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A relaxed hybrid shrinking iteration approach to solving generalized mixed equilibrium problems for totally quasi-ϕ-asymptotically nonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 63 (2014)
An optimal existing method for the approximation of common fixed points of countable families of nonlinear operators is introduced, by which a relaxed hybrid shrinking iterative algorithm is developed for the class of totally quasi-ϕ-asymptotically nonexpansive mappings, and a strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces. Since there is no need to impose the uniformity assumption on the involved countable family of mappings and no need to compute a complex series at each step in the iteration process, the result is more widely applicable than those of other authors with related interests.
Throughout this paper we assume that E is a real Banach space with its dual , C is a nonempty closed convex subset of E and is the normalized duality mapping defined by
In the sequel, we use to denote the set of fixed points of a mapping T.
Definition 1.1 
A mapping is said to be totally quasi-ϕ-asymptotically nonexpansive, if and there exist nonnegative real sequences , with (as ) and a strictly increasing continuous function with such that(1.1)
where denotes the Lyapunov functional defined by
It is obvious from the definition of ϕ that
(2) A countable family of mappings said to be uniformly quasi-ϕ-asymptotically nonexpansive, if and there exists a nonnegative real sequence with (as ) such that
(3) A countable family of mappings said to be uniformly totally quasi-ϕ-asymptotically nonexpansive, if and there exist nonnegative real sequences , with (as ) and a strictly increasing continuous function with such that
(4) A mapping is said to be uniformly L-Lipschitz continuous, if there exists a constant such that
Let be a bifunction, a real valued function and a nonlinear mapping. The so-called generalized mixed equilibrium problem (GMEP) is to find an such that
whose set of solutions is denoted by Ω. The equilibrium problem is an unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Many authors have proposed some useful methods to solve the EP (equilibrium problem), GEP (generalized equilibrium problem), MEP (mixed equilibrium problem), and GMEP. Concerning the weak and strong convergence of iterative sequences to a common element of the set of solutions for the GMEP, the set of solutions for variational inequality problems, and the set of common fixed points for relatively nonexpansive mappings, quasi-ϕ-nonexpansive mappings, quasi-ϕ-asymptotically nonexpansive mappings and total quasi-ϕ-asymptotically nonexpansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (see, for example, [2–17] and the references therein).
In 2010, Qin et al.  proposed the following shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of quasi-ϕ-nonexpansive mappings in the framework of Banach spaces:
where is the generalized projection (see (2.1)) of E onto .
In 2011, Saewan and Kumam  introduced a modified new hybrid projection method to find a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in an uniformly smooth and strictly convex Banach spaces E with the Kadec-Klee property:
However, it is obviously a quite strong condition that the involved mappings are assumed to be a countable family of uniformly -quasi-ϕ-asymptotically nonexpansive ones, which is a special case of totally quasi-ϕ-asymptotically nonexpansive mappings (see ). In addition, the accurate computation of the series at each step of the iteration process is not easily attainable, which will lead to gradually increasing errors.
Inspired and motivated by the studies mentioned above, by using a special way of choosing the indices, we propose a relaxed hybrid shrinking iteration scheme for approximating common fixed points of a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem for solving the generalized mixed equilibrium problems under suitable conditions, namely, there is no need to assume uniformity for the totally quasi-ϕ-asymptotic property of the involved mappings, and no need to compute complex series in the iteration process. The results extend and improve those of other authors with related interests.
We say that a Banach space E is strictly convex if the following implication holds for :
It is also said to be uniformly convex if for any , there exists a such that
It is well known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if
exists for each . E is said to be uniformly smooth if the limit (2.3) is attained uniformly for .
Following Alber , the generalized projection is defined by
Lemma 2.1 
Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
for all and ;
If and , then , ;
For , if and only if .
Remark 2.2 The following basic properties for a Banach space E can be found in Cioranescu .
If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E;
If E is reflexive and strictly convex, then is norm-weak-continuous;
If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto;
A Banach space E is uniformly smooth if and only if is uniformly convex;
Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence , if and , then as .
Lemma 2.3 
Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex subset of E. Let and be two sequences in C such that and , where ϕ is the function defined by (1.2), then .
Lemma 2.4 
Let E and C be the same as in Lemma 2.3. Let be a closed and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , and a strictly increasing continuous function such that and . If , then the fixed point set of T is a closed and convex subset of C.
Lemma 2.5 
Let E be a real uniformly convex Banach space and let be the closed ball of E with center at the origin and radius . Then for any for any sequence and for any sequence of positive numbers with , there exists a continuous strictly increasing convex function with such that such that for any positive integer , the following hold:
and for all ,
Assume that, to obtain the solution of GMEP, the function is convex and lower semicontinuous, the nonlinear mapping is continuous and monotone, and the bifunction satisfies the following conditions:
(A2) θ is monotone, i.e., ;
(A4) the mapping is convex and lower semicontinuous.
Lemma 2.6 
Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty closed convex subset of E. Let be a continuous and monotone mapping, a lower semicontinuous and convex function, and a bifunction satisfying the conditions (A1)-(A4). Let and . Then, the following hold:
There exists an such that
A mapping is defined by
Then, the mapping has the following properties:
a firmly nonexpansive-type mapping, i.e.,
Ω is a closed convex set of C;
, , ,
where denotes the set of asymptotic fixed points of , i.e.,
Lemma 2.7 
The unique solutions to the positive integer equation
where denotes the maximal integer that is not larger than x.
3 Main results
Recall that a mapping T on a Banach space is closed if and as , then .
Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed convex subset of E. Let be a bifunction satisfying the conditions (A1)-(A4), a continuous and monotone mapping, and a lower semicontinuous and convex function. Let be a countable family of closed and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , satisfying and (as and for each ) and a sequence of strictly increasing and continuous functions satisfying condition (1.1). Assume that each is uniformly -Lipschitz continuous with for each . Let be a sequence in for some and be a sequence in . Let be the sequence generated by
where and is the generalized projection of E onto ; and are the solutions to the positive integer equation: (, ), that is, for each , there exist unique and such that
If and is bounded, then converges strongly to .
Proof Two functions and are defined by
By Lemma 2.6, we know that the function τ satisfies the conditions (A1)-(A4) and has the property (i)-(v). Therefore, (3.1) can be rewritten as
We divide the proof into several steps.
F and () both are closed and convex subsets in C.
In fact, it follows from Lemma 2.4 that each is a closed and convex subset of C, so is F. In addition, with (=C) being closed and convex, we may assume that is closed and convex for some . In view of the definition of ϕ we have
where and . This shows that is closed and convex.
(II) G is a subset of .
It is obvious that . Suppose that for some . Since , by Lemma 2.6, it is easily shown that is quasi-ϕ-nonexpansive. Hence, for any , it follows from (1.4) that
Furthermore, it follows from Lemma 2.5 that for any ,
Substituting (3.4) into (3.3) and simplifying it, we have
This implies that , and so .
In fact, since , from Lemma 2.1(2) we have , . Again since , we have , . It follows from Lemma 2.1(1) that for each and for each ,
which implies that is bounded, so is . Since for all , and , we have . This implies that is nondecreasing, hence the limit
Since E is reflexive, there exists a subsequence of such that as . Since is closed and convex and , this implies that is weakly closed and for each . In view of , we have
Since the norm is weakly lower semicontinuous, we have
This implies that , and so as . Since , by virtue of the Kadec-Klee property of E, we obtain
Since is convergent, this, together with , shows that . If there exists some subsequence of such that as , then from Lemma 2.1(1) we have
that is, and so
is a member of F.
Set for each . Note that , and whenever for each . For example, by Lemma 2.7 and the definition of , we have and . Then we have
Note that , i.e., as . It follows from (3.6) and (3.7) that
Since , it follows from (3.1), (3.6), and (3.8) that
Since as , it follows from (3.9) and Lemma 2.3 that
Note that and whenever for each . From (3.5), for any , we have
This shows that . In view of the property of g, we have
In addition, () implies that . From Remark 2.2(ii) it yields that, as ,
Again since for each , as ,
this, together with (3.11) and the Kadec-Klee property of E, shows that
We use the assumptions that for each , is uniformly -Lipschitz continuous. Noting again that , i.e., for all , we then have
From (3.12) and as we have and , i.e., . It then follows that, for each ,
In view of the closedness of , it follows from (3.12) that , i.e., for each , and hence .
is also a member of G.
Since , it follows from (3.1) and (3.6) that
Since as , by virtue of Lemma 2.1 we have
This, together with (3.10), shows that and . By the assumption that for some , we have
Since , , by condition (A1), we have
By the assumption that the mapping is convex and lower semicontinuous, letting in (3.17), from (3.15) and (3.16), we have , . For any and any , set . Then since . By conditions (A1) and (A4), we have
Dividing both sides of the above equation by t, we have , . Letting , from condition (A3), we have , , i.e., and so .
, and so as .
Put . Since and , we have , . Then
which implies that since , and hence as . This completes the proof. □
We now provide a nontrivial family of mappings satisfying the conditions of Theorem 3.1.
Example 3.2 Let with the standard norm and . Let be a countable family of mappings defined by
We first show that is uniformly L-Lipschitzian. If and , then
The rest is trivial. Second, we claim that is a family of closed and totally quasi-ϕ-asymptotically nonexpansive mappings. In fact, for any and , we have, for all ,
where , , and . Note that , that is, for each .
Next, we define a bifunction satisfying the conditions (A1)-(A4) by
Let and . Then the set of solutions Ω to the generalized mixed equilibrium problem for θ, A and ψ is obviously . Since and F is bounded, it follows from Theorem 3.1 that the sequence defined by (3.1) converges strongly to .
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The author is very grateful to the referees for their useful suggestions, by which the contents of this article has been improved. This study is supported by the National Natural Science Foundation of China (Grant No. 11061037) and the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2013A02).
The author declares that they have no competing interests.
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Deng, W. A relaxed hybrid shrinking iteration approach to solving generalized mixed equilibrium problems for totally quasi-ϕ-asymptotically nonexpansive mappings. Fixed Point Theory Appl 2014, 63 (2014). https://doi.org/10.1186/1687-1812-2014-63
- totally quasi-ϕ-asymptotically nonexpansive mappings
- generalized mixed equilibrium problems
- generalized projections