Journal of Applied Mathematics
Volume 2012, Issue 1 154968
Research Article
Open Access

A New Iterative Scheme for Solving the Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

Rabian Wangkeeree, 

Corresponding Author

Rabian Wangkeeree

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand nu.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

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Pakkapon Preechasilp, 

Pakkapon Preechasilp

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand nu.ac.th

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First published: 29 May 2012
https://doi.org/10.1155/2012/154968
Citations: 2
Academic Editor: Zhenyu Huang

Abstract

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

1. Introduction

Let H be a real Hilbert space and C a nonempty closed convex subset of H with inner product 〈·, ·〉. Recall that a mapping S : C → C is called nonexpansive if ∥Sx − Sy∥ ≤ ∥x − y∥ for all x, y ∈ C. The set of all fixed points of S is denoted by F(S) = {x ∈ C : x = Sx}. A mapping g : C → C is a contraction on C if there is a constant α ∈ (0,1) such that ∥g(x) − g(y)∥ ≤ α∥x − y∥ for all x, y ∈ C. We use ΠC to denote the collection of all contractions on C. Note that each g ∈ ΠC has a fixed unique fixed point in C. A linear bounded operator A is strongly positive if there is a constant with property for all x ∈ H.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [1–4] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space:
()
where A is a linear bounded operator, C is the fixed point set of a nonexpansive mapping T, and b is a given point in H. Let H be a real Hilbert space. Recall that a linear bounded operator B is strongly positive if there is a constant with property
()
Recently, Marino and Xu [5] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [6]:
()
where A is a strongly positive bounded linear operator on H. They proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.3) converges strongly to the unique solution of the variational inequality:
()
which is the optimality condition for the minimization problem:
()
where h is a potential function for γg (i.e., h′(x) = γg(x) for x ∈ H).
A mapping B of C into H is called monotone if 〈Bx − By, x − y〉 ≥ 0 for all x, y ∈ C. The variational inequality problem is to find such that
()
The set of solutions of variational inequality is denoted by VI (C, B). A mapping B : C → H is called inverse-strongly monotone if there exists a positive real number β such that
()
For such a case, B is β-inverse-strongly monotone. If B is a β-inverse-strongly monotone mapping of C to H, then it is obvious that B is (1/β)-Lipschitz continuous. In 2009, Klin-eam and Suantai [7] introduced the following general iterative method:
()
where PC is the metric projection of H onto C, g is a contraction, A is a strongly positive linear bounded operator, B is a β-inverse strongly monotone mapping, {αn} ⊂ (0,1), and {λn} ⊂ [a, b] for some a, b with 0 < a < b < 2β. They proved that under certain appropriate conditions imposed on {αn} and {λn}, the sequence generated by (1.8) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping (say ) which solves the following variational inequality:
()
We recall the following well-known definitions. A bifunction f : C × C → ℝ is called
  • (a)

    monotone on C if

()
  • (b)

    pseudomonotone on C if

()
  • (c)

    Lipschitz-type continuous on C with two constants c1 > 0 and c2 > 0 if

()
We consider the following equilibrium problems: find such that
()
The set of solution of problem (1.13) is denoted by EP (f, C). If f(x, y)∶ = 〈Fx, y − x〉 for all x, y ∈ C, where F is a mapping from C to H, then problem EP (f, C) reduces to the variational inequalities (1.6). It is well known that problem EP (f, C) covers many important problems in optimization and nonlinear analysis as well as it has found many applications in economic, transportation, and engineering.
For solving the common element of the set of fixed points of a nonexpansive mapping and the solution set of equilibrium problems, S. Takahashi and W. Takahashi [8] introduced the following viscosity approximation method:
()
where {αn} ⊂ [0,1] and {rn} ⊂ (0, ∞). They showed that under certain conditions over {αn} and {rn}, sequences {xn} and {yn} converge strongly to z ∈ F(S)∩EP (F), where z = PF(S)∩EP (F)g(z).

In this paper, inspired and motivated by Klin-eam and Suantai [7] and S. Takahashi and W. Takahashi [8], we introduce the new algorithm for solving the common element of the set of fixed points of a nonexpansive mapping, the solution set of equilibrium problems, and the solution set of the variational inequality problems for an inverse strongly monotone mapping. Let f be monotone, Lipschitz-type continuous on C with two constants c1 > 0 and c2 > 0, A a strongly linear bounded operator, and B a β-inverse strongly monotone mapping. Let g : C → C be a contraction with coefficient α such that and S : C → C a nonexpansive mapping. The algorithm is now described as follows.

Step 1 (initialization). Choose positive sequences {αn} ⊂ (0,1) and {λn} ⊂ [c, d] for some c, d ∈ (0, 1/L), where L = max {2c1, 2c2} and for some a, b with 0 < a < b < 2β.

Step 2 (solving convex problems). For a given point x0 = x ∈ C and set n∶ = 0, we solve the following two strongly convex problems:

()

Step 3 (iteration n). Compute

()
where PC is the metric projection of H onto C. Increase n by 1 and go to Step 1.

We show that under some control conditions the sequences {xn}, {yn}, and {tn} defined by (1.15) and (1.16) converge strongly to a common element of solution set of monotone, Lipschitz-type continuous equilibrium problems, and the set of fixed points of nonexpansive mappings which is a unique solution of the variational inequality problem (1.6).

2. Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C × C → ℝ be a bifunction. For solving the mixed equilibrium problem, let us assume the following conditions for a bifunction f : C × C → ℝ:
  • (A1)

    f(x, x) = 0 for all x ∈ C;

  • (A2)

    f is Lipschitz-type continuous on C;

  • (A3)

    f is monotone on C;

  • (A4)

    for each x ∈ C, f(x, ·) is convex and subdifferentiable on C;

  • (A5)

    f is upper semicontinuous on C.

The metric (nearest point) projection PC from a Hilbert space H to a closed convex subset C of H is defined as follows: given x ∈ H, PCx is the only point in C such that ∥x − PCx∥ − inf {∥x − y∥ : y ∈ C}. In what follows lemma can be found in any standard functional analysis book.

Lemma 2.1. Let C be a closed convex subset of a real Hilbert space H. Given x ∈ H and y ∈ C, then

  • (i)

    y = PCx if and only if 〈x − y, y − z〉 ≥ 0 for all z ∈ C,

  • (ii)

    PC is nonexpansive,

  • (iii)

    for all x, y ∈ H,

  • (iv)

    〈x − PCx, PCx − y〉 for all x ∈ H and y ∈ C.

Using Lemma 2.1, one can show that the variational inequality (1.6) is equivalent to a fixed point problem.

Lemma 2.2. The point u ∈ C is a solution of the variational inequality (1.6) if and only if u satisfies the relation u = PC(u − λBu) for all λ > 0.

A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H, u ∈ Tx, and v ∈ Ty imply 〈x − y, u − v〉 ≥ 0. A monotone mapping T : H → 2H is maximal if the graph G(T) of T is not property contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, u) ∈ H × H,   〈x − y, u − v〉 ≥ 0 for every (y, v) ∈ G(T) implies u ∈ Tx. Let B be an inverse-strongly monotone mapping of C to H, let NCv be normal cone to C at v ∈ C, that is, NCv = {w ∈ H : 〈v − u, w〉 ≥ 0,   for  all  u ∈ C}, and define
()
Then T is a maximal monotone and 0 ∈ Tv if and only if v ∈ VI (C, B) [9].

Now we collect some useful lemmas for proving the convergence results of this paper.

Lemma 2.3 (see [10].)Let C be a nonempty closed convex subset of a real Hilbert space H and h : C → ℝ be convex and subdifferentiable on C. Then x* is a solution to the following convex problem:

()
if and only if 0 ∈ ∂h(x*) + NC(x*), where ∂h(·) denotes the subdifferential of h and NC(x*) is the (outward) normal cone of C at x* ∈ C.

Lemma 2.4 (see [11], Lemma  3.1.)Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C × C → ℝ be a pseudomonotone, Lipschitz-type continuous bifunction with constants c1 > 0 and c2 > 0. For each x ∈ C, let f(x, ·) be convex and subdifferentiable on C. Suppose that the sequences {xn}, {yn}, and {tn} are generated by Scheme (1.15) and p ∈ EP (f). Then

()

Lemma 2.5 (see [12].)Let C be a closed convex subset of a Hilbert space H and let S : C → C be a nonexpansive mapping such that F(S) ≠ ∅. If a sequence {xn} in C such that xn⇀z and xn − Sxn → 0, then z = Sz.

Lemma 2.6 (see [5].)Assume that A is a strongly positive linear bounded operator on a Hilbert space H with coefficient and 0 < ρ ≤ ∥A∥−1, then .

In the following, we also need the following lemma that can be found in the existing literature [3, 13].

Lemma 2.7 (see [3], Lemma  2.1.)Let {an} be a sequence of non-negative real number satisfying the following property:

()
where {γn}⊆(0,1) and {βn}⊆ℝ such that and limsup n→∞ βn ≤ 0. Then {an} converges to zero, as n → ∞.

3. Main Theorems

In this section, we prove the strong convergence theorem for solving a common element of solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed points of nonexpansive mappings.

Theorem 3.1. Let H be a real Hilbert space, and let C be a closed convex subset of H. Let f : C × C → ℝ be a bifunction satisfying (A1)–(A5), let B : C → H be a β-inverse strongly monotone mapping, let A be a strongly positive linear bounded operator of H into itself with coefficient such that ∥A∥ = 1 and let g : C → C be a contraction with coefficient α(α ∈ (0,1)). Assume that . Let S be a nonexpansive mapping of C into itself such that Ω∶ = F(S)∩EP (f)∩VI (C, B) ≠ ∅. Let the sequences {xn}, {yn}, and {tn} be generated by (1.15) and (1.16), where {αn} ⊂ (0,1), for some a, b ∈ (0,2β), and {λn} ⊂ [c, d] for some c, d ∈ (0, 1/L), where L = max {2c1, 2c2}. Suppose that the following conditions are satisfied:

  • (B1)

    lim n→∞ αn = 0;

  • (B2)

    ;

  • (B3)

    ;

  • (B4)

    .

Then the following holds.
  • (i)

    PΩ(I − A + γg) is a contraction on C; hence there exists q ∈ C such that q = PΩ(I − A + γg)(q), where PΩ is the metric projection of H onto C.

  • (ii)

    The sequences {xn}, {yn}, and {tn} converge strongly to the same point q.

Proof. For any x, y ∈ H, we have

()
Banach’s contraction principle guarantees that PΩ(γg + (I − A)) has a unique fixed point, say q ∈ H. That is, q = PΩ(γg + (I − A))(q). By Lemma 2.1(i), we obtain that
()

The proof of (ii) is divided into several steps.

Step 1. is nonexpansive mapping. Indeed, since B is a β-strongly monotone mapping and , for all x, y ∈ C, we have

()

Step 2. We show that {xn} is a bounded sequence. Put for all n ≥ 0. Let p ∈ Ω; we have

()
By Lemma 2.4, we have
()
By induction, we get that
()
Hence {xn} is bounded, and then {wn},   {yn}, and {tn} are also bounded.

Step 3. We show that

()
Since f(x, ·) is convex on C for each x ∈ C, applying Lemma 2.3, we see that tn =   argmin{(1/2)∥t − xn∥+λnf(yn, t) : t ∈ C} if and only if
()
where NC(x) is the (outward) normal cone of C at x ∈ C. This implies that , where w ∈ ∂2f(yn, tn) and . By the definition of the normal cone NC, we have
()
and so
()
Substituting t = tn+1 ∈ C into (3.10), we get that
()
Since f(x, ·) is subdifferentiable on C and w ∈ ∂2f(yn, tn), we have
()
From (3.11) and (3.12), we obtain that
()
By the similar way, we also have
()
It follows from (3.13) and (3.14) and f is Lipschitz-type continuous and monotone, we get
()
and hence
()
Thus, we have
()
where and M ≥ sup n≥0γ∥g(xn−1)∥ + ∥AS(tn−1)∥. Using (B3), (B4), and Lemma 2.7, we have lim n→∞∥xn+1 − xn∥ = 0.

Step 4. We show that

()
Indeed, for each p ∈ Ω, applying Lemma 2.4, we have
()
where
()
It then follows that
()
as n → ∞. Hence
()
By the similar way, also
()
From (3.22) and (3.23), we can conclude that
()

Step 5. We show that

()
From (1.15), we get that
()
and hence
()
Since αn → 0, we get that ∥Btn − Bp∥ → 0, as n → ∞. By Lemma 2.1(iii), we have
()
which implies that
()
Again from (3.29), we have
()
This implies that
()
Since αn → 0 and ∥Btn − Bp∥ → 0, we obtain that
()
From (1.16), we have
()
Since
()
from (3.7), (3.24), (3.32), and (3.33), we obtain that ∥tn − Stn∥ → 0, as n → ∞. Moreover, we get that
()
Since
()
it implies that
()
Since
()
we obtain that
()

Step 6. We show that

()
Indeed, we choose a subsequence of {wn} such that
()
Since {wn} is bounded, there exists a subsequence of which converges weakly to p. Next we show that p ∈ Ω.

We prove that p ∈ F(S). We may assume without loss of generality that . Since ∥wn − Swn∥ → 0, we obtain . Since ∥xn − Sxn∥ → 0,   ∥xn − wn∥ → 0 and by Lemma 2.5, we have p ∈ F(S).

We show that p ∈ EP (f). From Steps 4 and 6, we have that

()
Since yn is the unique solution of the convex minimization problem
()
we have
()
It follows that
()
where z ∈ ∂2f(xn, yn) and zn ∈ NC(yn). By the definition of the normal cone NC, we get that
()
On the other hand, since f(xn, ·) is subdifferentiable on C and z ∈ ∂2f(xn, yn), we have
()
Combining (3.47) with (3.46), we have
()
Hence
()
Thus, using {λ ⊂ [a, b] ⊂ (0, 1/L)} and the upper semicontinuity of f, we have
()
Hence p ∈ EP (f).

We show that p ∈ VI (C, B). Let

()
where NCv is normal cone to C at v ∈ C. Then T is a maximal monotone operator. Let (v, u) ∈ G(T). Since u − Bv ∈ NCv and wn ∈ C, we have 〈v − yn, u − Bv〉 ≥ 0. On the other hand, by Lemma 2.1(iv) and from , we have
()
and hence . Therefore, we get that
()
This implies that 〈v − p, u〉 ≥ 0 as k → ∞. Since T is maximal monotone, we have p ∈ T−10 and hence p ∈ VI (C, B).

From (a), (b), and (c), we obtain that p ∈ Ω. This implies that

()

Step 7. We show that xn → q. We observe that

()
Since {xn}, {g(xn)}, and ∥Swn∥ are bounded, we can take a constant M > 0 such that
()
This implies that
()
where σn = 2〈Swn − q, γg(q) − Aq〉 + Mαn. From (3.40), we have limsup n→∞σn ≤ 0. Applying Lemma 2.7 to (3.57), we obtain that xn → q as n → ∞. This completes the proof.

If we put γ = 1 and A = I in Theorem 3.1, we immediately obtain the following corollary.

Corollary 3.2. Let H be a real Hilbert space, and let C be a closed convex subset of H. Let f : C × C → ℝ be a bifunction satisfying (A1)–(A5), let B : C → H be a β-inverse strongly monotone mapping, and let g : C → C be a contraction with coefficient α  (α ∈ (0,1)). Assume that . Let S be a nonexpansive mapping of C into itself such that Ω : = F(S)∩EP (f)∩VI (C, B) ≠ ∅. Let the sequences {xn}, {yn}, and {tn} be generated by

()
where {αn} ⊂ (0,1), for some a, b ∈ (0,2β), and {λn} ⊂ [c, d] for some c, d ∈ (0, 1/L), where L = max {2c1, 2c2}. Suppose that the following conditions are satisfied:
  • (B1)

    lim n→∞αn = 0;

  • (B2)

    ;

  • (B3)

    ;

  • (B4)

    .

Then the following holds.
  • (i)

    PΩg is a contraction on C; and hence there exists q ∈ C such that q = PΩg(q), where PΩ is the metric projection of H onto C.

  • (ii)

    The sequences {xn}, {yn}, and {tn} converge strongly to the same point q which is the unique solution in the Ω to the following variational inequality:

()

If we put g ≡ u in the previous corollary, we get the following corollary.

Corollary 3.3. Let H be a real Hilbert space, and let C be a closed convex subset of H. Let f : C × C → ℝ be a bifunction satisfying (A1)–(A5), and let B : C → H be a β-inverse strongly monotone mapping. Assume that . Let S be a nonexpansive mapping of C into itself such that Ω : = F(S)∩EP (f)∩VI (C, B) ≠ ∅. Let the sequences {xn}, {yn}, and {tn} be generated by

()
where {αn} ⊂ (0,1), for some a, b ∈ (0,2β), and {λn} ⊂ [c, d] for some c, d ∈ (0, 1/L), where L = max {2c1, 2c2}. Suppose that the following conditions are satisfied:
  • (B1)

    lim n→∞ αn = 0;

  • (B2)

    ;

  • (B3)

    ;

  • (B4)

    .

Then the sequences {xn}, {yn}, and {tn} converge strongly to the same point q, where q = PΩu, which is the unique solution in the Ω to the following variational inequality:
()

4. Deduced Theorems

Let C be a nonempty closed convex subset of a real Hilbert space H with inner product 〈·, ·〉. Let F be a nonlinear mapping from C into H. Recall that the function F is called
  • (a)

    strongly monotone on C if there exists β > 0 such that

()
  • (b)

    monotone on C if

()
  • (c)

    pseudomonotone on C if

()

Remark 4.1. Notice that if F is L-Lipschitz on C, then for each x, y ∈ C,   f(x, y) = 〈F(x), y − x〉 is Lipschitz-type continuous with constants c1 = c2 = L/2 on C. Indeed,

()
Thus f is Lipschitz-type continuous on C.

Let f : C × C → ℝ be defined by f(x, y) = 〈F(x), y − x〉, where F : C → H. Thus, by Algorithm  (1.15), we get the following:

()
Similarly, we also obtain that tn = PC(xn − λnF(yn)). Applying Theorem 3.1, we obtain the convergence theorem for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set VI (C, B).

Corollary 4.2. Let H be a real Hilbert space, and let C be a closed convex subset of H. Let F : C → H be a monotone, L-Lipschitz continuous mapping, let B : C → H be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient such that ∥A∥ = 1, and let g : C → C be a contraction with coefficient α  (α ∈ (0,1)). Assume that . Let S be a nonexpansive mapping of C into itself such that Ω = F(S)∩EP (f)∩VI (C, B) ≠ ∅. Let the sequence {xn}, {yn}, and {tn} be generated by

()
where {αn} ⊂ (0,1), for some a, b ∈ (0,2β), and {λn} ⊂ [c, d] for some c, d ∈ (0, 1/L). Suppose that the following conditions are satisfied:
  • (B1)

    lim n→∞ αn = 0;

  • (B2)

    ;

  • (B3)

    ;

  • (B4)

    .

Then the sequences {xn}, {yn}, and {tn} converge strongly to the same point q, where q = PΩ(I − A + γg)(q).

Coflict of Interests

The authors declare that they have no conflict interests.

    Authors’ Contribution

    All authors read and approved the final paper.

      Acknowledgments

      The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. Finally, the first author is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.

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