Research  Open  Published:
Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasiϕnonexpansive mappings
Fixed Point Theory and Applicationsvolume 2011, Article number: 10 (2011)
Abstract
We consider a hybrid projection method for finding a common element in the fixed point set of an asymptotically quasiϕnonexpansive mapping and in the solution set of an equilibrium problem. Strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which has the KadecKlee property.
2000 Mathematics subject classification: 47H05, 47H09, 47H10, 47J25
1. Introduction and Preliminaries
Let E be a real Banach space, E* the dual space of E and C a nonempty closed convex subset of E. Let f be a bifunction from C × C to ℝ, where ℝ denotes the set of real numbers.
In this paper, we consider the following equilibrium problem. Find p ∈ C such that
We denote EP(f) the solution set of the equilibrium problem (1.1). That is,
Given a mapping Q : C → E*, let
Then p ∈ EP(f) if and only if p is a solution of the following variational inequality problem. Find p such that
Numerous problems in physics, optimization and economics reduce to find a solution of (1.1) (see [1–4]). Let T : C → C be a mapping.
The mapping T is said to be asymptotically regular on C if for any bounded subset K of C,
The mapping T is said to be closed if for any sequence {x_{ n } } ⊂ C such that
and
then Tx_{0} = y_{0}.
A point x ∈ C is a fixed point of T provided Tx = x. In this paper, we denote F(T) the fixed point set of T and denote → and ⇀ the strong convergence and weak convergence, respectively.
Recall that the mapping T is said to be nonexpansive if
T is said to be quasinonexpansive if F(T) ≠ Ø and
T is said to be asymptotically nonexpansive if there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 as n → ∞ such that
T is said to be asymptotically quasinonexpansive if F(T) ≠ Ø and there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 as n → ∞ such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] in 1972. They proved that if C is nonempty bounded closed and convex then every asymptotically nonexpansive selfmapping T on C has a fixed point in uniformly convex Banach spaces. Further, the fixed point set of T is closed and convex.
Recently, many authors considered the problem of finding a common element in the set of fixed points of a nonexpansive mapping and in the set of solutions of the equilibrium problem (1.1) based on iterative methods in the framework of real Hilbert spaces; see, for instance [4, 6–14] and the references therein. However, there are few results presented in Banach spaces.
In this paper, we will consider the problem in a Banach space. Before proceeding further, we give some definitions and propositions in Banach spaces.
Let E be a Banach space with the dual E*. We denote by J the normalized duality mapping from E to 2^{E*}defined by
where 〈•,•〉 denotes the generalized duality pairing.
A Banach space E is said to be strictly convex if for all x, y ∈ E with x = y = 1 and x ≠ y. It is said to be uniformly convex if lim_{n→∞}x_{ n } y_{ n } = 0 for any two sequences {x_{ n }} and {y_{ n }} in E such that x_{ n } = y_{ n } = 1 and
Let U_{ E } = {x ∈ E : x = 1} be the unit sphere of E. Then the Banach space E is said to be smooth provided
exists for each x, y ∈ U_{ E } . It is said to be uniformly smooth if the limit (1.3) is attained uniformly for x, y ∈ U_{ E } . It is well known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. It is also well known that if E is uniformly smooth if and only if E* is uniformly convex.
Recall that a Banach space E has the KadecKlee property [15–17], if for any sequence {x_{ n } } ⊂ E and x ∈ E with x_{ n } ⇀ x and x_{ n }  → x, then x_{ n }  x → 0 as n → ∞. It is well known that if E is a uniformly convex Banach space, then E has the KadecKlee property.
As we all know that if C is a nonempty closed convex subset of a Hilbert space H and P_{ C } : H → C is the metric projection of H onto C, then P_{ C } is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [18] recently introduced a generalized projection operator Π _{ C } in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that, in a Hilbert space H, (1.4) is reduced to ϕ(x, y) = xy^{2} , x, y ∈ H. The generalized projection Π _{ C } : E → C is a mapping that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(x, y), that is, , where is the solution to the minimization problem
The 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, [15, 17–19]). We know that Π _{ C } = P_{ C } in Hilbert spaces. It is obvious from the definition of function ϕ that
Remark 1.1. Let E be 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 (1.5), we have x = y. This implies that 〈x, Jy〉 = x^{2} = Jy^{2}. From the definition of J, we have Jx = Jy. Therefore, we have x = y (see [15, 17]).
Let C be a nonempty closed convex subset of E and T a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of T[20] if C contains a sequence {x_{ n } } which converges weakly to p such that
The set of asymptotic fixed points of T will be denoted by .
A mapping T from C into itself is said to be relatively nonexpansive [21–23] if and
for all x ∈ C and p ∈ F(T).
The mapping T is said to be relatively asymptotically nonexpansive [24] if and there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 as n → ∞ such that
for all x ∈ C, p ∈ F(T) and n ≥ 1. The asymptotic behavior of a relatively nonexpansive mapping was studied in [21–23].
The mapping T is said to be ϕnonexpansive if
for all x, y ∈ C.
The mapping T is said to be quasiϕnonexpansive [25–27] if F(T) ≠ ∅ and
for all x ∈ C and p ∈ F(T).
The mapping T is said to be asymptotically ϕnonexpansive if there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 as n → ∞ such that
for all x, y ∈ C.
The mapping T is said to be asymptotically quasiϕnonexpansive [27, 28] if F(T) ≠ ∅ and there exists a sequence {k_{ n } } ⊂ [0, ∞) with k_{ n } → 1 as n → ∞ such that
for all x ∈ C, p ∈ F(T) and n ≥ 1.
Remark 1.2. The class of (asymptotically) quasiϕnonexpansive mappings is more general than the class of relatively (asymptotically) nonexpansive mappings which requires the restriction: . In the framework of Hilbert spaces, (asymptotically) quasiϕnonexpansive mappings is reduced to (asymptotically) quasinonexpansive mappings (cf. [29–32]).
We assume that f satisfies the following conditions for studying the equilibrium problem (1.1).
(A1): f(x, x) = 0∀x ∈ C;
(A2): f is monotone, i.e., f(x, y) + f(y, x) ≤ 0∀x, y ∈ C;
(A3): lim sup_{t↓0}f (tz + (1  t)x, y) ≤ f(x, y)∀x, y, z ∈ C;
(A4): for each x ∈ C, y α f(x, y) is convex and weakly lower semicontinuous.
Recently, Takahashi and Zembayshi [33] considered the problem of finding a common element in the fixed point set of a relatively nonexpansive mapping and in the solution set of the equilibrium problem (1.1) (cf. [32]).
Theorem TZ. ([33]) Let E be a uniformly smooth and uniformly convex Banach space and let C be a nonempty closed convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4) and let T be a relatively nonexpansive mapping from C into itself such that F(T) ∩ EP(f) ≠ Ø. Let {x_{ n } } be a sequence generated by
for every n ≥ 0, where J is the duality mapping on E, {α_{ n } } ⊂ [0, 1] satisfies
and {r_{ n }} ⊂ [a, ∞) for some a > 0. Then {x_{ n }} converges strongly to ∏_{F(T)∩EP(f)}x, where ∏_{F(T)∩EP(f)}is the generalized projection of E onto F (T) ∩ EP (f ).
Very recently, Qin et al. [25] further improved Theorem TZ by considering shrinking projection methods which were introduced in [34] for quasiϕnonexpansive mappings in a uniformly convex and uniformly smooth Banach space.
Theorem QCK. [25]Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Ban ach space E. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4) and let T : C → C be a closed quasiϕnonexpansive mappings such that. Let {x_{ n }} be a sequence generated in the following manner:
where J is the duality mapping on E and {α_{ n } } is a sequence in [0, 1] satisfying
and {r_{ n } } ⊂ [a, ∞) for some a > 0. Then {x_{ n } } converges strongly to.
In this paper, we considered the problem of finding a common element in the fixed point set of an asymptotically quasiϕnonexpansive mapping which is an another generalization of asymptotically nonexpansive mappings in Hilbert spaces and in the solution set of the equilibrium problem (1.1). The results presented this paper mainly improve the corresponding results announced in [33].
In order to prove our main results, we need the following lemmas.
Lemma 1.3. [18]Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E. Then x_{0} = ∏_{ C }x if and only if
Lemma 1.4. [18]Let E be a reflexive, strictly convex and smooth Banach space, C a nonempty closed convex subset of E and x ∈ E. Then
Lemma 1.5. Let E be a strictly convex and smooth Banach space, C a nonempty closed convex subset of E and T : C → C a quasi ϕ nonexpansive mapping. Then F(T) is a closed convex subset of C.
Proof. Let {p_{ n } } be a sequence in F(T ) with p_{ n } → p as n → ∞. Then we have to prove that p ∈ F(T) for the closedness of F(T). From the definition of T, we have
which implies that ϕ(p_{ n } , Tp) → 0 as n → ∞. Note that
Letting n → ∞ in the above equality, we see that ϕ(p, Tp) = 0. This shows that p = Tp.
Next, we show that F(T) is convex. To end this, for arbitrary p_{1}, p_{2} ∈ F (T), t ∈ (0, 1), putting p_{3} = tp_{1} + (1  t)p_{2}, we prove that Tp_{3} = p_{3}. Indeed, from the definition of ϕ, we see that
This implies that p_{3} ∈ F (T ). This completes the proof.
Now we will improve the above Lemma 1.6 as follows.
Lemma 1.6. Let E be a uniformly smooth and strictly convex Banach space which has the KadecKlee property, C a nonempty closed convex subset of E and T : C → C a closed and asymptotically quasi ϕ nonexpansive mapping. Then F(T) is a closed convex subset of C.
Proof. It is easy to check that the closedness of F(T) can be deduced from the closedness of T. We mainly show that F(T) is convex. To end this, for arbitrary p_{1}, p_{2} ∈ F(T), t ∈ (0, 1), putting p_{3} = tp_{1} + (1  t)p_{2}, we prove that Tp_{3} = p_{3}.
Indeed, from the definition of ϕ, we see that
This implies that
From (1.5), we see that
It follows that
This shows that the sequence {J(T^{n}p_{3})}is bounded. Note that E* is reflexive; we may, without loss of generality, assume that J(T^{n}p_{3}) ⇀ e* ∈ E*. In view of the reflexivity of E, we have J(E) = E*. This shows that there exists an element e ∈ E such that Je = e*. It follows that
Taking lim inf_{n→ ∞}on the both sides of above equality, we obtain that
This implies that p_{3} = e, that is, Jp_{3} = e*. It follows that J(T^{n}p_{3}) ⇀ Jp_{3} ∈ E*.
In view of the KadecKlee property of E* and (1.9), we have
Note that J^{1} : E* → E is demicontinuous, we see that T^{n} p_{3} ⇀ p_{3}. By virtue of the KadecKlee property of E and (1.8), we have T^{n}p_{3} → p_{3} as n → ∞. Hence
as n → ∞. In view of the closedness of T, we can obtain that p_{3} ∈ F (T). This shows that F(T) is convex. This completes of proof
Lemma 1.7. [35, 36]Let E be a smooth and uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous and convex function g : [0, 2r] → R such that g(0) = 0 and
for all x, y ∈ B_{ r } = {x ∈ E : x ≤ r} and t ∈ [0, 1].
Lemma 1.8. Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4). Let r > 0 and x ∈ E. Then we have the followings.
(a): ([1]) There exists z ∈ C such that
(b): (Refs. [25, 33]) Define a mapping T_{ r } : E → C by
Then the following conclusions hold:
(1): S_{ r } is singlevalued;
(2): S_{ r } is a firmly nonexpansivetype mapping, i.e., for all x, y ∈ E,
(3): F(S_{ r } ) = EP)(f);
(4): S_{ r } is quasi ϕ nonexpansive;
(5): ϕ(q, S_{ r }x) + ϕ(S_{ r }x, x) ≤ ϕ (q, x), ∀q ∈ F(S_{ r } );
(6): EP(f) is closed and convex.
2. Main Results
Theorem 2.1. Let E be a uniformly smooth and strictly convex Banach space which has the KadecKlee property and C a nonempty closed convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4) and T : C → C a closed and asymptotically quasiϕnonexpansive mapping. Assume that T is asymptotically regular on C andis nonempty and bounded. Let {x_{ n } } be a sequence generated in the following manner:
wherefor each n ≥ 1, {α_{ n }} is a real sequence in [0, 1] such that lim inf_{n→ ∞}α_{ n }(1  α_{ n }) > 0, {r_{ n }} is a real sequence in [a, ∞), where a is some positive real number and J is the duality mapping on E. Then the sequence {x_{ n }} converges strongly to, whereis the generalized projection from E onto.
Proof. First, we show that C_{ n } is closed and convex by induction on n ≥ 1. It is obvious that C_{1} = C is closed and convex. Suppose that C_{ m } is closed and convex for some integer m. For z ∈ C_{ m } , we see that ϕ(z, u_{ m } ) ≤ ϕ(z, x_{ m } ) + (k_{ m } 1)M_{ m } is equivalent to
It is easy to see that C_{m+1}is closed and convex. This proves that C_{ n } is closed and convex for each n ≥ 1. This in turn shows that x_{0} is well defined. Putting , we from Lemma 1.8 see that is quasiϕnonexpansive.
Now, we are in a position to prove that for each n ≥ 1. Indeed, is obvious. Suppose that for some positive integer m. Then, , we have
which shows that w ∈ C_{m+1}. This implies that for each n ≥ 1.
On the other hand, it follows from Lemma 1.4 that
for each and for each n ≥ 1. This shows that the sequence ϕ(x_{ n }, x_{0}) is bounded. From (1.5), we see that the sequence {x_{ n }} is also bounded. Since the space is reflexive, we may, without loss of generality, assume that x_{ n } ⇀ p. Not that C_{ n } is closed and convex for each n ≥ 1. It is easy to see that p ∈ C_{ n } for each n ≥ 1. Note that
It follows that
This implies that
Hence, we have x_{ n }  → p as n → ∞. In view of the KadecKlee property of E, we obtain that x_{ n } → p as n → ∞.
Next, we show that p ∈ F(T). By the construction of C_{ n }, we have that C_{n+1}⊂ C_{ n }and . It follows that
Letting n → ∞, we obtain that ϕ(x_{n+1}, x_{ n }) → 0. In view of x_{n+1}∈ C_{n+1}, we have
It follows that
From (1.5), we see that
It follows that
This implies that {Ju_{ n } } is bounded. Note that E is reflexive and E* is also reflexive. We may assume that Ju_{ n } → x* ∈ E*. In view of the reflexivity of E, we see that J(E) = E*. This shows that there exists an x ∈ E such that Jx = x*. It follows that
Taking lim inf_{n→∞}the both sides of above equality yields that
That is, p = x, which in turn implies that x* = Jp. It follows that Ju_{ n } → Jp ∈ E*. From (2.4) and E* has the KadecKlee property, we obtain that
Note that J^{1} : E* → E is demicontinuous. It follows that u_{ n } → p. From (2.3) and E has the KadecKlee property, we obtain that
Note that
It follows that
Since J is uniformly normtonorm continuous on any bounded sets, we have
Let r = sup_{n≥0}{x_{ n }, T^{n}x_{ n }}. Since E is uniformly smooth, we know that E* is uniformly convex. In view of Lemma 1.7, we see that
It follows that
On the other hand, we have
It follows from (2.6) and (2.7) that
In view of lim_{n→∞}(k_{ n } 1) M_{ n } = 0 and (2.8) and the assumption lim inf_{n→∞}α_{ n }(1  α_{ n }) > 0, we see that
It follows from the property of g that
Since x_{ n } → p as n →∞ and J : E → E* is demicontinuous, we obtain that Jx_{ n } → Jp ∈ E*. Note that
This implies that Jx_{ n }  → Jp as n → ∞. Since E* has the KadecKlee property, we see that
Note that
From (2.9) and (2.10), we obtain at
Note that J^{1} : E* → E is demicontinuous. It follows that T^{n}x_{ n } → p. On the other hand, we have
In view of (2.11), we obtain that T^{n}x_{ n }  → p as n → ∞. Since E has the KadecKlee property, we obtain that
Note that
It follows from the asymptotic regularity of T and (2.12) that
That is, TT^{n}x_{ n }  p → 0 as n → ∞: It follows from the closedness of T that Tp = p:
Next, we show that p ∈ EF(f): From (2.1), we have
In view of and Lemma 1.8, we obtain
It follows from (2.8) that
From (1.5), we see that u_{ n }   y_{ n }  → 0 as n → ∞. In view of u_{ n } → p as n → ∞, we have
It follows that
Since E* is reflexive, we may assume that Jy_{ n } → q*∈ E*: In view of J(E) = E*, we see that there exists q ∈ E such that Jq = q*. It follows that
Taking lim inf_{n→∞}the both sides of above equality yields that
That is, p = q, which in turn implies that q* = Jp. It follows that Jy_{ n } → Jp ∈ E*. From (2.16) and E* has the KadecKlee property, we obtain that
Note that J^{1} : E* → E is demicontinuous. It follows that y_{ n } → p. From (2.15) and E has the KadecKlee property, we obtain that
Note that
It follows from (2.5) and (2.17) that
Since J is uniformly normtonorm continuous on any bounded sets, we have
From the assumption r_{ n } ≥ a, we see that
In view of , we see that
It follows from the condition (A 2) that
By taking the limit as n → ∞ in the above inequality, we from conditions (A 4) and (2.19) obtain that
For 0 < t < 1 and y ∈ C, define y_{ t } = ty + (1  t)p. It follows that y_{ t } ∈ C, which yields that f(y_{ t }, p) ≤ 0. It follows from conditions (A 1) and (A 4) that
That is,
Letting t ↓ 0, from condition (A 3), we obtain that f(p, y) ≥ 0 ∀y ∈ C: This implies that p ∈ EP(f). This shows that .
Finally, we prove that . From , we see that
Since for each n ≥ 1, we have
Letting n → ∞ in (2.20), we see that
In view of Lemma 1.3, we can obtain that . This completes the proof.
Remark 2.2. Theorem 2.1 improves Theorem QCK in the following aspects:

(a)
From a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which has the KadecKlee property;

(b)
From a quasiϕnonexpansive mapping to an asymptotically quasiϕnonexpansive mapping.
From the definition of quasiϕnonexpansive mappings, we see that every quasiϕnonexpansive mapping is asymptotically quasiϕnonexpansive with the constant sequence {1}. From the proof of Theorem 2.1, we have the following results immediately.
Corollary 2.3. Let E be a uniformly smooth and strictly convex Banach space which has the KadecKlee property and C a nonempty closed convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4) and T : C → C a closed and quasi ϕ nonexpansive mapping. Assume thatis nonempty.
Let {x_{ n } } be a sequence generated in the following manner:
where {α_{ n }} is a real sequence in [0, 1] such that lim inf_{n→∞}α_{ n }(1  α_{ n }) > 0, {r_{ n }} is a real sequence in [a, ∞), where a is some positive real number and J is the duality mapping on E. Then the sequence {x_{ n }} converges strongly to, whereis the generalized projection from E onto.
Remark 2.4. Corollary 2.3 improves Theorem TZ in the following aspects.

(a)
For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which has the KadecKlee property (note that every uniformly convex Banach space has the KadecKlee property).

(b)
For the mappings, we extend the mapping from a relatively nonexpansive mapping to a quasiϕnonexpansive mapping (we remove the restriction , where denotes the asymptotic fixed point set).

(c)
For the algorithms, we remove the set "W_{ n } " in Theorem TZ.
References
 1.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math Student 1994, 63: 123–145.
 2.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 2005, 6: 117–136.
 3.
Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. J Optim Theory Appl 2003, 117: 245–266. 10.1023/A:1023627606067
 4.
Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035
 5.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc 1972, 35: 171–174. 10.1090/S00029939197202985003
 6.
Ceng LC, Yao JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl Mathe Comput 2008, 198: 729–741. 10.1016/j.amc.2007.09.011
 7.
Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. J Math Anal Appl 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041
 8.
Kim JK, Huang NJ: Approximation methods of solutions for equilibrium problem in Hilbert spaces. Dyn Syst Appl 2008, 17: 503–508.
 9.
Kim JK, Cho SY, Qin X: Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Mathematica Scientia 2011,31B(5):1–17.
 10.
Kim JK, Cho SY, Qin X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J Ineq Appl 2010,2010(312602):1–18.
 11.
Qin X, Shang M, Su Y: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal 2008, 69: 3897–3909. 10.1016/j.na.2007.10.025
 12.
Qin X, Cho SY, Kim JK: On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings. Optimization 2011, 1–17.
 13.
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036
 14.
Wattanawitoon K, Kumam P: Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasinonexpansive mappings. Nonlinear Anal 2009, 3: 11–20.
 15.
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht; 1990.
 16.
Hudzik H, Kowalewski W, Lewicki G: Approximative compactness and full rotundity in MusielakOrlicz spaces and LorentzOrlicz spaces. Zeitschrift fuer Analysis and ihre Anwendungen 2006, 25: 163–192.
 17.
Takahashi W: Nonlinear Functional Analysis. YokohamaPublishers; 2000.
 18.
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Marcel Dekker, New York; 1996.
 19.
Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamer Math J 1994, 4: 39–54.
 20.
Reich S: A weak convergence theorem for the alternating method with Bregman distance. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Marcel Dekker, New York; 1996.
 21.
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J Appl Anal 2001, 7: 151–174. 10.1515/JAA.2001.151
 22.
Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer Funct Anal Optim 2003, 24: 489–508. 10.1081/NFA120023869
 23.
Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operatorswith applications to feasibility and optimization. Optimzation 1996, 37: 323–339. 10.1080/02331939608844225
 24.
Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer Funct Anal Optim 2007, 28: 1197–1215. 10.1080/01630560701766627
 25.
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J Comput Appl Math 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
 26.
Qin X, Cho YJ, Kang SM, Zhou H: Convergence of a modified Halperntype iteration algorithm for quasi ϕ nonexpansive mappings. Appl Math Lett 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015
 27.
Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi ϕ asymptotically nonexpansive mappings. J Appl Math Comput
 28.
Cho YJ, Qin X, Kang SM: Strong convergence of the modified Halperntype iterative algorithms in Banach spaces. An St Univ Ovidius Constanta, Ser Math 2009, 17: 51–68.
 29.
Kim JK, Sahu D, Nam YM: Convergence theorem for fixed points of nearly uniformly LLipschitzian asymptotically generalized phihemicontractive mappings. Nonlinear Anal TMA 2009,71(12):2833–2838. 10.1016/j.na.2009.06.091
 30.
Kim JK, Nam YM, Sim JY: Convergence theorem of implicit iterative sequences for a finite family of asymptotically quasinonexpansive type mappings. Nonlinear Anal TMA 2009,71(12):2839–2848. 10.1016/j.na.2009.06.090
 31.
Saewan S, Kumam P: Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasiasymptotically nonexpansive mappings. Abst Appl Anal 2010, 2010: 22. (Article ID 357120)
 32.
Saewan S, Kumam P: A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasinonexpansive mappings for generalized mixed equilibrium and variational inequality problems. Abst Appl Anal 2010, 2010: 31. (Article ID 123027)
 33.
Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal 2009, 70: 45–57. 10.1016/j.na.2007.11.031
 34.
Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J Math Anal Appl 2008, 341: 276–286. 10.1016/j.jmaa.2007.09.062
 35.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal 1991, 16: 1127–1138. 10.1016/0362546X(91)90200K
 36.
Zǎlinescu C: On uniformly convex functions. J Math Anal Appl 1983, 95: 344–374. 10.1016/0022247X(83)901129
Acknowledgements
This work was supported by the Kyungnam University Foundation Grant 2010.
Author information
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 Asymptotically quasiϕnonexpansive mapping
 Relatively nonexpansive mapping
 Generalized projection
 Equilibrium problem
 Lower semicontinuous