- Open Access
Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces
© Khan et al.; licensee Springer. 2014
Received: 24 September 2013
Accepted: 19 February 2014
Published: 6 March 2014
In this paper, we propose and analyze an explicit type algorithm for finding a common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two countable families of total quasi-φ-asymptotically nonexpansive mappings in a Banach space E. As an application of our result, we suggest a framework for finding a common solution of a finite family of generalized equilibrium problems and common zeros of two finite families of maximal monotone operators on E.
MSC:47H05, 47H10, 47H15, 47J25, 49M05.
1 Introduction and preliminaries
where stands for the duality product.
- (i)if , then problem (1.1) reduces to the following equilibrium problem :
- (ii)if , then problem (1.1) reduces to the classical variational inequality problem :
The equilibrium problem provides a unified approach to finding a solution of a large number of problems arising in physics, optimization, economics and fixed point problems . Moreover, the generalized equilibrium problem addresses monotone inclusion problems, variational inequality problems, minimization problems and vector equilibrium problem [2–4]. Since an algorithmic construction plays a key role in solving nonlinear equations in various fields of investigation, numerous implicit and explicit algorithms have been developed for the approximate solution of nonlinear equations as well as for the approximation of fixed points of various mappings [5–7].
strictly convex if for all with , we have ;
- (ii)uniformly convex if for any , there exists such that
A Banach space E is said to have the Kadec-Klee property if for any sequence in E with and , we have . Note that every uniformly convex Banach space E is strictly convex and enjoys the Kadec-Klee property but the converse is not true.
for and . The norm of E is said to be Gâteaux differentiable if exists for each and in this case E is smooth.
It is remarked that the set-valued mapping J is nonempty, closed and convex in a real Banach space whereas J is single-valued in a reflexive, strictly convex and smooth Banach space. Furthermore, , the inverse of the normalized duality mapping J, is also a duality mapping in a uniformly convex and smooth Banach space. Both J and are uniformly norm-to-norm continuous on each bounded subset of E and , respectively. If E is reflexive and strictly convex, then is norm-to-weak continuous. For more details, see [8, 9].
Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty, closed and convex subset of E. Then, for arbitrarily fixed , there exists a unique point such that . Following the notation of , we let and call a generalized projection onto C. Note that the generalized projection operator coincides with the metric projection in a Hilbert space.
A point is said to be an asymptotic fixed point  of if there exists a sequence such that and . The set of all asymptotic fixed points of T is denoted by .
nonexpansive if for all ;
relatively nonexpansive if and for all and ;
quasi-φ-nonexpansive if and for all and ;
quasi-φ-asymptotically nonexpansive if there exists a real sequence with ; and such that for all , and ;
- (v)total quasi-φ-asymptotically nonexpansive if there exist nonnegative real sequences and with , and such that
where is a strictly increasing continuous function with .
It is worth mentioning that the class of total quasi-φ-asymptotically nonexpansive mappings properly contains the mappings defined in (i)-(iv), but the converse is not true.
Recently, numerous attempts have been made to guarantee strong convergence through explicit and implicit algorithms for finding a common solution of the set of fixed points of (relatively nonexpansive, quasi-φ-nonexpansive, quasi-φ-asymptotically nonexpansive, total quasi-φ-asymptotically nonexpansive) mappings and the set of solutions of equilibrium problems; see [12–21] and the references cited therein.
where J is the duality mapping on E and is the generalized projection from E onto C. They proved that the sequence generated by (1.2) converges strongly to under some appropriate conditions.
where J, T, S and are as in (1.2). The authors showed that the sequence generated by (1.3) converges strongly to under some appropriate conditions.
In 2009, Wattanawitton and Kumam  approximated a common solution for a pair of relatively quasi-nonexpansive mappings and an equilibrium problem. Recently, Qin et al.  established strong convergence results for a pair of asymptotically quasi-φ-nonexpansive mappings in a Banach space. It is worth mentioning that the hybrid algorithms proposed in [12, 18, 22] are computationally complex. Therefore, it is natural to have improved and computationally simpler counterparts.
Quite recently, Zuo et al.  proposed a hybrid algorithm for total quasi-φ-asymptotically nonexpansive mappings and established strong convergence results in a Banach space. Moreover, they characterized such strong convergence results by using the notion of Mosco convergence; see also . Inspired and motivated by the work of Takahashi and Zembayashi , Chang et al.  and Zuo et al. , we aim to introduce and analyze a general algorithm based on the shrinking projection method for finding a common element of the set of common solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two countable families of total quasi-φ-asymptotically nonexpansive mappings. We also characterize the set of common solutions for families of total quasi-φ-asymptotically nonexpansive mappings and equilibrium problems in terms of Mosco convergence.
We now introduce the notion of Mosco convergence.
Let be a sequence of nonempty closed convex subsets of a reflexive Banach space E. We denote the set of all strong limit points of by , that is, if and only if there exists such that converges strongly to x and that for all n. Similarly, we define the set of all weak subsequential limit points by ; if and only if there exist a subsequence of and a sequence such that converges weakly to y and for all i. If satisfies , then we say that converges to in the sense of Mosco and we write . By definition, it always holds that . Therefore, to prove , it suffices to show that . One of the simplest examples of Mosco convergence is a decreasing sequence with respect to inclusion. The Mosco limit of such a sequence is . For more details, we refer to [23, 24].
For a relation between a sequence of closed convex sets and the corresponding generalized projections, we state the following lemma which plays a key role in our main result.
Lemma 1.1 (, Theorem 2.2)
Let E be a smooth, reflexive and strictly convex Banach space having the Kadec-Klee property. Let be a sequence of nonempty closed subset of E. If exists and is nonempty, then converges strongly to for each .
The following two results can be found as Remark 7.3 in .
The following well-known results are also needed in the sequel for the development of our main result.
Lemma 1.4 (, Proposition 2)
Let E be a uniformly convex and smooth Banach space and let , be two sequences of E. If and either or is bounded, then .
Lemma 1.5 (, Lemma 1.4)
for all and with .
(A1) for all ;
(A2) f is monotone, i.e., for all ;
(A3) for all ;
(A4) is convex and lower semicontinuous for all .
Lemma 1.7 (, Lemma 2.8)
is closed and convex;
- (3)is firmly nonexpansive-type mapping, i.e.,
2 Main results
Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the Kadec-Klee property. Let be two countable families of uniformly L-Lipschitzian and uniformly total quasi-φ-asymptotically nonexpansive mappings with sequences , and , , respectively. Let , , be a finite family of bifunctions such that (here the modN function takes values in ).
where and .
Theorem 2.1 Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the Kadec-Klee property and let be two countable families of uniformly L-Lipschitzian and uniformly total quasi-φ-asymptotically nonexpansive mappings with sequences and , where for each , and . Let , , be a finite family of bifunctions satisfying (A1)-(A4) such that . Let for some and satisfying:
Assume that . Then the sequence generated by (2.1) converges strongly to , where is the generalized projection of E onto F.
where . This shows that ; consequently, . By induction, we also get that for all with . Since and is a nonempty closed convex subset of E, hence both and are well defined.
for each .
So is nondecreasing and bounded. This implies that exists.
Hence Lemma 1.1 asserts that converges to .
In what follows, we show that:
Step 1. ;
Step 2. ;
Step 3. .
Moreover, it yields that , ; consequently, we have , . So we infer that .
Reasoning as above, one can also show that .
From and (A4), we obtain for all . Let for and . Then and hence . From (A1) and (A4), we have . Thus, . From (A3), we have for all . Hence . In a similar fashion, we have some such that and . Therefore, and hence .
Proof of Step 3. Lemma 1.1 asserts that the sequence converges to . Let and F is a nonempty closed convex subset of . Therefore . It suffices to show that . For this, we reason as follows:
Since E has the Kadec-Klee property, we have that .
The arbitrariness of implies that converges strongly to . □
Remark 2.2 If and are finite families in Theorem 2.1, then its conclusion can be strengthened as follows:
The sequence generated by (2.1) converges strongly to some if and only if , where and .
Since F is closed, this implies . Now, as in the proof of Step 3 in Theorem 2.1, we have . □
To proceed further, we need the following concept.
As an application of Theorem 2.1 to the case of finite families of mappings and , we obtain the following new strong convergence result.
Theorem 2.3 Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the Kadec-Klee property and let be two finite families of uniformly L-Lipschitzian and uniformly total quasi-φ-asymptotically nonexpansive mappings with sequences and , where for each , and . Let , , be a finite family of bifunctions satisfying (A1)-(A4) such that . Let for some and satisfying
Assume that and satisfy condition (I) and . Then the sequence generated by (2.1) converges strongly to some , where .
This implies that . Since f is nondecreasing and , we have . The rest of the proof follows from Remark 2.2 and is, therefore, omitted. □
As another application of Theorem 2.1, we establish a result for finding a common element in the set of solutions of a finite family of generalized equilibrium problems and the set of common zeros of two finite families of maximal monotone operators on a Banach space.
First we recall some preliminary concepts as follows.
A point satisfying is called a zero of T and the set of all such points is denoted by . T is said to be maximal monotone if T has no monotone extension. Equivalently, T is maximal monotone if the graph of T, i.e., , is not properly contained in the graph of any other monotone operator (cf. , p.264).
Let E be a strictly convex, reflexive and smooth Banach space and let be a maximal monotone operator. For a positive real number r, we can define a single-valued mapping by for each . This mapping is called resolvent of T for . It is known that if T is maximal monotone, then for each . Moreover, is a closed convex subset of E. For each , we can define Yosida approximation of T by for all . We know that ; for more details, see .
Theorem 2.4 Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the Kadec-Klee property. Let , , be two finite families of maximal monotone operators and let and be the corresponding finite families of resolvents of and , respectively, where . Let , , be a finite family of bifunctions satisfying (A1)-(A4). Let , and be three sequences in such that and satisfying
converges strongly to , where is the generalized projection of E onto F and .
for all and . Therefore, from the maximality of , we obtain that . On the other hand, and , so we have for each . Similarly, we can show that and for each . Moreover, the resolvent of (resp. of ) with satisfies for all and (resp. for all and ). Hence and are relatively nonexpansive mappings. Since the class of relatively nonexpansive mappings is properly contained in the class of total quasi-φ-asymptotically nonexpansive mappings, for a finitely many mappings case, one can derive the desired result from Theorem 2.1. □
Algorithm (2.1) is comparatively more general and computationally simpler than the algorithms which appeared in [12, 18] and , respectively, in the context of (two) countable families of total quasi-φ-asymptotically nonexpansive mappings.
Theorem 2.1 improves and extends [, Theorem 3.1], , [, Theorem 3.1] and [, Theorem 3.1] for two countable families of total quasi-φ-asymptotically nonexpansive mappings and a finite family of generalized equilibrium problems.
Theorem 2.4 improves and extends [, Theorem 4.1] for two finite families of maximal monotone operators. Moreover, our algorithm (2.14) is computationally simpler in comparison with algorithm (4.3) in [, Theorem 3.1].
Theorem 2.4 improves and extends [, Theorem 4.1] for two finite families of maximal monotone operators in the more general domain of a strictly convex, reflexive and smooth Banach space E.
In the context of an approximate common solution for an equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings, algorithms (2.1) and (2.14) are more general and computationally simpler in comparison with algorithms (3.1) and (4.2), respectively, in .
Algorithm (2.14) is computationally simpler in comparison with algorithms (3.6) in  and (3.1) in . Moreover, Theorem 2.4 improves [, Theorem 3.2] and [, Theorem 3.1] by removing the and conditions in the corresponding algorithms.
Theorem 2.4 is an analogue of [, Theorem 3.1] for the approximation of a common zero for a finite family of maximal monotone operators.
We are very grateful to anonymous referees for their critical and useful comments. The author MAA Khan gratefully acknowledges the support of Higher Education Commission (HEC) of Pakistan. The authors H Fukhar-ud-din and AR Khan are grateful to King Fahd University of Petroleum and Minerals for supporting the research project IN121037.
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