Abstract
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.
Similar content being viewed by others
1 Introduction
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.
A mapping T is said to be asymptotically nonexpansive if there exists a sequence with as such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] 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.
A mapping T is said to be asymptotically nonexpansive in the intermediate sense (see [2]) if it is continuous and the following inequality holds:
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 [3]). 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.
Iterative approximation of a fixed point for asymptotically nonexpansive mappings in Hilbert or Banach spaces has been studied extensively by many authors (see [4–6] and the references therein).
Let E be a smooth Banach space. The function defined by
is studied by Alber [7]. It follows from the definition of ϕ that
Remark 1.1
-
(i)
If E is a reflexive, strictly convex and smooth Banach space, then for , if and only if .
-
(ii)
If E is a real Hilbert space, then .
Let E be reflexive, strictly convex and smooth Banach space. The generalized projection mapping, introduced by Alber [7], is a mapping that assigns to an arbitrary point the minimum point of the functional , that is, , where is is the solution to the minimization problem
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 [8]) if and for all and .
Recently, Matsushita and Takahashi [9] 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.
Theorem 1.1 Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself and let be a sequence of real numbers such that and . Suppose that is given by
where J is the normalized duality mapping on E. If is nonempty, then converges strongly to .
In [10], Hao introduced the following iterative scheme for approximating a fixed point of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space E: , , ,
where .
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.
2 Preliminaries
Let E be a real Banach space with the norm and let be the dual space of E. The normalized duality mapping is defined by
By the Hahn-Banach theorem, is nonempty.
A Banach space E is called strictly convex if for all with , where is the unit sphere of E. A Banach space E is called smooth if the limit
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.
Remark 2.1 The basic properties of a Banach space E related to the normalized duality mapping J are as follows (see [11]):
-
(1)
If E is a smooth Banach space, then J is single-valued and semicontinuous;
-
(2)
If E is a uniformly smooth Banach space, then J is uniformly norm-to-norm continuous on each bounded subset of E;
-
(3)
If E is a uniformly smooth Banach space, then E is smooth and reflexive;
-
(4)
If E is a reflexive and strictly convex Banach space, then is norm-weak∗-continuous;
-
(5)
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.
Definition 2.1 A mapping is said to be
-
(1)
quasi-ϕ-nonexpansive if and
for all and ;
-
(2)
asymptotically quasi-ϕ-nonexpansive in the intermediate sense if and
put
Remark 2.2 From the definition of asymptotically quasi-ϕ-nonexpansiveness in the intermediate sense, it is obvious that as and
Recall that T is said to be asymptotically regular on C if for any bounded subset K of C,
Definition 2.2 A mapping is said to be closed if for any sequence with and , .
Following Alber [7], the generalized projection is defined by
In 2006, Wu and Huang [12] introduced a generalized f-projection operator in a Banach space, which extends the definition of the generalized projection . Let be a functional defined as follows:
for all , where ρ is a positive number and is proper, convex, and lower semicontinuous. From the definition of G, it is easy to see the following properties:
-
(i)
is convex and continuous with respect to when y is fixed;
-
(ii)
is convex and lower semicontinuous with respect to y when is fixed.
Definition 2.3 ([13])
Let E be a real smooth Banach space and let C be a nonempty closed and convex subset of E. We say that is a generalized f-projection operator if
Lemma 2.1 ([14])
Let E be a Banach space and be a lower semicontinuous and convex function. Then there exist and such that
for all .
Lemma 2.2 ([13])
Let E be a reflexive smooth Banach space and let C be a nonempty closed convex subset of E. The following statements hold:
-
(1)
is a nonempty closed convex subset of C for all ;
-
(2)
For all , if and only if
for all ;
-
(3)
If E is strictly convex, then is a single-valued mapping.
Let θ be a bifunction from to ℝ, where ℝ denotes the set of real numbers. The equilibrium problem is to find such that
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 ;
(A3) for all ,
(A4) for all , is convex and lower semicontinuous.
Lemma 2.3 ([15])
Let C be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E and let θ be a bifunction from to ℝ satisfying the conditions (A1)-(A4). For all and , define a mapping as follows:
Then the following conclusions hold:
-
(1)
is single-valued;
-
(2)
is a firmly nonexpansive-type mapping, i.e., for all ,
-
(3)
is closed and convex;
-
(4)
is quasi-ϕ-nonexpansive;
-
(5)
, .
Lemma 2.4 ([10])
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 ([13])
Let E be a real reflexive smooth Banach space and let C be a nonempty closed and convex subset of E. Then, for any and ,
for all .
3 Main results
Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let θ be a bifunction from to ℝ satisfying the conditions (A1)-(A4). Let be a closed and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C, is nonempty, and is bounded. Let be a convex and lower semicontinuous function with and . Let be a sequence in and , be sequences in satisfying the following conditions:
-
(i)
;
-
(ii)
;
-
(iii)
.
Let be a sequence generated by
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 .
In fact, it is obvious that is closed and convex. Suppose that is closed and convex for some . For , we see that . It follows that , where . Notice that
and
The above inequalities are equivalent to
and
Multiplying t and on both sides of (3.2) and (3.3), respectively, we obtain
Hence we have
This implies that is closed and convex for all . This shows that is well defined.
Step 2. We show that for all .
For , we have . Now, assume that for some . Let . Since T is asymptotically quasi-ϕ-nonexpansive with intermediate sense, we have from Remark 2.2 and Lemma 2.3 that
which shows that . This implies that and so for all .
Step 3. We prove that is bounded and exists.
By Lemma 2.1, we have the result that there exist and such that
Since , it follows that
For all and , we have
This implies that the sequence is bounded and so is . From (1.2) and Lemma 2.5, we obtain
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.
By (3.4), we obtain
Since is bounded and E is reflexive, we may assume that as . Since is closed and convex, we find that . From the weak lower semicontinuity of the norm and , we obtain
which implies that . From Lemma 2.5, we obtain
Hence we have . In view of the Kadec-Klee property of E, we find that
And we have
Since J is uniformly norm-to-norm continuous, it follows that
From and (3.1), we have
This is equivalent to the following:
Due to (3.5), (3.7), the assumption (ii), and Remark 2.2, we have
By (1.2), it follows that
as . Since J is uniformly norm-to-norm continuous, we obtain
as . This implies that is bounded in . Since is reflexive, we assume that as . In view of , there exists such that . This implies that . We have
Taking on both sides of the equality above, this yields
which shows that and so . It follows from (3.9) and the Kadec-Klee property of that as . Since is norm-weak-continuous, we have
From (3.8), (3.10), and the Kadec-Klee property of E, we have
On the other hand, we see from the weak lower semicontinuity of the norm that
which implies that
By (3.6) and (3.11), we obtain . The uniform continuity of J on bounded sets gives
Now, using the definition of ϕ, we have, for all ,
From (3.13), we obtain
as . By (3.12), it follows that
Hence, for any , it follows from the convexity of and Lemma 2.3 that
From (3.12), (3.14), (3.15), Remark 2.2, and the assumption (ii), we obtain
From Lemma 2.3, we see that for any and ,
Taking on both sides of the inequality above, we have
From (1.2), we have as . By (3.8), we have
as , and so
as . That is, is bounded in . Since is reflexive, we can assume that as . In view of , there exists such that . It follows that
Taking on both sides of the equality above, it follows that
From Remark 1.1, , i.e., . It follows that as . From (3.17) and the Kadec-Klee property of , we have
as . Since is norm-weak∗-continuous, as . From (3.16) and the Kadec-Klee property of E, we have
Step 5. We show that .
By Step 4, we get
The uniform continuity of J on bounded sets gives
From the assumption and (3.18), we see that as . But from (A2) and (3.1), we note that
and hence
which implied that for all . Put for all and . Then we get and . Therefore, from (A1) and (A4), we obtain
Thus, for all . Furthermore, as , we have from (A3) that for all . This implies that .
Finally, we show that . In view of , we find that
Hence we have
From the assumptions (ii), (iii), and (3.13), we have
Notice that
This implies from (3.19) that
The demicontinuity of implies that as . We have
With the aid of (3.20), we see that . Since E has the Kadec-Klee property, we find that
Since
we find from (3.21) and the asymptotic regularity of T that
i.e., as . It follows from the closedness of T that . So, and hence .
Step 6. We show that and so as .
Since ℱ is a closed convex set, it follows from Lemma 2.2 that is single-valued, which is denoted by . By the definition of and , we also have
for all . By the definition of G, we know that for any , is convex and lower semicontinuous with respect to u and so
From the definition of and , we conclude that
and as . This completes the proof. □
Remark 3.1
-
(i)
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 [10] 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.
Corollary 3.1 Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let θ be a bifunction from to ℝ satisfying the conditions (A1)-(A4). Let be a closed and quasi-ϕ-nonexpansive mapping. Assume that is nonempty. Let be a convex and lower semicontinuous function with and . Let be a sequence in and , be sequences in satisfying the following conditions:
-
(i)
;
-
(ii)
;
-
(iii)
.
Let be a sequence generated by
where is a real sequence in for some and is the generalized f-projection operator. Then converges strongly to .
Remark 3.2
-
(i)
By Remark 3.1, Theorem 3.1 extends Theorem 2.1 of Hao [10].
-
(ii)
Theorem 3.1 generalizes Theorem 3.1 of Matsushita and Takahashi [9] in the following respects:
-
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;
-
(iii)
in view of the mappings and the frame work of the spaces, Theorem 3.1 generalizes and improves Theorem 3.1 of Ma et al. [16], Theorem 3.1 of Qin et al. [17], Theorem 3.1 of Qing and Lv [18] and Theorem 3.1 of Saewan [19].
We now provide a nontrivial family of mappings satisfying the conditions of Theorem 3.1.
Example 3.1 Let with the standard norm and . Let be a mapping defined by
We first show that T is an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense with . In fact, for , we have
and
Therefore, we have
Next, we define a bifunction satisfying the conditions (A1)-(A4) by
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 .
Author’s contributions
JUJ conceived of the study, its design, and its coordination. The author read and approved the final manuscript.
References
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, 65(2):169–179.
Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 1974, 17: 339–346. 10.1007/BF02757136
Chidume CE, Ofoedu EU, Zegeye H: Strong and weak convergence theorem for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2003, 280: 364–374. 10.1016/S0022-247X(03)00061-1
Górnicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carol. 1989, 30: 249–252.
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. Lecture Notes in Pure and Appl. Math. 178. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.
Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 284613
Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007
Hao Y: Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl. 2013., 2013: Article ID 227
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.
Wu KQ, Huang NJ: The generalized f -projection operator with an application. Bull. Aust. Math. Soc. 2006, 73: 307–317. 10.1017/S0004972700038892
Li X, Huang N, O’Regan D: Strong convergence theorems for relative nonexpansive mappings in Banach spaces with applications. Comput. Math. Appl. 2010, 60: 1322–1331. 10.1016/j.camwa.2010.06.013
Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.
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
Ma Z, Wang L, Chang SS: Strong convergence theorem for quasi- ϕ -asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. J. Inequal. Appl. 2013., 2013: Article ID 306
Qin X, Cho SY, Wang L: Algorithms for treating equilibrium and fixed point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 308
Qing Y, Lv S: A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl. 2013., 2013: Article ID 305
Saewan S: Strong convergence theorem for total quasi- ϕ -asymptotically nonexpansive mappings in a Banach space. Fixed Point Theory Appl. 2013., 2013: Article ID 297
Acknowledgements
The author is grateful to the anonymous referees for useful suggestions, which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jeong, J.U. Convergence theorems for equilibrium problem and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl 2014, 199 (2014). https://doi.org/10.1186/1687-1812-2014-199
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-199