Strong convergence theorems for Bregman W-mappings with applications to convex feasibility problems in Banach spaces
- Eskandar Naraghirad^{1}Email author and
- Sara Timnak^{1}
https://doi.org/10.1186/s13663-015-0395-1
© Naraghirad and Timnak 2015
Received: 31 January 2015
Accepted: 4 August 2015
Published: 20 August 2015
Abstract
In this paper we introduce new modified Mann iterative processes for computing fixed points of an infinite family of Bregman W-mappings in reflexive Banach spaces. Let \(W_{n}\) be the Bregman W-mapping generated by \(S_{n},S_{n-1},\ldots,S_{1}\) and \(\beta_{n,n},\beta_{n,n-1},\ldots,\beta_{n,1}\). We first express the set of fixed points of \(W_{n}\) as the intersection of fixed points of \(\{S_{i}\}_{i=1}^{n}\). As a consequence, we show that \(W_{n}\) is a Bregman weak relatively nonexpansive mapping if \(S_{i}\) is a Bregman weak relatively nonexpansive mapping for each \(i=1,2,\ldots,n\). When specialized to the fixed point set of a Bregman nonexpansive type mapping T, the required sufficient condition \(\tilde{F}(T)=F(T)\) is less restrictive than the usual condition \(\hat{F}(T)=F(T)\) which is based on the demiclosedness principle. We then prove some strong convergence theorems for these mappings. Some application of our results to convex feasibility problem is also presented. Our results improve and generalize many known results in the current literature.
Keywords
MSC
1 Introduction
Let C be a nonempty, closed and convex subset of a smooth Banach space E, let T be a mapping from C into itself. A point \(p\in C\) is said to be an asymptotic fixed point [11] of T if there exists a sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in C which converges weakly to p and \(\lim_{n\to\infty} \Vert x_{n} -Tx_{n}\Vert =0\). We denote the set of all asymptotic fixed points of T by \(\hat{F}(T)\). A point \(p\in C\) is called a strong asymptotic fixed point of T if there exists a sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in C which converges strongly to p and \(\lim_{n\to\infty} \Vert x_{n}-Tx_{n}\Vert =0\). We denote the set of all strong asymptotic fixed points of T by \(\tilde{F}(T)\).
- (1)
\(F(T)\) is nonempty;
- (2)
\(\phi(u,Tx)\leq\phi(u,x)\), \(\forall u\in F(T)\), \(x\in C\);
- (3)
\(\hat{F}(T)=F(T)\).
In 2005, Matsushita and Takahashi [12] proved the following strong convergence theorem for relatively nonexpansive mappings in a Banach space.
Theorem 1.1
1.1 Some facts about gradients
Let \(A:E\to2^{E^{*}}\) be a set-valued mapping. We define the domain and range of A by \(\operatorname {dom}A=\{x\in E:Ax\neq{\O}\}\) and \(\operatorname {ran}A=\bigcup_{x\in E} Ax\), respectively. The graph of A is denoted by \(G(A)=\{(x,x^{*})\in E\times E^{*}:x^{*}\in Ax\}\). The mapping \(A\subset E\times E^{*}\) is said to be monotone [15] if \(\langle x-y,x^{*}-y^{*} \rangle\geq0\) whenever \((x,x^{*}),(y,y^{*})\in A\). It is also said to be maximal monotone [16] if its graph is not contained in the graph of any other monotone operator on E. If \(A\subset E\times E^{*}\) is maximal monotone, then we can show that the set \(A^{-1}0=\{z\in E:0\in Az\}\) is closed and convex.
1.2 Some facts about Legendre functions
- (i)
Essentially smooth if ∂g is both locally bounded and single-valued on its domain.
- (ii)
Essentially strictly convex if \((\partial g)^{-1}\) is locally bounded on its domain and g is strictly convex on every convex subset of \(\operatorname {dom}\partial g\).
- (iii)
Legendre if it is both essentially smooth and essentially strictly convex (for more details, we refer to [19]).
Examples of Legendre functions are given in [21, 22]. The most notable example of a Legendre function is \(\frac{1}{s}\Vert \cdot \Vert ^{s}\) (\(1< s<\infty\)), where the Banach space E is smooth and strictly convex and, in particular, a Hilbert space.
1.3 Some facts about Bregman distances
1.4 Some facts about uniformly convex functions
1.5 Some facts about resolvents
Examples and some important properties of such operators are discussed in [28].
1.6 Some facts about Bregman quasi-nonexpansive mappings
- (1)
\(F(T)\) is nonempty;
- (2)
\(D_{g}(p,Tv)\leq D_{g}(p,v)\), \(\forall p\in F(T)\), \(v\in C\);
- (3)
\(\hat{F}(T)=F(T)\).
- (1)
\(F(T)\) is nonempty;
- (2)
\(D_{g}(p,Tv)\leq D_{g}(p,v)\), \(\forall p\in F(T)\), \(v\in C\);
- (3)
\(\tilde{F}(T)=F(T)\).
The theory of fixed points with respect to Bregman distances has been studied in the last ten years and much intensively in the last four years. For some recent articles on the existence of fixed points for Bregman nonexpansive type mappings, we refer the readers to [20–22, 27, 28]. But it is worth mentioning that, in all the above results for Bregman nonexpansive type mappings, the assumption \(\hat{F}(T)=F(T)\) is imposed on the map T. So, the following question arises naturally in a Banach space setting.
Question 1.1
Is it possible to obtain strong convergence of modified Mann-type schemes to a common fixed point of an infinite family of Bregman W-mappings \(\{S_{j}\}_{j\in \mathbb{N}}\) without imposing the assumption \(\hat{F}(S_{j})=F(S_{j})\) on \(S_{j}\)?
In this paper we introduce new modified Mann iterative processes for computing fixed points of an infinite family of Bregman W-mappings in reflexive Banach spaces. Let \(W_{n}\) be the Bregman W-mapping generated by \(S_{n},S_{n-1},\ldots,S_{1}\) and \(\beta_{n,n},\beta_{n,n-1},\ldots,\beta_{n,1}\). We first express the set of fixed points of \(W_{n}\) as the intersection of fixed points of \(\{S_{i}\}_{i=1}^{n}\). As a consequence, we show that \(W_{n}\) is a Bregman weak relatively nonexpansive mapping if \(S_{i}\) is a Bregman weak relatively nonexpansive mapping for each \(i=1,2,\ldots,n\). We then prove some strong convergence theorems for these mappings. Some application of our results to convex feasibility problem is also presented. No assumption \(\hat{F}(T)=F(T)\) is imposed on the mapping T. Consequently, the above question is answered in the affirmative in a reflexive Banach space setting. Our results improve and generalize many known results in the current literature; see, for example, [8, 9, 12, 30–33].
2 Preliminaries
In this section, we begin by recalling some preliminaries and lemmas which will be used in the sequel.
The following definition is slightly different from that in Butnariu and Iusem [13].
Definition 2.1
([14])
- (1)
g is continuous, strictly convex and Gâteaux differentiable;
- (2)
the set \(\{y\in E:D_{g}(x,y)\leq r\}\) is bounded for all \(x\in E\) and \(r>0\).
The following lemma follows from Butnariu and Iusem [13] and Zălinescu [26].
Lemma 2.1
- (1)
\(\nabla g:E\to E^{*}\) is one-to-one, onto and norm-to-weak ^{∗} continuous;
- (2)
\(\langle x-y,\nabla g(x)-\nabla g(y) \rangle=0\) if and only if \(x=y\);
- (3)
\(\{x\in E:D_{g}(x,y)\leq r\}\) is bounded for all \(y\in E\) and \(r>0\);
- (4)
\(\operatorname {dom}g^{*}=E^{*}, g^{*}\) is Gâteaux differentiable and \(\nabla g^{*}=(\nabla g)^{-1}\).
We know the following two results from [26].
Theorem 2.1
- (1)
g is strongly coercive and uniformly convex on bounded subsets of E;
- (2)
\(\operatorname {dom}g^{*}=E^{*}, g^{*}\) is bounded on bounded subsets and uniformly smooth on bounded subsets of \(E^{*}\);
- (3)
\(\operatorname {dom}g^{*}=E^{*}, g^{*}\) is Fréchet differentiable and \(\nabla g^{*}\) is uniformly norm-to-norm continuous on bounded subsets of \(E^{*}\).
Theorem 2.2
- (1)
g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;
- (2)
\(g^{*}\) is Fréchet differentiable and \(\nabla g^{*}\) is uniformly norm-to-norm continuous on bounded subsets of \(E^{*}\);
- (3)
\(\operatorname {dom}g^{*}=E^{*}, g^{*}\) is strongly coercive and uniformly convex on bounded subsets of \(E^{*}\).
The following result was proved in [29].
Lemma 2.2
The following result was first proved in [19] (see also [14]).
Lemma 2.3
- (1)
\(D_{g}(x,\nabla g^{*}(x^{*}))=V_{g}(x,x^{*})\) for all \(x\in E\) and \(x^{*}\in E^{*}\).
- (2)
\(V_{g}(x,x^{*})+\langle\nabla g^{*}(x^{*})-x,y^{*} \rangle\leq V_{g}(x,x^{*}+y^{*})\) for all \(x\in E\) and \(x^{*},y^{*}\in E^{*}\).
The following result was proved in [29].
Lemma 2.4
- (i)For any \(x,y\in B_{r}\) and \(\alpha\in(0,1)\),$$g\bigl(\alpha x+(1-\alpha)y\bigr)\leq \alpha g(x)+(1-\alpha)g(y)-\alpha(1- \alpha)\rho_{r}\bigl(\Vert x-y\Vert \bigr). $$
- (ii)For any \(x,y\in B_{r}\),$$\rho_{r}\bigl(\Vert x-y\Vert \bigr)\leq D_{g}(x,y). $$
- (iii)If, in addition, g is bounded on bounded subsets and uniformly convex on bounded subsets of E then, for any \(x\in E\), \(y^{*},z^{*}\in B_{r}\) and \(\alpha\in(0,1)\),$$V_{g}\bigl(x,\alpha y^{*}+(1-\alpha)z^{*}\bigr)\leq\alpha V_{g} \bigl(x,y^{*}\bigr)+(1-\alpha)V_{g}\bigl(x,z^{*}\bigr)-\alpha(1-\alpha) \rho^{*}_{r}\bigl(\bigl\Vert y^{*}-z^{*}\bigr\Vert \bigr). $$
The following result was proved in [29].
Lemma 2.5
Now we prove the following important result.
Proposition 2.1
- (i)
\(F(W_{n})=\bigcap_{i=1}^{n}F(S_{i})\);
- (ii)
for every \(k=1,2,\ldots,n\), \(x\in C\) and \(z\in F(W_{n})\), \(D_{g}(z,U_{n,k}x)\leq D_{g}(z,x)\) and \(D_{g}(z,S_{k}U_{n,k+1}x)\leq D_{g}(z,x)\);
- (iii)
for every \(n\in\Bbb{N}\), \(W_{n}\) is a Bregman weak relatively nonexpansive mapping.
Proof
(i) It is clear that \(\bigcap_{i=1}^{n}F(S_{i})\subset F(W_{n})\). For the converse inclusion, take any \(w\in \bigcap_{i=1}^{n}F(S_{i})\) and \(z\in F(W_{n})\).
Next we prove the following convex combination of Bregman weak relatively nonexpansive mappings in a Banach space.
Proposition 2.2
- (i)
\(\bigcap_{n=1}^{\infty}F(T_{n})=F\);
- (ii)
for every \(n\in\Bbb{N}\), \(x\in C\) and \(z\in F\), \(D_{g}(z,T_{n}x)\leq D_{g}(z,x)\);
- (iii)
for every \(n\in\Bbb{N}\), \(T_{n}\) is a Bregman weak relatively nonexpansive mapping.
Proof
3 Strong convergence theorems
In this section, we prove strong convergence theorems in a reflexive Banach space. We start with the following simple lemma which was proved in [35].
Lemma 3.1
Let E be a reflexive Banach space and \(g:E\to\mathbb{R}\) be a convex, continuous, strongly coercive and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of E. Let \(T:C\to C\) be a Bregman quasi-nonexpansive mapping. Then \(F(T)\) is closed and convex.
Theorem 3.1
Proof
We divide the proof into several steps.
Step 1. We show that \(C_{n}\) is closed and convex for each \(n\in\mathbb{N}\cup\{0\}\).
Step 2. We claim that \(F\subset C_{n}\) for all \(n\in\mathbb {N}\cup\{0\}\).
Step 3. We prove that \(\{x_{n}\}_{n\in \mathbb{N}}, \{y_{n}\}_{n\in\mathbb{N}}\) and \(\{W_{n}x_{n}\}_{n\in \mathbb{N}}\) are bounded sequences in C.
Step 4. We show that \(x_{n}\to u\) for some \(u\in F\), where \(u=\operatorname {proj}^{g}_{F}x\).
The function g is bounded on bounded subsets of E and, thus, ∇g is also bounded on bounded subsets of \(E^{*}\) (see, for example, [13] for more details). This implies that the sequences \(\{\nabla g(x_{n})\}_{n\in\mathbb{N}}\), \(\{\nabla g(y_{n})\}_{n\in\mathbb{N}}\) and \(\{\nabla g(W_{n}x_{n}):n\in \mathbb{N}\cup\{0\}\}\) are bounded in \(E^{*}\).
Theorem 3.2
Remark 3.1
- (1)
For the structure of Banach spaces, we extend the duality mapping to a more general case, that is, a convex, continuous and strongly coercive Bregman function which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets.
- (2)
For the mappings, we extend the mapping from a relatively nonexpansive mapping to a countable family of Bregman W-mappings. We remove the assumption \(\hat{F}(T)=F(T)\) on the mapping T and extend the result to a countable family of Bregman weak relatively nonexpansive mappings, where \(\hat{F}(T)\) is the set of asymptotic fixed points of the mapping T.
- (3)
For the algorithm, we remove the set \(W_{n}\) in Theorem 1.1.
The following result was proved in [29].
Lemma 3.2
Let E be a reflexive Banach space and \(g:E\to\mathbb{R}\) be a strongly coercive Bregman function which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets of E. Let A be a maximal monotone operator from E to \(E^{*}\) such that \(A^{-1}(0)\neq{\O}\). Let \(r>0\) and \(\operatorname {Res}^{g}_{rA}=(\nabla g+rA)^{-1}\nabla g\) be the g-resolvent of A. Then \(\operatorname {Res}^{g}_{rA}\) is a Bregman weak relatively nonexpansive mapping.
As an application of our main result, we include a concrete example in support of Theorem 3.1. Using Theorem 3.1, we obtain the following strong convergence theorem for maximal monotone operators.
Theorem 3.3
- (1)
\(\liminf_{n\to\infty}\alpha_{n}(1-\alpha_{n})>0\);
- (2)
\(0\leq\beta_{n}<1\) for all \(n\in\mathbb{N}\cup\{0\}\) and \(\liminf_{n\to \infty}\beta_{n}<1\).
Proof
Below we include a nontrivial example of an infinite family of Bregman weak relatively nonexpansive mappings in order to reconstruct a Bregman W-mapping in the setting of Hilbert spaces.
Example 3.1
4 Applications to convex feasibility problems
Let \(\{D_{n}\}_{n\in\Bbb{N}}\) be a family of nonempty, closed and convex subsets of a Banach space E. The convex feasibility problem is to find an element in the assumed nonempty intersection \(\bigcap_{n=1}^{\infty}D_{n}\) (see [36]). In the following, we prove a strong convergence theorem concerning convex feasibility problems in a reflexive Banach space.
Theorem 4.1
Proof
5 Numerical example
In this section, in order to demonstrate the effectiveness, realization and convergence of algorithm of Theorem 3.1, we consider the following simple example.
Example 5.1
In this section, we give some numerical experiment results (based on Matlab) as follows.
6 Conclusion
This table shows the values of the sequence \(\pmb{\{x_{n}\}_{n\in\Bbb{N}}}\) on 30th iteration steps (initial value \(\pmb{x_{0}=1}\) )
n | \(\boldsymbol{x_{n}}\) | \(\boldsymbol{u_{n}}\) | \(\boldsymbol{y_{n}}\) |
---|---|---|---|
1 | 1.000000000000000e+000 | 5.000000000000000e−001 | 2.500000000000000e−001 |
2 | 6.250000000000000e−001 | 3.125000000000000e−001 | 1.562500000000000e−001 |
3 | 3.906250000000000e−001 | 1.953125000000000e−001 | 9.765625000000000e−002 |
4 | 2.441406250000000e−001 | 1.220703125000000e−001 | 6.103515625000000e−002 |
5 | 1.525878906250000e−001 | 7.629394531250000e−002 | 3.814697265625000e−002 |
6 | 9.536743164062500e−002 | 4.768371582031250e−002 | 2.384185791015625e−002 |
7 | 5.960464477539063e−002 | 2.980232238769531e−002 | 1.490116119384766e−002 |
8 | 3.725290298461914e−002 | 1.862645149230957e−002 | 9.313225746154785e−003 |
9 | 2.328306436538696e−002 | 1.164153218269348e−002 | 5.820766091346741e−003 |
10 | 1.455191522836685e−002 | 7.275957614183426e−003 | 3.637978807091713e−003 |
11 | 9.094947017729282e−003 | 4.547473508864641e−003 | 2.273736754432321e−003 |
12 | 5.684341886080802e−003 | 2.842170943040401e−003 | 1.421085471520200e−003 |
13 | 3.552713678800501e−003 | 1.776356839400251e−003 | 8.881784197001252e−004 |
14 | 2.220446049250313e−003 | 1.110223024625157e−003 | 5.551115123125783e−004 |
15 | 1.387778780781446e−003 | 6.938893903907228e−004 | 3.469446951953614e−004 |
16 | 8.673617379884036e−004 | 4.336808689942018e−004 | 2.168404344971009e−004 |
17 | 5.421010862427522e−004 | 2.710505431213761e−004 | 1.355252715606881e−004 |
18 | 3.388131789017201e−004 | 1.694065894508601e−004 | 8.470329472543003e−005 |
19 | 2.117582368135751e−004 | 1.058791184067875e−004 | 5.293955920339377e−005 |
20 | 1.323488980084844e−004 | 6.617444900424221e−005 | 3.308722450212111e−005 |
21 | 8.271806125530277e−005 | 4.135903062765138e−005 | 2.067951531382569e−005 |
22 | 5.169878828456423e−005 | 2.584939414228212e−005 | 1.292469707114106e−005 |
23 | 3.231174267785264e−005 | 1.615587133892632e−005 | 8.077935669463161e−006 |
24 | 2.019483917365790e−005 | 1.009741958682895e−005 | 5.048709793414475e−006 |
25 | 1.262177448353619e−005 | 6.310887241768094e−006 | 3.155443620884047e−006 |
26 | 7.888609052210119e−006 | 3.944304526105059e−006 | 1.972152263052530e−006 |
27 | 4.930380657631324e−006 | 2.465190328815662e−006 | 1.232595164407831e−006 |
28 | 3.081487911019577e−006 | 1.540743955509789e−006 | 7.703719777548944e−007 |
29 | 1.925929944387236e−006 | 9.629649721936179e−007 | 4.814824860968089e−007 |
30 | 1.203706215242023e−006 | 6.018531076210113e−007 | 3.009265538105056e−007 |
Declarations
Acknowledgements
The authors would like to thank the referees and the editor for sincere evaluation and constructive comments which improved the paper considerably.
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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