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A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems
Fixed Point Theory and Applications volume 2011, Article number: 32 (2011)
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.
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) ≤ ηx  y for all x, y ∈ C. In addition, let D : C → H be a nonlinear mapping, φ : C → ℝ ∪ {+∞} be a realvalued 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)
βinversestrongly monotone if there exists a constant β > 0 such that

(3)
A setvalued 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 linearbounded 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 ξinversestrongly monotone mapping, Takahashi and Toyoda [6] introduced the following iterative scheme:
where B is a ξinversestrongly 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 x_{1} = 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: x_{1} ∈ 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 βinversestrongly 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_{ C }x, 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_{ C }x ⇔ 〈x  x*, y  x*〉 ≤ 0, ∀y ∈ C;

(ii)
P_{ C } : H → C is nonexpansive, that is, P_{ C }x  P_{ C }y ≤ x  y, ∀x, y ∈ H;

(iii)
P_{ C } is firmly nonexpansive, that is, P_{ C }x  P_{ C }y^{2} ≤ 〈P_{ C }x  P_{ C }y, x  y〉, ∀x, y ∈ H;

(iv)
tx + (1  t)y^{2} = tx^{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 βinverse 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 ξinverse strongly monotone. If 0 < α_{ n } ≤ 2ξ, then S  α_{ n }BS is a nonexpansive mapping in H.
Proof. For any x, y ∈ C and 0 < α_{ n } ≤ 2ξ, we have
Hence, S  α_{ n }BS 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_{ C }w_{1}be the normal cone to C at w_{1} ∈ C, that is, N_{ C }w_{1} = {w ∈ H : 〈w_{1}  w_{2}, w〉 ≥ 0, ∀w_{2} ∈ C} and define a mapping Q on C by
Then, Q is maximal monotone and 0 ∈ Qw_{1}if and only if w_{1} ∈ 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 linearbounded 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 linearbounded operator on H with coefficientand 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 mappingas follows:
Then, the following properties hold:

(i)
For each x ∈ H, ;

(ii)
is singlevalued;

(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 to ℝ 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, βinverse 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 linearbounded operator withand. 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 βinverse strongly monotone and 0 ≤ λ_{ n } ≤ 2β. Then, we have
Let z_{ n } = PC(Su_{ n }  α_{ n }BSu_{ n } ) and S  α_{ n }BS be a nonexpansive mapping. Then, we have from Lemma 2.4 that
and
Similarly, and let w_{ n } = P_{ C } (Sy_{ n }  α_{ n }BSy_{ 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_{n1}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_{ n }BS 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_{ n }BSy_{ n } ) and S  α_{ n }BS 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_{ n }BSy_{ 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_{ n }BS 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 ξinverse strongly monotone and by condition (C4), we have
Then, Q is maximal monotone. Let (q_{1}, q_{2}) ∈ G(Q). Since q_{2}  Bq_{1} ∈ N_{ C }q_{1} and w_{ n } ∈ C, we have 〈q_{1}  w_{ n } , q_{2} Bq_{1}〉 ≥ 0. On the other hand, from w_{ n } = P_{ C } (Sy_{ n }  α_{ n }BSy_{ n } ), we have
that is,
Therefore, we obtain
Noting that as i → ∞, we obtain
Since Q is maximal monotone, we obtain that q ∈ Q^{1}0, 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 ξinversestrongly 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 linearbounded operator withand. 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. □
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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 NRUCSEC Project No. 54000267. Helpful comments by anonymous referees are also acknowledged.
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Onjaiuea, N., Jaiboon, C. & Kumam, P. A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems. Fixed Point Theory Appl 2011, 32 (2011). https://doi.org/10.1186/16871812201132
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DOI: https://doi.org/10.1186/16871812201132