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
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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.
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Acknowledgements
This work was supported by the Kyungnam University Foundation Grant 2010.
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Kim, J.K. Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasiϕnonexpansive mappings. Fixed Point Theory Appl 2011, 10 (2011). https://doi.org/10.1186/16871812201110
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DOI: https://doi.org/10.1186/16871812201110