Abstract
An iterative algorithm is considered for variational inequalities, generalized equilibrium problems and fixed point problems. Strong convergence of the proposed iterative algorithm is obtained in the framework Hilbert spaces. Mathematical subject classification: 47H05, 47H09, 47J25, 47N10.
generalized equilibrium problem; variational inequality; fixed point; nonexpansive mapping
Some results on variational inequalities and generalized equilibrium problems with applications
Xiaolong QinI; Sun Young ChoII; Shin Min KangIII
IDepartment of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
IIDepartment of Mathematics, Gyeongsang National University, Jinju 660-701, Korea
IIIDepartment of Mathematics and the RINS, Gyeongsang National University Jinju 660-701, Korea. E-mail: smkang@gnu.ac.kr
ABSTRACT
An iterative algorithm is considered for variational inequalities, generalized equilibrium problems and fixed point problems. Strong convergence of the proposed iterative algorithm is obtained in the framework Hilbert spaces.
Mathematical subject classification: 47H05, 47H09, 47J25, 47N10.
Keywords: generalized equilibrium problem, variational inequality, fixed point, nonexpansive mapping.
1 Introduction and preliminaries
Let H be a real Hilbert space, whose inner product and norm are denoted by ‹·, ·› and ║·║, respectively. Let C be a nonempty closed and convex subset of H and PC be the projection of H onto C.
Let , S, A, T be nonlinear mappings. Recall the following definitions:
(1) : C → C is said to be α-contractive if there exists a constant α ∈ (0,1) such that
(2) S : C → C is said to be nonexpansive if
Throughout this paper, we use F(S) to denote the set of fixed points of the mapping S.
(3) A : C → H is said to be monotone if
(4) A : C → H is said to be inverse-strongly monotone if there exists δ > 0 such that
Such a mapping A is also called δ-inverse-strongly monotone. We know that if S : C → C is nonexpansive, then A = I S is -inverse-strongly monotone; see [1, 21] for more details.
(5) A set-valued mapping T : H → 2H is said to be monotone if for all x, y ∈ H, ∈ Tx and g ∈ Ty ⇒ ‹x y, g› > 0. A monotone mapping T : H → 2H is maximal if the graph of G(T) of T is not properly contained in the graph of any other monotone mapping. It isknown that a monotone mapping T is maximal if and only if for (x, ) ∈ H × H, ‹x y, g› > 0 for every (y, g) ∈ G(T) implies that ∈ Tx. Let A be a monotone mapping of C into H and let NCυ be the normal cone to C at υ ∈ C, i.e., NCυ = {w ∈ H : ‹υ u, w› > 0, ∀u ∈ C} and define
Then T is maximal monotone and 0 ∈ Tυ if and only if ‹Aυ, u υ› > 0, ∀u ∈ C; see [16] for more details.
Recall that the classical variational inequality is to find an x ∈ C such that
In this paper, we use VI (C, A) to denote the solution set of the variational inequality (1.1). For given z ∈ H and u ∈ C, we see that
holds if and only if u = PCz. It is known that projection operator PC satisfies
One can see that the variational inequality (1.1) is equivalent to a fixed point problem. An element u ∈ C is a solution of the variational inequality (1.1) if and only if u ∈ C is a fixed point of the mapping PC(I λA)u, where λ > 0 is a constant and I is the identity mapping. This can be seen from the following. u ∈ C is a solution of the variational inequality (1.1), this is,
which is equivalent to
where λ > 0 is a constant. This implies from (1.2) that u = PC(I λA)u, that is, u is a fixed point of the mapping PC(I λA). This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.
Let A : C → H be a δ-inverse-strongly monotone mapping and F be a bifunction of C × C into , where denotes the set of real numbers. We consider the following generalized equilibrium problem:
In this paper, the set of such an x ∈ C is denoted by EP(F, A), i.e.,
Next, we give some special cases of the generalized equilibrium problem (1.3).
(I) If A ≡ 0, the zero mapping, then the problem (1.3) is reduced to the following equilibrium problem:
In this paper, the set of such an x ∈ C is denoted by EP(F), i.e.,
(II) If F ≡ 0, then the problem (1.3) is reduced to the classical variational inequality (1.1).
In 2005, Iiduka and Takahashi [8] considered the classical variational inequality (1.1) and a single nonexpansive mapping. To be more precise, they obtained the following results.
Theorem IT. Let C be a closed convex subset of a real Hilbert space H. Let A be anα-inverse-strongly monotone mapping of C into H and S be a nonexpansive mapping of C into itself such that F(S)∩VI(C, A) ≠ . Suppose that x1 = x ∈ C and {xn} is given by
where {αn} is a sequence in [0,1) and {λn} is a sequence in [0,2α]. If {αn} and {λn} are chosen so that {λn} ⊂ [a, b] for some a, b with 0 < a < b < 2α,
then {xn} converges strongly to PF(S)∩VI(C, A)x.
On the other hand, we see that the problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games and others; see, for instance, [2, 5, 9]. Recently, many authors considered iterative methods for the problems (1.3) and (1.4), see [3-7, 11-15, 18, 20, 22, 24] for more details.
To study the equilibrium problems (1.3) and (1.4), we may assume that F satisfies the following conditions:
(A1) F(x, x) = 0 for all x ∈ C;
(A2) F is monotone, i.e., F(x, y) + F(y, x) < 0 for all x, y ∈ C;
(A3) for each x, y, z ∈ C,
(A4) for each x ∈ C, y → F(x, y) is convex and weakly lower semi-continuous.
Put (x, y) = F(x, y) + ‹Ax, y x› for each x, y ∈ C. It is not hard tosee that also confirms (A1)-(A4).
In 2007, Takahashi and Takahashi [20] introduced the following iterative method
where is a α-contraction, T is a nonexpansive mapping. They considered the problem of approximating a common element of the set of fixed points of a single nonexpansive mapping and the set of solutions of the equilibrium problem (1.4). Strong convergence theorems of the iterative algorithm (1.6) are established in a real Hilbert space.
Recently, Takahashi and Takahashi [22] further considered the generalized equilibrium problem (1.3). They obtained the following result in a real Hilbert space.
Theorem TT. Let C be a closed convex subset of a real Hilbert space H and F : C × C → be a bi-function satisfying (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C into H and S be a non-expansive mapping of C into itself such that F(S) ∩ EP(F, A) ≠ . Let u ∈ C and x1∈ C and let {zn} ⊂ C and {xn} ⊂ C be sequences generated by
where {αn} ⊂ [0,1], {βn} ⊂ [0,1] and {rn} ⊂ [0,2α] satisfy
Then, {xn} converges strongly to z = PF(S)∩EP(F, A)u.
Very recently, Chang, Lee and Chan [5] introduced a new iterative method for solving equilibrium problem (1.4), variational inequality (1.1) and the fixed point problem of nonexpansive mappings in the framework of Hilbert spaces. More precisely, they proved the following theorem.
Theorem CLC. Let H be a real Hilbert space, C be a nonempty closed convex subset of H and F be a bifunction satisfying the conditions (A1)-(A4). Let A : C → H be an α-inverse-strongly monotone mapping and {Si : C → C} be a family of infinitely nonexpansive mappings with F ∩ VI(C, A) ∩ EP(F) ≠ , where F := F(Si) and : C → C be a ξ-contractive mapping. Let {xn}, {yn} {kn} and {un} be sequences defined by
where {Wn : C → C} is the sequence defined by (1.9), {αn}, {βn} and {γn} are sequences in [0,1], {λn} is a sequence in [a, b] ⊂ (0,2α) and {rn} is a sequence in (0, ∞). If the following conditions are satisfied:
(1) αn + βn+ γn = 1;
(2) limn→∞αn = 0; αn = ∞;
(3) 0 < lim infn→∞βn< lim supn→∞βn < 1;
(4) lim infn→∞rn > 0; |rn+1 rn| < ∞;
(5) limn→∞|λn+1 λn| = 0,
then {xn} and {un} converge strongly to z ∈ F ∩ VI(C, A) ∩ EP(F).
In this paper, motivated and inspired by the research going on in this direction, we introduce a general iterative method for finding a common element of the set of solutions of generalized equilibrium problems, the set of solutions of variational inequalities, and the set of common fixed points of a family ofnonexpansive mappings in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results of Ceng and Yao [3, 4], Chang Lee and Chan [5], Iiduka and Takahashi [8], Qin, Shang and Zhou [12], Su, Shang and Qin [18], Takahashi and Takahashi [20, 22], Yao and Yao [25] and many others.
In order to prove our main results, we need the following definitions and lemmas.
A space X is said to satisfy Opial condition [10] if for each sequence {xn} in X which converges weakly to point x ∈ X, we have
Lemma 1.1 ([2]). Let C be a nonempty closed convex subset of H and F : C × C → be a bifunction satisfying (A1)-(A4). Then, for any r > 0 and x ∈ H, there exists z ∈ C such that
Lemma 1.2 ([2], [7]). Suppose that all the conditions in Lemma 1.1 are satisfied. For any give r > 0 define a mapping Tr : H → C as follows:
then the following conclusions hold:
(1) Tr is single-valued;
(2) Tr is firmly nonexpansive, i.e., for any x,y ∈ H,
(3) F(Tr) = EP(F);
(4) EP(F) is closed and convex.
Lemma 1.3 ([23]). Assume that {αn} is a sequence of nonnegative real numbers such that
where {γn} is a sequence in (0,1) and {δn} is a sequence such that
Then limn→∞αn = 0.
Definition 1.4 ([19]). Let {Si : C → C} be a family of infinitely nonexpansive mappings and {γi} be a nonnegative real sequence with 0 < γi < 1, ∀i > 1. For n > 1 define a mapping Wn: C → C as follows:
Such a mapping Wn is nonexpansive from C to C and it is called a W-mapping generated by Sn, Sn-1, ..., S1 and γn, γn-1, ..., γ1.
Lemma 1.5 ([19]). Let C be a nonempty closed convex subset of a Hilbert space H, {Si : C → C} be a family of infinitely nonexpansive mappings with F(Si) ≠ and {γi} be a real sequence such that 0 < γi< l < 1, ∀i > 1. Then
(1) Wn is nonexpansive and F(Wn) = F(Si), for each n > 1;
(2) for each x ∈ C and for each positive integer k, the limit limn→∞Un,k exists.
(3) the mapping W : C → C defined by
is a nonexpansive mapping satisfying F(W) = F(Si) and it is called the W-mapping generated by S1, S2, ... andγ1, γ2, ....
Lemma 1.6 ([5]). Let C be a nonempty closed convex subset of a Hilbert space H, {Si : C → C} be a family of infinitely nonexpansive mappings with F(Si) ≠ and {γi} be a real sequence such that 0 < γi< l < 1, ∀i > 1. If K is any bounded subset of C, then
Throughout this paper, we always assume that 0 < γi< l < 1, ∀i > 1.
Lemma 1.7 ([17]). Let {xn} and {yn} be bounded sequences in a Hilbert space H and {βn} be a sequence in [0,1] with
Suppose that xn+1 = (1 βn)yn + βnxn for all n > 0 and
Then limn→∞║yn xn║ = 0.
2 Main results
Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C × C to which satisfies (A1)-(A4). Let A1 : C → H be a δ1-inverse-strongly monotone mapping, A2 : C → H be a δ2-inverse-strongly monotone mapping, A3 : C → H be a δ3-inverse-strongly monotone mapping and {Si: C → C} be a family of infinitely nonexpansive mappings. Assume that Ω := FP ∩ EP(F, A3) ∩ VI ≠ , where FP = F(Si) and VI = VI(C, A1) ∩ VI(C, A2). Let : C → C be an α-contraction. Let x1∈ C and {xn} be a sequence generated by
where {Wn : C → C} is the sequence generated in (1.9), {αn}, {βn} and {γn} are sequences in (0,1) such that αn + βn + γn = 1 for each n > 1 and {rn}, {λn} and {ηn} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
(R1) 0 < a < ηn< b < 2δ1, 0 < a' < λn< b' < 2δ2, 0 < < rn < < 2δ3, ∀n > 1;
(R2) limn→∞αn = 0 and αn = ∞;
(R3) 0 < lim infn→∞βn< lim supn→∞βn < 1;
(R4) limn→∞(λn λn+1) = limn→∞(ηn ηn+1) = limn→∞(rn rn+1) = 0.
Then the sequence {xn} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:
Proof. First, we show, for each n > 1, that the mappings I ηnA1, I λnA2 and I rnA3 are nonexpansive. Indeed, for ∀x, y ∈ C, we obtain from the restriction (R1) that
which implies that the mapping I ηnA1 is nonexpansive, so are I λnA2and I rnA3 for each n > 1. Note that un can be re-written as un = (I rnA3)xn for each n > 1. Take x* ∈ Ω. Noticing that x* = PC(I ηnA1)x* = PC(I - λnA2)x* = (I rnA3)x*, we have
On the other hand, we have
It follows from (2.2) and (2.3) that
which yields that
From the algorithm (2.1) and (2.5), we arrive at
By simple inductions, we obtain that
which gives that the sequence {xn} is bounded, so are {yn}, {zn} and {un}. Without loss of generality, we can assume that there exists a bounded set K ⊂ C such that
Notice that un+1 = (I - rn+1A3)xn+1 and un = (I rnA3)xn, wesee from Lemma 1.2 that
and
Let y = un in (2.7) and y = un+1 in (2.8). By adding up these two inequalities and using the assumption (R2), we obtain that
Hence, we have
This implies that
It follows that
where M1 is an appropriate constant such that
From the nonexpansivity of PC, we also have
Substituting (2.9) into (2.10), we arrive at
In a similar way, we can obtain that
Combining (2.11) with (2.12), we see that
where M2 is an appropriate constant such that
Letting
we see that
It follows that
On the other hand, we have
where K is the bounded subset of C defined by (2.6). Substituting (2.13) into (2.16), we arrive at
which combines with (2.15) yields that
In view of the restriction (R2), (R3) and (R4), we obtain from Lemma 1.6 that
Hence, we obtain from Lemma 1.7 that
In view of (2.14), we have
Thanks to the restriction (R3), we see that
For any x* ∈ Ω, we see that
Note that
Substituting (2.19) into (2.18), we arrive at
This implies that
By virtue of the restrictions (R1) and (R2), we obtain from (2.17) that
Next, we show that
Indeed, by using (2.18), we obtain that
On the other hand, we have
Substituting (2.23) into (2.22), we arrive at
This in turn gives that
In view of the restrictions (R1) and (R2), we obtain from (2.17) that (2.21) holds.
On the other hand, we see from (2.22) that
It follows that
This implies that
In view of the restrictions (R1), (R2) and (R3), we see from (2.17) that
On the other hand, we see from Lemma 1.2 that
This in turn implies that
Combining (2.24) with (2.26), we arrive at
It follows that
Thanks to the restrictions (R2) and (R3), we see from (2.17) and (2.25) that
In view of the firm nonexpansivity of PC, we see that
which implies that
Substituting (2.28) into (2.22), we arrive at
from which it follows that
In view of the restrictions (R2) and (R3), we obtain from (2.17) and (2.21) that
In a similar way, we can obtain that
Note that
It follows that
In view of the restrictions (R2) and (R3), we obtain from (2.17) that
Notice that
From (2.27), (2.29), (2.30) and (2.31), we arrive at
Next, we prove that
where z = PΩ(z). To see this, we choose a subsequence {} of {xn} such that
Since {} is bounded, there exists a subsequence {} of {} which converges weakly to w. Without loss of generality, we may assume that
w. On the other hand, we have
It follows from (2.27), (2.29) and (2.30) that
Therefore, we see that
w. First, we prove that w ∈ VI(C, A1). For the purpose, let T be the maximal monotone mapping defined by:
For any given (x, y) ∈ G(T), hence y A1x ∈ NC. Since yn ∈ C, by the definition of NC, we have
Notice that
It follows that
and hence
From the monotonicity of A1, we see that
Since
w and A1 is Lipschitz continuous, we obtain from (2.30) that ‹x w, y› > 0. Notice that T is maximal monotone, hence 0 ∈ Tw. This shows that w ∈ VI(C, A1). It follows from (2.27) and (2.29), we also have
Therefore, we obtain
w. Similarly, we can prove w ∈ VI(C, A2). That is, w ∈ VI = VI(C, A2) ∩ VI(C, A1).Next, we show that w ∈ FP = F(Si). Suppose the contrary, w ∉ FP, i.e., Ww ≠ w. Since
w, we see from Opial condition that
On the other hand, we have
From Lemma 1.6, we obtain from (2.32) that limn→∞║Wyn yn║ = 0, which combines with (2.36) yields that that
This derives a contradiction. Thus, we have w ∈ FP.
Next, we show that w ∈ EP(F, A3). It follows from (2.27) that un w. Since un = (I - r A3)xn, for any y ∈ C, we have
From the assumption (A2), we see that
Replacing n by ni, we arrive at
Putting yt = ty + (1 t)w for any t ∈ (0,1] and y ∈ C, we see that yt ∈ C. It follows from (2.37) that
In view of the monotonicity of A3, (2.27) and the restriction (R1), we obtain from the assumption (A4) that
From the assumptions (A1) and (A4), we see that
from which it follows that
It follows from the assumption (A3) that w ∈ EP(F, A3). On the other hand, we see from (2.33) that
Finally, we show that xn → z, as n → ∞. Note that
which implies that
From the restriction (R2), we obtain from Lemma 1.3 that limn→∞║xn z║ = 0. This completes the proof.
□
Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C × C to which satisfies (A1)-(A4). Let A1: C → H be a δ1-inverse-strongly monotone mapping, A2 : C → H be a δ2-inverse-strongly monotone mapping and {Si : C → C} be a family of infinitely nonexpansive mappings. Assume that Ω := FP ∩ EP(F) ∩ VI ≠ , where FP = F(Si) and VI = VI(C, A1) ∩ VI(C, A2). Let : C → C be an α-contraction. Let x1∈ C and {xn} be a sequence generated by
where {Wn : C → C} is the sequence generated in (1.9), {αn}, {βn} and {γn} are sequences in (0,1) such that αn + βn + γn = 1 for each n > 1 and {rn}, {λn} and {ηn} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
(R1) 0 < a < ηn < b < 2δ1, 0 < a' < λn < b' < 2δ2, 0 < < rn < < 2δ3, ∀n > 1;
(R2) limn→∞αn = 0 and αn = ∞;
(R3) 0 < lim infn→∞βn< lim supn→∞βn < 1;
(R4) limn→∞(λn λn+1) = limn→∞(ηn ηn+1) = limn→∞(rn rn+1) = 0.
Then the sequence {xn} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:
Proof. Putting A3≡ 0, we see that
for all δ ∈ (0, ∞). We can conclude from Theorem 2.1 the desired conclusion easily. This completes the proof.
□
Remark 2.3. If A1≡ A2 and λn≡ ηn, then Corollary 2.2 is reduced to Theorem 3.1 of Chang et al. [5]. If A2≡ 0, (x) ≡ e ∈ C a arbitrary fixed point and Si≡ I, the identity mapping, then Corollary 2.2 is reduced to Theorem 3.1 of Plubtieng and Punpaeng [11].
Corollary 2.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let A1 : C → H be a δ1-inverse-strongly monotone mapping, A2 : C → H be a δ2-inverse-strongly monotone mapping, A3 : C→ H be a δ3-inverse-strongly monotone mapping and {Si : C → C} be a family of infinitely nonexpansive mappings. Assume that Ω := FP ∩ EP(F, A3) ∩ VI ≠ , where FP = F(Si) and VI = VI(C, A1) ∩ VI(C, A2). Let : C → C be an α-contraction. Let x1∈ C and {xn} be a sequence generated by
where {Wn : C → C} is the sequence generated in (1.9), {αn}, {βn} and {γn} are sequences in (0,1) such that αn + βn + γn = 1 for each n > 1 and {rn}, {λn} and {ηn} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
(R1) 0 < a < ηn< b < 2δ1, 0 < a' < λn< b' < 2δ2, 0 < < rn < < 2δ3, ∀n > 1;
(R2) limn→∞αn = 0 and αn = ∞;
(R3) 0 < lim infn→∞βn< lim supn→∞βn < 1;
(R4) limn→∞(λn λn+1) = limn→∞(ηn ηn+1) = limn→∞(rn rn+1) = 0.
Then the sequence {xn} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:
Proof. Putting F ≡ 0, we see that
is equivalent to
This implies that
From the proof of Theorem 2.1, we can conclude the desired conclusion immediately. This completes the proof.
□
Remark 2.5. Corollary 2.4 includes Theorem 3.1 of Yao and Yao [25] as a special case, see [25] for more details.
As some applications of our main results, we can obtain the following results.
Recall that a mapping T : C → C is said to be a k-strict pseudo-contraction if there exists a constant k ∈ [0,1) such that
Note that the class of k-strict pseudo-contractions strictly includes the class of nonexpansive mappings.
Put A = I - T, where T : C → C is a k-strict pseudo-contraction. Then A is -inverse-strongly monotone; see [1] for more details.
Corollary 2.6. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C × C to which satisfies (A1)-(A4). Let T1 : C → C be a k1-inverse-strongly monotone mapping, T2 : C → C be a k2-inverse-strongly monotone mapping, T3 : C → C be a k3-inverse-strongly monotone mapping and {Si: C → C} be a family of infinitely nonexpansive mappings. Assume that Ω: = FP ∩ EP(F, I T3) ∩ VI ≠ , where FP = F(Si) and VI = F(T1) ∩ F(T2). Let : C → C be an α-contraction. Let x1∈ C and {xn} be a sequence generated by
where {Wn : C → C} is the sequence generated in (1.9), {αn}, {βn} and {γn} are sequences in (0,1) such that αn+ βn + γn = 1 for each n > 1 and {rn}, {λn} and {ηn} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
(R1) 0 < a < ηn< b < (1 k1), 0 < a' < λn< b' < (1 k2), 0 < < rn < < (1 k3), ∀n > 1;
(R2) limn→∞αn = 0 and αn = ∞;
(R3) 0 < lim infn→∞βn< lim supn→∞βn < 1;
(R4) limn→∞(λn λn+1) = limn→∞(ηn ηn+1) = limn→∞(rn rn+1) = 0.
Then the sequence {xn} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:
and
□
3 Conclusion
The iterative process (2.1) presented in this paper which can be employed to approximate common elements in the solution set of the generalized equilibrium problem (1.3), in the solution set of the classical variational inequality (1.1) and in the common fixed point set of a family nonexpansive mappings is general.It is of interest to improve the main results presented in this paper to the framework of real Banach spaces.
Acknowledgments. The authors are extremely grateful to the editor and the referees for useful suggestions that improve the contents of the paper.
Received: 08/IV/09.
Accepted: 07/XII/09.
#CAM-90/09.
- [1] F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space J. Math. Anal. Appl., 20 (1967), 197-228.
- [2] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems Math. Stud., 63 (1994), 123-145.
- [3] L.C. Ceng and J.C. Yao, Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings Appl. Math. Comput., 198 (2008), 729-741.
- [4] L.C. Ceng and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities Math. Meth. Oper. Res., 67 (2008), 375-390.
- [5] S.S. Chang, H.W.J. Lee and C.K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization Nonlinear Anal., 70 (2009), 3307-3319.
- [6] V. Colao, G. Marino and H.K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems J. Math. Anal. Appl., 344 (2008), 340-352.
- [7] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces J. Nonlinear Convex Anal., 6 (2005), 117-136.
- [8] H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings Nonlinear Anal., 61 (2005), 341-350.
- [9] A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems J. Nonlinear Convex Anal., 9 (2008), 37-43.
- [10] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings Bull. Amer. Math. Soc., 73 (1967), 595-597.
- [11] S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings Appl. Math. Comput., 197 (2008), 548-558.
- [12] X. Qin, M. Shang and H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces Appl. Math. Comput., 200 (2008), 242-253.
- [13] X. Qin, M. Shang and Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces Nonlinear Anal., 69 (2008), 3897-3909.
- [14] X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems Math. Comput. Modelling, 48 (2008), 1033-1046.
- [15] X. Qin, Y.J. Cho and S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces J. Comput. Appl. Math., 225 (2009), 20-30.
- [16] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators Trans. Amer. Math. Soc., 149 (1970), 75-88.
- [17] T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochne integrals J. Math. Anal. Appl., 305(2005), 227-239.
- [18] Y. Su, M. Shang and X. Qin, An iterative method of solution for equilibrium and optimization problems Nonlinear Anal., 69 (2008), 2709-2719.
- [19] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications Taiwanese J. Math., 5 (2001), 387-404.
- [20] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces J. Math. Anal. Appl., 331 (2007), 506-515.
- [21] W. Takahashi, Nonlinear Functional Analysis Yokohama Publishers, Yokohama (2000).
- [22] S. Takahashi and W. Takahahsi, Strong convergence theorem of a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space Nonlinear Anal., 69 (2008), 1025-1033.
- [23] H.K. Xu, Iterative algorithms for nonlinear operators J. London Math. Soc., 66 (2002), 240-256.
- [24] Y. Yao, M.A. Noor and Y.C. Liou, On iterative methods for equilibrium problems Nonlinear Anal., 70 (2009), 497-509.
- [25] Y. Yao and J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings Appl. Math. Comput., 186 (2007), 1551-1558.
Publication Dates
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Publication in this collection
22 Nov 2010 -
Date of issue
2010
History
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Received
08 Apr 2009 -
Accepted
07 Dec 2009