Hybrid shrinking iterative solutions to convex feasibility problems for countable families of relatively nonexpansive mappings and a system of generalized mixed equilibrium problems
© Deng and Qian; licensee Springer. 2014
Received: 22 February 2014
Accepted: 13 June 2014
Published: 22 July 2014
We propose a new hybrid shrinking iterative scheme for approximating common elements of the set of solutions to convex feasibility problems for countable families of relatively nonexpansive mappings of a set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of other authors, in which the involved mappings consist of just finitely many ones.
MSC:47H09, 47H10, 47J25.
where denotes the metric projection from a Hilbert space H onto a closed convex subset K of H and proved that the sequence converges strongly to . A projection onto the intersection of two half-spaces is computed by solving a linear system of two equations with two unknowns (see [, Section 3]).
whose set of solutions is denoted by .
The equilibrium problem is a 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.
and proved that the sequence converges strongly to .
Inspired and motivated by the studies mentioned above, in this paper, we use a modified hybrid iteration scheme for approximating common elements of the set of solutions to convex feasibility problem for a countable families of relatively nonexpansive mappings, of set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of the authors, in which the involved mappings consist of just finitely many ones.
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 .
Lemma 2.2 
for all .
Lemma 2.3 
Let E be a uniformly convex and smooth Banach space and let and be two sequences of E. If , where ϕ is the function defined by (1.4), and either or is bounded, then .
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 a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping 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 2.5 
for all and with .
Lemma 2.6 
where denotes the maximal integer that is not larger than x.
3 Main results
, ; ;
; and ;
Then converges strongly to , where is the generalized projection from C onto F.
and () both are closed and convex subsets in C.
(II) F is a subset of .
where is a continuous strictly convex function with .
It then immediately follows from (3.31) and (3.32) that for each and hence .
which implies that since , and hence as . This completes the proof. □
Remark 3.2 Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. We first give an example  of how to compute such a projection onto the intersection of two half-spaces.
converges strongly to .
Since the algorithm (3.1) involves the projection onto the intersection of two convex sets not necessarily half-spaces, we next give an example  to explain and illustrate how the projection is calculated in the general convex case.
for with initial values and for . If , then converges to , where , .
Note Another iterative method termed HAAR (Haugazeau-like Averaged Alternating Reflections) for finding the projection onto intersection of finitely many closed convex sets in a Hilbert space can be found in [, Remark 3.4(iii)].
The so-called convex feasibility problem for a family of mappings is to find a point in the nonempty intersection .
Note Although the problem mentioned above is indeed a convex feasibility problem, it is mainly referred to the finite case.
Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty, closed, convex subset of E. Let be a sequence of -inverse strongly monotone mappings, a sequence of lower semi-continuous and convex functions, and a sequence of bifunctions satisfying the conditions:
(A2) θ is monotone, i.e., ;
(A4) the mapping is convex and lower semicontinuous.
whose set of common solutions is denoted by , where denotes the set of solutions to generalized mixed equilibrium problem for , , and .
is a sequence of single-valued mappings;
is a sequence of closed relatively nonexpansive mappings;
, ; ;
; and ;
and satisfies the equation (, ). Then converges strongly to , which is some common solution to the convex feasibility problem for and a system of generalized mixed equilibrium problems for .
The authors are 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).
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