- Research
- Open Access
Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators
- Xiaolong Qin^{1, 2} and
- Jen-Chih Yao^{2}Email author
https://doi.org/10.1186/s13660-016-1163-4
© Qin and Yao 2016
Received: 12 August 2016
Accepted: 4 September 2016
Published: 22 September 2016
Abstract
Zero point problems of two accretive operators and fixed point problems of a nonexpansive mappings are investigated based on a Mann-like iterative algorithm. Weak convergence theorems are established in a Banach space.
Keywords
MSC
1 Introduction and preliminaries
The modulus of convexity of E is the function \(\delta_{E}(\epsilon ):(0,2]\rightarrow[0,1]\) defined by \(\delta_{E}(\epsilon)=\inf\{1-\frac{\Vert x+t\Vert }{2}:\Vert x\Vert =\Vert y\Vert =1,\Vert x-y\Vert \geq \epsilon\}\). Recall that E is said to be uniformly convex if \(\delta_{E}(\epsilon )>0\) for any \(\epsilon\in(0,2]\). Let \(p>1\). We say that E is p-uniformly convex if there exists a constant \(c_{q}>0\) such that \(\delta_{E}(\epsilon)\geq c_{p}\epsilon^{p}\) for any \(\epsilon\in(0,2]\).
Let \(B_{E}=\{x\in E: \Vert x\Vert =1\}\). The norm of E is said to be Gâteaux differentiable if the limit \(\lim_{t\rightarrow0}(\Vert x+ty\Vert -\Vert x\Vert )/t \) exists for each \(x,y\in B_{E}\). In this case, E is said to be smooth. The norm of E is said to be uniformly Gâteaux differentiable if for each \(y\in B_{E}\), the limit is attained uniformly for all \(x\in B_{E}\). The norm of E is said to be Fréchet differentiable if for each \(x\in B_{E}\), the limit is attained uniformly for all \(y\in B_{E}\). The norm of E is said to be uniformly Fréchet differentiable if the limit is attained uniformly for all \(x,y\in B_{E}\).
Typical examples of both uniformly convex and uniformly smooth Banach spaces are \(L^{p}\), where \(p>1\). To be more precise, \(L^{p}\) is mini\(\{p,2\} \)-uniformly smooth for every \(p>1\). It is well known that E is p-uniformly convex if and only if \(E^{*}\) is q-uniformly smooth, where p and q satisfy the relation \(\frac{1}{p}+\frac{1}{q}=1\).
For a multi-valued accretive operator A, we can define a nonexpansive single valued mapping \(J_{r}^{A}:R(I+rA)\rightarrow D(A)\) by \(J_{r}^{A}=(I+rA)^{-1} \) for each \(r>0\), which is called the resolvent operator of A.
The convex feasibility problem asks to find a point in the intersection of convex sets. This is an important problem in mathematics and engineering; see, e.g., [3–6] and the references therein. Oftentimes, the convex sets are given as fixed point sets of projections or (more generally) averaged nonexpansive operators. In this paper, we will focus our attention on the problem of finding a common element in \(\operatorname{Fix}(T)\cap(A+B)^{-1}(0)\), where T is a nonexpansive mapping, A is an α-inverse strongly accretive operator and B is an m-accretive operator, in the framework of uniformly convex and q-uniformly smooth Banach spaces. The problem is quite general in the sense that it includes: split feasibility problems, convexly constrained linear inverse problems, fixed point problems, variational inequalities, convexly constrained minimization problems, and Nash equilibrium problems in noncooperative games, as special cases; see, for instance, [7–12] and the references therein. Recently, mean valued iterative algorithms have been introduced by many authors to investigate this problem; see, for instance, [13–18] and the references therein. Related work can also be found, e.g., in [19–22]. However, there is little work in the existing literature in the setting of Banach spaces. The aim of this paper is to establish a weak convergence theorem in the framework of Banach spaces based on a Mann-like iterative algorithm. Applications are also provided to support the main results of this article.
In order to obtain our main results, we also need the following lemmas.
Lemma 1.1
Proof
Lemma 1.2
[1]
Lemma 1.3
[1]
Lemma 1.4
[23]
Lemma 1.5
[24]
Let E be a real uniformly convex Banach space, and let T be a nonexpansive mapping on E. Then \(I-T\) is demiclosed at zero.
Lemma 1.6
[25]
Let E be a real uniformly convex Banach space. Let \(E^{*}\) the dual space of E such that it has the Kadec-Klee property. Suppose that \(\{x_{n}\}\) is a bounded sequence such that \(\lim_{n\rightarrow\infty} \Vert (1-a)p_{1}-p_{2}+ax_{n}\Vert \) exists for all \(a\in[0,1]\) and \(p_{1},p_{2}\in\omega_{w}(x_{n})\), where \(\omega _{w}(x_{n}):\{x:\exists x_{n_{i}}\rightharpoonup x\}\) denotes the weak ω-limit set of \(\{x_{n}\}\) Then \(\omega_{w}(x_{n})\) is a singleton.
2 Main results
Theorem 2.1
Let E be a real uniformly convex and q-uniformly smooth Banach space with constant \(K_{q}\). Let \(B:D(B)\subset E\rightarrow2^{E}\) be an m-accretive operator, \(A:E\rightarrow E\) an α-inverse strongly accretive operator and \(T:E\rightarrow E\) a nonexpansive mapping such that \(\operatorname{Fix}(T)\cap (A+B)^{-1}(0)\neq\emptyset\). Let \(\{r_{n}\}\) be a positive number sequence and let \(\{\alpha_{n}\}\) be a real number sequence in \((0,1)\) such that \(\{\alpha_{n}\}\subset[\alpha,\alpha ']\), where \(0<\alpha<\alpha'<1\) and \(\{r_{n}\}\subset[r,r']\), where \(0< r< r'<(\frac{q\alpha}{K_{q}})^{\frac{1}{q-1}}\). Let \(\{x_{n}\}\) be a sequence generated in the following manner: \(x_{0}\in E\) and \(x_{n+1}=\alpha_{n}Tx_{n}+(1-\alpha_{n})(I+r_{n}B)^{-1}(x_{n}-r_{n}Ax_{n})\), \(\forall n\geq0\), Then \(\{x_{n}\}\) converges weakly to some point in \(\operatorname{Fix}(T)\cap (A+B)^{-1}(0)\).
Proof
Note that, in the framework of Hilbert spaces, the concept of monotonicity coincides with the concept of accretivity. Next, we apply our main results to solve variational inequality problems and minimizer problems of convex functions in the framework of Hilbert spaces.
Corollary 2.2
Let H be a real Hilbert space. Let C be a nonempty closed and convex subset of E and let \(\operatorname{Proj}_{C}^{H}\) be the metric projection from H onto C. Let A an α-inverse strongly monotone operator on H and T a nonexpansive mapping on C such that \(\operatorname{Fix}(T)\cap \operatorname{VI}(C,A)\neq\emptyset\). Let \(\{r_{n}\}\) be a positive number sequence and let \(\{\alpha_{n}\}\) be a real number sequence in \((0,1)\) such that \(\{\alpha_{n}\}\subset[\alpha,\alpha ']\), where \(0<\alpha<\alpha'<1\) and \(\{r_{n}\}\subset[r,r']\), where \(0< r< r'<2\alpha\). Let \(\{x_{n}\}\) be a sequence generated in the following manner: \(x_{0}\in C\) and \(x_{n+1}=\alpha_{n}Tx_{n}+(1-\alpha_{n})\operatorname{Proj}_{C}^{H}(x_{n}-r_{n}Ax_{n})\), \(\forall n\geq0\). Then \(\{x_{n}\}\) converges weakly to some point in \(\operatorname{Fix}(T)\cap \operatorname{VI}(C,A)\).
Now, we are in a position to consider the problem of finding minimizers of proper lower semicontinuous convex functions. For a proper lower semicontinuous convex function \(g:H\rightarrow(-\infty,\infty]\), the subdifferential mapping ∂g of g is defined by \(\partial g(x)=\{x^{*}\in H:g(x)+\langle y-x,x^{*}\rangle\leq g(y),\forall y\in H\}\), \(\forall x\in H\). Rockafellar [26] proved that ∂g is a maximal monotone operator. It is easy to verify that \(0\in\partial g(v)\) if and only if \(g(v)=\min_{x\in H} g(x)\).
Corollary 2.3
Let H be a real Hilbert space. Let \(g:H\rightarrow(-\infty,\infty]\) be a proper convex and lower semicontinuous function and let \(T:H\rightarrow H\) be a nonexpansive mapping such that \(\operatorname{Fix}(T)\cap(\partial g)^{-1}(0)\neq\emptyset\). Let \(\{r_{n}\}\) be a positive number sequence and let \(\{\alpha_{n}\}\) be a real number sequence in \((0,1)\) such that \(\{\alpha_{n}\}\subset[\alpha,\alpha ']\), where \(0<\alpha<\alpha'<1\) and \(\{r_{n}\}\subset[r,r']\), where \(0< r< r'<(\frac{q\alpha}{K_{q}})^{\frac{1}{q-1}}\). Let \(\{x_{n}\}\) be a sequence generated in the following manner: \(x_{0}\in H\) and \(x_{n+1}=\alpha_{n}Tx_{n}+(1-\alpha_{n})y_{n}\), \(\forall n\geq0\), where \(y_{n}=\min_{z\in H}\{g(z)+\frac{\Vert z-x_{n}+e_{n}\Vert ^{2}}{2r_{n}}\}\). Then \(\{x_{n}\}\) converges weakly to some point in \(\operatorname{Fix}(T)\cap (A+B)^{-1}(0)\).
Proof
Since \(g:H\rightarrow(-\infty,\infty]\) is a proper convex and lower semicontinuous function, we see that subdifferential ∂g of g is maximal monotone. Putting \(A=0\), we have \(y_{n}=\arg\min_{z\in H}\{g(z)+\frac{\Vert z-x_{n}\Vert ^{2}}{2r_{n}}\} \) is equivalent to \(0\in\partial g(y_{n})+\frac{1}{r_{n}}(y_{n}-x_{n})\). Hence, we have \(x_{n}\in y_{n}+r_{n}\partial g(y_{n})\). By use of Theorem 2.1, we find the desired conclusion immediately. □
Declarations
Acknowledgements
The first author was supported by the National Natural Science Foundation of China under Grant No. 11401152. The second author was partially supported by the Grant MOST 103-2923-E-039-001-MY3.
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
References
- Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127-1138 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Kato, T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 19, 508-520 (1967) MathSciNetView ArticleMATHGoogle Scholar
- Bauschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011) View ArticleMATHGoogle Scholar
- Censor, Y, Zenios, SA: Parallel Optimization. Oxford University Press, Oxford (1997) MATHGoogle Scholar
- Combettes, PL: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155-270 (1996) View ArticleGoogle Scholar
- Iiduka, H: Iterative algorithm for solving triple-hierarchical constrained optimization problem. J. Optim. Theory Appl. 48, 580-592 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Aoyama, K, Iiduka, H, Takahashi, W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006, Article ID 35390 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Cho, SY, Qin, X, Wang, L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014, Article ID 94 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Guan, WB, Song, W: The generalized forward-backward splitting method for the minimization of the sum of two functions in Banach spaces. Numer. Funct. Anal. Optim. 36, 867-886 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kamimura, S, Takahashi, W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 8, 361-374 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Liu, M, Chang, SS: An iterative method for equilibrium problems and quasi-variational inclusion problems. Nonlinear Funct. Anal. Appl. 14, 619-638 (2009) MathSciNetMATHGoogle Scholar
- Qin, X, Cho, SY, Wang, L: Iterative algorithms with errors for zero points of m-accretive operators. Fixed Point Theory Appl. 2013, Article ID 148 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Bin Dehaish, BA, Qin, X, Latif, A, Bakodah, H: Weak and strong convergence of algorithms for the sum of two accretive operators with applications. J. Nonlinear Convex Anal. 16, 1321-1336 (2015) MathSciNetMATHGoogle Scholar
- Cho, SY, Latif, A, Qin, X: Regularization iterative algorithms for monotone and strictly pseudocontractive mappings. J. Nonlinear Sci. Appl. 9, 3909-3919 (2016) MathSciNetMATHGoogle Scholar
- Duca, PM, Muu, LD: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65, 1855-1866 (2016). doi:10.1080/02331934.2016.1195831 MathSciNetView ArticleGoogle Scholar
- Hecai, Y: On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces. Fixed Point Theory Appl. 2013, Article ID 155 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Qin, X, Cho, SY, Wang, L: A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl. 2014, Article ID 75 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Qin, X, Bin Dehaish, BA, Cho, SY: Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces. J. Nonlinear Sci. Appl. 9, 2789-2797 (2016) MathSciNetMATHGoogle Scholar
- Yao, Y, Liou, YC, Yao, JC: Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction. Fixed Point Theory Appl. 2015, Article ID 127 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Yao, Y, Agarwal, RP, Postolache, M, Liou, YC: Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 2014, Article ID 183 (2014) MathSciNetView ArticleGoogle Scholar
- Yao, Y, Agarwal, RP, Liou, YC: Iterative algorithms for quasi-variational inclusions and fixed point problems of pseudocontractions. Fixed Point Theory Appl. 2014, Article ID 82 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Yao, Y, Postolache, M, Liou, YC: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl. 2013, Article ID 201 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math. 32, 107-116 (1979) MathSciNetView ArticleMATHGoogle Scholar
- Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041-1044 (1965) MathSciNetView ArticleMATHGoogle Scholar
- Falset, JG, Kaczor, W, Kuczumow, T, Reich, S: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 43, 377-401 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Rockafellar, RT: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97-116 (1976) MathSciNetView ArticleMATHGoogle Scholar