Abstract
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.
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1 Introduction
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. A point p in C is said to be an asymptotic fixed point of T if C contains a sequence which converges weakly to p such that the . The set of asymptotic fixed points of T will be denoted by . A mapping is said to be nonexpansive if
A mapping is said to be relatively nonexpansive if and
where denotes the Lyapunov functional defined by
It is obvious from the definition of ϕ that
and
The asymptotic behavior of a relatively nonexpansive mapping was studied in [1–4]. In 1953, Mann [5] introduced the iteration as follows: a sequence is defined by
where the initial element is arbitrary and is a sequence of real numbers in . The Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [6]. In an infinite-dimensional Hilbert space, a Mann iteration can yield only weak convergence (see [7, 8]). Attempts to modify the Mann iteration method (1.8) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [9] proposed the following modification of Mann iteration method (1.8) for a nonexpansive mapping T from C into itself in a Hilbert space: from an arbitrary ,
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 [[10], Section 3]).
Let be a bifunction, a real-valued function, and a nonlinear mapping. The so-called generalized mixed equilibrium problem is to find an such that
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.
In 2007, Plubtieng and Ungchittrakool [11] established strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the following hybrid method in mathematical programming:
Their results extended and improved the corresponding ones announced by Nakajo and Takahashi [9], Martinez-Yanes and Xu [12], and Matsushita and Takahashi [4].
Recently, Su and Qin [13] modified iteration (1.9), the so-called monotone CQ method for nonexpansive mapping, as follows: from an arbitrary ,
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.
2 Preliminaries
We say that E is strictly convex if the following implication holds for :
It is also said to be uniformly convex if for any , there exists 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 [14], the generalized projection is defined by
Lemma 2.1 [14]
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:
-
(1)
for all and .
-
(2)
If and , then , .
-
(3)
For , if and only if .
Lemma 2.2 [15]
Let E be a uniformly convex and smooth Banach space and let . Then there exists a continuous, strictly increasing, and convex function such that and
for all .
Lemma 2.3 [16]
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 .
Remark 2.4 The following basic properties for a Banach space E can be found in Cioranescu [17].
-
(i)
If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E.
-
(ii)
If E is reflexive and strictly convex, then is norm-weak-continuous.
-
(iii)
If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping is single valued, one-to-one, and onto.
-
(iv)
A Banach space E is uniformly smooth if and only if is uniformly convex.
-
(v)
Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence , if and , then as .
Lemma 2.5 [18]
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 there exists a continuous strictly increasing convex function with such that
for all and with .
Lemma 2.6 [19]
The unique solutions to the positive integer equation
are
where denotes the maximal integer that is not larger than x.
3 Main results
Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty, closed, convex subset of E. Let and be two sequences of relatively nonexpansive mappings with . Let be the sequence generated by
where , , , and are sequences in satisfying
-
(1)
, ; ;
-
(2)
; and ;
and is the solution to the positive integer equation (, ), that is, for each , there exists a unique such that
Then converges strongly to , where is the generalized projection from C onto F.
Proof We divide the proof into several steps.
-
(I)
and () both are closed and convex subsets in C.
This follows from the fact that is equivalent to
(II) F is a subset of .
In fact, we note by [[4], Proposition 2.4] that for each , and are closed convex sets and so is F. It is clear that . Suppose that for some . For any , by the convexity of , we have
and then
This implies that . It follows from and Lemma 2.1(2) that
Particularly,
and hence , which yields . By induction, .
-
(III)
.
In view of and the definition of , we also have
This implies that
It follows from Lemma 2.2 that
Since J is uniformly norm-to-norm continuous on bounded sets, we have
and
This implies that
From (3.10) and , we have . Since is also uniformly norm-to-norm continuous on bounded sets, we obtain
From we have . Since is bounded, and for any . We also find that , and are bounded, and then there exists an such that . Therefore Lemma 2.5 is applicable and we observe that
That is,
where is a continuous strictly convex function with .
Let be any subsequence of . Since is bounded, there exists a subsequence of such that for any ,
From (1.6) we have
for some appropriate constant . Since
it follows that
From (3.3), we have
and hence . By (3.15), we observe that, as ,
Since , it follows that . By the properties of the mapping g, we have . Since is also uniformly norm-to-norm continuous on bounded sets, we obtain
and then . Next, we note by the convexity of and (1.7) that, as ,
since . By Lemma 2.3, we have and hence
as . Moreover, we observe that
as .
-
(IV)
as .
It follows from the definition of and Lemma 2.1(2) that . Since , we have
Therefore, is nondecreasing. Using and Lemma 2.1(1), we have
for all and for all , that is, is bounded. Then
In particular, by (1.5), the sequence is bounded. This implies that is bounded. Note again that and for any positive integer k, . By Lemma 2.1(1),
By Lemma 2.2, we have, for with ,
where is a continuous, strictly increasing, and convex function with . Then the properties of the function g show that is a Cauchy sequence in C, so there exists such that
Now, set for each . Note that and whenever . By Lemma 2.6 and the definition of , we have and . Then it follows from (3.15) and (3.24) that
It then immediately follows from (3.31) and (3.32) that for each and hence .
Put . Since and , we have , . Then
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 [20] of how to compute such a projection onto the intersection of two half-spaces.
Let H be a Hilbert space and suppose that satisfies
Set
and
In [21], Haugazeau introduced the operator Q as an explicit description of the projector onto the intersection of the two half-spaces defined in (3.34). He proved in [21] that the sequence defined by and
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 [22] to explain and illustrate how the projection is calculated in the general convex case.
Dykstra’s algorithm Let be closed and convex subsets of . For any and , the sequences are defined by the following recursive formulas:
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 [[20], Remark 3.4(iii)].
4 Applications
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:
(A1) ;
(A2) θ is monotone, i.e., ;
(A3) ;
(A4) the mapping is convex and lower semicontinuous.
A system of generalized mixed equilibrium problems for , and is to find an such that
whose set of common solutions is denoted by , where denotes the set of solutions to generalized mixed equilibrium problem for , , and .
Define a countable family of mappings with as follows:
where , , . It has been shown by Zhang [23] that
-
(1)
is a sequence of single-valued mappings;
-
(2)
is a sequence of closed relatively nonexpansive mappings;
-
(3)
.
Theorem 4.1 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 relatively nonexpansive mappings and be a sequence of mappings defined by (4.2) with . Let be the sequence generated by
where , , and are sequences in satisfying
-
(1)
, ; ;
-
(2)
; 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 .
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Acknowledgements
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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Deng, WQ., Qian, S. Hybrid shrinking iterative solutions to convex feasibility problems for countable families of relatively nonexpansive mappings and a system of generalized mixed equilibrium problems. Fixed Point Theory Appl 2014, 148 (2014). https://doi.org/10.1186/1687-1812-2014-148
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DOI: https://doi.org/10.1186/1687-1812-2014-148