- Open Access
Krasnosel’skiĭ-Mann-Opial type iterative solution of m-accretive operator equation and its stability in arbitrary Banach spaces
© Xu et al.; licensee Springer. 2014
- Received: 20 August 2013
- Accepted: 31 January 2014
- Published: 14 February 2014
Let X be a Banach space. Suppose that is a Lipschitz accretive operator. The objective of this note is to discuss simultaneously the existence and uniqueness of solution of the equation for any given , and its convergence, estimate of convergent rate, and stability of Krasnosel’skiĭ-Mann-Opial type iterative solution . 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 where is a solution of .
MSC:47H06, 47H10, 47H17.
- accretive operator
- m-accretive operator
- iterative solution
- Krasnosel’skiĭ-Mann-Opial type iterative procedure
Throughout this paper, X is assumed to be a real Banach space.
for all and . An accretive operator A is said to be m-accretive if for all (or, equivalently, for some) , where I stands for the identity operator on X.
for all . To pick , we have . It follows that for the equation there exists a unique solution .
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 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 , the operator also is nonexpansive. Moreover, as has been pointed out by Krasnosel’skiĭ  for and by Schaefer  for an arbitrary λ, if X is uniformly convex and S has at least one fixed point in C, then the operator is asymptotically regular. For any and any , Opial proved that the iterative sequence (or ) is weakly convergent to a fixed point of S under some assumptions (see, e.g., Theorem 3 of ).
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 .
In 1986, Chidume [8, 9] proved that the Mann iterative sequence converges strongly to a solution of the equation where A is a Lipschitz accretive operator defined on the Hilbert space H or the space . The result was generalized by Ding and Deng , that is, they proved that the Ishikawa iterative sequence converges strongly to a solution of the equation where A is a Lipschitz accretive operator defined on p-uniformly smooth Banach space X. Zhu  proved that the Mann iterative sequence converges strongly to the unique solution of the equation under slightly different conditions where is a Lipschitz m-accretive operator and 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 ), where A is Lipschitz m-accretive and 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 for any given , 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 is said to be dissipative if and only if is accretive and S is called m-dissipative if is m-accretive (see, e.g., Barbu ). Some related results of the equation are also proved in [9, 10, 14] where and T is a Lipschitzian dissipative operator on X.
To set the framework we recall some basic notations as follows.
Definition 1.3 
Let be an operator. Suppose that the iterative sequence defined by , and converges strongly to . Furthermore, suppose that is a sequence in given by where is arbitrary sequence in X. Then the iteration procedure is said to be stable with respect to S (or, simply, S-stable) if implies that .
Lemma 1.4 
where for all and (as ), then we have .
We now prove the following propositions.
Proposition 2.1 If is a Lipschitz accretive operator with Lipschitz constant L (), then the sequence is defined by (1.8) is an iterative solution of (1.3), where the iterative parameter .
This completes the proof. □
for any .
for all .
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 of (1.3) is S-stable.
That is, the Krasnosel’skiĭ-Mann-Opial type iterative solution is S-stable.
This completes the proof. □
For the Krasnosel’skiĭ-Mann-Opial type iterative solution of , we study simultaneously the existence, uniqueness, convergence, convergence rate estimate, and the stability.
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(2.24)
It is convenient in simplifying the computation and obtaining the convergent acceleration.
The authors would like to thank the referees for his/her invaluable comments and suggestions, which improved this work significantly.
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