Abstract
A modified Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space.
MSC:47H06, 47H09, 47J25, 65J15.
Fixed Point Theory and Applications volume 2013, Article number: 347 (2013)
A modified Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space.
MSC:47H06, 47H09, 47J25, 65J15.
In this paper, we are concerned with the problem of finding zero points of accretive operators. Interest in accretive operators stems mainly from their firm connection with equations of evolution, and this is an important class of nonlinear operators. It is known that many physically significant problems can be modelled by initial value problems of the form
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 of t, then (1.1) is reduced to
whose solutions correspond to the equilibrium points of (1.1). An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial value problem (1.1) is solvable if A is locally Lipschitz and accretive on E. One of the most popular techniques for solving zero points of accretive operators goes back to the work of Browder [2]. One of the basic ideas in the case of a Hilbert space H is reducing the above equation (1.2) to a fixed point problem of the operator {R}_{A}:H\to {2}^{H} defined by {R}_{A}={(I+A)}^{-1}, which is called the classical resolvent of A.
The paper is organized in the following way. In Section 2, we present the preliminaries that are needed in our work. In Section 3, a modified Mann iteration with computational errors is presented. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space. In Section 4, applications of the main results are discussed.
Let E be a real Banach space E and let {E}^{\ast} be the dual space of E. Let \u3008\cdot ,\cdot \u3009 denote the pairing between E and {E}^{\ast}. The normalized duality mapping J:E\to {2}^{{E}^{\ast}} is defined by
for all x\in E. Let {U}_{E}=\{x\in E:\parallel x\parallel =1\}. E is said to be smooth or is said to have a Gâteaux differentiable norm if the limit {lim}_{t\to 0}\frac{\parallel x+ty\parallel -\parallel x\parallel}{t} exists for each x,y\in {U}_{E}. E is said to have a uniformly Gâteaux differentiable norm if for each y\in {U}_{E}, the limit is attained uniformly for all x\in {U}_{E}. E is said to be uniformly smooth or is said to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for x,y\in {U}_{E}. In the sequel, we use j to denote the single-valued normalized duality mapping. It is 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.
Recall that a closed convex subset C of a Banach space E is said to have the normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K, i.e., sup\{\parallel x-y\parallel :y\in K\}<d(K), where d(K) is the diameter of K. It is well known that a closed convex subset of a uniformly convex Banach space has the normal structure and a compact convex subset of a Banach space has the normal structure; see [3] for more details.
Let T:C\to C be a mapping. Recall that T is said to be contractive if there exits a constant \alpha \in (0,1) such that
For such a case, we also call T an α-contraction. T is said to be nonexpansive if
Let D be a nonempty subset of C. Let Q:C\to D. Q is said to be contraction if {Q}^{2}=Q; sunny if for each x\in C and t\in (0,1), we have Q(tx+(1-t)Qx)=Qx; sunny nonexpansive retraction if Q is sunny, nonexpansive, and contraction. K 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 [4], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Let E be a smooth Banach space and C be a nonempty subset of E. Let Q:E\to C be a retraction and j be the normalized duality mapping on E. Then the following are equivalent:
Q is sunny and nonexpansive;
{\parallel Qx-Qy\parallel}^{2}\le \u3008x-y,j(Qx-Qy)\u3009, \mathrm{\forall}x,y\in E;
\u3008x-Qx,j(y-Qx)\u3009\le 0, \mathrm{\forall}x\in E, y\in C.
Krasnoselski-Mann iteration generates a sequence \{{x}_{n}\} in the following manner:
It is known that the Krasnoselski-Mann iteration only has weak convergence even for nonexpansive mappings in infinite-dimensional Hilbert spaces; for more details, see [5] and the references therein. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energy \parallel {x}_{n}-x\parallel of the error between the iterate {x}_{n} and the solution x eventually becomes arbitrarily small. To improve the weak convergence of Krasnoselski-Mann iterative process, different modified Mann iterations have been considered; see [6–30] and the references therein.
Let I denote the identity operator on E. An operator A\subset E\times E with domain D(A)=\{z\in E:Az\ne \mathrm{\varnothing}\} 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 A{x}_{i}, i=1,2, there exists j({x}_{1}-{x}_{2})\in J({x}_{1}-{x}_{2}) such that \u3008{y}_{1}-{y}_{2},j({x}_{1}-{x}_{2})\u3009\ge 0. An accretive operator A is said to be m-accretive if R(I+rA)=E for all r>0. In a real Hilbert space, an operator A is m-accretive if and only if A is maximal monotone. In this paper, we use {A}^{-1}(0) to denote the set of zeros of A.
For an accretive operator A, we can define a nonexpansive single-valued mapping {J}_{r}:R(I+rA)\to D(A) by {J}_{r}={(I+rA)}^{-1} for each r>0, which is called the resolvent of A.
One of classical methods of studying the problem 0\in Ax, where A\subset E\times E is an accretive operator, is the following:
where {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1} and \{{r}_{n}\} is a sequence of positive real numbers.
The following iteration also has been extensively investigated:
where \{{\alpha}_{n}\} is a real number sequence in (0,1), \{{r}_{n}\} is a positive real number sequence, and {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1}. It is known that the sequence \{{x}_{n}\} generated in the above iteration converges strongly to a zero point of A in a Banach space under some restrictions imposed on \{{\alpha}_{n}\} and \{{r}_{n}\}.
Chen et al. [29] investigated the following iteration:
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are real number sequences in (0,1), \{{r}_{n}\} is a positive real number sequence, and {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1}. They proved that the sequence \{{x}_{n}\} generated in the above iteration converges strongly to a zero point of A in a Banach space; for more details, see [29] and the references therein.
We also remark that the viscosity approximation method was first introduced by Moudafi [31] in the framework of Hilbert spaces. Moudafi proved that the desired solution is not only a fixed point of nonlinear mappings but a solution to some variational inequality; for more details, see [31] and the references therein.
Recently, Qin et al. [20] investigated the iteration (2.4) with double computational errors and established a strong convergence theorem in a real reflexive Banach space with the uniformly Gâteaux differentiable norm; for more details, see [20] and the references therein. Different regularization methods recently have been investigated for treating zero points of accretive operators. In this paper, a modified Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space. The results mainly improve the corresponding results in Qin and Su [6], Hao [15], Qin et al. [20] and Chen et al. [29].
In order to state our main results, we also need the following lemmas.
Lemma 2.1 [20]
Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure, and C be a nonempty closed convex subset of E. Let S:C\to C be a nonexpansive mapping with a fixed point, and f:C\to C be a fixed contraction with the coefficient \alpha \in (0,1). Let \{{x}_{t}\} be a sequence generated by the following {x}_{t}=tf({x}_{t})+(1-t)S{x}_{t}, where t\in (0,1). Then \{{x}_{t}\} converges strongly as t\to 0 to a fixed point {x}^{\ast} of T, which is the unique solution in F(T) to the following variational inequality
Lemma 2.2 [32]
Let \{{x}_{n}\} and \{{y}_{n}\} be bounded sequences in a Banach space E, and \{{\beta}_{n}\} be a sequence in (0,1) with 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1. Suppose that {x}_{n+1}=(1-{\beta}_{n}){y}_{n}+{\beta}_{n}{x}_{n}, \mathrm{\forall}n\ge 1 and
Then {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}-{x}_{n}\parallel =0.
Lemma 2.3 [33]
Let \{{a}_{n}\}, \{{b}_{n}\}, and \{{c}_{n}\} be three nonnegative real sequences satisfying
where \{{t}_{n}\} is a sequence in (0,1). Assume that the following conditions are satisfied:
{\sum}_{n=0}^{\mathrm{\infty}}{t}_{n}=\mathrm{\infty} and {b}_{n}=o({t}_{n});
{\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}.
Then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
Lemma 2.4 [34]
Let E be a Banach space and let A be an m-accretive operator. For \lambda >0, \mu >0, and x\in E, we have {J}_{\lambda}x={J}_{\mu}(\frac{\mu}{\lambda}x+(1-\frac{\mu}{\lambda}){J}_{\lambda}x), where {J}_{\lambda}={(I+\lambda A)}^{-1} and {J}_{\mu}={(I+\mu A)}^{-1}.
Theorem 3.1 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let A be an m-accretive operator in E such that C:=\overline{D(A)} is convex and has the normal structure. Let f:C\to C be an α-contraction. Let \{{x}_{n}\} be a sequence generated in the following manner: {x}_{0}\in C and
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are real number sequences in (0,1), \{{e}_{n}\} is a sequence in E, \{{r}_{n}\} is a positive real number sequence, and {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1}. Assume that {A}^{-1}(0) is not empty and the above control sequences satisfy the following restrictions:
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1;
{\sum}_{n=1}^{\mathrm{\infty}}\parallel {e}_{n}\parallel <\mathrm{\infty};
{r}_{n}\ge r for each n\ge 1 and {lim}_{n\to \mathrm{\infty}}|{r}_{n}-{r}_{n+1}|=0.
Then the sequence \{{x}_{n}\} converges strongly to \overline{x}, which is the unique solution to the following variational inequality \u3008f(\overline{x})-\overline{x},j(p-\overline{x})\u3009\le 0, \mathrm{\forall}p\in {A}^{-1}(0).
Proof First, we prove that \{{x}_{n}\} is bounded. Fixing p\in {A}^{-1}(0), we see that
It follows that
This proves that the sequence \{{x}_{n}\} is bounded. If {r}_{n}\ge {r}_{n+1}, we see from Lemma 2.4 that
where {M}_{2} is an appropriate constant such that {M}_{2}\ge {sup}_{n\ge 1}\{\parallel {J}_{{r}_{n+1}}({x}_{n}+{e}_{n+1})-({x}_{n+1}+{e}_{n+2})\parallel \}. Put {z}_{n}=\frac{{x}_{n+1}-{\beta}_{n}{x}_{n}}{1-{\beta}_{n}}; that is,
Note that
It follows that
Substituting (3.1) into (3.3), we arrive at
In view of restrictions (a), (b) and (c), we find that
By virtue of Lemma 2.2, one obtains that
It follows from (3.1) that {x}_{n+1}-{x}_{n}=(1-{\beta}_{n})({z}_{n}-{x}_{n}). In view of restriction (c), we find from (3.4) that
If {r}_{n}\le {r}_{n+1}, we can prove that (3.5) still holds. On the other hand, we have
It follows that
In view of restriction (b), one finds that
Notice that
Since {\sum}_{n=1}^{\mathrm{\infty}}\parallel {e}_{n}\parallel <\mathrm{\infty}, from (3.6) we obtain that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{J}_{{r}_{n}}{x}_{n}\parallel =0. Taking a fixed number k such that r>k>0, we arrive at
Since
we therefore find that
Now, we are in a position to prove that {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\u3008\overline{x}-f(\overline{x}),j({x}_{n}-\overline{x})\u3009\le 0, where \overline{x}={lim}_{t\to 0}{z}_{t}, and {z}_{t} solves the fixed point equation
Therefore, we see that
This implies that
It follows from (3.8) that
Since {z}_{t}\to \overline{x} as t\to 0 and from the fact that j is strong to weak^{∗} uniformly continuous on bounded subsets of E, we find that
It follows that
For any \u03f5>0, there exists \kappa >0 such that \mathrm{\forall}t\in (0,\kappa ) the following inequality holds
This implies that
Note that ϵ is arbitrary. In view of (3.9), we see that {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\u3008f(\overline{x})-\overline{x},j({x}_{n}-\overline{x})\u3009\le 0. This implies that
Finally, we show that {x}_{n}\to \overline{x} as n\to \mathrm{\infty}. Notice that
It follows that
where {d}_{n}=\parallel {e}_{n+1}\parallel (2\parallel {x}_{n}-\overline{x}\parallel +\parallel {e}_{n+1}\parallel ). In view of restrictions (a) and (c), we know that {\sum}_{n=1}^{\mathrm{\infty}}{d}_{n}<\mathrm{\infty}. Put {a}_{n}={\parallel {x}_{n}-\overline{x}\parallel}^{2}, {t}_{n}={\alpha}_{n}, and {c}_{n}={d}_{n}. In view of Lemma 2.3, we find the desired conclusion. □
If f(x)=u, where u is a fixed element in C, for any x\in C, we find the following result.
Corollary 3.2 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let A be an m-accretive operator in E such that C:=\overline{D(A)} is convex and has the normal structure. Let \{{x}_{n}\} be a sequence generated in the following manner: {x}_{0}\in C and
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are real number sequences in (0,1), \{{r}_{n}\} is a positive real number sequence, and {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1}. Assume that {A}^{-1}(0) is not empty and the above control sequences satisfy the following restrictions:
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1;
{r}_{n}\ge r for each n\ge 1 and {lim}_{n\to \mathrm{\infty}}|{r}_{n}-{r}_{n+1}|=0.
Then the sequence \{{x}_{n}\} converges strongly to \overline{x}={Q}_{{A}^{-1}(0)}u.
In this section, we consider solutions of variational inequalities. Let C be a nonempty, closed, and convex subset of a Banach space E. Let A:C\to {E}^{\ast} be a single-valued monotone operator which is hemicontinuous; that is, continuous along each line segment in C with respect to the weak^{∗} topology of {E}^{\ast}. Consider the following variational inequality problem of finding a point x\in C such that
In this section, we use \mathit{VI}(C,A) to denote the solution set of the variational inequality involving A. The symbol {N}_{C}(x) stands for the normal cone for C at a point x\in C; that is,
Theorem 4.1 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let A:C\to {E}^{\ast} be a single-valued, monotone, and hemicontinuous operator. Assume that \mathit{VI}(C,A) is not empty and C has the normal structure. Let f:C\to C be an α-contraction. Let \{{x}_{n}\} be a sequence generated in the following manner: {x}_{0}\in C and
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are real number sequences in (0,1), \{{e}_{n}\} is a sequence in E, \{{r}_{n}\} is a positive real number sequence, and {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1}. Assume that {A}^{-1}(0) is not empty and the above control sequences satisfy the following restrictions:
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1;
{r}_{n}\ge r for each n\ge 1 and {lim}_{n\to \mathrm{\infty}}|{r}_{n}-{r}_{n+1}|=0.
Then the sequence \{{x}_{n}\} converges strongly to \overline{x}, which is the unique solution to the following variational inequality \u3008f(\overline{x})-\overline{x},j(p-\overline{x})\u3009\le 0, \mathrm{\forall}p\in {A}^{-1}(0).
Proof Define a mapping T\subset E\times {E}^{\ast} by
By Rockafellar [35], we know that T is maximal monotone, and {T}^{-1}(0)=\mathit{VI}(C,A). For each {r}_{n}>0 and {x}_{n}\in E, we see that there exists a unique {x}_{{r}_{n}}\in D(T) such that {x}_{n}\in {x}_{r}+{r}_{n}T({x}_{{r}_{n}}), where {x}_{{r}_{n}}={(I+{r}_{n}T)}^{-1}{x}_{n}. Notice that
which is equivalent to
that is, -A{y}_{n}+\frac{1}{{r}_{n}}({x}_{n}-{y}_{n})\in {N}_{C}({y}_{n}). This implies that {y}_{n}={(I+{r}_{n}T)}^{-1}{x}_{n}. Following the proof of Theorem 3.1, we can immediately conclude the desired conclusion. □
If f(x)=u, where u is a fixed element in C, for any x\in C, we find the following result.
Corollary 4.2 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let A:C\to {E}^{\ast} be a single-valued, monotone, and hemicontinuous operator. Assume that \mathit{VI}(C,A) is not empty and C has the normal structure. Let \{{x}_{n}\} be a sequence generated in the following manner: {x}_{0}\in C and
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are real number sequences in (0,1), \{{e}_{n}\} is a sequence in E, \{{r}_{n}\} is a positive real number sequence, and {J}_{{r}_{n}}={(I+{r}_{n}A)}^{-1}. Assume that {A}^{-1}(0) is not empty and the above control sequences satisfy the following restrictions:
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1;
{r}_{n}\ge r for each n\ge 1 and {lim}_{n\to \mathrm{\infty}}|{r}_{n}-{r}_{n+1}|=0.
Then the sequence \{{x}_{n}\} converges strongly to \overline{x}={Q}_{{A}^{-1}(0)}u.
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This work is supported by the Department of Hebei Education (Z2013110). The authors are grateful to Professor Jin Wang and the anonymous reviewers for useful suggestions which improved the contents of the article.
The authors declare that they have no competing interests.
All authors contributed equally to this manuscript. All authors read and approved the final manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Song, J., Chen, M. A modified Mann iteration for zero points of accretive operators. Fixed Point Theory Appl 2013, 347 (2013). https://doi.org/10.1186/1687-1812-2013-347
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DOI: https://doi.org/10.1186/1687-1812-2013-347