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Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 59 (2014)
Abstract
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
Let E be a real Banach space with the norm and be its dual. Let C be a nonempty subset of E and be a mapping. We denote by the set of fixed points of T. We symbolize weak convergence and strong convergence of a sequence in E as and , respectively. Let (the set of reals) be a bifunction and be a nonlinear mapping. A generalized equilibrium problem is to find the set
where stands for the duality product.
Note that:
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(i)
if , then problem (1.1) reduces to the following equilibrium problem :
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(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 [1]. 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].
Recall that a Banach space E is said to be:
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(i)
strictly convex if for all with , we have ;
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(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.
Furthermore, define by
for and . The norm of E is said to be Gâteaux differentiable if exists for each and in this case E is smooth.
The normalized duality mapping is defined by
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].
The Lyapunov functional is defined by
It is obvious from the definition of φ that for all . In a real Hilbert space, for all . For details, see [5, 10].
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 [5], 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 [11] of if there exists a sequence such that and . The set of all asymptotic fixed points of T is denoted by .
Recall that a mapping is:
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(i)
nonexpansive if for all ;
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(ii)
relatively nonexpansive if and for all and ;
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(iii)
quasi-φ-nonexpansive if and for all and ;
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(iv)
quasi-φ-asymptotically nonexpansive if there exists a real sequence with ; and such that for all , and ;
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(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.
A mapping is said to be uniformly L-Lipschitzian if
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.
In 2008, Takahashi and Zembayashi [21] introduced an explicit algorithm based on the shrinking projection method for finding a common solution of the set of fixed points of a relatively nonexpansive mapping T and the set of solutions of an equilibrium problem. Their algorithm reads as follows:
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.
In 2010, Chang et al. [12] proved a strong convergence theorem for finding a common element of the set of solutions for generalized equilibrium problem (1.1) and the set of common fixed points for a pair of relatively nonexpansive mappings in Banach spaces. Their algorithm reads as follows:
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 [22] approximated a common solution for a pair of relatively quasi-nonexpansive mappings and an equilibrium problem. Recently, Qin et al. [18] 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. [16] 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 [23]. Inspired and motivated by the work of Takahashi and Zembayashi [21], Chang et al. [12] and Zuo et al. [16], 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 ([23], 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 [5].
Lemma 1.2 Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E, let and let . Then if and only if
Lemma 1.3 Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let . Then
The following well-known results are also needed in the sequel for the development of our main result.
Lemma 1.4 ([25], 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 ([6], Lemma 1.4)
Let E be a uniformly convex Banach space and let be a closed ball in E. Then there exists a continuous strictly increasing convex function with such that
for all and with .
For solving the equilibrium problem, let us assume that the bifunction f satisfies the following conditions (cf. [2, 26]):
(A1) for all ;
(A2) f is monotone, i.e., for all ;
(A3) for all ;
(A4) is convex and lower semicontinuous for all .
The following result is stated as Lemma 2.7 in [21] (see also [2]).
Lemma 1.6 Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let be a bifunction satisfying (A1)-(A4), let and . Then there exists such that
Lemma 1.7 ([21], Lemma 2.8)
Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E. Let be a bifunction satisfying (A1)-(A4). For and , define a mapping by
for all . Then the following hold:
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(1)
is closed and convex;
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(2)
is single-valued;
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(3)
is firmly nonexpansive-type mapping, i.e.,
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(4)
.
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 ).
Algorithm
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:
(C1) .
Assume that . Then the sequence generated by (2.1) converges strongly to , where is the generalized projection of E onto F.
Proof For each , define by . Then coincides with the classical equilibrium problem and satisfies (A1)-(A4). Now, we show that algorithm (2.1) is well defined. By Lemma 1.7(1) we have that F is closed and convex. Next we show that is closed and convex. Clearly, is closed and convex. Suppose that is closed and convex for (set of naturals). For each , we observe that
This implies that is closed and convex. This implies, inductively, that is closed and convex for all . Now we prove that for each . Obviously, . Suppose that for some . It follows from Lemma 1.6 that and is relatively nonexpansive. Hence, for any , we have
That is,
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.
Note that is nondecreasing. In fact, from the definition of , we conclude that and ; hence
Moreover, from Lemma 1.3 we get that
for each .
So is nondecreasing and bounded. This implies that exists.
Let (a set of positive integers). Then Lemma 1.3 implies
Letting , we have . By Lemma 1.4, we have
Hence is Cauchy. Since C is a closed subset of the Banach space E, we can assume that there exists a point such that
Note that is a decreasing sequence of closed convex subsets of E with is nonempty. That is,
Hence Lemma 1.1 asserts that converges to .
In what follows, we show that:
Step 1. ;
Step 2. ;
Step 3. .
Proof of Step 1. As , so . It follows from (2.3) and the fact that , we get for all . Again by Lemma 1.4, we have
Observe that the following implication of the triangle inequality
yields that
As J is uniformly norm-to-norm continuous on bounded sets, so we have
Moreover, it follows from (2.4) and (2.6) that
From (2.2), we know that for all and for all . So, by Lemma 1.3, we have
Letting in the above estimate and using (2.7), we have for all ; consequently, Lemma 1.4 asserts that
From (2.6) and (2.9), we have
Hence, we conclude that
On the other hand, from Lemma 1.5, we have
Since , re-arranging the terms of the above estimate and simplifying, we get
Reasoning as above and then utilizing (2.6), we get that
Note that g is a continuous function and , so we have
In view of estimate (2.4), we conclude that . Hence, from (2.11) we obtain
Since is uniformly norm-to-norm continuous, (2.11) implies that
Observe that
So, we conclude from (2.4) and (2.13) that
Since each is uniformly L-Lipschitzian, we have
Hence, from (2.3) and (2.13), we obtain that
Moreover, it yields that , ; consequently, we have , . So we infer that .
Reasoning as above, one can also show that .
Proof of Step 2. We first show that , where for some . In view of estimate (2.9) and (C1), i.e., , we observe that
From , for all and , we have
Using (A2), the above estimate yields 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 and , we have
Since the norm is weakly lower semicontinuous, we have
From the definition of , we have . Hence . Therefore, we have
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 .
Proof The necessity is obvious. Conversely, suppose that . Since is bounded and exists (established above). Moreover, Lemma 1.3 and the following estimate
imply when . Hence, Lemma 1.4 asserts that . Thus, is a Cauchy sequence. Therefore, there exists a point such that as . Thus,
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.
Let f be a nondecreasing self-mapping on with and for all . Let and be two finite families of total quasi-φ-asymptotically nonexpansive mappings on C with . Then the two families are said to satisfy condition (I) on C if
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
(C1) .
Assume that and satisfy condition (I) and . Then the sequence generated by (2.1) converges strongly to some , where .
Proof It follows from Theorem 2.1 that
Since and satisfy condition (I), so we have either
or
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 multi-valued operator is said to be monotone if for any and with and it holds that
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. [27], 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 [9].
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
(C1) .
Assume that . Then the sequence generated by
converges strongly to , where is the generalized projection of E onto F and .
Proof The first part of this proof - (resp. ) are relatively nonexpansive mappings for each - is essentially due to Matsushita and Takahashi (cf. [[27], p.265]) which we include for completeness. Note that for each . Let . Then there exists such that and . Since , we have
It follows from and the monotonicity of that
for all and . Letting , we have
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. □
Remark 2.5 It is worth to mention that Theorem 2.1 and Theorem 2.4 improve and generalize various results available in the current literature. In particular, we highlight some significant features of both these theorems as follows.
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(i)
Algorithm (2.1) is comparatively more general and computationally simpler than the algorithms which appeared in [12, 18] and [22], respectively, in the context of (two) countable families of total quasi-φ-asymptotically nonexpansive mappings.
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(ii)
Theorem 2.1 improves and extends [[12], Theorem 3.1], [16], [[21], Theorem 3.1] and [[22], Theorem 3.1] for two countable families of total quasi-φ-asymptotically nonexpansive mappings and a finite family of generalized equilibrium problems.
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(iii)
Theorem 2.4 improves and extends [[12], 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 [[12], Theorem 3.1].
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(iv)
Theorem 2.4 improves and extends [[28], 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.
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(v)
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 [28].
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(vi)
Algorithm (2.14) is computationally simpler in comparison with algorithms (3.6) in [29] and (3.1) in [30]. Moreover, Theorem 2.4 improves [[29], Theorem 3.2] and [[30], Theorem 3.1] by removing the and conditions in the corresponding algorithms.
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(vii)
Theorem 2.4 is an analogue of [[17], Theorem 3.1] for the approximation of a common zero for a finite family of maximal monotone operators.
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Acknowledgements
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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Khan, M.A.A., Fukhar-ud-din, H. & Khan, A.R. Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces. Fixed Point Theory Appl 2014, 59 (2014). https://doi.org/10.1186/1687-1812-2014-59
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DOI: https://doi.org/10.1186/1687-1812-2014-59