New hybrid shrinking projection algorithm for common fixed points of a family of countable quasi-Bregman strictly pseudocontractive mappings with equilibrium and variational inequality and optimization problems
- Yongchun Xu^{1} and
- Yongfu Su^{2}Email author
https://doi.org/10.1186/s13663-015-0347-9
© Xu and Su 2015
Received: 9 April 2015
Accepted: 9 June 2015
Published: 26 June 2015
Abstract
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al. (Fixed Point Theory Appl. 2014:231, 2014). These new results improve and extend the previously known ones in the literature.
Keywords
MSC
1 Introduction
Recently, many authors have studied further new hybrid iterative schemes in the framework of real Banach spaces; for instance, see [6–8]. Qin and Wang [9] have introduced a new class of mappings which are asymptotically quasi-nonexpansive with respect to the Lyapunov functional (cf. [10]) in the intermediate sense. By using the shrinking projection method, Hao [11] has proved a strong convergence theorem for an asymptotically quasi-nonexpansive mapping with respect to the Lyapunov functional in the intermediate sense.
In 1967, Bregman [12] discovered an elegant and effective technique for using of the so-called Bregman distance function (see Section 2) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique is 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, and for computing fixed points of nonlinear mappings.
Many authors have studied iterative methods for approximating fixed points of mappings of nonexpansive type with respect to the Bregman distance; see [13–17]. In [18], the author introduced a new class of nonlinear mappings which is an extension of asymptotically quasi-nonexpansive mappings with respect to the Bregman distance in the intermediate sense and proved the strong convergence theorems for asymptotically quasi-nonexpansive mappings with respect to Bregman distances in the intermediate sense by using the shrinking projection method.
Recently, Zegeye and Shahzad [19] have proved a strong convergence theorem for the common fixed point of a finite family of right Bregman strongly nonexpansive mappings in a reflexive Banach space. Alghamdi et al. [20] proved a strong convergence theorem for the common fixed point of a finite family of quasi-Bregman nonexpansive mappings. Pang et al. [21] proved weak convergence theorems for Bregman relatively nonexpansive mappings. Shahzad and Zegeye [22] proved a strong convergence theorem for multivalued Bregman relatively nonexpansive mappings, while Zegeye and Shahzad [23] proved a strong convergence theorem for a finite family of Bregman weak relatively nonexpansive mappings.
Motivated and inspired by the above works, in 2015 Ugwunnadi et al. [24] proved a new strong convergence theorem for a finite family of closed quasi-Bregman strictly pseudocontractive mappings and a system of equilibrium problems in a real reflexive Banach space.
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: (1) the notion of uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al. [24]. These new results improve and extend the previously known ones in the literature.
2 Preliminaries
Throughout this paper, we assume that E is a real reflexive Banach space with the dual space of \(E^{*}\) and \(\langle\cdot,\cdot\rangle\) is the pairing between E and \(E^{*}\).
Proposition 2.1
([25])
- (i)
\(\operatorname{ran} \partial f =E^{*}\) and \(\partial f^{*} =(\partial f)^{-1}\) is bounded on bounded subsets of \(E^{*}\);
- (ii)
f is strongly coercive.
Proposition 2.2
([27])
If a function \(f : E\rightarrow R\) is convex, 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 \(E^{*}\).
Proposition 2.3
([27])
- (i)
f is strongly coercive and uniformly convex on bounded subsets of E;
- (ii)
\(f^{*}\) is Fréchet differentiable and \(\nabla f^{*}\) is uniformly norm-to-norm continuous on bounded subsets of \(\operatorname{dom} f^{*} = E^{*}\).
- (L1)
the interior of the domain of f, \(\operatorname{int}\operatorname{dom} f\), is nonempty, f is Gâteaux differentiable, and \(\operatorname{dom} \nabla f = \operatorname{int}\operatorname{dom} f\);
- (L2)
the interior of the domain of \(f^{*}\), \(\operatorname{int}\operatorname{dom} f^{*}\), is nonempty, \(f^{*}\) is Gâteaux differentiable, and \(\operatorname{dom} \nabla f^{*} = \operatorname{int}\operatorname{dom} f^{*}\).
Proposition 2.4
([14])
Let \(f : E \rightarrow (-\infty, +\infty]\) be a Legendre function such that \(\nabla f^{*}\) is bounded on bounded subsets of \(\operatorname{int}\operatorname{dom} f^{*}\). Let \(x\in \operatorname{int}\operatorname{dom} f\). If the sequence \(\{D_{f}(x,x_{n})\}\) is bounded, then the sequence \(\{x_{n}\}\) is also bounded.
Proposition 2.5
([14])
- (i)
the function \(W^{f}(\cdot,x)\) is convex for all \(x\in \operatorname{dom} f\);
- (ii)
\(W^{f}(\nabla f(x),y)=D_{f}(y,x)\) for all \(x\in \operatorname{int}\operatorname{dom} f\) and \(y\in \operatorname{dom} f\).
Proposition 2.6
([30])
Let \(f : E \rightarrow(-\infty, +\infty]\) be a convex function whose domain contains at least two points. If f is lower semi-continuous, then f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets.
Proposition 2.7
([32])
Let \(f : E\rightarrow R\) be a totally convex function. If \(x \in E\) and the sequence \(\{D_{f}(x_{n},x)\}\) is bounded, then the sequence \(\{x_{n}\}\) is also bounded.
Proposition 2.8
([24])
A function \(f : E\rightarrow[0,+\infty)\) is totally convex on bounded subsets of E if and only if it is sequentially consistent.
Proposition 2.9
([33])
Let \(f : E\rightarrow R\) be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of domf. Then \(\operatorname{proj}_{C}^{f}(x)\) exists uniquely for all \(x \in \operatorname{int}\operatorname{dom} f\).
- (i)
If E is a Hilbert space, then the Bregman projection is reduced to the metric projection onto C.
- (ii)If E is a smooth Banach space, then the Bregman projection is reduced to the generalized projection \(\Pi_{C}(x)\) which is defined bywhere ϕ is the Lyapunov functional (cf. [10]) defined by$$\Pi_{C}(x)=\operatorname{argmin} \bigl\{ \phi(y,x): y\in C\bigr\} , $$for all \(y,x \in E\).$$\phi(y,x)=\|y\|^{2}-2\langle y, Jx\rangle+\|x\|^{2} $$
Proposition 2.10
([31])
- (i)
The vector \(x^{*}\) is the Bregman projection of x onto C.
- (ii)The vector \(x^{*}\) is the unique solution z of the variational inequality$$\bigl\langle z-y, \nabla f(x)-\nabla f(z)\bigr\rangle \geq0, \quad \forall y \in C. $$
- (iii)The vector \(x^{*}\) is the unique solution z of the inequality$$D_{f}(y,z)+D_{f}(z,x)\leq D_{f}(y,x), \quad \forall y \in C. $$
In recent years, the following notions have been presented by some authors.
A point \(p \in C\) is said to be asymptotic fixed point of a map T if there exists a sequence \(\{x_{n}\}\) in C which converges weakly to p such that \(\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n}\|= 0\). We denote by \(\widehat{F}(T)\) the set of asymptotic fixed points of T. A point \(p \in C\) is said to be strong asymptotic fixed point [34] of a mapping T if there exists a sequence \(\{x_{n}\}\) in C which converges strongly to p such that \(\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n}\|= 0\). We denote by \(\widetilde{F}(T)\) the set of strong asymptotic fixed points of T. Let \(f : E \rightarrow R\), a mapping \(T : C \rightarrow C\) is said to be Bregman relatively nonexpansive [17] if \(F(T)=\widehat{F}(T)\) and \(D_{f}(p, T(x))\leq D_{f}(p, x)\) for all \(x \in C\) and \(p \in F(T)\). The mapping \(T : C \rightarrow C\) is said to be Bregman weak relatively nonexpansive if \(F(T)=\widetilde{F}(T)\) and \(D_{f}(p, T(x))\leq D_{f}(p, x)\) for all \(x \in C\) and \(p \in F(T)\). The mapping \(T : C \rightarrow C\) is said to be quasi-Bregman relatively nonexpansive [24] if \(F(T)\neq\emptyset\) and \(D_{f}(p, T(x))\leq D_{f}(p, x)\) for all \(x \in C\) and \(p \in F(T)\). In [24] quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A mapping \(T : C \rightarrow C\) is said to be right quasi-Bregman relatively nonexpansive [24] if \(F(T)\neq\emptyset\) and \(D_{f}( T(x),p)\leq D_{f}(x, p)\) for all \(x \in C\) and \(p \in F(T)\).
In [24], authors presented the definition of quasi-Bregman strictly pseudocontractive mapping. In this paper, we extend this definition to the quasi-Bregman pseudocontractive mapping as follows.
Definition 2.11
In this paper, we will use the following definition.
Definition 2.12
([34])
Let C be a nonempty, closed, and convex subset of E. Let \(\{T_{n}\}\) be a sequence of mappings from C into itself with a nonempty common fixed point set \(F=\bigcap_{n=1}^{\infty}F(T_{n})\). \(\{T_{n}\}\) is said to be uniformly closed if for any convergent sequence \(\{z_{n}\} \subset C\) such that \(\|T_{n}z_{n}-z_{n}\|\rightarrow0\) as \(n\rightarrow\infty\), the limit of \(\{z_{n}\}\) belongs to F.
The next lemmas have been proved in [24], which is useful for the results of [24], but in this paper we do not use Lemma 2.13 and Lemma 2.15.
Lemma 2.13
([24])
Lemma 2.14
([24])
Let \(f : E \rightarrow R\) be a Legendre function which is uniformly Fréchet differentiable on bounded subsets of E, let C be a nonempty, closed, and convex subset of E, and let \(T : C \rightarrow C\) be a quasi-Bregman strictly pseudocontractive mapping with respect to f. Then \(F(T)\) is closed and convex.
Lemma 2.15
([24])
Problem (2.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, e.g., [31, 32].
- (A1)
\(F(x,x)=0\) for all \(x\in C\),
- (A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\), for all \(x,y\in C\),
- (A3)
for all \(x,y,z\in C\), \(\limsup_{t\downarrow0}F(tz+(1-t)x,y)\leq F(x,y)\),
- (A4)
for all \(x\in C\), \(F(x,\cdot)\) is convex and lower semi-continuous.
Lemma 2.16
Let E be a reflexive Banach space and let \(f : E \rightarrow R\) be a Legendre function. Let C be a nonempty, closed, and convex subset of E and let \(F : C \times C \rightarrow R\) be a bifunction satisfying (A1)-(A4). For \(r> 0\), let \(T_{r} : E \rightarrow C\) be the mapping defined by (2.5). Then \(\operatorname{dom} T_{r} =E\).
Lemma 2.17
- (i)
\(T_{r}\) is single-valued.
- (ii)\(T_{r}\) is a firmly nonexpansive-type mapping, i.e., for all \(x,y\in E\),$$\bigl\langle T_{r}x-T_{r}y, \nabla f(T_{r}x)- \nabla f(T_{r}y) \bigr\rangle \leq\bigl\langle T_{r}x-T_{r}y, \nabla f(x)-\nabla f(y) \bigr\rangle . $$
- (iii)
\(F(T_{r})=\widehat{F}(T_{r})=\mathit{EP}(F)\).
- (iv)
\(\mathit{EP}(F)\) is closed and convex.
- (v)
\(D_{f}(p, T_{r}x)+D_{f}(T_{r}x,x)\leq D_{f}(p,x)\), \(\forall p\in \mathit{EP}(F)\), \(\forall x \in E\).
Lemma 2.18
- (i)
\(K_{r}\) is single-valued.
- (ii)\(K_{r}\) is a firmly nonexpansive-type mapping, i.e., for all \(x,y\in E\),$$\bigl\langle K_{r}x-K_{r}y, \nabla f(K_{r}x)- \nabla f(K_{r}y) \bigr\rangle \leq\bigl\langle K_{r}x-K_{r}y, \nabla f(x)-\nabla f(y) \bigr\rangle . $$
- (iii)
\(F(K_{r})=\widehat{F}(K_{r})=\mathit{EP}\).
- (iv)
EP is closed and convex.
- (v)
\(D_{f}(p, K_{r}x)+D_{f}(K_{r}x,x)\leq D_{f}(p,x)\), \(\forall p\in \mathit{EP}(F)\), \(\forall x \in E\).
Proof
From [36] we have the following conclusion.
Theorem 2.19
From Theorem 2.19, we know that the generalized duality mapping \(J_{p}: E\rightarrow E^{*}\) is a monotone operator. It is well known that if E is also smooth and 2-uniformly convex, the normalized duality mapping \(J=J_{2}: E\rightarrow E^{*}\) is a single-valued monotone operator.
3 Main results
We now prove the following theorem.
Theorem 3.1
Proof
We divide the proof into six steps.
Step 3. We show that \(\{x_{n}\}\) converges to a point \(p \in C\).
Step 4. We show that the limit of \(\{x_{n}\}\) belongs to \(\bigcap_{n=1}^{\infty}F(T_{n})\).
Step 5. We show that the limit of \(\{x_{n}\}\) belongs to \(\mathit{EP}_{j}\) for all \(j=1,2,3, \ldots ,m\).
Remark 3.2
Theorem 3.1 includes the following three special cases.
4 Examples
Conclusion 4.1
\(\{T_{n}\}\) has a unique common fixed point 0, that is, \(F=\bigcap_{n=1}^{\infty}F(T_{n})=\{0\}\) for all \(n\geq0\).
Proof
The conclusion is obvious. □
Conclusion 4.2
\(\{T_{n}\}\) is a uniformly closed family of countable quasi-Bregman \((2n+1)\)-pseudocontractive mappings.
Proof
Example 4.3
Example 4.4
5 The mistakes in the result of Ugwunnadi et al. [24]
In [24], from page 10, line −3 to page 11, line 2, there exists a mistake ratiocination as follows.
Mistake ratiocination 1
In fact, the authors attempt taking \(w=x_{n+1}\) in (3.6) and (3.7) to get the (∗). This is an obvious mistake since (3.6) and (3.7) are right for only \(w \in F\), but \(x_{n+1}\) does not belong to F. Therefore, the definition of an iterative sequence \(\{x_{n}\}\) must be modified so that \(\lim_{n\rightarrow\infty} D_{f}(x_{n+1},x_{n})=0\) implies \(\lim_{n\rightarrow\infty} D_{f}(x_{n+1},y_{n})=0\).
In [24], page 12, line 3, there exists another mistake ratiocination as follows.
Mistake ratiocination 2
In fact, we are proving that \(p\in \mathit{EP}_{j}\) for any \(j=1,2,3,\ldots,m\), therefore, if we do not know whether \(p\in \mathit{EP}_{j}\), then the above inequalities are not right since if \(p\in \mathit{EP}_{j}\), the above inequalities are right. In this paper, we have overcome these shortcomings by modifying the iterative scheme.
Declarations
Acknowledgements
This project is supported by the National Natural Science Foundation of China under grant (11071279).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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