- Open Access
Convergence theorems for equilibrium problem and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense
© Jeong; licensee Springer. 2014
Received: 28 April 2014
Accepted: 2 September 2014
Published: 24 September 2014
In this paper, we introduce an iterative process which converges strongly to a common element of the set of fixed points of an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense and the solution set of generalized equilibrium problem in Banach spaces. Our theorems improve, generalize, and extend several results recently announced.
MSC:47H05, 47H09, 47H10.
Let E be a real Banach space with the dual space . Let C be a nonempty closed convex subset of E. Let be a nonlinear mapping. We denote by the set of fixed points of T.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk  in 1972. In uniformly convex Banach spaces, they proved that if C is nonempty, bounded, closed, and convex, then every asymptotically nonexpansive self-mapping T on C has a fixed point. Further, the fixed point set of T is closed and convex.
If and (1.1) holds for all , , then T is called asymptotically quasi-nonexpansive in the intermediate sense. It is well known that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self-mapping of C which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point (see ). It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.
If E is a reflexive, strictly convex and smooth Banach space, then for , if and only if .
If E is a real Hilbert space, then .
A point p in C is said to be an asymptotic fixed point of T if C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by . A mapping T is called relatively nonexpansive (see ) if and for all and .
Recently, Matsushita and Takahashi  proved strong convergence theorems for approximation of fixed points of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. More precisely, they proved the following theorem.
where J is the normalized duality mapping on E. If is nonempty, then converges strongly to .
Motivated and inspired by the works mentioned above, in this paper, we introduce a new iterative scheme of the generalized f-projection operator for finding a common element of the set of fixed points of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and the solution set of generalized equilibrium problem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
By the Hahn-Banach theorem, is nonempty.
exists for each . It is also called uniformly smooth if the limit exists uniformly for all . In this paper, we denote the strong convergence and weak convergence of a sequence by and , respectively.
If E is a smooth Banach space, then J is single-valued and semicontinuous;
If E is a uniformly smooth Banach space, then J is uniformly norm-to-norm continuous on each bounded subset of E;
If E is a uniformly smooth Banach space, then E is smooth and reflexive;
If E is a reflexive and strictly convex Banach space, then is norm-weak∗-continuous;
E is a uniformly smooth Banach space if and only if is uniformly convex.
Recall that a Banach space E has the Kadec-Klee property if for any sequence and with and , then as . It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.
- (1)quasi-ϕ-nonexpansive if and
for all and ;
- (2)asymptotically quasi-ϕ-nonexpansive in the intermediate sense if and
Definition 2.2 A mapping is said to be closed if for any sequence with and , .
is convex and continuous with respect to when y is fixed;
is convex and lower semicontinuous with respect to y when is fixed.
Definition 2.3 ()
Lemma 2.1 ()
for all .
Lemma 2.2 ()
is a nonempty closed convex subset of C for all ;
- (2)For all , if and only if
If E is strictly convex, then is a single-valued mapping.
for all . The set of solutions of (2.1) is denoted by .
For solving the equilibrium problem for a bifunction , let us assume that θ satisfies the following conditions:
(A1) for all ;
(A2) θ is monotone; i.e., for all ;
(A4) for all , is convex and lower semicontinuous.
Lemma 2.3 ()
- (2)is a firmly nonexpansive-type mapping, i.e., for all ,
is closed and convex;
Lemma 2.4 ()
Let E be a reflexive, strictly convex and smooth Banach space such that both E and have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let be a closed and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Then is a closed convex subset of C.
Lemma 2.5 ()
for all .
3 Main results
where , is a real sequence in for some and is the generalized f-projection operator. Then converges strongly to .
Proof It follows from Lemma 2.3 and Lemma 2.4 that ℱ is a closed convex subset of C, so that is well defined for any .
We split the proof into six steps.
Step 1. We first show that is nonempty, closed, and convex for all .
This implies that is closed and convex for all . This shows that is well defined.
Step 2. We show that for all .
which shows that . This implies that and so for all .
Step 3. We prove that is bounded and exists.
This shows that is nondecreasing. It follows from the boundedness that exists.
Step 4. Next, we prove that , , and as , where is some point in C.
Step 5. We show that .
Thus, for all . Furthermore, as , we have from (A3) that for all . This implies that .
i.e., as . It follows from the closedness of T that . So, and hence .
Step 6. We show that and so as .
and as . This completes the proof. □
If , then and .
- (ii)If we take , , , and for all , then the iterative scheme (3.1) reduces to the following scheme:
where , which is the algorithm introduced by Hao  and an improvement to (1.3).
If T is quasi-ϕ-nonexpansive, then Theorem 3.1 is reduced to following without the boundedness of and the asymptotically regularity of T.
where is a real sequence in for some and is the generalized f-projection operator. Then converges strongly to .
from the relatively nonexpansive mapping to the asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense;
from a uniformly convex and uniformly smooth Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property;
We now provide a nontrivial family of mappings satisfying the conditions of Theorem 3.1.
Then the set of solutions to the equilibrium problem for θ is obviously . Since and is bounded, it follows from Theorem 3.1 that the sequence defined by (3.1) converges strongly to .
JUJ conceived of the study, its design, and its coordination. The author read and approved the final manuscript.
The author is grateful to the anonymous referees for useful suggestions, which improved the contents of the article.
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