Hybrid iterative algorithm for finite families of countable Bregman quasi-Lipschitz mappings with applications in Banach spaces
- Minjiang Chen^{1},
- Jianzhi Bi^{1} and
- Yongfu Su^{2}Email author
https://doi.org/10.1186/s13660-015-0731-3
© Chen et al. 2015
Received: 27 March 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 method for finding a common element of the set EP of solutions of a generalized equilibrium problem, the common fixed point set F of finite uniformly closed families of countable Bregman quasi-Lipschitz mappings in reflexive Banach spaces. It is proved that under appropriate conditions, the sequence generated by the hybrid shrinking projection method converges strongly to some point in \(\mathit{EP} \cap F\). Relative examples are given. Strong convergence theorems are proved. The application for Bregman asymptotically quasi-nonexpansive mappings is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasi-Lipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the application for Bregman asymptotically quasi-nonexpansive mappings is also given; (4) because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This 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 authors has 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 has 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.
The purpose of this paper is to introduce and consider a new hybrid shrinking projection method for finding a common element of the set EP of solutions of a generalized equilibrium problem, the common fixed point set F of finite uniformly closed families of countable Bregman quasi-Lipschitz mappings in reflexive Banach spaces. It is proved that under appropriate conditions, the sequence generated by the hybrid shrinking projection method converges strongly to some point in \(\mathit{EP} \cap F\). Relative examples are given. Strong convergence theorems are proved. The application for Bregman asymptotically quasi-nonexpansive mappings is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasi-Lipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the application for Bregman asymptotically quasi-nonexpansive mappings is also given; (4) because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This 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 \(\langle\cdot,\cdot\rangle\) the pairing between E and \(E^{*}\).
Proposition 2.1
([19])
- (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
([20])
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
([20])
- (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
([25])
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
([27])
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
([28])
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
([29])
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
([26])
- (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 this paper, we present the following definition.
Definition 2.11
Bregman quasi-Lipschitz mappings are a more generalized class than the class of Bregman quasi-mappings. On the other hand, this class also contains the relatively quasi-Lipschitz mappings and quasi-Lipschitz mappings. Therefore, Bregman quasi-Lipschitz mappings are very important in the nonlinear analysis and fixed point theory and applications.
Definition 2.12
Let C be a nonempty, closed, and convex subset of E. Let \(\{T_{n}\}\) be 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.
In Section 4, we will give two examples of a uniformly closed family of countable Bregman quasi-Lipschitz mappings.
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 2.13
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 2.14
- (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)
\(F(T_{r})=\hat{F}(T_{r})=\mathit{EP}(F)\).
- (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 2.15
- (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)
\(F(K_{r})=\hat{F}(K_{r})=\mathit{EP}\).
- (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
3 Main results
Theorem 3.1
Proof
Firstly, we prove that F is closed. Let \(\{p_{n}\}\subset F\), \(p_{n}\rightarrow p\) as \(n\rightarrow\infty\), then \(\|T_{n}p_{n}-p_{n}\|=0\rightarrow0\) as \(n\rightarrow\infty\). Since \(\{T_{n}\}\) is uniformly closed, we know that \(p\in F\). Hence F is closed.
Next we will prove the main strong convergence theorem for the finite families of countable Bregman quasi-Lipschitz mappings by using a new hybrid projection scheme. In this scheme, we will use some detailed technology.
Theorem 3.2
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 F.
Step 5. We show that the limit of \(\{x_{n}\}\) belongs to EP.
Definition 3.3
If we choose \(f (x) =\frac{1}{2} \|x\|^{2}\) for all \(x \in E\), then Theorem 3.2 reduces to the following corollary.
Corollary 3.4
4 Example
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 Bregman quasi-Lipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L_{n}=\lim_{n\rightarrow\infty}\frac{n+1}{n}=1\).
Proof
Example 4.3
Example 4.4
5 Application
Lemma 5.1
Proof
Next we give an application of Theorem 3.2 to find the fixed point of Bregman asymptotically quasi-nonexpansive mappings.
Theorem 5.2
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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