Approximation of a zero point of monotone operators with nonsummable errors
- Takanori Ibaraki^{1}Email author
https://doi.org/10.1186/s13663-016-0535-2
© Ibaraki 2016
Received: 30 November 2015
Accepted: 29 March 2016
Published: 11 April 2016
Abstract
In this paper, we study an iterative scheme for two different types of resolvents of a monotone operator defined on a Banach space. These resolvents are generalizations of resolvents of a monotone operator in a Hilbert space. We obtain iterative approximations of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications.
Keywords
MSC
1 Introduction
On the other hand, Kimura [10] introduced the following iterative scheme for finding a fixed point of nonexpansive mappings by the shrinking projection method with error in a Hilbert space:
Theorem 1.1
(Kimura [10])
We remark that the original result of the theorem above deals with a family of nonexpansive mappings, and the shrinking projection method was first introduced by Takahashi et al. [11]. This result was extended to more general Banach spaces by Kimura [12] (see also Ibaraki and Kimura [13]).
In this paper, we study the shrinking projection method with error introduced by Kimura [10] (see also [12, 14]). We obtain an iterative approximation of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications.
2 Preliminaries
- (1)
\(Jx \ne\emptyset\) for each \(x \in E\);
- (2)
if E is reflexive, then J is surjective;
- (3)
if E is smooth, then the duality mapping J is single valued.
- (4)
if E is strictly convex, then J is one-to-one and satisfies that \(\langle x-y, x^{*}-y^{*} \rangle> 0\) for each \(x,y \in E\) with \(x \neq y\), \(x^{*} \in Jx\) and \(y^{*} \in Jy\);
- (5)
if E is reflexive, smooth, and strictly convex, then the duality mapping \(J_{*}: E^{*} \to E\) is the inverse of J, that is, \(J_{*} = J^{-1}\);
- (6)
if E uniformly smooth, then the duality mapping J is uniformly norm to norm continuous on each bounded set of E.
Let E be a reflexive and strictly convex Banach space and let C be a nonempty closed convex subset of E. It is well known that for each \(x\in E\) there exists a unique point \(z \in C\) such that \(\Vert x-z\Vert =\min\{\Vert x-y\Vert : y \in C\}\). Such a point z is denoted by \(P_{C} x\) and \(P_{C}\) is called the metric projection of E onto C. The following result is well known; see, for instance, [16].
Lemma 2.1
The following results describe the relation between the set of fixed points and that of asymptotic fixed points for each type of mapping.
Lemma 2.2
(Aoyama-Kohsaka-Takahashi [19])
Let E be a smooth Banach space, let C be a nonempty closed convex subset of E and let \(T: C \to E\) be a mapping of type (P). If \(F(T)\) is nonempty, then \(F(T)\) is closed and convex and \(F(T)=\hat{F}(T)\).
Lemma 2.3
(Kohsaka-Takahashi [18])
Let E be a strictly convex Banach space whose norm is uniformly Gâteaux differentiable, let C be a nonempty closed convex subset of E and let \(T: C \to E\) be a mapping of type (Q). If \(F(T)\) is nonempty, then \(F(T)\) is closed and convex and \(F(T)=\hat{F}(T)\).
In 1984, Tsukada [20] proved the following theorem for the metric projections in a Banach space. For the exact definition of Mosco limit \(\mathrm {M}\text{-}\!\lim _{n} C_{n}\), see [21].
Theorem 2.4
(Tsukada [20])
Let E be a reflexive and strictly convex Banach space and let \(\{C_{n}\}\) be a sequence of nonempty closed convex subsets of E. If \(C_{0} =\mathrm {M}\text{-}\!\lim _{n} C_{n}\) exists and is nonempty, then for each \(x \in E\), \(\{P_{C_{n}}x\}\) converges weakly to \(P_{C_{0}}x\), where \(P_{C_{n}}\) is the metric projection of E onto \(C_{n}\). Moreover, if E has the Kadec-Klee property, the convergence is in the strong topology.
One of the simplest example of the sequence \(\{C_{n}\}\) satisfying the condition in this theorem above is a decreasing sequence with respect to inclusion; \(C_{n+1}\subset C_{n}\) for each \(n\in \mathbb {N}\). In this case, \(\mathrm {M}\text{-}\!\lim C_{n} =\bigcap_{n=1}^{\infty} C_{n}\) (see [7, 12, 21, 22] for more details).
- (1)
\((\Vert x\Vert -\Vert y\Vert )^{2} \leq V(x,y) \leq(\Vert x\Vert +\Vert y\Vert )^{2}\) for each \(x,y \in E\);
- (2)
\(V(x,y) + V(y,x) = 2 \langle x-y, Jx-Jy \rangle\) for each \(x,y \in E\);
- (3)
\(V(x,y) = V(x,z) + V(z,y) + 2 \langle x-z, Jz-Jy \rangle\) for each \(x,y,z \in E\);
- (4)
if E is additionally assumed to be strictly convex, then \(V(x,y)=0\) if and only if \(x=y\).
Lemma 2.5
(Kamimura-Takahashi [23])
Let E be a smooth and uniformly convex Banach space and let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences in E such that either \(\{x_{n}\}\) or \(\{y_{n}\}\) is bounded. If \(\lim_{n} V(x_{n}, y_{n})=0\), then \(\lim_{n} \Vert x_{n}-y_{n} \Vert =0\).
The following results show the existence of mappings \(\underline{g}_{r}\) and \(\overline{g}_{r}\), related to the convex structures of a Banach space E. These mappings play important roles in our result.
Theorem 2.6
(Xu [24])
- (i)if E is uniformly convex, then there exists a continuous, strictly increasing, and convex function \(\underline{g}_{r}:[0,2r] \to\mathopen[0,\infty\mathclose[\) with \(\underline{g}_{r}(0)=0\) such thatfor all \(x,y\in B_{r}\) and \(\alpha\in[0,1]\);$$\bigl\Vert \alpha x +(1-\alpha) y\bigr\Vert ^{2} \leq\alpha \Vert x\Vert ^{2}+(1-\alpha)\Vert y \Vert ^{2} -\alpha(1- \alpha)\underline{g}_{r}\bigl(\Vert x-y \Vert \bigr) $$
- (ii)if E is uniformly smooth, then there exists a continuous, strictly increasing, and convex function \(\overline{g}_{r}:[0,2r] \to\mathopen[0,\infty\mathclose[\) with \(\overline{g}_{r}(0)=0\) such thatfor all \(x,y\in B_{r}\) and \(\alpha\in[0,1]\).$$\bigl\Vert \alpha x +(1-\alpha) y\bigr\Vert ^{2} \geq\alpha \Vert x\Vert ^{2}+(1-\alpha)\Vert y \Vert ^{2} -\alpha(1- \alpha)\overline{g}_{r}\bigl(\Vert x-y \Vert \bigr) $$
Theorem 2.7
(Kimura [12])
3 Approximation theorem for the resolvents of type (P)
In this section, we discuss an iterative scheme of resolvents of a monotone operator defined on a Banach space. Let E be a reflexive, smooth, and strictly convex Banach space. An operator \(A \subset E \times E^{*}\) with domain \(D(A)=\{ x \in E: Ax \ne\emptyset\}\) and range \(R(A)=\bigcup\{Ax: x \in D(A)\}\) is said to be monotone if \(\langle x-y, x^{*}-y^{*} \rangle\geq0\) for any \((x, x^{*}), (y, y^{*}) \in A\). A monotone operator A is said to be maximal if \(A=B\) whenever \(B \subset E \times E^{*}\) is a monotone operator such that \(A \subset B\). We denote by \(A^{-1}0\) the set \(\{z\in D(A): 0\in Az\}\).
- (1)
\(P_{r}\) is mapping of type (P) from C into \(D(A)\);
- (2)
\((P_{r} x, A_{r}x)\in A\) for all \(x \in C\);
- (3)
\(\Vert A_{r}x \Vert \leq \vert Ax \vert :=\inf\{\Vert x^{*}\Vert : x^{*} \in Ax\}\) for all \(x \in D(A)\);
- (4)
\(F(P_{r})=A^{-1}0\).
Theorem 3.1
Proof
4 Approximation theorem for the resolvents of type (Q)
- (1)
\(Q_{r}\) is mapping of type (Q) from C into \(D(A)\);
- (2)
\((Jx-JQ_{r}x)/r \in AQ_{r} x\) for all \(x \in C\);
- (3)
\(F(Q_{r})=A^{-1}0\).
Lemma 4.1
Proof
We obtain an approximation theorem for a zero point of a monotone operator in a smooth and uniformly convex Banach space by using the resolvent of type (Q).
Theorem 4.2
Proof
5 Applications
Corollary 5.1
Proof
Corollary 5.2
Proof
Next, we study the approximation of fixed points for mappings of type (P) and (Q). Before show our applications, we need the following results.
Lemma 5.3
([17])
Let E be a reflexive, smooth, and strictly convex Banach space, let C be a nonempty subset of E, let \(T:C\to E\) be a mapping, and let \(A_{T}\subset E\times E^{*}\) be an operator defined by \(A_{T}=J(T^{-1}-I)\). Then T is of mapping of type (P) if and only if \(A_{T}\) is monotone. In this case \(T=(I+J^{-1}A_{T})^{-1}\).
Lemma 5.4
([31])
Let E be a reflexive, smooth, and strictly convex Banach space, let C be a nonempty subset of E and let \(T:C\to E\) be a mapping, and let \(A_{T}\subset E\times E^{*}\) be an operator defined by \(A_{T}=JT^{-1}-J\). Then T is a mapping of type (Q) if and only if \(A_{T}\) is monotone. In this case \(T=(J+A_{T})^{-1}J\).
As a direct consequence of Theorems 3.1 and 4.2, we can show the following corollaries.
Corollary 5.5
Proof
Corollary 5.6
Declarations
Acknowledgements
The author is supported by Grant-in-Aid for Young Scientific (B) No. 24740075 from the Japan Society for the Promotion of Science.
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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