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Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces
Journal of Inequalities and Applications volume 2013, Article number: 119 (2013)
Abstract
In this paper, a shrinking projection algorithm based on the prediction correction method for equilibrium problems and weak Bregman relatively nonexpansive mappings is introduced and investigated in Banach spaces, and then the strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field.
MSC:26B25, 47H09, 47J05, 47J25.
1 Introduction
In this paper, without other specifications, let R be the set of real numbers, C be a nonempty, closed and convex subset of a real reflexive Banach space E with the dual space . The norm and the dual pair between and E are denoted by and , respectively. Let be a proper convex and lower semicontinuous function. Denote the domain of f by domf, i.e., . The Fenchel conjugate of f is the function defined by . Let be a nonlinear mapping. Denote by the set of fixed points of T. A mapping T is said to be nonexpansive if for all .
In 1994, Blum and Oettli [1] firstly studied the equilibrium problem: finding such that
where is functional. Denote the set of solutions of the problem (1.1) by . Since then, various equilibrium problems have been investigated. It is well known that equilibrium problems and their generalizations have been important tools for solving problems arising in the fields of linear or nonlinear programming, variational inequalities, complementary problems, optimization problems, fixed point problems and have been widely applied to physics, structural analysis, management science and economics etc. One of the most important and interesting topics in the theory of equilibria is to develop efficient and implementable algorithms for solving equilibrium problems and their generalizations (see, e.g., [2–8] and the references therein). Since the equilibrium problems have very close connections with both the fixed point problems and the variational inequalities problems, finding the common elements of these problems has drawn many people’s attention and has become one of the hot topics in the related fields in the past few years (see, e.g., [9–16] and the references therein).
In 1967, Bregman [17] discovered an elegant and effective technique for using of the so-called Bregman distance function (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique has been applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings and so on (see, e.g., [18–24] and the references therein). In 2005, Butnariu and Resmerita [25] presented Bregman-type iterative algorithms and studied the convergence of the Bregman-type iterative method of solving some nonlinear operator equations.
Recently, by using the Bregman projection, Reich and Sabach [26] presented the following algorithms for finding common zeroes of maximal monotone operators () in a reflexive Banach space E, respectively:
and
where , is an error sequence in E with and is the Bregman projection with respect to f from E onto a closed and convex subset C. Further, under some suitable conditions, they obtained two strong convergence theorems of maximal monotone operators in a reflexive Banach space. Reich and Sabach [7] also studied the convergence of two iterative algorithms for finitely many Bregman strongly nonexpansive operators in a Banach space. In [15], Reich and Sabach proposed the following algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators () in a reflexive Banach space E if :
Under some suitable conditions, they proved that the sequence generated by (1.2) converges strongly to and applied the result to the solution of convex feasibility and equilibrium problems.
Very recently, Chen et al. [27] introduced the concept of weak Bregman relatively nonexpansive mappings in a reflexive Banach space and gave an example to illustrate the existence of a weak Bregman relatively nonexpansive mapping and the difference between a weak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansive mapping. They also proved the strong convergence of the sequences generated by the constructed algorithms with errors for finding a fixed point of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings under some suitable conditions.
This paper is devoted to investigating the shrinking projection algorithm based on the prediction correction method for finding a common element of solutions to the equilibrium problem (1.1) and fixed points to weak Bregman relatively nonexpansive mappings in Banach spaces, and then the strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions.
2 Preliminaries
Let be a nonlinear mapping. A point is called an asymptotic fixed point of T (see [28]) if C contains a sequence which converges weakly to ω such that . A point is called a strong asymptotic fixed point of T (see [28]) if C contains a sequence which converges strongly to ω such that . We denote the sets of asymptotic fixed points and strong asymptotic fixed points of T by and , respectively.
Let be a sequence in E; we denote the strong convergence of to by . For any , the right-hand derivative of f at x in the direction is defined by
f is called Gâteaux differentiable at x if, for all , exists. In this case, coincides with , the value of the gradient of f at x. f is called Gâteaux differentiable if it is Gâteaux differentiable for any . f is called Fréchet differentiable at x if this limit is attained uniformly for . We say that f is uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for and .
The Legendre function is defined in [18]. From [18], if E is a reflexive Banach space, then f is the Legendre function if and only if it satisfies the following conditions (L1) and (L2):
(L1) The interior of the domain of f, , is nonempty, f is Gâteaux differentiable on and ;
(L2) The interior of the domain of , , is nonempty, is Gâteaux differentiable on and .
Since E is reflexive, we know that (see [29]). This, by (L1) and (L2), implies the following equalities:
and
By Bauschke et al. [[18], Theorem 5.4], the conditions (L1) and (L2) also yield that the functions f and are strictly convex on the interior of their respective domains. From now on we assume that the convex function is Legendre.
We first recall some definitions and lemmas which are needed in our main results.
Assumption 2.1 Let C be a nonempty, closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let satisfy the following conditions (C1)-(C4):
(C1) for all ;
(C2) H is monotone, i.e., for all ;
(C3) for all ,
(C4) for all , is convex and lower semicontinuous.
Let be a Gâteaux differentiable and convex function. The function defined by
is called the Bregman distance with respect to f.
Remark 2.1 [15]
The Bregman distance has the following properties:
-
(1)
the three point identity, for any and ,
-
(2)
the four point identity, for any and ,
Definition 2.2 [17]
Let be a Gâteaux differentiable and convex function. The Bregman projection of onto the nonempty, closed and convex set is the necessarily unique vector satisfying the following:
Remark 2.2
-
(1)
If E is a Hilbert space and for all , then the Bregman projection is reduced to the metric projection of x onto C;
-
(2)
If E is a smooth Banach space and for all , then the Bregman projection is reduced to the generalized projection (see [11, 28]), which is defined by
where , J is the normalized duality mapping from E to .
Let C be a nonempty, closed and convex set of domf. The operator with is called:
-
(1)
quasi-Bregman nonexpansive if
-
(2)
Bregman relatively nonexpansive if and
-
(3)
Bregman firmly nonexpansive if
or, equivalently,
-
(4)
a weak Bregman relatively nonexpansive mapping with if and
Definition 2.4 [4]
Let be functional. The resolvent of H is the operator defined by
Definition 2.5 [21]
Let be a convex and Gâteaux differentiable function. f is called:
-
(1)
totally convex at if its modulus of total convexity at x, that is, the function defined by
is positive whenever ;
-
(2)
totally convex if it is totally convex at every point ;
-
(3)
totally convex on bounded sets if is positive for any nonempty bounded subset B of E and , where the modulus of total convexity of the function f on the set B is the function defined by
The function is called:
-
(1)
cofinite if ;
-
(2)
coercive if ;
-
(3)
sequentially consistent if for any two sequences and in E such that is bounded,
Lemma 2.1 [[26], Proposition 2.3]
If is Fréchet differentiable and totally convex, then f is cofinite.
Lemma 2.2 [[25], Theorem 2.10]
Let be a convex function whose domain contains at least two points. Then the following statements hold:
-
(1)
f is sequentially consistent if and only if it is totally convex on bounded sets;
-
(2)
If f is lower semicontinuous, then f is sequentially consistent if and only if it is uniformly convex on bounded sets;
-
(3)
If f is uniformly strictly convex on bounded sets, then it is sequentially consistent and the converse implication holds when f is lower semicontinuous, Fréchet differentiable on its domain and the Fréchet derivative ∇f is uniformly continuous on bounded sets.
Lemma 2.3 [[30], Proposition 2.1]
Let be uniformly Fréchet differentiable and bounded on bounded subsets of E. Then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of .
Lemma 2.4 [[26], Lemma 3.1]
Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is also bounded.
Lemma 2.5 [[26], Proposition 2.2]
Let be a Gâteaux differentiable and totally convex function, and let C be a nonempty, closed convex subset of E. Suppose that the sequence is bounded and any weak subsequential limit of belongs to C. If for any , then converges strongly to .
Lemma 2.6 [[27], Proposition 2.17]
Let be the Legendre function. Let C be a nonempty, closed convex subset of and be a quasi-Bregman nonexpansive mapping with respect to f. Then is closed and convex.
Lemma 2.7 [[27], Lemma 2.18]
Let be Gâteaux differentiable and proper convex lower semicontinuous. Then, for all ,
where and with .
Lemma 2.8 [[25], Corollary 4.4]
Let be Gâteaux differentiable and totally convex on . Let and be a nonempty, closed convex set. If , then the following statements are equivalent:
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(1)
the vector is the Bregman projection of x onto C with respect to f;
-
(2)
the vector is the unique solution of the variational inequality:
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(3)
the vector is the unique solution of the inequality:
Lemma 2.9 [[7], Lemmas 1 and 2]
Let be a coercive Legendre function. Let C be a nonempty, closed and convex subset of . Assume that satisfies Assumption 2.1. Then the following results hold:
-
(1)
is single-valued and ;
-
(2)
is Bregman firmly nonexpansive;
-
(3)
is a closed and convex subset of C and ;
-
(4)
for all and for all ,
Lemma 2.10 [[31], Proposition 5]
Let be a Legendre function such that is bounded on bounded subsets of . Let . If is bounded, then the sequence is bounded too.
3 Main results
In this section, we will introduce a new shrinking projection algorithm based on the prediction correction method for finding a common element of solutions to the equilibrium problem (1.1) and fixed points to weak Bregman relatively nonexpansive mappings in Banach spaces, and then the strong convergence of the sequence generated by the proposed algorithm is proved under some suitable conditions.
Let and be the sequences in such that and . We propose the following shrinking projection algorithm based on the prediction correction method.
Algorithm 3.1 Step 1: Select an arbitrary starting point , let and .
Step 2: Given the current iterate , calculate the next iterate as follows:
Theorem 3.1 Let C be a nonempty, closed and convex subset of a real reflexive Banach space E, be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on a bounded subset of E, and be bounded on bounded subsets of . Let satisfy Assumption 2.1 and be a weak Bregman relatively nonexpansive mapping such that . Then the sequence generated by Algorithm 3.1 converges strongly to the point , where is the Bregman projection of C onto .
To prove Theorem 3.1, we need the following lemmas.
Lemma 3.1 Assume that for all . Then the sequence is bounded.
Proof Since for all , it follows from Lemma 2.8 that and so, by , we have
Let . It follows from Lemma 2.8 that
and so
Therefore, is bounded. Moreover, is bounded and so are , , . This completes the proof. □
Lemma 3.2 Assume that for all . Then the sequence is a Cauchy sequence.
Proof By the proof of Lemma 3.1, we know that is bounded. It follows from (3.2) that exists. From for all and Lemma 2.8, one has
and so . Therefore, we have
Since f is totally convex on bounded subsets of E, by Definition 2.6, Lemma 2.2 and (3.3), we obtain
Thus is a Cauchy sequence and so . This completes the proof. □
Lemma 3.3 Assume that for all . Then the sequence converges strongly to a point in .
Proof From Lemma 3.2, the sequence is a Cauchy sequence. Without loss of generality, let . Since f is uniformly Fréchet differentiable on bounded subsets of E, it follows from Lemma 2.2 that ∇f is norm-to-norm uniformly continuous on bounded subsets of E. Hence, by (3.4), we have
and so
Since , we have
It follows from and that is bounded and
By Lemma 2.10, is bounded. Hence, and so
Taking into account , we obtain
and so as . For any , from Lemma 2.9, we get
By the three point identity of the Bregman distance, one has
Since f is a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on a bounded subset of E, it follows from Lemma 2.3 that f is continuous on E and ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of . Therefore, we have
that is, . Furthermore, one has and thus
Since as , we have as . Further, in the light of and Definition 2.4, it follows that, for each ,
and hence, combining this with Assumption 2.1,
Consequently, one can conclude that
For any and , let . It follows from Assumption 2.1 that and
and so . Moreover, one has
which shows that .
Next, we prove that . Note that
This implies that
Letting in (3.7), it follows from and that
Moreover, we have that . This together with implies that . In view of , one has . Therefore, the sequence generated by Algorithm 3.1 converges strongly to a point in . This completes the proof. □
Now, we prove Theorem 3.1 by using lemmas.
Proof of Theorem 3.1 From Lemmas 2.6 and 2.9, it follows that is a nonempty, closed and convex subset of E. Clearly, and are closed and convex and so are closed and convex for all .
Now, we show that for all . Take arbitrarily. Then
which implies that and so for all .
Next, we prove that for all . Obviously, (). Suppose that for all . In view of , it follows from Lemma 2.8 that
Moreover, one has
and so, for each ,
This implies that . To sum up, we have and so
This together with yields that is a nonempty, closed convex subset of C for all . Thus is well defined and, from both Lemmas 3.2 and 3.3, the sequence is a Cauchy sequence and converges strongly to a point of .
Finally, we now prove that . Since , it follows from that
Thus, by Lemma 2.5, we have as . Therefore, the sequence converges strongly to the point . This completes the proof. □
Remark 3.1 (1) If for all , then the weak Bregman relatively nonexpansive mapping is reduced to the weak relatively nonexpansive mapping defined by Su et al. [32], that is, T is called a weak relatively nonexpansive mapping if the following conditions are satisfied:
where for all and J is the normalized duality mapping from E to ;
-
(2)
If for all , then Algorithm 3.1 is reduced to the following iterative algorithm.
Algorithm 3.2 Step 1: Select an arbitrary starting point , let and .
Step 2: Given the current iterate , calculate the next iterate as follows:
-
(3)
Particularly, if , then Algorithm 3.2 is reduced to the following iterative algorithm.
Algorithm 3.3 Step 1: Select an arbitrary starting point , let and .
Step 2: Given the current iterate , calculate the next iterate as follows:
-
(4)
If for all , then, by Algorithm 3.3, we can get the following modified Mann iteration algorithm for the equilibrium problem (1.1).
Algorithm 3.4 Step 1: Select an arbitrary starting point , let and .
Step 2: Given the current iterate , calculate the next iterate as follows:
If for all , then, by Theorem 3.1 and Remark 3.1, the following results hold.
Corollary 3.1 Let C be a nonempty, closed convex subset of a real reflexive Banach space E. Suppose that satisfies Assumption 2.1 and is a weak relatively nonexpansive mapping such that . Then the sequence generated by Algorithm 3.2 converges strongly to the point , where is the generalized projection of C onto .
Corollary 3.2 Let C be a nonempty, closed convex subset of a real reflexive Banach space E. Let be a weak relatively nonexpansive mapping such that . Then the sequence generated by Algorithm 3.3 converges strongly to the point , where is the generalized projection of C onto .
Corollary 3.3 Let C be a nonempty, closed convex subset of a real reflexive Banach space E. Suppose that satisfies Assumption 2.1 such that . Then the sequence generated by Algorithm 3.4 converges strongly to the point , where is the generalized projection of C onto .
Remark 3.2
-
(1)
It is well known that any closed and firmly nonexpansive-type mapping (see [11, 33]) is a weak Bregman relatively nonexpansive mapping whenever for all . If for all and E is a uniformly convex and uniformly smooth Banach space, then Corollary 3.2 improves [[11], Corollary 3.1];
-
(2)
If for all and E is a uniformly convex and uniformly smooth Banach space, then Corollary 3.2 is reduced to [[32], Theorem 3.1];
-
(3)
If for all , , for all and E is a uniformly convex and uniformly smooth Banach space, then Corollary 3.1 improves [[11], Theorem 4.1].
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Acknowledgements
The authors are indebted to the referees and the associate editor for their insightful and pertinent comments on an earlier version of the work. The second author (Jiawei Chen) was supported by the Natural Science Foundation of China and the Fundamental Research Fund for the Central Universities, the third author (Yeol Je Cho) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).
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Agarwal, R.P., Chen, JW. & Cho, Y.J. Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces. J Inequal Appl 2013, 119 (2013). https://doi.org/10.1186/1029-242X-2013-119
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DOI: https://doi.org/10.1186/1029-242X-2013-119