Abstract
Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of , which is the unique solution of a variational inequality.
AMS subject classifications: 49J40; 47J20; 47H09.
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1 Introduction
The theory of equilibrium problems has played an important role in the study of a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimization, and has numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis and mechanics. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be productive and innovative. It is remarkable that the variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems [1, 2].
Let H be a real Hilbert space. Throughout this paper, we write x n ⇀ x to indicate that the sequence {x n } converges weakly to x. The x n → x indicates that {x n } converges strongly to x. Let C be a nonempty closed convex subset of H and Θ be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for Θ: C × C → R is to find such that
The set of solutions of problem (1.1) is denoted by EP(Θ). Given a mapping T: C → H, let Θ (x, y) = 〈Tx, y - x〉 for all x, y ∈ C. Then, z ∈ EP(Θ) if and only if 〈Tz, y - z〉 ≥ 0 for all y ∈ C. Numerous problems in physics, optimization, and economics reduce to finding a solution of problem (1.1). Equilibrium problems have been studied extensively [2–18]. Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP(Θ) is nonempty and derived a strong convergence theorem. Very recently, Peng and Yao [4] introduced the following generalized mixed equilibrium problem of finding such that
where A: H → H is a nonlinear mapping, φ: C → R is a function and Θ: C × C → R is a bifunction. The set of solutions of problem (1.2) is denoted by GMEP.
In particular, whenever A = 0, problem (1.2) reduces to the following mixed equilibrium problem of finding such that
which was considered by Ceng and Yao [5]. The set of solutions of this problem is denoted by MEP.
Whenever φ = 0, problem (1.2) reduces to the following generalized equilibrium problem of finding such that
which was introduced and studied by Takahashi and Takahashi [13]. The set of solutions of problem (1.3) is denoted by GEP. Obviously, the generalized equilibrium problem covers the equilibrium problem as a special case. It is assumed in [4] that Θ: C ×C → R is a bifunction satisfying conditions (H1)-(H4) and φ: C → R is a lower semicontinuous and convex function with restriction (A1) or (A2), where
(H1) Θ (x, x) = 0, ∀x ∈ C;
(H2) Θ is monotone, i.e., Θ (x, y) + Θ (y, x) ≤ 0, ∀x, y ∈ C;
(H3) for each y ∈ C, x ↦ Θ (x, y) is weakly upper semicontinuous;
(H4) for each x ∈ C, y ↦ Θ (x, y) is convex and lower semicontinuous;
(A1) for each x ∈ H and r > 0, there exist a bounded subset D x ⊂ C and y x ∈ C such that for any z ∈ C \ D x ,
(A2) C is a bounded set.
It is worth pointing out that, related iterative methods for solving fixed point problems, variational inequalities and optimization problems can be found in [19–35].
Recall that a ρ-Lipschitzian mapping T: C → H is a mapping on C such that
where ρ ≥ 0 is a constant. In particular, if ρ ∈ [0, 1) then T is called a contraction on C; if ρ = 1 then T is called a nonexpansive mapping on C. Denote the set of fixed points of T by Fix(T). It is well known that if C is a nonempty bounded closed convex subset of H and S: C → C is nonexpansive, then Fix(S) ≠ Ø. Let P C be the metric projection of H onto C, that is, for every point x ∈ H, there exists a unique nearest point of C, denoted by P C x, such that ǀǀ x - P C x ǀǀ ≤ ǀǀ x - y ǀǀ for all y ∈ C. Recall also that a mapping A of C into H is called
-
(i)
monotone if
-
(ii)
η-strongly monotone if there exists a constant η > 0 such that
-
(iii)
δ-inverse strongly monotone if there exists a constant δ > 0 such that
Furthermore, let A be a strongly positive bounded linear operator on H, that is, there exists a constant such that
1.1 The W-mappings
The concept of W-mappings was introduced in Atsushiba and Takahashi [22]. It is very useful in establishing the convergence of iterative methods for computing a common fixed point of nonlinear mappings (see, for instance, [23, 25, 27]).
Let λn,1, λn,2..., λ n, N ∈ (0, 1], n ≥ 1. Given the nonexpansive mappings S1, S2,..., S N on H, Atsushiba and Takahashi defines, for each n ≥ 1, mappings Un,1, Un,2,..., U n, N by
The W n is called the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Note that Nonexpansivity of S i implies the nonexpansivity of W n .
Colao et al. [14] introduced an iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space H. Moreover, they proved the strong convergence of the proposed iterative algorithm.
1.2 Theorem CMX
(See [[14], Theorem 3.1]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a finite family of nonexpansive mappings on H, A a strongly positive bounded linear operator on H with coefficient and f an α-contraction on H for some α ∈ (0, 1). Moreover, let {α n } be a sequence in (0, 1), a sequence in [a, b] with 0 < a ≤ b < 1, {r n } a sequence in (0, ∞) and γ and β two real numbers such that 0 < β < 1 and . Let Θ: C × C → R be a bifunction satisfying assumptions (H1)-(H4) and . For every n ≥ 1, let W n be the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Given x1 ∈ H arbitrarily, suppose the sequences {x n } and {u n } are generated iteratively by
where the sequences {α n }, {r n } and the finite family of sequences satisfy the conditions:
-
(i)
limn→∞α n = 0 and ;
-
(ii)
lim infn→∞r n > 0 and limn→∞r n /rn+1= 1 (or limn→∞ǀrn+1- r n ǀ = 0);
-
(iii)
limn→∞ǀλ n, i - λn- 1, iǀ = 0 for every i ∈ {1,..., N}.
Then both {x n } and {u n } converge strongly to , which is the unique fixed point of the composite mapping , i.e.,
Very recently, Yao et al. [10] relaxed the β in Colao, Marino and Xu's iterative scheme (1.6) by a sequence of {β n }. They showed that if with additional condition 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 holds, then the sequences {x n } and {u n } generated by (1.6) (but now with β n in the place of β) still converge strongly to , which is the unique fixed point of the composite mapping , i.e.,
1.3 Hybrid steepest-descent method
Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, and let T: H → H be nonexpansive such that Fix(T) ≠ Ø. Yamada [20] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem: finding such that
This method generates a sequence {x n } via the following iterative scheme:
where 0 < μ < 2η/κ2, the initial guess x0 ∈ H is arbitrary and the sequence {λ n } in (0, 1) satisfies the conditions:
A key fact in Yamada's argument is that, for small enough λ > 0, the mapping
is a contraction, due to the κ-Lipschitz continuity and η-strong monotonicity of F.
1.4 Our hybrid model
In this paper, assume Θ: C × C → R is a bifunction satisfying assumptions (H1)-(H4) and φ: C → R is a lower semicontinuous and convex function with restriction (A1) or (A2). Let the mapping A: H → H be δ-inverse strongly monotone, and be a finite family of nonexpansive mappings on H such that . Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ2 and 0 ≤ γρ < τ, where . By combining Yamada's hybrid steepest-descent method [20] and Colao, Marino and Xu's hybrid viscosity approximation method [14] (see also [10]), we propose the following hybrid iterative method for finding a common element of the set of solutions of generalized mixed equilibrium problem (1.2) and the set of fixed points of finitely many nonexpansive mappings , that is, for given x1 ∈ H arbitrarily, let {x n } and {u n } be generated iteratively by
where {a n }, {β n } ⊂ (0, 1), {r n } ⊂ (0, 2δ], with 0 < a ≤ b < 1, and W n is the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . We shall prove that under quite mild hypotheses, both sequences {x n } and {u n } converge strongly to , where is a unique solution of the variational inequality:
Compared with Theorem 3.2 of Yao et al. [10], our Theorem 3.1 improves and extends their Theorem 3.2 [10] in the following aspects:
-
(i)
The contraction f: H → H with coefficient ρ ∈ (0, 1) in [[10], Theorem 3.2] is extended to the case of general Lipschitzian mapping f on H with constant ρ ≥ 0.
-
(ii)
The strongly positive bounded linear operator A: H → H with coefficient in [[10], Theorem 3.2] is extended to the case of general κ-Lipschitzian and η-strongly monotone operator F: H → H with constants κ, η > 0.
-
(iii)
The equilibrium problem in [[10], Theorem 3.2] is extended to the case of generalized mixed equilibrium problem (1.2). Obviously, the problem (1.2) is more complicated than their problem (1.1).
-
(iv)
The hybrid viscosity approximation method in [[10], Theorem 3.2] (see also [[14], Theorem 3.1]) is extended to develop our iterative method by virtue of Yamada's hybrid steepest-descent method [20].
2 Preliminaries
Let H be a real Hilbert space with inner product 〈·,·〉, and norm ǀǀ · ǀǀ. Let C be a nonempty closed convex subset of H. Recall that the metric (or nearest point) projection from H onto C is the mapping P C : H → C which assigns to each point x ∈ H the unique point P C x ∈ C satisfying the property
In order to prove our main results in the next section, we need the following lemmas and propositions.
Lemma 2.1 (See [36]). Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C, we then have
-
(i)
z = P C x if and only if 〈 x - z, y - z 〉 ≤ 0, ∀y ∈ C.
-
(ii)
z = P C x if and only if ǀǀ x - z ǀǀ2 ≤ ǀǀ x - y ǀǀ2 - ǀǀ y - z ǀǀ2, ∀ y ∈ C.
-
(iii)
〈P C x - P C y, x - y 〉 ≥ ǀǀ P C x - P C y ǀǀ2, ∀x, y ∈ H.
Consequently, P C is nonexpansive and monotone.
Lemma 2.2 (See [5]). Let C be a nonempty closed convex subset of H. Let Θ: C×C → R be a bifunction satisfying conditions (H1)-(H4) and let φ: C → R be a lower semicontinuous and convex function. For r > 0 and x ∈ H, define a mapping as follows:
for all x ∈ H. Assume that either (A1) or (A2) holds. Then the following assertions hold:
(i) for each x ∈ H and is single-valued;
-
(ii)
is firmly nonexpansive, i.e., for any x, y ∈ H,
-
(iii)
;
-
(iv)
MEP(Θ, φ)is closed and convex.
Remark 2.1. If φ = 0, then is rewritten as additionally, then .
Lemma 2.3 (See [21]). Let {x n } and {y 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. Suppose xn+1= (1 - β n )y n +β n x n for all integers n ≥ 0 and lim supn→∞(ǀǀyn+1- y n ǀǀ - ǀǀxn+1- x n ǀǀ) ≤ 0. Then, limn→∞ǀǀy n - x n ǀǀ = 0.
Proposition 2.1 (See [[6], Proposition 2.1]). Let C, H, Θ, φ and be as in Lemma 2.2. Then the following inequality holds:
for all s, t > 0 and x ∈ H.
Lemma 2.4 (See [19]). Let {a n } be a sequence of nonnegative numbers satisfying the condition
where {δ n }, {σ n } are sequences of real numbers such that
-
(i)
{δ n } ⊂ [0, 1] and , or equivalently,
-
(ii)
lim supn→∞σ n ≤ 0, or
(ii)' is convergent.
Then limn→∞a n = 0.
We will need the following result concerning the W-mapping W n generated by S1,..., S N and λn,1, λn,2,..., λ n, N in (1.5).
Proposition 2.2 (See [23]). Let C be a nonempty closed convex subset of a Banach space X. Let S1, S2,..., S N be a finite family of nonexpansive mappings of C into itself such that , and let λn,1, λn,2,..., λ n, N be real numbers such that 0 < λ n, i ≤ b < 1 for i = 1, 2,..., N. For any n ≥ 1, let W n be the W-mapping of C into itself generated by S1,..., S N and λn,1,..., λ n, N . If X is strictly convex, then .
Proposition 2.3 (See [[14], Lemma 2.8]). Let C be a nonempty convex subset of a Banach space. Let be a finite family of nonexpansive mappings of C into itself and be sequences in [0, 1] such that λ n, i → λ i (i = 1,..., N). Moreover for every integer n ≥ 1, let W and W n be the W-mappings generated by S1,..., S N and λ1,..., λ N and S1,..., S N and λn,1,..., λ n, N respectively. Then for every x ∈ C, it follows that
The following two lemmas are the immediate consequences of the inner product on H.
Lemma 2.5. For all x, y ∈ H, there holds the inequality
Lemma 2.6 (See [36]). For all x, y, z ∈ H and α, β, γ ∈ [0, 1] with α + β + γ = 1, there holds the equality
The following lemma plays a crucial role in proving strong convergence of our iterative schemes.
Lemma 2.7 (See [[19], Lemma 3.1]). Let λ be a number in (0, 1] and let μ > 0. Let F: H → H be an operator on H such that, for some constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone. Associating with a nonexpansive mapping T: H → H, define the mapping Tλ: H → H by
Then Tλ is a contraction provided μ < 2η/κ2, that is,
where .
Remark 2.2. Put , where I is the identity operator of H. Then we have μ < 2η/κ2 = 4. Also, put μ = 2. Then it is easy to see that and
In particular, whenever λ > 0, we have Tλx: = Tx - λμF(Tx) = (1 - λ) Tx.
3 Iterative scheme and strong convergence
In this section, based on Yamada's hybrid steepest-descent method [20] and Colao, Marino and Xu's hybrid viscosity approximation method [14] (see also [10]), we introduce a hybrid iterative method for finding a common element of the set of solutions of generalized mixed equilibrium problem (1.2) and the set of fixed points of finitely many nonexpansive mappings in a real Hilbert space. Moreover, we derive the strong convergence of the proposed iterative algorithm to a common solution of problem (1.2) and the fixed point problem of finitely many nonexpansive mappings.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ: C × C → R be a bifunction satisfying assumptions (H1)-(H4) and φ: C → R be a lower semicontinuous and convex function with restriction (A1) or (A2). Let the mapping A: H → H be δ-inverse strongly monotone, and be a finite family of nonexpansive mappings on H such that . Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ- Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ2 and 0 ≤ γρ < τ, where . Suppose {a n } and {β n } are two sequences in (0, 1), {r n } is a sequence in (0, 2δ] and is a sequence in [a, b] with 0 < a ≤ b < 1. For every n ≥ 1, let W n be the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Given x1 ∈ H arbitrarily, suppose the sequences {x n } and {u n } are generated iteratively by
where the sequences {a n }, {β n }, {r n } and the finite family of sequences satisfy the conditions:
-
(i)
limn→∞α n = 0 and ;
-
(ii)
0 < lim infn→∞β n ≤ lim supn→∞β n < 1;
-
(iii)
0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1- r n ) = 0;
-
(iv)
limn→∞(λn+1, i- λ n, i ) = 0 for all i = 1, 2,..., N.
Then both {x n } and {u n } converge strongly to , where is a unique solution of the variational inequality:
Proof. Let . Note that F: H → H is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H is a ρ-Lipschitzian mapping with constant ρ ≥ 0. Then, we have
where , and hence
for all x, y ∈ H. Since 0 ≤ γρ < τ ≤ 1, it is known that 1 - (τ - γρ) ∈ [0, 1). Therefore, Q(I - μF + γf) is a contraction of H into itself, which implies that there exists a unique element x* ∈ H such that .
From the definition of , we know that . Take arbitrarily. Since , A is δ-inverse strongly monotone and 0 < r n ≤ 2δ, we deduce that, for any n ≥ 1,
First we will prove that both {x n } and {u n } are bounded.
Indeed, taking into account the control conditions (i) and (ii), we may assume, without loss of generality, that α n ≤ 1 - β n for all n ≥ 1. Now, by Proposition 2.2 we have p ∈ Fix(W n ).
Then utilizing Lemma 2.7, from (3.1) and (3.3) we obtain
It follows from (3.4) and induction that
Therefore {x n } is bounded. We also obtain that {u n }, {Ax n }, {W n u n } and {f (x n )} are all bounded. We shall use M to denote the possible different constants appearing in the following reasoning.
Next, we show that ǀǀ xn+1- x n ǀǀ → 0.
Indeed, set xn+1= β n x n + (1 - β n )z n for all n ≥ 1. Then from the definition of z n we obtain
It follows that
From (1.5), since S i and U n, i for all i = 1, 2,..., N are nonexpansive,
Again, from (1.5),
Therefore, we have
and then
Substituting (3.8) into (3.6), we have
On the other hand, utilizing the δ-inverse strongly monotonicity of A we have
Since and , we get
Using (3.9) and (3.11) in (3.5), we get
Note that 0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1- r n ) = 0. Then utilizing Proposition 2.1 we have
Consequently, it follows from (3.13) and conditions (i), (iii), (iv) that
Hence by Lemma 2.3 we have
Consequently
From (3.11), (3.13), (3.14) and condition (iii) we have
Since xn+1= a n γ f(x n ) + β n x n + ((1 - β n )I - a n μF)W n u n , we have
that is
It follows that
On the other hand, from (3.3) and (3.4) we get
and hence
Obviously, conditions (i), (ii), (iii) guarantee that α n → 0, 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 and 0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ. Thus from ǀǀ x n - xn+1ǀǀ → 0 we conclude that
Note that is firmly nonexpansive. Hence we have
which implies that
Therefore, utilizing Lammas 2.5 and 2.7 we deduce from (3.17) that
Then we have
So, from (3.14)-(3.16) and α n → 0, we have
Since
we also have
Next, let us show that
where is a unique solution of the variational inequality (3.2). To show this, we can choose a subsequence of {u n } such that
Since is bounded, there exists a subsequence {u ij } of which converges weakly to w. Without loss of generality, we may assume that . From ǀǀW n u n - u n ǀǀ → 0, we obtain . Now we show that w ∈ GMEP. From , we know that
From (H2) it follows that
Replacing n by n i , we have
Put u t = ty + (1 - t)w for all t ∈ (0, 1] and y ∈ C. Then, we have u t ∈ C. So, from (3.18) we have
Since , we have . Further, from the monotonicity of A, we have . So, from (H4), the weakly lower semicontinuity of and , we have
as i → ∞. From (H1), (H4) and (3.19), we also have
and hence
Letting t → 0, we have, for each y ∈ C,
This implies that w ∈ GMEP.
We shall show . To see this, we observe that we may assume (by passing to a further subsequence if necessary)
Let W be the W-mapping generated by S1,..., S N and λ1,..., λ N . Then by Proposition 2.3, we have, for every x ∈ H,
Moreover, from Proposition 2.2 it follows that . Assume that ; then w ≠ Ww. Since w ∈ GMEP, in terms of ǀǀ x n - W n u n ǀǀ → 0 and Opial's property of a Hilbert space, we conclude from (3.20) that
due to the δ-inverse strong monotonicity of A. This is a contradiction. So, we get . Therefore . Since , we have
Finally, we prove that {x n } and {u n } converge strongly to x*. From (3.1), utilizing Lemmas 2.5 and 2.7 we have
This implies that
where M1 = sup{ǀǀx n - p ǀǀ2: n ≥ 1}, and . It is easy to see that and . Hence, by Lemma 2.4, the sequence {x n } converges strongly to x*. Consequently, we can obtain from ǀǀx n - u n ǀǀ → 0 that {u n } also converges strongly to x*. This completes the proof. □
Remark 3.1.
-
(i)
The new technique of argument is applied to derive our Theorem 3.1. For instance, Lemma 2.7 for deriving the convergence of hybrid steepest-descent method plays an important role in proving the strong convergence of the sequences {x n }, {u n } in our Theorem 3.1. In addition, utilizing Proposition 2.1 and rn+1- r n → 0 we can obtain .
-
(ii)
In order to show , the proof of Theorem 3.2 [10] directly asserts that ǀǀu n - W n u n ǀǀ→ 0 (n → ∞) implies for all n. Actually, this assertion seems impossible under their assumptions imposed on . However, following Colao, Marino and Xu's Step 7 of the proof in [[14], Theorem 3.1] and utilizing Proposition 2.3 (i.e., Lemma 2.8 in [14]), we successively derive by the condition with 0 < a ≤ b < 1.
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A: H → H be δ-inverse strongly monotone, Θ: C × C → R be a bifunction satisfying assumptions (H1)-(H4) and φ: C → R be a lower semicontinuous and convex function with restriction (A1) or (A2) such that GMEP ≠ Ø. Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ2 and 0 ≤ γρ < τ, where . Suppose {α n } and {β n } are two sequences in (0, 1) and {r n } is a sequence in (0, 2δ]. Given x1 ∈ H arbitrarily, suppose the sequences {x n } and {u n } are generated iteratively by
where the sequences {α n }, {β n }, {r n } satisfy the conditions:
-
(i)
limn→∞α n = 0 and ;
-
(ii)
0 < lim infn →∞β n ≤ lim supn →∞β n < 1;
-
(iii)
0 < lim infn →∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1- r n ) = 0.
Then both {x n } and {u n } converge strongly to x* ∈ GMEP, where x* = P GMEP (I - μF + γ f)x*.
Proof. Put S i x = x for all i = 1, 2,..., N and x ∈ H and take the finite family of sequences in [a, b] with 0 < a ≤ b < 1 such that limn→∞(λn+1, i- λ n, i ) = 0 for all i = 1, 2,..., N. In this case, the W-mapping W n generated by S1,..., S N and λn,1, λn,2,..., λ n, N , is the identity mapping I of H. It is easy to see that all conditions of Theorem 3.1 are satisfied. Thus, the desired result follows from Theorem 3.1. □
Theorem 3.3. Let H be a real Hilbert space. Let be a finite family of nonexpansive mappings on H such that . Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ2 and 0 ≤ γρ < τ, where . Suppose {α n } and {β n } are two sequences in (0, 1) and is a sequence in [a, b] with 0 < a ≤ b < 1. For every n ≥ 1, let W n be the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Given x1 ∈ H arbitrarily, let {x n } be a sequence generated by
where the sequences {α n }, {β n } and the finite family of sequences satisfy the conditions:
-
(i)
limn→∞α n = 0 and ;
-
(ii)
0 < lim infn →∞β n ≤ lim supn →∞β n < 1;
-
(iii)
limn→∞(λn+1, i- λ n, i ) = 0 for all i = 1, 2,..., N.
Then {x n } converges strongly to , where .
Proof. Put C = H and r n = 1, and take Θ(x, y) = 0, Ax = 0 and φ(x) = 0 for all x, y ∈ H. Then Θ: H × H → R is a bifunction satisfying assumptions (H1)-(H4) and φ: H → R is a lower semicontinuous and convex function with restriction (A1). Moreover the mapping A: H → H is δ-inverse strongly monotone for any . In this case, from Theorem 3.1 we deduce that u n = x n , 0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1-r n ) = 0. Beyond question, all conditions of Theorem 3.1 are satisfied. Therefore the conclusion follows. □
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Acknowledgements
Lu-Chuan Ceng was partially supported by National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Leading Academic Discipline Project of Shanghai Normal University (DZL707), Sy-Ming Guu was partially supported by the grant NSC 97-2221-E-155-041-MY3, and Jen-Chih Yao was partially supported by the grant NSC 99-2221-E-037-007-MY3.
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LC conceived of the study and drafted the manuscript initially. SM participated in its design, coordination and finalized the manuscript. JC outlined the scope and design of the study. All authors read and approved the final manuscript.
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Ceng, LC., Guu, SM. & Yao, JC. Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. Fixed Point Theory Appl 2012, 92 (2012). https://doi.org/10.1186/1687-1812-2012-92
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DOI: https://doi.org/10.1186/1687-1812-2012-92