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Krasnosel’skiĭ-Mann-Opial type iterative solution of m-accretive operator equation and its stability in arbitrary Banach spaces
- Yuguang Xu^{1},
- Zeqin Liu^{2} and
- Zuhua Liu^{1, 3}Email author
https://doi.org/10.1186/1687-1812-2014-41
© Xu et al.; licensee Springer. 2014
Received: 20 August 2013
Accepted: 31 January 2014
Published: 14 February 2014
Abstract
Let X be a Banach space. Suppose that $A:X\to X$ is a Lipschitz accretive operator. The objective of this note is to discuss simultaneously the existence and uniqueness of solution of the equation $x+Ax=f$ for any given $f\in X$, and its convergence, estimate of convergent rate, and stability of Krasnosel’skiĭ-Mann-Opial type iterative solution $\{{x}_{n}\}\subseteq X$. If an iterative parameter is selected suitably then the iterative procedure converges strongly to a unique solution of the equation and the iterative process is stable in arbitrary Banach space without any convexity or reflexivity. In particular, if A is nonexpansive then an estimate of the convergence rate can be written as $\parallel {x}_{n+1}-q\parallel \le {(\frac{17}{18})}^{n+1}\parallel {x}_{0}-q\parallel $ where $q\in X$ is a solution of $x+AX=f$.
MSC:47H06, 47H10, 47H17.
Keywords
- accretive operator
- m-accretive operator
- iterative solution
- stability
- Krasnosel’skiĭ-Mann-Opial type iterative procedure
1 Introduction and preliminaries
Throughout this paper, X is assumed to be a real Banach space.
for all $x,y\in D(A)$ and $r>0$. An accretive operator A is said to be m-accretive if $R(I+\lambda A)=X$ for all (or, equivalently, for some) $\lambda >0$, where I stands for the identity operator on X.
for all $r>0$. To pick $r=1$, we have $q={q}^{\ast}$. It follows that for the equation $x+Ax=f$ there exists a unique solution $q\in X$.
It is well known that the approximative solution of operator equation is closely related with the iterative sequence convergence of a fixed point of the mapping. Therefore a brief history on iterative methods is reviewed in the following.
In 1922, using the Picard iterative method, Banach’s fixed point theorem was obtained where $S:D(S)\subseteq X\to X$ is a contractive type operator. For a given nonexpansive mapping S of a closed convex set C of a Banach space X into itself and $\lambda \in (0,1)$, the operator ${S}_{\lambda}=\lambda I+(1-\lambda )S$ also is nonexpansive. Moreover, as has been pointed out by Krasnosel’skiĭ [4] for $\lambda =1/2$ and by Schaefer [5] for an arbitrary λ, if X is uniformly convex and S has at least one fixed point in C, then the operator ${S}_{\lambda}$ is asymptotically regular. For any ${x}_{0}\in C$ and any $\lambda \in (0,1)$, Opial proved that the iterative sequence $\{{S}_{\lambda}^{n+1}{x}_{0}\}$ (or $\{\lambda {x}_{n}+(1-\lambda )S{x}_{n}\}$) is weakly convergent to a fixed point of S under some assumptions (see, e.g., Theorem 3 of [6]).
In this note, we need an iterative method, the so-called Krasnosel’skiĭ-Mann-Opial type iterative procedure which is defined as follows.
where the iterative parameter $\alpha \in (0,1)$.
In 1986, Chidume [8, 9] proved that the Mann iterative sequence converges strongly to a solution of the equation $x+Ax=f$ where A is a Lipschitz accretive operator defined on the Hilbert space H or the space ${L}_{p}$. The result was generalized by Ding and Deng [10], that is, they proved that the Ishikawa iterative sequence converges strongly to a solution of the equation $x+Ax=f$ where A is a Lipschitz accretive operator defined on p-uniformly smooth Banach space X. Zhu [11] proved that the Mann iterative sequence converges strongly to the unique solution of the equation $x+Ax=f$ under slightly different conditions where $A:D(A)\subseteq X\to X$ is a Lipschitz m-accretive operator and $D(A)$ is an open subset of a uniformly smooth Banach space X. Recently, Chidume and Osilike further extended the above results to the Mann iterative sequence (see, e.g., Theorem 5 of [12]), where A is Lipschitz m-accretive and $D(A)$ is a closed subset of a real Banach space X which is both uniformly convex and p-uniformly smooth.
The objective of this note is to discuss simultaneously the existence and uniqueness of a solution of the equation $x+Ax=f$ for any given $f\in X$, and its convergence, estimate of convergent rate, and stability of a Krasnosel’skiĭ-Mann-Opial type iterative solution. If the iterative parameter is selected suitably then the iterative procedure converges strongly to a unique solution of the equation and the iterative process is stable in arbitrary Banach space without any convexity or reflexivity.
Remark 1.2 A class of operators closely related to the class of accretive operators is the class of dissipative operators. An operator $S:D(A)\subseteq X\to X$ is said to be dissipative if and only if $(-S)$ is accretive and S is called m-dissipative if $(-S)$ is m-accretive (see, e.g., Barbu [13]). Some related results of the equation $x-\lambda Tx=f$ are also proved in [9, 10, 14] where $\lambda >0$ and T is a Lipschitzian dissipative operator on X.
To set the framework we recall some basic notations as follows.
Definition 1.3 [15]
Let $S:X\to X$ be an operator. Suppose that the iterative sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset X$ defined by ${x}_{n+1}=f(S,{x}_{n})$, and $\{{x}_{n}\}$ converges strongly to $q\in F(S)=\{x\in X:Sx=x\}\ne \mathrm{\varnothing}$. Furthermore, suppose that $\{{\u03f5}_{n}\}$ is a sequence in $(0,\mathrm{\infty})$ given by ${\u03f5}_{n}=\parallel {z}_{n+1}-f(S,{z}_{n})\parallel $ where ${\{{z}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is arbitrary sequence in X. Then the iteration procedure $\{{x}_{n}\}$ is said to be stable with respect to S (or, simply, S-stable) if ${lim}_{n\to \mathrm{\infty}}{\u03f5}_{n}=0$ implies that ${lim}_{n\to \mathrm{\infty}}{z}_{n}=q$.
Lemma 1.4 [16]
where ${\u03f5}_{n}\ge 0$ for all $n\ge 0$ and ${\u03f5}_{n}\to 0$ (as $n\to \mathrm{\infty}$), then we have ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$.
2 Iterative solution of m-accretive operator equation and its stability
We now prove the following propositions.
Proposition 2.1 If $A:X\to X$ is a Lipschitz accretive operator with Lipschitz constant L ($1\le L$), then the sequence $\{{x}_{n}\}$ is defined by (1.8) is an iterative solution of (1.3), where the iterative parameter ${\alpha}_{0}\in (0,1/{(L+1)}^{2})$.
This completes the proof. □
for any $n\ge 0$.
for all $n\ge 0$.
In the sequel, we need to discuss the stability of the iterative solution of (1.3).
Proposition 2.4 If the conditions of Proposition 2.1 are satisfied then the Krasnosel’skiĭ-Mann-Opial type iterative solution $\{{x}_{n}\}$ of (1.3) is S-stable.
That is, the Krasnosel’skiĭ-Mann-Opial type iterative solution is S-stable.
This completes the proof. □
- (1)
For the Krasnosel’skiĭ-Mann-Opial type iterative solution of $x+Ax=f$, we study simultaneously the existence, uniqueness, convergence, convergence rate estimate, and the stability.
- (2)
The operator may not be strongly accretive, ϕ-strongly accretive or uniformly accretive. The cases that A is uniformly accretive, ϕ-strongly accretive or strongly accretive, as regards (1.2) and (1.3), have been studied in [17, 19–21] and [22, 23];
- (3)By the way, the iterative parameter α does not depend on any geometric structure of space X and on any properties of the operators, but the selection of the parameter is to deal with the convergent rate of the iterative sequence. In this paper, a prototype of the iteration parameter is$\alpha =\frac{1}{2{(L+1)}^{2}}.$(2.24)
It is convenient in simplifying the computation and obtaining the convergent acceleration.
Declarations
Acknowledgements
The authors would like to thank the referees for his/her invaluable comments and suggestions, which improved this work significantly.
Authors’ Affiliations
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