Some results on an infinite family of accretive operators in a reflexive Banach space
- Sheng-Hua Wang^{1}Email author and
- Po Zhang^{1}
https://doi.org/10.1186/s13663-014-0252-7
© Wang and Zhang; licensee Springer. 2015
Received: 12 August 2014
Accepted: 15 December 2014
Published: 1 February 2015
Abstract
The purpose of this article is to investigate a Halpern-like proximal point algorithm for common zero points of an infinite family of accretive operators. Possible computational errors are taken into account. Strong convergence theorems are established in a reflexive Banach space.
Keywords
accretive operator fixed point resolvent nonexpansive mapping zero1 Introduction
The class of accretive operators is an important class of nonlinear operators. Interest in accretive operators stems mainly from their firm connection with equations of evolutions. It is well known that many physically significant problems can be modeled by initial value problems of the following form: \(x'(t)+Ax(t)=0\), \(x(0)=x_{0}\) where A is an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave or Schrödinger equations. If \(x(t)\) is dependent on t, then the above problem is reduced to \(Au=0 \) whose solutions correspond to the equilibrium points of the initial value problem. An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial value problem is solvable if A is locally Lipschitz and accretive on E. One of the most popular techniques for solving zero points of accretive operators is the proximal point algorithm, which was proposed by Martinet [2, 3] and generalized by Rockafellar [4, 5].
Halpern algorithm is efficient to study fixed points of nonexpansive mappings. The advantage of Halpern algorithm for nonexpansive mappings is that strong convergence is guaranteed without any compact assumptions or projections involved. Recently Halpern-like proximal point algorithms have been extensively studied by many authors; see [6–22] and the references therein.
In this article, we investigate common zeros of an infinite family of accretive operators based on a Halpern-like proximal point algorithm. Strong convergence theorems are established in a reflexive and strictly convex Banach space which has a weakly continuous duality mapping.
2 Preliminaries
Let \(U_{E}=\{x\in E: \Vert x\Vert =1\}\). The norm of E is said to be Gâteaux differentiable if the limit \(\lim_{t\rightarrow0}\frac{\Vert x+ty\Vert -\Vert x\Vert }{t} \) exists for each \(x,y\in U_{E}\). In this case, E is said to be smooth. The norm of E is said to be uniformly Gâteaux differentiable if for each \(y\in U_{E}\), the limit is attained uniformly for all \(x\in U_{E}\). The norm of E is said to be Fréchet differentiable if for each \(x\in U_{E}\), the limit is attained uniformly for all \(y\in U_{E}\). The norm of E is said to be uniformly Fréchet differentiable if the limit is attained uniformly for all \(x,y\in U_{E}\). It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E. It is well known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm to weak^{∗} continuous on each bounded subset of E.
- (1)
contraction if \(Q_{D}^{2}=Q_{D}\);
- (2)
sunny if for each \(x\in C\) and \(t\in(0,1)\), we have \(Q_{D} (tx+(1-t)Q_{D}x )=Q_{D}x\);
- (3)
sunny nonexpansive retraction if \(Q_{D}\) is sunny, nonexpansive, and it is a contraction.
D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [24], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
- (1)
\(Q_{C}\) is sunny and nonexpansive;
- (2)
\(\Vert Q_{C}x-Q_{C}y\Vert ^{2}\leq\langle x-y,J(Q_{C}x-Q_{C}y)\rangle\), \(\forall x,y\in E\);
- (3)
\(\langle x-Q_{C}x,J(y-Q_{C}x)\rangle\leq0\), \(\forall x\in E\), \(y\in C\).
It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction \(Q_{C}\) is coincident with the metric projection from E onto C. Let C be a nonempty closed convex subset of a smooth Banach space E, let \(x\in E\) and let \(x_{0}\in C\). Then we have from the above that \(x_{0}=Q_{C}x\) if and only if \(\langle x-x_{0}, J(y-x_{0})\rangle\leq0\) for all \(y\in C\), where \(Q_{C}\) is a sunny nonexpansive retraction from E onto C.
Let C be a nonempty, closed, and convex subset of E. Let \(S:C\rightarrow C\) be a mapping. In this paper, we use \(F(S)\) to denote the set of fixed points of S. Recall that S is said to be α-contractive iff there exists a constant \(\alpha\in [0,1)\) such that \(\Vert Sx-Sy\Vert \leq\alpha \Vert x-y\Vert \), \(\forall x,y\in C\). S is said to be nonexpansive iff \(\Vert Sx-Sy\Vert \leq \Vert x-y\Vert \), \(\forall x,y\in C\). It is well known that many nonlinear problems can be reduced to the search for fixed points of nonexpansive mappings, for example, equilibrium problems, saddle point problems, and variational inequalities. Let K be a nonempty closed and convex subset of a smooth Banach space E. Recall the following variational inequality. Find a point \(u\in C\) such that \(\langle Au,J(v-u)\rangle\geq0\), \(\forall v\in C\). This problem is connected with fixed point problems of nonexpansvie mappings. From [25], we know that this variational inequality problem is equivalent to fixed point problems of nonlinear mapping \(Q_{K}(I-rA)\), where I is the identity mapping and r is a positive real number.
Let I denote the identity operator on E. An operator \(A\subset E\times E\) with domain \(D(A)=\{z\in E:Az\neq\emptyset\}\) and range \(R(A)=\bigcup\{Az:z\in D(A)\}\) is said to be accretive if for each \(x_{i}\in D(A)\) and \(y_{i}\in Ax_{i}\), \(i=1,2\), there exists \(j(x_{1}-x_{2})\in J(x_{1}-x_{2})\) such that \(\langle y_{1}-y_{2},j(x_{1}-x_{2})\rangle\geq0\). An accretive operator A is said to be m-accretive if \(R(I+rA)=E\) for all \(r>0\). In this paper, we use \(A^{-1}(0)\) to denote the set of zero points of A. For an accretive operator A, we can define a single-valued mapping \(J_{r}:R(I+rA)\rightarrow D(A)\) by \(J_{r}=(I+rA)^{-1} \) for each \(r>0\), which is called the resolvent of A.
Next, we give lemmas which play important roles in the article.
Lemma 2.1
[26]
Let E be a reflexive Banach space which has a weakly continuous duality map \(J_{\varphi}(x)\) with gauge φ. Let C be nonempty, closed, and convex subset of E. Let \(f:C\rightarrow C\) be an α-contractive mapping and let \(T: C\rightarrow C\) be a nonexpansive mapping. Let \(x_{t}\in C\) be the unique fixed point of the mapping \(tf+(1-t)T\), where \(t\in(0,1)\). Then T has a fixed point if and only if \(\{x_{t}\}\) remains bounded as \(t\rightarrow0^{+}\), and in this case, \(\{x_{t}\}\) converges as \(t\rightarrow0^{+}\) strongly to a fixed point \(\bar{x}\) of T, where \(\bar{x}\) is the unique solution to the following variational inequality: \(\langle f(\bar{x})-\bar{x},J_{\varphi}(p-\bar{x})\rangle\leq0\), \(\forall p\in\bigcap_{m=1}^{N} A_{m}^{-1}(0)\).
Lemma 2.2
[27]
Let C be a closed convex subset of a strictly convex Banach space E. Let \(S_{m}:C\rightarrow C\) be a nonexpansive mapping for each \(m\geq1\). Let \(\{\delta_{m}\}\) be a real number sequence in \((0,1)\) such that \(\sum_{m=1}^{\infty}\delta_{m}=1\). Suppose that \(\bigcap_{m=1}^{\infty}F(S_{m})\) is nonempty. Then the mapping \(\sum_{m=1}^{\infty}\delta_{m} S_{m}\) is nonexpansive with \(F(\sum_{m=1}^{\infty}\delta_{m} S_{m})=\bigcap_{m=1}^{\infty}F(S_{m})\).
The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [28].
Lemma 2.3
- (i)For all \(x, y \in E\), the following inequality holds:$$\Phi\bigl(\Vert x+y\Vert \bigr)\leq\Phi\bigl(\Vert x\Vert \bigr)+\bigl\langle y, J_{\varphi}(x+y)\bigr\rangle . $$
- (ii)
Assume that a sequence \(\{x_{n}\}\) in E converges weakly to a point \(x\in E\).
Lemma 2.4
[29]
Lemma 2.5
[30]
Let \(\{a_{n}\}\), \(\{b_{n}\}\), \(\{c_{n}\}\), and \(\{ e_{n}\}\) be three nonnegative real sequences satisfying \(b_{n+1}\leq(1-a_{n})b_{n}+a_{n}c_{n}+e_{n}\), \(\forall n\geq n_{0}\), where \(n_{0}\) is some positive integer, \(\{a_{n}\}\) is a number sequence in \((0,1)\) such that \(\sum_{n=n_{0}}^{\infty}a_{n}=\infty\), \(\{c_{n}\}\) is a number sequence such that \(\limsup_{n\rightarrow\infty} c_{n}\leq0\) and \(\sum_{n=n_{0}}^{\infty}e_{n}=\infty\). Then \(\lim_{n\rightarrow\infty} a_{n}=0\).
3 Main results
Theorem 3.1
- (1)
\(\alpha_{n}+\beta_{n}+\gamma_{n}=\alpha_{n}'+\beta_{n}'+\gamma _{n}'=\sum_{i=1}^{\infty}\delta_{n,i}=1\);
- (2)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
- (3)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
- (4)
\(\sum_{n=1}^{\infty}\gamma_{n}'<\infty\), \(\lim_{n\rightarrow\infty}\delta_{n,i}=\delta_{i}\in(0,1)\).
Proof
The proof is split into four steps.
Step 1. Show that \(\{x_{n}\}\) is bounded.
Step 2. Show that \(\lim_{n\rightarrow\infty} \Vert x_{n+1}-x_{n}\Vert = 0\).
Step 3. Show that \(\limsup_{n\rightarrow\infty}\langle f(x)-x,J_{\varphi}(x_{n}-x)\rangle\leq0\).
Step 4. Show that \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\).
In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following result.
Corollary 3.2
- (1)
\(\alpha_{n}+\beta_{n}+\gamma_{n}=\alpha_{n}'+\beta_{n}'+\gamma _{n}'=\sum_{i=1}^{\infty}\delta_{n,i}=1\);
- (2)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
- (3)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
- (4)
\(\sum_{n=1}^{\infty}\gamma_{n}'<\infty\), \(\lim_{n\rightarrow\infty}\delta_{n,i}=\delta_{i}\in(0,1)\).
For a single accretive operator, Theorem 3.1 is reduced to the following result.
Corollary 3.3
- (1)
\(\alpha_{n}+\beta_{n}+\gamma_{n}=\alpha_{n}'+\beta_{n}'+\gamma_{n}'=1\);
- (2)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
- (3)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
- (4)
\(\sum_{n=1}^{\infty}\gamma_{n}'<\infty\).
Finally, we investigate the Ky Fan inequality, which is also known as the equilibrium problem [31].
- (A1)
\(F(x,x)=0\) for all \(x\in C\);
- (A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\) for all \(x,y\in C\);
- (A3)for each \(x,y,z\in C\),$$\limsup_{t\downarrow0}F\bigl(tz+(1-t)x,y\bigr)\leq F(x,y); $$
- (A4)
for each \(x\in C\), \(y\mapsto F(x,y)\) is convex and weakly lower semi-continuous.
Lemma 3.4
[31]
- (a)
\(J_{r}\) is single-valued;
- (b)
\(J_{r}\) is firmly nonexpansive;
- (c)
\(F(J_{r})=EP(F)\);
- (d)
\(EP(F)\) is closed and convex.
Theorem 3.5
- (1)
\(\alpha_{n}+\beta_{n}+\gamma_{n}=\alpha_{n}'+\beta_{n}'+\gamma _{n}'=\sum_{i=1}^{\infty}\delta_{n,i}=1\);
- (2)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
- (3)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
- (4)
\(\sum_{n=1}^{\infty}\gamma_{n}'<\infty\), \(\lim_{n\rightarrow\infty}\delta_{n,i}=\delta_{i}\in(0,1)\).
Proof
Remark
The framework of the ILA
ILA: | Ishikawa-like algorithm (for equilibrium problem ( 3.7 )) |
---|---|
Step 0: | Choose \(x_{1}\in C\), \(\alpha_{1},\beta_{1},\gamma_{1},\alpha _{1}',\beta_{1}',\gamma_{1}'\in[0,1]\). Set n: = 1. |
Step 1: | Given \(x_{n}\in C\). Choose \(\alpha_{n},\beta_{n},\gamma_{n},\alpha _{n}',\beta_{n}',\gamma_{n}'\in[0,1]\) and compute \(x_{n+1}\in C\) as \(\begin{array}[t]{l@{\quad}l} F(z_{n},z)+\frac{1}{r}\langle z-z_{n},z_{n}-x_{n}\rangle\geq0,&\forall z\in C,\\ y_{n}=\alpha_{n}'x_{n}+\beta_{n}'z_{n}+\gamma_{n}'e_{n},&\\ x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}y_{n}. & \end{array}\) |
Update n: = n + 1 and go to Step 1. |
Declarations
Acknowledgements
The authors thank the anonymous reviewers for useful suggestions which improved the contents of the article. The authors also thank the Fundamental Research Funds for the Central Universities (2014ZD44).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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