Strong convergence theorems for Bregman quasi-strict pseudo-contractions in reflexive Banach spaces with applications
- Zi-Ming Wang^{1}Email author
Received: 3 January 2015
Accepted: 25 May 2015
Published: 17 June 2015
Abstract
In this paper, a simple iterative algorithm is introduced for finding a fixed point of a Bregman quasi-strict pseudo-contraction. Furthermore, strong convergence results are established in a reflexive Banach space. Finally, the solution of equilibrium problem, variational inequality, and zero point problem of maximal monotone operator are considered as applications.
Keywords
MSC
1 Introduction
Since nonexpansive fixed point theory can be applied to the solution of diverse problems such that solving variational inequality problems, equilibrium problems, and convex feasibility problems, strict pseudo-contractions have more powerful applications than nonexpansive mappings in solving inverse problems [2]. In recent years, construction of an iterative algorithm for seeking fixed points of nonexpansive mappings and strict pseudo-contractions has extensively been investigated; see [1, 3–12] and the references therein. It is well known that, in an infinite-dimensional Hilbert space, the normal Mann iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. So, in order to get strong convergence for nonexpansive mappings and strict pseudo-contractive mappings, one has to modify the normal Mann’s iterative algorithm such as the so-called hybrid projection iteration method.
When one tries to extend this theory to general Banach spaces, some difficulties must be encountered because many of the useful examples of nonexpansive mappings in Hilbert spaces are no longer nonexpansive in Banach spaces, for example, the resolvent \(R_{A}:=(I+A)^{-1}\) of a maximal monotone mapping \(A:H\rightarrow2^{H}\) and the metric projection \(P_{C}\). In this connection, Alber [13] introduced a generalized projection operator \(\Pi_{C}\) in Banach spaces which is an analogue of the metric projection in Hilbert spaces. Recently, applying the generalized projection operator in reflexive, strictly convex and smooth Banach spaces with some property, Zhou and Gao [14] introduced a modified hybrid projection iterative algorithm and proved a strong convergence theorem for a closed quasi-strict pseudo-contraction which is an extension of strict pseudo-contractive mappings and relatively nonexpansive mappings [15–17]. In fact, there are several ways to overcome these difficulties. Another way is to use the Bregman distance instead of the norm, Bregman (quasi-)nonexpansive mappings instead of the (quasi-)nonexpansive mappings and the Bregman projection instead of the metric projection.
In recent years, many authors focused attention on constructing the fixed point of Bregman nonlinear operators by utilizing the Bregman distance and the Bregman projection, see [18–23] and the references therein. In 2014, Zegeye and Shahzad [21] investigated an iterative scheme for a Bregman relatively nonexpansive mapping. Very recently, Ugwunnadi et al. [23] introduced the concept of Bregman quasi-strict pseudo-contraction and proved the strong convergence by using hybrid Bregman projection iterative algorithm for Bregman quasi-strict pseudo-contractions.
Motivated and inspired by the above works, in this paper we aim to propose a new simple hybrid Bregman projection iterative algorithm for a Bregman quasi-strict pseudo-contraction and to prove strong convergence results in the framework of reflexive Banach spaces. The results presented in this paper improve the known corresponding results announced in the literature sources listed in this work.
2 Preliminaries
In this section, we collect some preliminaries and lemmas which will be used to prove our main results.
A point \(p\in C\) is said to be an asymptotic fixed point of a mapping T if C contains a sequence \(\{x_{n}\}\) 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 a strong asymptotic fixed point of a mapping T if C contains a sequence \(\{x_{n}\}\) 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.
- (a)
- (b)T is said to be relatively weak nonexpansive if \(\widetilde {F}(T)=F(T)\neq\emptyset\) and$$ \phi(p,Tx)\leq\phi(p,x),\quad \forall x\in C, p\in F(T). $$(2.6)
- (c)T is said to be hemi-relatively nonexpansive if \(F(T)\neq\emptyset \) and$$ \phi(p,Tx)\leq\phi(p,x), \quad\forall x\in C, p\in F(T). $$(2.7)
- (d)T is said to be quasi-ϕ-strictly pseudo-contractive [14] if \(F(T)\neq\emptyset\) and there exists a constant \(k\in[0,1)\) such that$$ \phi(p,Tx)\leq\phi(p,x)+k\phi(x,Tx),\quad \forall x\in C, p\in F(T). $$(2.8)
Remark 2.1
- (1)
The class of relatively nonexpansive mappings is included by the class of relatively weak nonexpansive mappings. In fact, for any mapping \(T:C\rightarrow C\), we have \(F(T)\subset\widetilde{F}(T)\subset\widehat {F}(T)\). Therefore, if T is a relatively nonexpansive mapping, then \(F(T)=\widetilde{F}(T)=\widehat{F}(T)\).
- (2)
The class of relatively weak nonexpansive mappings is contained by the class of hemi-relatively nonexpansive mappings. Hemi-relatively nonexpansive mappings do not require \(F(T)=\widetilde{F}(T)\).
- (3)
The class of quasi-ϕ-strict pseudo-contractions is more general than the class of hemi-relatively nonexpansive mappings. In fact, a hemi-relatively nonexpansive mapping is a quasi-ϕ-strict pseudo-contraction with \(k=0\).
- (i)(The three point identity): for any \(x\in \operatorname{dom} f\) and \(y, z\in \operatorname{int} \operatorname{dom} f\),$$ D_{f}(x,y)+D_{f}(y,z)-D_{f}(x,z)=\bigl\langle \triangledown f(z)-\triangledown f(y),x-y\bigr\rangle ; $$(2.10)
- (ii)(The four point identity): for any \(y, w\in \operatorname{dom} f\) and \(x, z\in \operatorname{int} \operatorname{dom} f\),$$ D_{f}(y,x)-D_{f}(y,z)-D_{f}(w,x)+D_{f}(w,z)= \bigl\langle \triangledown f(z)-\triangledown f(x),y-w\bigr\rangle . $$(2.11)
It should be observed that if E is a smooth Banach space, setting \(f(x)=\|x\|^{2}\) for all \(x\in E\), we have \(\triangledown f(x)=2Jx\) for all \(x\in E\). Hence \(D_{f}(x,y)\) reduces to the Lyapunov function \(\phi (x,y)=\|2\|^{2}-2\langle x,Jy\rangle+\|y\|^{2}\) for all \(x, y\in E\) and the Bregman projection \(P^{f}_{C}(x)\) reduces to the generalized projection \(\Pi_{C}\) from E onto C. If E is a Hilbert space H, then \(D_{f}(x,y)\) becomes \(\phi(x,y)=\|x-y\|^{2}\) for \(x, y\in H\) and the Bregman projection \(P^{f}_{C}(x)\) becomes the metric projection \(P_{C}\) from E onto C.
Similarly to the metric projection in a Hilbert space, Bregman projections with respect to totally convex and differentiable functions have variational characterizations.
Lemma 2.2
(see Butnariu and Resmerita [26])
- (a)
The vector \(\hat{x}\) is the Bregman projection of x onto C with respect to f, i.e., \(z=P^{f}_{C}(x)\);
- (b)The vector \(\hat{x}\) is the unique solution of the variational inequality$$ \bigl\langle \triangledown f(x)-\triangledown f(z), z-y\bigr\rangle \geq0, \quad\forall y\in C; $$(2.13)
- (c)The vector \(\hat{x}\) is the unique solution of the inequality$$ D_{f}(y,z)+D_{f}(z,x)\leq D_{f}(y,x), \quad\forall y\in C. $$(2.14)
The function f is said to be essentially smooth if ∂f is both locally bounded and single-valued on its domain. It is called essentially strictly convex if \((\partial f)^{-1}\) is locally bounded on its domain and f is strictly convex on every convex subset of \(\operatorname{dom} \partial f\). f is said to be a Legendre if it is both essentially smooth and essentially strictly convex. When the subdifferential of f is single-valued, it coincides with the gradient \(\partial f=\triangledown f\), see [27].
- (i)
f is essentially smooth if and only if \(f^{*}\) is essentially strictly convex, see [28];
- (ii)
\((\partial f)^{-1}=\partial f^{*}\), see [29];
- (iii)
f is Legendre if and only if \(f^{*}\) is Legendre, see [28];
- (iv)
If f is Legendre, then ▽f is a bijection satisfying \(\triangledown f=(\triangledown f^{*})^{-1}\), \(\operatorname{ran} \triangledown f=\operatorname{dom} \triangledown f^{*}=\operatorname{int} \operatorname{dom} f^{*}\) and \(\operatorname{ran} \triangledown f^{*}=\operatorname{dom} \triangledown f=\operatorname{int} \operatorname{dom} f\), see [28].
The following result is useful in the next section.
Lemma 2.3
(see Bauschke et al. [28])
Suppose \(x\in E\) and \(y\in \operatorname{int} \operatorname{dom} f\). If f is essentially strictly convex, then \(D_{f}(x,y)=0\Leftrightarrow x=y\).
When E is a smooth and strictly convex Banach space, one important and interesting example of a Legendre function is \(f(x)=\frac{1}{p}\|x\| ^{p}\) (\(1< p<\infty\)). In this case the gradient ▽f of f coincides with the generalized duality mapping of E, i.e., \(\triangledown f=J_{p}\) (\(1< p<\infty\)). In particular, \(\triangledown f=I\), the identity mapping in Hilbert spaces.
The following lemmas will be useful in the proof of the next section.
Lemma 2.4
(see Butnariu and Iusem [30])
The function f is totally convex on bounded sets if and only if the function f is sequentially consistent.
Lemma 2.5
(see Reich and Sabach [18])
Let \(f:E\rightarrow\mathbb{R}\) be a Gâteaux differentiable and totally convex function. If \(x_{0}\in E\) and the sequence \(\{ D_{f}(x_{n},x_{0})\}\) is bounded, then the sequence \(\{x_{n}\}\) is bounded too.
Recall the following definitions.
Definition 2.6
- (1)T is said to be Bregman relatively nonexpansive if \(\widehat {F}(T)=F(T)\neq\emptyset\) and$$ D_{f}(p,Tx)\leq D_{f}(p,x),\quad \forall x\in C, p\in F(T). $$(2.20)
- (2)T is said to be Bregman weak relatively nonexpansive if \(\widetilde{F}(T)=F(T)\neq\emptyset\) and$$ D_{f}(p,Tx)\leq D_{f}(p,x),\quad \forall x\in C, p\in F(T). $$(2.21)
- (3)T is said to be Bregman quasi-nonexpansive if \(F(T)\neq\emptyset\) and$$ D_{f}(p,Tx)\leq D_{f}(p,x),\quad \forall x\in C, p\in F(T). $$(2.22)
- (4)T is said to be Bregman quasi-strictly pseudo-contractive [23] if there exists a constant \(k\in[0,1)\) and \(F(T)\neq\emptyset\) such that$$ D_{f}(p,Tx)\leq D_{f}(p,x)+kD_{f}(x,Tx), \quad\forall x \in C, p\in F(T). $$(2.23)
- (5)
A mapping \(T:C\rightarrow C\) is said to be closed if for any sequence \(\{x_{n}\}\subset C\) with \(x_{n}\rightarrow x\in C\) and \(Tx_{n}\rightarrow y\in C\) as \(n\rightarrow\infty\), then \(Tx=y\).
Remark 2.7
- (1)
Bregman relatively nonexpansive mappings, Bregman weak relatively nonexpansive mappings, Bregman quasi-nonexpansive mappings, and Bregman quasi-strict pseudo-contractions are more general than relatively nonexpansive mappings, relatively weak nonexpansive mappings, hemi-relatively nonexpansive mappings, and quasi-ϕ-strictly pseudo-contractions, respectively.
- (2)
The class of Bregman quasi-strictly pseudo-contractions is more general than the class of Bregman relatively nonexpansive mappings, the class of Bregman weak relatively nonexpansive mappings, and the class of Bregman quasi-nonexpansive mappings.
Now, we give some examples of Bregman quasi-strict pseudo-contractions.
Example 2.8
(see Reich and Sabach [31])
Example 2.9
Proof
Example 2.10
(see Ugwunnadi et al. [23])
Let \(E=\mathbb{R}\) and define \(T, f:[-1,0]\rightarrow\mathbb{R}\) by \(f(x)=x\) and \(Tx=2x\) for all \(x\in[-1,0]\). Then T is a Bregman quasi-strict pseudo-contraction but not a quasi-ϕ-strict pseudo-contraction.
3 Main results
In this section, we state and prove our main theorem.
Theorem 3.1
Proof
The proof is split into seven steps.
Step 1: Show that \(F(T)\) is closed and convex.
First, we prove that \(F(T)\) is closed. Let \(\{p_{n}\}\) be a sequence in \(F(T)\) with \(p_{n}\rightarrow p\) as \(n\rightarrow\infty\). One has \(Tp_{n}=p_{n}\rightarrow p\) as \(n\rightarrow\infty\). By the closedness of T, one has \(Tp=p\). This implies that \(F(T)\) is closed.
Step 2: Show that \(C_{n}\) is closed and convex for all \(n\geq1\) .
Step 3: Show that \(F(T)\subset C_{n}\) for all \(n\geq1\) .
Step 4: Show that \(\lim_{n\rightarrow\infty}D_{f}(x_{n},x_{0})\) exists.
On the other hand, noticing that \(x_{n}=P_{C_{n}}^{f}(x_{0})\) and \(x_{n+1}=P_{C_{n+1}}^{f}(x_{0})\in C_{n+1}\subset C_{n}\), one has \(D_{f}(x_{n},x_{0})\leq D_{f}(x_{n+1},x_{0})\) for all \(n\geq1\). This implies that \(\{D_{f}(x_{n},x_{0})\}\) is a nondecreasing sequence. Therefore, \(\lim_{n\rightarrow\infty}D_{f}(x_{n},x_{0})\) exists.
Step 5: Show that \(x_{n}\rightarrow\widehat{p}\) as \(n\rightarrow \infty\) .
Step 6: Show that \(\widehat{p}=T\widehat{p}\) .
Step 7: Show that \(\widehat{p}=P_{F(T)}^{f}(x_{0})\) .
Since the class of Bregman quasi-nonexpansive mappings is Bregman quasi-strict pseudo-contractive, the following corollary is obtained by using Theorem 3.1.
Corollary 3.2
Setting \(f(x)=\|x\|^{2}\) for all \(x\in E\), then \(\triangledown f(x)=2Jx\) for all \(x\in E\). Hence \(D_{f}(x,y)\) reduces to the Lyapunov function \(\phi(x,y)=\|2\|^{2}-2\langle x,Jy\rangle+\|y\|^{2}\) for all \(x, y\in E\), the Bregman projection \(P^{f}_{C}(x)\) reduces to the generalized projection \(\Pi_{C}\) from E onto C and the Bregman quasi-nonexpansive mapping reduces to the hemi-relatively nonexpansive mapping. So, by utilizing Corollary 3.2, the following corollary is obtained.
Corollary 3.3
Similar to Corollary 3.3, the following corollary can be obtained from Theorem 3.1.
Corollary 3.4
4 Applications
4.1 Application to equilibrium problems
- (A1)
\(g(x,x)=0\) for all \(x\in C\);
- (A2)
g is monotone, i.e., \(g(x,y)+g(y,x)\leq0\) for all \(x, y\in C\);
- (A3)for all \(x, y, z\in C\),$$ \limsup_{t\downarrow0}g\bigl(tz+(1-t)x,y\bigr)\leq g(x,y); $$(4.1)
- (A4)
for each \(x\in C\), \(g(x,\cdot)\) is convex and lower semicontinuous.
- (1)
\(\operatorname{Res}_{g}^{f}\) is single-valued;
- (2)
The set of fixed points of \(\operatorname{Res}_{g}^{f}\) is the solution set of the corresponding equilibrium problem, i.e., \(F(\operatorname{Res}_{g}^{f})=EP(g)\);
- (3)
\(\operatorname{Res}_{g}^{f}\) is a closed Bregman quasi-nonexpansive mapping, so is a closed Bregman quasi-strict pseudo-contraction.
Theorem 4.1
Proof
Since \(\operatorname{Res}_{g}^{f}\) is a closed Bregman quasi-strict pseudo-contraction, by applying Theorem 3.1, the sequence \(\{x_{n}\}\) converges strongly to \(\widehat{p}=P_{EP(g)}^{f}(x_{0})\). □
4.2 Application to variational inequality problems
The following result which points out the connection between the fixed point set of \(P_{C}^{f}\circ A^{f}\) and the solution set of the variational inequality corresponding to the Bregman inverse strongly monotone operator A was introduced by Reich and Sabach [31].
Lemma 4.2
Let \(f:E\rightarrow(-\infty,+\infty]\) be a Legendre and totally convex function which satisfies the range condition \(\operatorname{ran} (\triangledown f-A)\subseteq \operatorname{ran} (\triangledown f)\). Let \(A:E\rightarrow E^{*}\) be a Bregman inverse strongly monotone mapping. If C is a nonempty, closed, and convex subset of \((\operatorname{dom} A) \cap(\operatorname{int} \operatorname{dom} f)\), then (1) \(VI(C,A)=F(P_{C}^{f}\circ A^{f})\); (2) \(P_{C}^{f}\circ A^{f}\) is a Bregman relatively nonexpansive mapping, so is a closed Bregman quasi-strict pseudo-contraction.
Theorem 4.3
4.3 Application to zero point problem of maximal monotone operators
Let E be a real reflexive Banach space, \(A:E\rightarrow2^{E^{*}}\) be a maximal monotone operator. The problem of finding an element \(x\in E\) such that \(0^{*}\in Ax\) is very important in optimization theory and related fields.
From Example 2.8, we know that \(\operatorname{Res}_{A}^{f}\) is a closed Bregman quasi-strict pseudo-contraction. So the following result is obtained easily by applying Theorem 3.1.
Theorem 4.4
5 Numerical examples
In this section, we use a numerical example to demonstrate the convergence of Theorem 3.1.
Declarations
Acknowledgements
The author is grateful to the referees and the editor for their careful reading and for their valuable comments and suggestions which led to the present form of the paper. This research is supported by the Project of Shandong Province Higher Educational Science and Technology Program (grant No. J15LI51, No. J14LI51) and the Science and Technology Research of Education Department Program of Jiangxi Province (grant No. GJJ14759).
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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