Abstract

In the setting of Hilbert spaces, we introduce a relaxed hybrid steepest descent method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of a variational inequality for an inverse strongly monotone mapping and the set of solutions of generalized mixed equilibrium problems. We prove the strong convergence of the method to the unique solution of a suitable variational inequality. The results obtained in this article improve and extend the corresponding results.

AMS (2000) Subject Classification: 46C05; 47H09; 47H10.

Keywords:

relaxed hybrid steepest descent method; inverse strongly monotone mappings; nonexpansive mappings; generalized mixed equilibrium problem

1. Introduction

Let H be a real Hilbert space, C be a nonempty closed convex subset of H and let P_C be the metric projection of H onto the closed convex subset C. Let S : C → C be a nonexpansive mapping, that is, ||Sx - Sy|| ≤ ||x - y|| for all x, y ∈ C. We denote by F(S) the set fixed point of S. If C ⊂ H is nonempty, bounded, closed and convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty; see, for example, [1,2]. A mapping f : C → C is a contraction on C if there exists a constant η ∈ (0, 1) such that ||f(x) - f(y)|| ≤ η||x - y|| for all x, y ∈ C. In addition, let D : C → H be a nonlinear mapping, φ : C → ℝ ∪ {+∞} be a real-valued function and let F : C × C → ℝ be a bifunction such that C ∩ dom φ ≠ ∅, where ℝ is the set of real numbers and dom φ = {x ∈ C : φ(x) < +∞}.

The generalized mixed equilibrium problem for finding x ∈ C such that

The set of solutions of (1.1) is denoted by GMEP(F, φ, D), that is,

We find that if x is a solution of a problem (1.1), then x ∈ dom φ.

If D = 0, then the problem (1.1) is reduced into the mixed equilibrium problem which is denoted by MEP(F, φ).

If φ = 0, then the problem (1.1) is reduced into the generalized equilibrium problem which is denoted by GEP(F, D).

If D = 0 and φ = 0, then the problem (1.1) is reduced into the equilibrium problem which is denoted by EP(F).

If F = 0 and φ = 0, then the problem (1.1) is reduced into the variational inequality problem which is denoted by VI(C, D).

The generalized mixed equilibrium problems include, as special cases, some optimization problems, fixed point problems, variational inequality problems, Nash equilibrium problems in noncooperative games, equilibrium problem, Numerous problems in physics, economics and others. Some methods have been proposed to solve the problem (1.1); see, for instance, [3,4] and the references therein.

Definition 1.1. Let B : C → H be nonlinear mappings. Then, B is called

(1) monotone if 〈Bx - By, x - y〉 ≥ 0, ∀x, y ∈ C,

(2) β-inverse-strongly monotone if there exists a constant β > 0 such that

(3) A set-valued mapping Q : H → 2^H is called monotone if for all x, y ∈ H, f ∈ Qx and g ∈ Qy imply 〈x- y, f - g〉 ≥ 0. A monotone mapping Q : H → 2^H is called maximal if the graph G(Q) of Q is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping Q is maximal if and only if for (x, f) Î H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) Î G(Q) implies f Î Qx.

A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping defined on a real Hilbert space H:

where F is the fixed point set of a nonexpansive mapping S defined on H and b is a given point in H.

A linear-bounded operator A is strongly positive if there exists a constant with the property

Recently, Marino and Xu [5] introduced a new iterative scheme by the viscosity approximation method:

They proved that the sequences {x_n} generated by (1.2) 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 γf.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequalities for a ξ-inverse-strongly monotone mapping, Takahashi and Toyoda [6] introduced the following iterative scheme:

where B is a ξ-inverse-strongly monotone mapping, {γ_n} is a sequence in (0, 1), and {α_n} is a sequence in (0, 2ξ). They showed that if F(S) ∩ VI(C, B) is nonempty, then the sequence {x_n} generated by (1.3) converges weakly to some z ∈ F(S) ∩ VI(C, B).

The method of the steepest descent, also known as The Gradient Descent, is the simplest of the gradient methods. By means of simple optimization algorithm, this popular method can find the local minimum of a function. It is a method that is widely popular among mathematicians and physicists due to its easy concept.

For finding a common element of F(S) ∩ VI(C, B), let S : H → H be nonexpansive mappings, Yamada [7] introduced the following iterative scheme called the hybrid steepest descent method:

where x₁ = x ∈ H, {α_n} ⊂ (0, 1), B : H → H is a strongly monotone and Lipschitz continuous mapping and μ is a positive real number. He proved that the sequence {x_n} generated by (1.4) converged strongly to the unique solution of the F(S) ∩ VI(C, B).

On the other hand, for finding an element of F(S) ∩ VI(C, B) ∩ EP(F), Su et al. [8] introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces: x₁ ∈ H

where α_n ⊂ [0, 1) and r_n ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they prove {x_n} and {u_n} converge strongly to the same point z ∈ F(S) ∩ VI(C, B) ∩ EP(F), where z = P_{F(S)∩VI(C,B) ∩ EP(F)}f(z).

For finding a common element of F(S) ∩ GEP(F, D), let C be a nonempty closed convex subset of a real Hilbert space H. Let D be a β-inverse-strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself, Takahashi and Takahashi [9] introduced the following iterative scheme:

where {α_n} ⊂ [0, 1], {γ_n} ⊂ [0, 1] and {r_n} ⊂ [0, 2β] satisfy some parameters controlling conditions. They proved that the sequence {x_n} defined by (1.6) converges strongly to a common element of F(S) ∩ GEP(F, D).

Recently, Chantarangsi et al. [10] introduced a new iterative algorithm using a viscosity hybrid steepest descent method for solving a common solution of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem in a real Hilbert space. Jaiboon [11] suggests and analyzes an iterative scheme based on the hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problems for inverse strongly monotone mappings in Hilbert spaces.

In this article, motivated and inspired by the studies mentioned above, we introduce an iterative scheme using a relaxed hybrid steepest descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problems for inverse strongly monotone mapping in a real Hilbert space. Our results improve and extend the corresponding results of Jung [12] and some others.

2. Preliminaries

Throughout this article, we always assume H to be a real Hilbert space, and let C be a nonempty closed convex subset of H. For a sequence {x_n}, the notation of x_n ⇀ x and x_n → x means that the sequence {x_n} converges weakly and strongly to x, respectively.

For every point x ∈ H, there exists a unique nearest point in C, denoted by P_Cx, such that

Such a mapping P_C from H onto C is called the metric projection.

The following known lemmas will be used in the proof of our main results.

Lemma 2.1. Let H be a real Hilbert spaces H. Then, the following identities hold:

(i) for each x ∈ H and x* ∈ C, x* = P_Cx ⇔ 〈x - x*, y - x*〉 ≤ 0, ∀y ∈ C;

(ii) P_C : H → C is nonexpansive, that is, ||P_Cx - P_Cy|| ≤ ||x - y||, ∀x, y ∈ H;

(iii) P_C is firmly nonexpansive, that is, ||P_Cx - P_Cy||² ≤ 〈P_Cx - P_Cy, x - y〉, ∀x, y ∈ H;

(iv) ||tx + (1 - t)y||² = t||x||² + (1 - t)||y||² - t(1 - t)||x - y||², ∀t ∈ [0, 1], ∀x, y ∈ H;

(v) ||x + y||² ≤ ||x||² + 2〈y, x + y〉.

Lemma 2.2. [2]Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let B be a mapping of C into H. Let x* ∈ C. Then, for λ > 0,

where P_C is the metric projection of H onto C.

Lemma 2.3. [2]Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. Let β > 0, and let A : C → H be β-inverse strongly monotone. If 0 < ϱ ≤ 2β, then I -ϱA is a nonexpansive mapping of C into H, where I is the identity mapping on H.

Lemma 2.4. Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, let S : C → C be a nonexpansive mapping, and let B : C → H be a ξ-inverse strongly monotone. If 0 < α_n ≤ 2ξ, then S - α_nBS is a nonexpansive mapping in H.

Proof. For any x, y ∈ C and 0 < α_n ≤ 2ξ, we have

Hence, S - α_nBS is a nonexpansive mapping of C into H. □

Lemma 2.5. [13]Let B be a monotone mapping of C into H and let N_Cw₁ be the normal cone to C at w₁ ∈ C, that is, N_Cw₁ = {w ∈ H : 〈w₁ - w₂, w〉 ≥ 0, ∀w₂ ∈ C} and define a mapping Q on C by

Then, Q is maximal monotone and 0 ∈ Qw₁ if and only if w₁ ∈ VI(C, B).

Lemma 2.6. [14]Each Hilbert space H satisfies Opial's condition, that is, for any sequence {x_n} ⊂ H with x_n ⇀ x, the inequality

holds for each y ∈ H with y ≠ x.

Lemma 2.7. [5]Let C be a nonempty closed convex subset of H and let f be a contraction of H into itself with coefficient η ∈ (0, 1) and A be a strongly positive linear-bounded operator on H with coefficient . Then, for ,

That is, A - γ f is strongly monotone with coefficient .

Lemma 2.8. [5]Assume A to be a strongly positive linear-bounded operator on H with coefficient and 0 < ρ ≤ ||A||^-1. Then, .

For solving the generalized mixed equilibrium problem and the mixed equilibrium problem, let us give the following assumptions for the bifunction F, the function φ and the set C:

(H1) F(x, x) = 0, ∀x ∈ C;

(H2) F is monotone, that is, F(x, y) + F(y, x) ≤ 0 ∀x, y ∈ C;

(H3) for each y ∈ C, x α F(x, y) is weakly upper semicontinuous;

(H4) for each x ∈ C, y α F(x, y) is convex;

(H5) for each x ∈ C, y α F(x, y) is lower semicontinuous;

(B1) for each x ∈ H and λ > 0, there exist abounded subset G_x ⊆ C and y_x ∈ C such that for any z ∈ C \n G_x,

(B2) C is a bounded set.

Lemma 2.9. [15]Let C be a nonempty closed convex subset of H. Let F : C ×C → ℝ be a bifunction satisfies (H1)-(H5), and let φ : C → ℝ∪{+∞} be a proper lower semi continuous and convex function. Assume that either (B1) or (B2) holds. For λ > 0 and x ∈ H, define a mapping as follows:

Then, the following properties hold:

(i) For each x ∈ H, ;

(ii) is single-valued;

(iii) is firmly nonexpansive, that is, for any x, y ∈ H,

(iv) ;

(v) MEP(F, φ) is closed and convex.

Lemma 2.10. [16]Assume {a_n} to be a sequence of nonnegative real numbers such that

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

(1) ,

(2) or

Then, lim_{n →∞} a_n = 0.

3. Main results

In this section, we are in a position to state and prove our main results.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be bifunction from C × C toℝ satisfying (H1)-(H5), and let φ : C → ℝ ∪ {+∞} be a proper lower semicontinuous and convex function with either (B1) or (B2). Let B, D be two ξ, β-inverse strongly monotone mapping of C into H, respectively, and let S : C → C be a nonexpansive mapping. Let f : C → C be a contraction mapping with η ∈ (0, 1), and let A be a strongly positive linear-bounded operator with and . Assume that Θ := F (S) ∩ VI(C, B) ∩ GMEP(F, φ, D) ≠ ∅. Let {x_n}, {y_n} and {u_n} be sequences generated by the following iterative algorithm:

where {δn} and {β_n} are two sequences in (0, 1) satisfying the following conditions:

(C1) lim_{n →∞} β_n = 0 and ,

(C2) {δ_n} ⊂ [0, b], for some b ∈ (0, 1) and lim_{n →∞} |δ_n+1 - δ_n| = 0,

(C3) {λ_n} ⊂ [c, d] ⊂ (0, 2β) and lim_{n →∞} |λ_n+1 - λ_n| = 0,

(C4) {α_n} ⊂ [e, g] ⊂ (0, 2ξ) and lim_{n →∞} |α_n+1 - α_n| = 0.

Then, {x_n} converges strongly to z ∈ Θ, which is the unique solution of the variational inequality

Proof. We may assume, in view of β_n → 0 as n → ∞, that β_n ∈ (0, ||A||^-1). By Lemma 2.8, we obtain , ∀_n ∈ ℕ.

We divide the proof of Theorem 3.1 into six steps.

Step 1. We claim that the sequence {x_n} is bounded.

Now, let p ∈ Θ. Then, it is clear that

Let , D be β-inverse strongly monotone and 0 ≤ λ_n ≤ 2β. Then, we have

Let z_n = PC(Su_n - α_nBSu_n) and S - α_nBS be a nonexpansive mapping. Then, we have from Lemma 2.4 that

and

Similarly, and let w_n = P_C(Sy_n - α_nBSy_n) in (3.4). Then, we can prove that

which yields that

This shows that {x_n} is bounded. Hence, {u_n}, {z_n}, {y_n}, {w_n}, {BSu_n}, {BSy_n}, {Az_n} and {f(x_n)} are also bounded.

We can choose some appropriate constant M > 0 such that

Step 2. We claim that lim_n→∞ ||x_n+1 - x_n|| = 0.

It follows from Lemma 2.9 that and for all n ≥ 1, and we get

and

Take y = u_n-1 in (3.8) and y = u_n in (3.7), and then we have

and

Adding the above two inequalities, the monotonicity of F implies that

and

Without loss of generality, let us assume that there exists c ∈ ℝ such that λ_n > c > 0, ∀n ≥ 1. Then, we have

and hence,

Since S - α_nBS is nonexpansive for each n ≥ 1, we have

Substituting (3.9) into (3.10), we obtain

From (3.1), we have

Substituting (3.11) into (3.12) yields

Since w_n = P_C(Sy_n - α_nBSy_n) and S - α_nBS is nonexpansive mapping, we have

Also, from (3.1) and (3.13), we have

Set and

Then, we have

From the conditions (C1)-(C4), we find that

Therefore, applying Lemma 2.10 to (3.16), we have

Step 3. We claim that lim_n→∞ ||Sw_n - w_n|| = 0.

For any p ∈ Θ and Lemma 2.4, we obtain

From (3.1) and (3.18), we have

From (3.1), (3.5), (3.19) and Lemma 2.1(iv), we have

It follows that

From condition (C1) and (3.17), we obtain

From w_n = PC(Sy_n - α_nBSy_n), (3.19) and Lemma 2.4, we have

Using (3.1), (3.19) and (3.23), we obtain

It follows that

From condition (C1), (3.17) and (3.22), we obtain

Since P_C is firmly nonexpansive, we have

Hence, we have

Using (3.24) and (3.28), we have

It follows that

From the condition (C1), (3.17), (3.22) and (3.26), we obtain

Note that

From (3.1) and (3.32), we can compute

It follows that

which implies that

In addition, from the firmly nonexpansivity of , we have

Hence, we obtain

Substituting (3.36) into (3.32) to get

and hence,

It follows that

This together with ||x_n+1 - x_n|| → 0, ||Dx_n - D_p|| → 0, β_n → 0 as n → ∞ and the condition on λ_n implies that

Consequently, from (3.17) and (3.40)

From (3.1) and condition (C1), we have

Since S - α_nBS is nonexpansive mapping(Lemma 2.4), we have

Next, we will show that ||x_n - y_n|| → 0 as n → ∞.

We consider x_n+1 - y_n = δ_n(w_n - y_n) = δn(w_n - z_n + z_n - y_n).

From (3.43), we have

From the condition (C2), (3.41) and (3.42), it follows that

From (3.17) and (3.45), we obtain

We observe that

Consequently, we obtain

Step 4. We prove that the mapping P_Θ(γf + (I - A)) has a unique fixed point.

Let f be a contraction of C into itself with coefficient η ∈ (0, 1). Then, we have

Since , it follows that P_Θ(γf + (I - A)) is a contraction of C into itself. Therefore, by the Banach Contraction Mapping Principle, it has a unique fixed point, say z ∈ C, that is,

Step 5. We claim that q ∈ F(S) ∩ VI(C, B) ∩ GMEP(F, φ, D).

First, we show that q ∈ F(S).

Assume q ∉ F(S). Since and q ≠ Sq, based on Opial's condition (Lemma 2.6), it follows that

This is a contradiction. Thus, we have q ∈ F(S).

Next, we prove that q ∈ GMEP(F, φ, D).

From Lemma 2.9 that for all n ≥ 1 is equivalent to

From (H2), we also have

Replacing n by n_i, we obtain

Let y_t = t_y + (1 - t)q for all t ∈ (0, 1] and y ∈ C. Since y ∈ C and q ∈ C, we obtain y_t ∈ C. Hence, from (3.49), we have

Since , i → ∞ we obtain . Furthermore, by the monotonicity of D, we have

Hence, from (H4), (H5) and the weak lower semicontinuity of φ, and , we have

From (H1), (H4) and (3.51), we also get

Dividing by t, we get

Letting t → 0 in the above inequality, we arrive that, for each y ∈ C,

This implies that q ∈ GMEP(F, φ, D).

Finally, we prove that q ∈ VI(C, B).

We define the maximal monotone operator:

Since B is ξ-inverse strongly monotone and by condition (C4), we have

Then, Q is maximal monotone. Let (q₁, q₂) ∈ G(Q). Since q₂ - Bq₁ ∈ N_Cq₁ and w_n ∈ C, we have 〈q₁ - w_n, q₂ - Bq₁〉 ≥ 0. On the other hand, from w_n = P_C(Sy_n - α_nBSy_n), we have

that is,

Therefore, we obtain

Noting that as i → ∞, we obtain

Since Q is maximal monotone, we obtain that q ∈ Q^-10, and hence q ∈ VI(C, B). This implies q ∈ Θ. Since z = P_Θ(γf + (I - A))(z), we have

On the other hand, we have

From (3.46) and (3.53), we obtain that

Step 6. Finally, we claim that x_n → z, where z = P_Θ(γf + (I - A))(z).

We note that

which implies that

On the other hand, we have

where K is an appropriate constant such that K ≥ sup_n≥1{||x_n - z||²}.

Set and . Then, we have

From the condition (C1) and (3.54), we see that

Therefore, applying Lemma 2.10 to (3.58), we get that {x_n} converges strongly to z ∈ Θ.

This completes the proof. □

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, let B be ξ-inverse-strongly monotone mapping of C into H, and let S : C → C be a nonexpansive mapping. Let f : C → C be a contraction mapping with η ∈ (0, 1), and let A be a strongly positive linear-bounded operator with and . Assume that Θ := F(S) ∩ VI(C, B) ≠ ∅. Let {x_n} and {y_n} be sequence generated by the following iterative algorithm:

where {δ_n} and {β_n} are two sequences in (0, 1) satisfying the following conditions:

(C1) lim_{n → ∞} β_n = 0 and ,

(C2) {δ_n} ⊂ [0, b], for some b ∈ (0, 1) and lim_{n → ∞} |δ_n+1 - δ_n| = 0,

(C3) {α_n} ⊂ [e, g] ⊂ (0, 2ξ) and lim_{n → ∞} |α_n+1 - α_n| = 0.

Then, {x_n} converges strongly to z ∈ Θ, which is the unique solution of the variational inequality

Proof. Put F(x, y) = φ = D = 0 for all x, y ∈ C and λ_n = 1 for all n ≥ 1 in Theorem 3.1, we get u_n = x_n. Hence, {x_n} converges strongly to z ∈ Θ, which is the unique solution of the variational inequality (3.59). □

Corollary 3.3. [12]Let C be a nonempty closed convex subset of a real Hilbert space H and let F be bifunction from C × C to ℝ satisfying (H1)-(H5). Let S : C → C be a nonexpansive mapping and let f : C → C be a contraction mapping with η ∈ (0, 1). Assume that Θ := F(S) ∩ EP(F) ≠ ∅. Let {x_n}, {y_n} and {u_n} be sequence generated by the following iterative algorithm:

where {δ_n} and {β_n} are two sequences in (0, 1) and {λ_n} ⊂ (0, ∞) satisfying the following conditions:

(C1) lim_{n → ∞} β_n = 0 and ,

(C2) {δ_n} ⊂ [0, b], for some b ∈ (0, 1) and lim_{n → ∞} |δ_n+1 - δ_n| = 0,

(C3) lim_{n → ∞} |λ_n+1 - λ_n| = 0.

Then, {x_n} converges strongly to z ∈ Θ.

Proof. Put φ = D = 0, γ = 1, A = I and α_n = 0 in Theorem 3.1. Then, we have P_C(Su_n) = Su_n and P_C(Sy_n) = Sy_n. Hence, {x_n} generated by (3.60) converges strongly to z ∈ Θ. □

4. Competing interests

The authors declare that they have no competing interests.

5. Authors' contributions

All authors contribute equally and significantly in this research work. All authors read and approved the final manuscript.

6. Acknowledgements

This research was partially supported by the Research Fund, Rajamangala University of Technology Rattanakosin. The first author was supported by the 'Centre of Excellence in Mathematics', the Commission on High Education, Thailand for Ph.D. program at King Mongkuts University of Technology Thonburi (KMUTT). The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute, the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5480206. The third author was supported by the NRU-CSEC Project No. 54000267. Helpful comments by anonymous referees are also acknowledged.

References