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
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