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Convergence theorems for uniformly quasiϕasymptotically nonexpansive mappings, generalized equilibrium problems, and variational inequalities
 Siwaporn Saewan^{1} and
 Poom Kumam^{1, 2}Email author
https://doi.org/10.1186/1029242X201196
© Saewan and Kumam; licensee Springer. 2011
Received: 28 June 2011
Accepted: 27 October 2011
Published: 27 October 2011
Abstract
In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings, the set of the variational inequality for an αinversestrongly monotone operator, and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a 2uniformly convex and uniformly smooth Banach space. The results presented in this article improve and extend the recent results of Zegeye [Nonlinear Anal. 72, 21362146 (2010)], Wattanawitoon and Kumam [Nonlinear Anal. Hybrid Syst. 3(1), 1120 (2009)] and many others.
2000 MSC: 47H05, 47H09, 47H10.
Keywords
1 Introduction and preliminaries
 (i)monotone if$\u3008AxAy,xy\u3009\ge 0,\phantom{\rule{1em}{0ex}}\forall x,y\in C,$
 (ii)αinversestrongly monotone if there exists a constant α > 0 such that$\u3008AxAy,xy\u3009\ge \alpha \parallel AxAy{\parallel}^{2},\phantom{\rule{1em}{0ex}}\forall x,y\in C.$
Remark 1.1. It is easy to see that an αinversestrongly monotone is monotone and $\frac{1}{\alpha}$Lipschitz continuous.
The above formulation (1.1) is more general than equilibrium problem (1.2) and cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem, and optimization problem, which can also be written in the form of an GEP(f, B). In other words, the EP(f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many articles have appeared in the literature on the existence of solutions of EP(f); see, for example, [1, 2] and references therein. Some solution methods have been proposed to solve the GEP(f, B) and EP(f); see, for example, [1, 3–13] and references therein.
existence and uniqueness of the operator Π _{ C } follows from the properties of the functional ϕ(x, y) and strict monotonicity of the mapping J (see, for example, [14–18]).
Recall that a point p in C is said to be an asymptotic fixed point of T[19] if C contains a sequence {x_{ n } } which converges weakly to p such that lim_{ n }→_{∞} x_{ n } Tx_{ n }  = 0. The set of asymptotic fixed points of T will be denoted by $\stackrel{\u0303}{F\left(T\right)}$. A mapping T is said to be ϕnonexpansive, if ϕ(Tx, Ty) ≤ ϕ(x, y) for x, y ∈ C.
A mapping T from C into itself is said to be relatively nonexpansive mapping[20–22] if
(R1) F(T) is nonempty;
(R2) ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T);
(R3) $\stackrel{\u0303}{F\left(T\right)}=F\left(T\right)$.
A mapping T is said to be relatively quasinonexpansive (or quasi ϕnonexpansive) if the conditions (R1) and (R2) are satisfied. The asymptotic behavior of a relatively nonexpansive mapping was studied in [23–25].
A mapping T is said to be quasi ϕasymptotically nonexpansive if F(T) ≠ ∅ and there exists a real sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 such that ϕ(p, T^{ n }x) ≤ k_{ nϕ }(p, x) for all n ≥ 1 x ∈ C and p ∈ F(T). We note that the class of relatively quasinonexpansive mappings is more general than the class of relatively nonexpansive mappings [23–27] which requires the strong restriction: $F\left(T\right)=\stackrel{\u0303}{F\left(T\right)}$.
A mapping T is said to be closed if for any sequence {x_{ n } } ⊂ C with x_{ n } → x and Tx_{ n } → y, then Tx = y. It is easy to know that each relatively nonexpansive mapping is closed.
A Banach space E is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all x, y ∈ E with x = y = 1 and x ≠ y. Let U = {x ∈ E : x = 1} be the unit sphere of E. Then, a Banach space E is said to be smooth if the limit ${lim}_{t\to 0}\frac{\parallel x+ty\parallel \parallel x\parallel}{t}$exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U. Let E be a Banach space. The modulus of convexity of E is the function δ : [0, 2] → [0, 1] defined by $\delta \left(\epsilon \right)=inf\left\{1\parallel \frac{x+y}{2}\parallel :x,y\in E,\parallel x\parallel =\parallel y\parallel =1,\parallel xy\parallel \ge \epsilon \right\}$. A Banach space E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0, 2]. Let p be a fixed real number with p ≥ 2. A Banach space E is said to be puniformly convex if there exists a constant c > 0 such that δ(ε) ≥ cε ^{ p } for all ε ∈ [0, 2]; see [28, 29] for more details. Observe that every puniform convex is uniformly convex. One should note that no a Banach space is puniform convex for 1 < p < 2. It is well known that a Hilbert space is 2uniformly convex, uniformly smooth. For each p > 1, the generalized duality mapping${J}_{p}:E\to {2}^{{E}^{*}}$ is defined by J_{ p }>(x) = {x* ∈ E* : 〈x, x*〉 = x^{ p }, x* = x^{p1}} for all x ∈ E. In particular, J = J_{2} is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identity mapping.
Remark 1.2. If E is a reflexive, strictly convex, and smooth Banach space, then for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y. It is sufficient to show that if ϕ(x, y) = 0, then x = y. From (1.4), we have x = y. This implies that 〈x, Jy〉 = x^{2} = Jy^{2}. From the definition of J, one has Jx = Jy. Therefore, we have x = y; see [16, 18] for more details.
 (i)
If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.
 (ii)
If E is a reflexive and strictly convex Banach space, then J ^{1} is normweak*continuous.
 (iii)
If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping $J:E\to {2}^{{E}^{*}}$ is singlevalued, onetoone, and onto.
 (iv)
A Banach space E is uniformly smooth if and only if E* is uniformly convex.
 (v)
Each uniformly convex Banach space E has the KadecKlee property, that is, for any sequence {x_{ n } } ⊂ E, if x_{ n } ⇀ x ∈ E and x_{ n }  → x, then x_{ n } → x.
They proved that {x_{ n } } converges strongly to Π_{F(T)}x_{0}, where Π_{F(T)}is the generalized projection from C onto F(T).
for every n = 1, 2, 3,..., where Π _{ C } is the generalized metric projection from E onto C, J is the duality mapping from E into E*, and {λ_{ n } } is a sequence of positive real numbers. They proved that the sequence {x_{ n } } generated by (1.9) converges weakly to some element of V I(A, C).
In [32, 33], Takahashi and Zembayashi studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Banach spaces. Wattanawitoon and Kumam [34] using the idea of Takahashi and Zembayashi [32] extend the notion from relatively nonexpansive mappings or ϕnonexpansive mappings to two relatively quasinonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasinonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces.
On the other hand, the block iterative method is a method which often used by many authors to solve the convex feasibility problem (see, [11, 35, 36], etc.). In 2008, Plubtieng and Ungchittrakool [37] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. In 2010, Chang et al. [38] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mapping, they obtain the strong convergence theorems in a Banach space.
In this article, motivated and inspired by the study of Chang et al. [38], Qin et al. [9], Takahashi and Zembayashi [32], Wattanawitoon and Kumam [34], and Zegeye [39], we introduce a new modified block hybrid projection algorithm for finding a common element of the set of the variational inequality for an αinversestrongly monotone operator, the set of solutions of the generalized equilibrium problems, and the set of common fixed points of an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings which more general than closed quasiϕnonexpansive mappings in the framework Banach spaces. The results presented in this article improve and generalize the main results of Chang et al. [38], Zegeye [39], Wattanawitoon and Kumam [34], and some wellknown results in the literature.
2 Basic results
We also need the following lemmas for the proof of our main results.
where J is the normalized duality mapping of E and 0 < c ≤ 1.
The best constant $\frac{1}{c}$ in lemma is called the puniformly convex constant of E.
where J_{ p } is the generalized duality mapping of E, and$\frac{1}{c}$is the puniformly convexity constant of E.
Lemma 2.3. (Kamimura and Takahashi[17]). Let E be a uniformly convex and smooth Banach space and let {x_{ n } } and {y_{ n } } be two sequences of E. If ϕ(x_{ n } , y_{ n } ) → 0 and either {x_{ n } } or {y_{ n } } is bounded, then x_{ n }  y_{ n }  → 0.
For solving the equilibrium problem for a bifunction f : C × C → ℝ, let us 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;
(A4) for each x ∈ C, y α f(x, y) is convex and lower semicontinuous.
 (1)
T _{ r } is singlevalued;
 (2)T_{ r } is a firmly nonexpansivetype mapping, for all x, y ∈ E,$\u3008{T}_{r}x{T}_{r}y,J{T}_{r}xJ{T}_{r}y\u3009\le \u3008{T}_{r}x{T}_{r}y,JxJy\u3009;$
 (3)
F(T_{ r } ) = GEP(f, B);
 (4)
GEP(f, B) is closed and convex.
for all x ∈ E and x* ∈ E*, that is, V(x, x*) = ϕ(x, J^{1}(x*)).
for all × ∈ E and x*, y* ∈ E*.
An operator $\mathcal{M}\subset E\times {E}^{*}$ is said to be monotone if 〈x  y, x*  y*〉 ≥ 0 whenever (x, x*), (y, y*) ∈ T. We denote the set {x ∈ E : 0 ∈ Tx} by ${\mathcal{M}}^{1}0$. A monotone $\mathcal{M}$ is said to be maximal if its graph $G\left(\mathcal{M}\right)=\left\{\left(x,y\right):y\in \mathcal{M}x\right\}$ is not property contained in the graph of any other monotone operator. If $\mathcal{M}$ is maximal monotone, then the solution set ${\mathcal{M}}^{1}0$ is closed and convex. Let E be a reflexive, strictly convex, and smooth Banach space, it is known that $\mathcal{M}$ is a maximal monotone if and only if $R\left(J+r\mathcal{M}\right)={E}^{*}$ for all r > 0. Define the resolvent of $\mathcal{M}$ by J_{ r }x = x_{ r }. In other words, ${J}_{r}={(J+r\mathcal{M})}^{1}J$ for all r > 0. J_{ r }is a singlevalued mapping from E to $D\left(\mathcal{M}\right)$. Also, ${\mathcal{M}}^{1}\left(0\right)=F\left({J}_{r}\right)$ for all r > 0, where F(J_{ r }) is the set of all fixed points of J_{ r }. Define, for r > 0, the Yosida approximation of $\mathcal{M}$ by ${\mathcal{M}}_{r}=\left(JJ{J}_{r}\right)\u2215r$. We know that ${\mathcal{M}}_{r}x\in \mathcal{M}\left({J}_{r}x\right)$ for all r > 0 and x ∈ E.
Then, $\mathcal{M}$is maximal monotone and${\mathcal{M}}^{1}0=VI\left(A,C\right)$.
Lemma 2.12. (Chang et al. [38]). Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty closed convex subset of E. Let T : C → C be a closed and quasi ϕasymptotically nonexpansive mapping with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1. Then, F (T ) is a closed convex subset of C.
3 Main results
In this section, we prove the new convergence theorems for finding the set of solutions of a general equilibrium problems, the common fixed point set of a family of closed and uniformly quasiϕasymptotically nonexpansive mappings, and the solution set of variational inequalities for an αinverse strongly monotone mapping in a 2uniformly convex and uniformly smooth Banach space.
where ζ_{ n }= sup_{q∈F}(k_{ n }1)ϕ(q, x_{ n }), {α_{n,i}} is sequence in [0, 1], {r_{ n }} ⊂ [d, ∞) for some d > 0 and {λ_{ n }} ⊂ [a, b] for some a,b with 0 < a < b < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If${\sum}_{i=0}^{\infty}{\alpha}_{n,i}=1$for all n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n }} converges strongly to p ∈ F, where p = Π_{ F }x_{0}.
Since lim_{n→∞}ϕ(x_{ n } , x_{0}) exists and we taking m, n → ∞, then we get ϕ(x_{ m } , x_{ n } ) → 0. From Lemma 2.3, we have lim_{n→∞}x_{ m } x_{ n }  = 0. Thus, {x_{ n } } is a Cauchy sequence and by the completeness of E and there exist a point p ∈ C such that x_{ n } → p as n → ∞.
Next, we will show that $p\in F:=GEP\left(f,B\right)\cap \left({\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\right)\cap VI\left(A,C\right).$
This implies that p ∈ GEP(f, B).
By (3.15) and (3.35), it yields that $\parallel {S}_{i}^{n+1}{z}_{n}{S}_{i}^{n}{z}_{n}\parallel \to 0$. From ${S}_{i}^{n}{z}_{n}\to p$, we have ${S}_{i}^{n+1}{z}_{n}\to p$, that is, ${S}_{i}{S}_{i}^{n}{z}_{n}\to p$. In view of closeness of S_{ i } , we have S_{ i }p = p, for all i ≥ 1. This imply that $p\in {\cap}_{i=1}^{\infty}F\left({S}_{i}\right)$
By Lemma 2.10, $\mathcal{M}$ is maximal monotone and ${\mathcal{M}}^{1}0=VI\left(A,C\right)$. Let $\left(v,w\right)\in G\left(\mathcal{M}\right)$. Since $w\in \mathcal{M}v=Av+{N}_{C}\left(v\right)$, we get w  Av ∈ N_{ C } (v).
where M = sup_{n≥1}v  z_{ n } . Take the limit as n → ∞ and (3.28), we obtain 〈v  p, w〉 ≥ 0. By the maximality of $\mathcal{M}$, we have $p\in {\mathcal{M}}^{1}0$, that is, p ∈ V I(A, C).
By Lemma 2.4, we can conclude that p = Π _{ F }x_{0} and x_{ n } → p as n → ∞. This completes the proof. □
If S_{ i } = S for each i ∈ ℕ, then Theorem 3.2 is reduced to the following corollary.
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)ϕ(q, x_{ n } ), {α_{ n } } is sequence in [0, 1], {r_{ n } } ⊂ [d, ∞) for some d > 0 and {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If lim inf_{n→∞}α_{ n }(1  α_{ n } ) > 0, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F } x_{0}.
For a special case that i = 1, 2, we can obtain the following results on a pair of quasiϕasymptotically nonexpansive mappings immediately from Theorem 3.2.
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)ϕ(q, x_{ n } ), {α_{ n } }, {β_{ n } } and {γ_{ n } } are sequences in [0, 1], {r_{ n } } ⊂ [d, ∞) for some d > 0 and {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If α_{ n } + β_{ n } + γ_{ n } = 1 for all n ≥ 0 and lim inf_{n→∞}α_{ n }β_{ n }> 0 and lim inf_{n→∞}α_{ n }γ_{ n }> 0, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
where {α_{n,i}} is sequence in [0, 1], {r_{ n }} ⊂ [d, ∞) for some d > 0 and {λ_{ n }} ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If${\sum}_{i=0}^{\infty}{\alpha}_{n,i}=1$for all n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Proof Since ${\left\{{S}_{i}\right\}}_{i=1}^{\infty}:C\to C$ is an infinite family of closed quasiϕnonexpansive mappings, it is an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings with sequence k_{ n } = 1. Hence, the conditions appearing in Theorem 3.2 F is a bounded subset in C and for each i ≥ 1, S_{ i } is uniformly L_{ i } Lipschitz continuous are of no use here. By virtue of the closeness of mapping S_{ i } for each i ≥ 1, it yields that p ∈ F(S_{ i } ) for each i ≥ 1, that is, $p\in {\cap}_{i=1}^{\infty}F\left({S}_{i}\right)$. Therefore, all the conditions in Theorem 3.2 are satisfied. The conclusion of Corollary 3.5 is obtained from Theorem 3.2 immediately. □
where {α_{n,i}} is sequence in [0, 1], {r_{ n } } ⊂ [d, ∞) for some d > 0 and {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/ 2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If α_{ i } ∈ (0, 1) such that${\sum}_{i=0}^{N}{\alpha}_{i}=1$, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Remark 3.7. Theorem 3.2, Corollaries 3.5 and 3.6 improve and extend the corresponding results in Wattanawitoon and Kumam [34] and Zegeye [39] in the following senses:

from a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family of quasiϕasymptotically mappings;

for the mappings, we extend the mappings from nonexpansive mappings, relatively quasinonexpansive mappings or quasiϕnonexpansive mappings and a finite family of closed relatively quasinonexpansive mappings to an infinite family of quasiϕasymptotically nonexpansive mappings.
4 Deduced theorems
where ζ_{ n }= sup_{q∈F}(k_{ n } 1)ϕ(q, x_{ n }), {α_{n,i}} is sequence in [0, 1], {r_{ n }} ⊂ [a, ∞) for some a > 0. If${\sum}_{i=0}^{\infty}{\alpha}_{n,i}=1$for all n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n }} converges strongly to p ∈ F, where p = Π_{ F }x_{0}.
Proof Put A ≡ 0 in Theorem 3.2. Then, we get that z_{ n } = x_{ n } . Thus, the method of proof of Theorem 3.2 gives the required assertion without the requirement that E be 2uniformly convex. □
If setting B ≡ 0 in Corollary 4.1, then we have the following corollary.
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)ϕ(q, x_{ n } ), {α_{n,i}} is sequence in [0, 1], {r_{ n } } ⊂ [a, ∞) for some a > 0. If${\sum}_{i=0}^{\infty}{\alpha}_{n,i}=1$for all n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n } } converges strongly to p ∈ F, where p = Π_{ F }x_{0}.
If setting f ≡ 0 in Corollary 4.1, then we obtain the following corollary.
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)ϕ(q, x_{ n }), {α_{n,i}} is sequence in [0, 1], {r_{ n }} ⊂ [a, ∞) for some a > 0. If${\sum}_{i=0}^{\infty}{\alpha}_{n,i}=1$for all n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n }} converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Remark 4.4. Corollaries 4.1{4.3 improve and extend the corresponding results in Zegeye [39] and Wattanawitoon and Kumam [34] in the sense from a finite family of closed relatively quasinonexpansive mappings and closed relatively quasinonexpansive mappings to more general than an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings.
5 Application to Hilbert spaces
If E = H, a Hilbert space, then E is 2uniformly convex (we can choose c = 1) and uniformly smooth real Banach space and closed relatively quasinonexpansive map reduces to closed quasinonexpansive map. Moreover, J = I, identity operator on H and Π _{ C } = P_{ C } , projection mapping from H into C. Thus, the following corollaries hold.
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)q  x_{ n } , {α_{n,i}} is sequence in [0, 1], {r_{ n } } ⊂ [a, ∞) for some a > 0 and {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < α/2. If${\sum}_{i=0}^{\infty}{\alpha}_{n,i}=1$for all n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Remark 5.2. Theorem 5.1 improve and extend the Corollary 3.7 in Zegeye [39] in the aspect for the mappings, we extend the mappings from a finite family of closed relatively quasinonexpansive mappings to more general an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings.
Declarations
Acknowledgements
Ms. Siwaporn Saewan would like to thank the Office of the Higher Education Commission, Thailand, for supporting by grant found under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree for this research. I thank Dr. Poom Kumam for clarifying several points in my research. Moreover, this research was supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand (under the Project no. RG154021).
Authors’ Affiliations
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