- Open Access
A relaxed hybrid shrinking iteration approach to solving generalized mixed equilibrium problems for totally quasi-ϕ-asymptotically nonexpansive mappings
© Deng; licensee Springer. 2014
Received: 20 August 2013
Accepted: 28 February 2014
Published: 14 March 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.
In the sequel, we use to denote the set of fixed points of a mapping T.
- (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)
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).
where is the generalized projection (see (2.1)) of E onto .
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.
exists for each . E is said to be uniformly smooth if the limit (2.3) is attained uniformly for .
Lemma 2.1 
for all and ;
If and , then , ;
For , if and only if .
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 
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 
- (1)There exists an such that
- (2)A mapping is defined by
- (ii)a firmly nonexpansive-type mapping, i.e.,
Ω is a closed convex set of C;
, , ,
Lemma 2.7 
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 .
If and is bounded, then converges strongly to .
F and () both are closed and convex subsets in C.
where and . This shows that is closed and convex.
(II) G is a subset of .
is a member of F.
is also a member of G.
, and so as .
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.
where , , and . Note that , that is, for each .
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 .
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).
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