

for all x, yC.

A mapping T with domain D(T) and range R(T) is called nonexpansive if
for all x, yD(T) and T is said to be κ-strict pseudo-contration if there exist κ ∈ [0, 1) such that
We know that the class of κ-strict pseudo-contractions includes class of nonexpansive mappings. If κ = 1, then T is said to be pseudo-contractive. T is strong pseudo-contraction if there exists a positive constant λ ∈ (0, 1) such that T + λI is pseudo-contractive. In a real Hilbert space H (1.5) is equivalent to
T is pseudo-contractive if and only if
Then, T is strongly pseudo-contractive, if there exists a positive constant λ ∈ (0, 1) such that

The class of κ-strict pseudo-contractions fall into the one between classes of nonexpansive mappings and pseudo-contractions, and the class of strong pseudo-contractions is independent of the class of κ-strict pseudo-contractions.

We denote by F(T) the set of fixed points of T. If CH is bounded, closed and convex and T is a nonexpansive mapping of C into itself, then F(T) is nonempty; for instance, see [5]. Recently, Tada and Takahashi [6] and Takahashi and Takahashi [7] considered iterative methods for finding an element of EP(G) ∩ F(T). Browder and Petryshyn [8] showed that if a κ-strict pseudo-contraction T has a fixed point in C, then starting with an initial x0C, the sequence {x n } generated by the recursive formula:
where α is a constant such that 0 < α < 1, converges weakly to a fixed point of T. Marino and Xu [9] extended Browder and Petryshyn's above mentioned result by proving that the sequence {x n } generated by the following Manns algorithm [10]:

converges weakly to a fixed point of T provided the control sequence Open image in new window satisfies the conditions that κ < α n < 1 for all n and Open image in new window .

Recently, in 2009, Qin et al. [11] introduced a general iterative method for finding a common element of EP(F, T), F(S), and F(D). They defined {x n } as follows:

where the mapping D : CC is defined by D(x) = P C (P C (x - ηBx) - λAP C (x - ηBx)), S k is the mapping defined by S k x = kx + (1 - k)Sx, ∀xC, S : CC is a κ-strict pseudo-contraction, and A, B : CH are a-inverse-strongly monotone mapping and b-inverse-strongly monotone mappings, respectively. Under suitable conditions, they proved strong convergence of {x n } defined by (1.9) to z = PEP(F, T)∩F(S) ∩F(D)u.

Let C be a nonempty convex subset of a real Hilbert space. Let T i , i = 1, 2, ... be mappings of C into itself. For each j = 1, 2, ..., let Open image in new window where I = [0, 1] and Open image in new window . For every n ∈ ℕ, we define the mapping S n : CC as follows:

This mapping is called S-mapping generated by T n , ..., T1 and α n , αn-1, ..., α1.

Question. How can we define an iterative method for finding an element in Open image in new window ?

In this article, motivated by Qin et al. [11], by using S-mapping, we introduce a new iteration method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Our iteration scheme is define as follows.

For u, x1C, let {x n } be generated by

For i = 1, 2, ..., N, let F i : C × C → ℝ be bifunction, A i : CH be α i -inverse strongly monotone and let G i : CC be defined by G i (y) = P C (I - λ i A i )y, ∀yC with (0, 1] ⊂ (0, 2 α i ) such that Open image in new window , where B is the K-mapping generated by G1, G2, ..., G N and β1, β2, ..., β N .

We prove a strong convergence theorem of purposed iterative sequence {x n } to a point Open image in new window and z is a solution of (1.10)

2 Preliminaries

In this section, we collect and provide some useful lemmas that will be used for our main result in the next section.

Let C be a closed convex subset of a real Hilbert space H, and let P C be the metric projection of H onto C i.e., so that for xH, P C x satisfies the property:

The following characterizes the projection P C .

Lemma 2.1 [5]. Given xH and yC. Then, P C x = y if and only if there holds the inequality
Lemma 2.2 [12]. Let {s n } be a sequence of nonnegative real number satisfying
where {α n }, {β n } satisfy the conditions

Then limn→∞s n = 0.

Lemma 2.3 [13]. Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n ∈ ℕ} be a sequence of nonexpansive mappings on C. Suppose Open image in new window is nonempty. Let {λ n } be a sequence of positive numbers with Open image in new window . Then, a mapping S on C defined by

for xC is well defined, nonexpansive and Open image in new window hold.

Lemma 2.4 [14]. Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : CC be a nonexpansive mapping. Then, I - S is demi-closed at zero.

Lemma 2.5 [15]. Let {x n } and {z n } be bounded sequences in a Banach space X and let {β n } be a sequence in 0[1] with 0 < lim infn→∞β n ≤ lim supn→∞β n < 1.

for all integer n ≥ 0 and

Then limn→∞||x n - z n || = 0.

For solving the equilibrium problem for a bifunction F : C × C → ℝ, let us assume that F satisfies the following conditions:

(A 1) F(x, x) = 0 ∀xC;

(A 2) F is monotone, i.e. F(x, y) + F(y, x) ≤ 0, ∀x, yC;

(A 3) ∀x, y, zC,

(A 4) ∀xC, yF(x, y) is convex and lower semicontinuous.

The following lemma appears implicitly in [1].

Lemma 2.6 [1]. Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C intosatisfying (A 1) - (A 4). Let r > 0 and xH. Then, there exists zC such that

for all xC.

Lemma 2.7 [16]. Assume that F : C × C → ℝ satisfies (A 1) - (A 4). For r > 0 and xH, define a mapping T r : HC as follows.

for all zH. Then, the following hold.

(1) T r is single-valued,

(2) T r is firmly nonexpansive i.e

(3) F(T r ) = EP (F );

(4) EP(F) is closed and convex.

Definition 2.1 [17]. Let C be a nonempty convex subset of real Banach space. Let Open image in new window be a finite family of nonexpanxive mappings of C into itself, and let λ1, ..., λ N be real numbers such that 0 ≤ λ i ≤ 1 for every i = 1, ..., N . We define a mapping K : CC as follows.

Such a mapping K is called the K-mapping generated by T1, ..., T N and λ1, ..., λ N .

Lemma 2.8 [17]. Let C be a nonempty closed convex subset of a strictly convex Banach space. Let Open image in new window be a finite family of nonexpanxive mappings of C into itself with Open image in new window and let λ1, ..., λ N be real numbers such that 0 < λ i < 1 for every i = 1, ..., N - 1 and 0 < λ N ≤ 1. Let K be the K-mapping generated by T1, ..., T N and λ1, ..., λ N . Then Open image in new window .

Lemma 2.9 [9]. Let C be a nonempty closed convex subset of a real Hilbert space H and S : CC be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition.

Lemma 2.10. Let C be a nonempty closed convex subset of a real Hilbert space. Let Open image in new window be κ i -strict pseudo-contraction mappings of C into itself with Open image in new window and κ = sup i κ i and let Open image in new window , where I = [0, 1], Open image in new window , Open image in new window , and Open image in new window for all j = 1, 2, .... For every n ∈ ℕ, let S n be S-mapping generated by T n , ..., T1 and α n , αn-1, ..., α1. Then, for every xC and k ∈ ℕ, limn→∞U n , k x exists.

Proof. Let xC and Open image in new window . Fix k ∈ ℕ, then for every n ∈ ℕ with nk,

where Open image in new window and Open image in new window

For any k, n, p ∈ ℕ, p > 0, nk, we have

Since a ∈ (0, 1), we have limn→∞a n = 0. From (2.5), we have that {U n , k x} is a Cauchy sequence. Hence lim n→∞Un,kx exists. □

For every k ∈ ℕ and xC, we define mapping U∞,kand S : CC as follows:

Such a mapping S is called S-mapping generated by T n , Tn-1, ... and α n , αn- 1, ...

Remark 2.11. For each n ∈ ℕ, S n is nonexpansive and limn→∞supxD||S n x - Sx|| = 0 for every bounded subset D of C. To show this, let x, yC and D be a bounded subset of C. Then, we have
Then, we have that S : CC is also nonexpansive indeed, observe that for each x, yC
By (2.8), we have
This implies that for m > n and xD,
By letting m → ∞, for any xD, we have
It follows that

Lemma 2.12. Let C be a nonempty closed convex subset of a real Hilbert space. Let Open image in new window be κ i -strict pseudo-contraction mappings of C into itself with Open image in new window and κ = supiκ i and let Open image in new window , where I = [0, 1], Open image in new window , Open image in new window and Open image in new window for all j = 1, .... For every n ∈ ℕ, let S n and S be S-mappings generated by T n , ..., T1 and α n , αn-1, ..., α1 and T n , Tn-1, ..., and α n , αn-1, ..., respectively. Then Open image in new window .

Proof. It is evident that Open image in new window . For every n, k ∈ ℕ, with nk, let x0F (S) and Open image in new window , we have
For k ∈ ℕ and (2.12), we have

as n → ∞. This implies that U, k x0 = x0, ∀k ∈ ℕ.

Again by (2.12), we have
as n → ∞. Hence

From U∞,kx0 = x0, ∀k ∈ ℕ, and (2.15), we obtain that T k x0 = x0, ∀k ∈ ℕ. This implies that Open image in new window . □

Lemma 2.13. Let C be a closed convex subset of Hilbert space H. Let A i : CH be mappings and let G i : CC be defined by G i (y) = P C (I - λ i A i )y with λ i > 0, ∀ i = 1, 2, ... N. Then Open image in new window if and only if Open image in new window .

Proof. For given Open image in new window , we have x* ∈ VI(C, A i ), ∀ i = 1, 2, ..., N. Since 〈A i x*, x - x*〉 ≥ 0, we have 〈λ i A i x*, x - x*〉 ≥ 0, ∀λ i > 0, i = 1, 2, ..., N. It follows that
Hence, x* = P C (I - λ i A i )x* = G i (x*), ∀xC, i = 1, 2, ..., N. Therefore, we have Open image in new window . For the converse, let Open image in new window ; then, we have for every i = 1, ..., N, x* = G i (x*) = P C (I - λ i A i )x*, ∀λ i > 0, i = 1, 2, ..., N. It implies that

Hence, 〈A i x*, x - x*〉 ≥ 0, ∀xC, so x* ∈ VI(C, A i ), ∀i = 1, 2, ..., N. Hence, Open image in new window .

3 Main results

Theorem 3.1. Let C be a closed convex subset of Hilbert space H. For every i = 1, 2, ..., N, let F i : C × C → ℝ be a bifunction satisfying (A1) - (A4), let A i : CH be α i -inverse strongly monotone and let G i : CC be defined by G i (y) = P C (I - λ i A i )y, ∀yC with λ i ∈ (0, 1] ⊂ (0, 2α i ). Let B : CC be the K-mapping generated by G1, G2, ..., G N and β1, β2, ..., β N where β i ∈ (0, 1), ∀i = 1, 2, 3, ..., N - 1, β N ∈ (0, 1] and let Open image in new window be κ i -strict pseudo-contraction mappings of C into itself with κ = sup i κ i and let , where I = [0, 1], Open image in new window , Open image in new window , and Open image in new window for all j = 1, 2, ... . For every n ∈ ℕ, let S n and S are S-mapping generated by T n , ..., T1 and ρ n , ρn - 1, ..., ρ1 and T n , Tn- 1, ..., and ρ n , ρn - 1, ..., respectively. Assume that Open image in new window . For every n ∈ ℕ, i = 1, 2, ..., N, let {x n } and Open image in new window be generated by x1, uC and

where {α n }, {β n }, {γ n }, {a n }, {b n }, {c n } ⊂ (0, 1), Open image in new window , and Open image in new window , satisfy the following conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window , Open image in new window , Open image in new window , with a, b, c ∈ (0, 1).

Then, the sequence {x n }, {y n }, Open image in new window , ∀i = 1, 2, ..., N, converge strongly to Open image in new window and z is a solution of (1.10).

Proof. First, we show that (I - λ i A i ) is nonexpansive mapping for every i = 1, 2, ..., N. For x, yC, we have

Thus, (I - λ i A i ) is nonexpansive, and so are B and G i , for all i = 1, 2, ..., N.

Now, we shall divide our proof into five steps.

Step 1. We shall show that the sequence {x n } is bounded. Since

By Lemma 2.7, we have Open image in new window .

Let Open image in new window . Then F(z, y) + 〈y - z, A i z〉 ≥ 0 ∀yC, so we have
Again by Lemma 2.7, we have Open image in new window , ∀i = 1, 2, ..., N. Since B is K-mapping generated by G1, G2, ..., G N and β1, β2, ..., β N and Open image in new window . By Lemma 2.8, we have Open image in new window . Since Open image in new window , we have zF(B). Setting e n = a n S n x n + b n Bx n + c n y n , ∀n ∈ ℕ, we have

By induction, we can prove that {x n } is bounded, and so are Open image in new window , {y n }, {Bx n } {S n x n }, {e n }.

Step 2. We will show that limn→∞||xn+1- x n || = 0. Let Open image in new window , and then we have
From definition of d n , we have
By definition of e n , we have
By (3.6) and (3.7), we have
From Remark 2.11 and conditions (i)-(iii), we have
From (3.5), (3.12) and Lemma 2.5, we have
We can rewrite (3.5) as
By (3.13) and (3.14), we have
Step. 3. Show that limn→∞||x n - e n || = 0. From (3.1), we have
It implies that
By conditions (i), (ii), and (3.15), we have
Step. 4. We show that lim supn→∞u - z, x n - z〉 ≤ 0, where Open image in new window . Let Open image in new window be a subsequence of {x n } such that
Without loss of generality, we may assume that Open image in new window converges weakly to some q in H. Next, we will show that
First, we define a mapping A : CC by

Since Open image in new window , we have Open image in new window . By Lemma 2.3, we have Open image in new window .

Next, we define Q : CC by
Again, by Lemma 2.3, we have
By (3.19), we have
By condition (iii), (3.20), and (2.11), we have
by (3.16) and (3.21), we have
From, (3.22), we have
By Lemma 2.4, we obtain that
From (3.17)

Step. 5. Finally, we show that limn→∞x n = z, where Open image in new window .

By nonexpansiveness of S n and B, we can show that ||e n - z|| ≤ ||x n - z||. Then,
It follows that

From Step 4, (3.26), and Lemma 2.2, we have limn→∞ x n = z, where Open image in new window . The proof is complete. □

4 Applications

From Theorem 3.1, we obtain the following strong convergence theorems in a real

Hilbert space:

Theorem 4.1. Let C be a closed convex subset of Hilbert space H. For every i = 1, 2, ..., N, let F i : C × C → ℝ be a bifunction satisfying (A1) - (A4) and let Open image in new window be κ i -strict pseudo-contraction mappings of C into itself with κ = sup i κ i and let Open image in new window , where I = [0, 1], Open image in new window , Open image in new window , and Open image in new window for all j = 2, ... .. For every n ∈ ℕ, let S n and S are S-mappings generated by T n , ..., T1 and ρ n , ρn - 1, ..., ρ1 and T n , Tn- 1, ..., and ρ n , ρn- 1, ..., respectively. Assume that Open image in new window . For every n∈ ℕ, i = 1, 2, ..., N, let {x n } and Open image in new window be generated by x1, uC and

where {α n }, {β n }, {γ n }, {a n }, {b n }, {c n } ⊂ (0, 1), Open image in new window , and Open image in new window , satisfy the following conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window , Open image in new window , Open image in new window , with a, b, c ∈ (0, 1),

Then, the sequence {x n }, {y n }, Open image in new window , ∀i = 1, 2, ..., N, converge strongly to Open image in new window , and z is solution of (1.10)

Proof. From Theorem 3.1, let A i ≡ 0; then we have G i (y) = P Cy = yyC. Then, we get Bx n = x n n ∈ ℕ. Then, from Theorem 3.1, we obtain the desired conclusion. □

Next theorem is derived from Theorem 3.1, and we modify the result of [11] as follows:

Theorem 4.2. Let C be a closed convex subset of Hilbert space H and let F : C × C → ℝ be a bifunction satisfying (A1)-(A4), let A : CH be α-inverse strongly monotone mapping, and let T be κ-strict pseudo-contraction mappings of C into itself. Define a mapping T κ by T κ x = κx + (1 - κ)Tx, ∀xC. Assume that Open image in new window . For every n ∈ ℕ, let {x n } and {v n } be generated by x1, uC and

where {α n }, {β n }, {γ n }, {a, b, c} ⊂ (0, 1), α n + β n + γ n = a + b + c = 1, and {r, λ} ⊂ (ς, τ) ⊂ (0, 2α) satisfy the following conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

Then, the sequence {x n } and {v n } converge strongly to Open image in new window .

Proof. From Theorem 3.1, choose N = 1 and let A1 = A, λ1 = λ. Then, we have B(y) = G1(y) = P C (I - λA)y, ∀yC. Choose Open image in new window , a = a n , b = b n , c = c n for all n ∈ ℕ, and let T κ S1 : CC be S-mapping generated by T1 and ρ1 with T1 = T and Open image in new window , and then we obtain the desired result from Theorem 3.1 □

Notes

Acknowledgements

The authors would like to thank Professor Dr. Suthep Suantai for his valuable suggestion in the preparation and improvement of this article.

References

  1. 1.
    Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math Stud 1994, 63: 123–145.MathSciNetGoogle Scholar
  2. 2.
    Moudafi A, Thera M: Proximal and Dynamical Approaches to Equilibrium Problems. In Lecture Notes in Economics and Mathematical Systems. Volume 477. Springer; 1999:187–201.Google Scholar
  3. 3.
    Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042MathSciNetCrossRefGoogle Scholar
  4. 4.
    Iiduka H, Takahashi W: Weak convergence theorem by Ces'aro means for nonexpansive mappings and inverse-strongly monotone mappings. J Nonlinear Convex Anal 2006, 7: 105–113.MathSciNetGoogle Scholar
  5. 5.
    Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yoko-hama; 2000.Google Scholar
  6. 6.
    Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. J Optim Theory Appl 2007, 133: 359–370. 10.1007/s10957-007-9187-zMathSciNetCrossRefGoogle Scholar
  7. 7.
    Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetCrossRefGoogle Scholar
  8. 8.
    Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J Math Anal Appl 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetCrossRefGoogle Scholar
  9. 9.
    Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mann WR: Mean value methods in iteration. Proc Am Math Soc 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3CrossRefGoogle Scholar
  11. 11.
    Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis: Real World Applications 2010, 11:, 2963–2972.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Xu HK: Iterative algorithms for nonlinear operators. J Lond Math Soc 2002, 66: 240–256. 10.1112/S0024610702003332CrossRefGoogle Scholar
  13. 13.
    Bruck RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans Am Math Soc 1973, 179: 251–262.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc Sympos Pure Math 1976, 18: 78–81.Google Scholar
  15. 15.
    Suzuki T: Strong convergence of Krasnoselskii and Manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MathSciNetCrossRefGoogle Scholar
  16. 16.
    Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 2005, 6: 117–136.MathSciNetGoogle Scholar
  17. 17.
    Kangtunyakarn A, Suantai S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal 2009, 71: 4448–4460. 10.1016/j.na.2009.03.003MathSciNetCrossRefGoogle Scholar

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© Kangtunyakarn; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut's Institute of Technology LadkrabangBangkokThailand

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