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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.
1 Introduction and preliminaries
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 .
2 Iterative solution of m-accretive operator equation and its stability
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.
- Browder FE: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Am. Math. Soc. 1967, 73: 875–882. 10.1090/S0002-9904-1967-11823-8View ArticleMathSciNetMATHGoogle Scholar
- Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508–520. 10.2969/jmsj/01940508View ArticleMATHGoogle Scholar
- Martin RH Jr.: A global existence theorem for autonomous differential equations in Banach spaces. Proc. Am. Math. Soc. 1970, 26: 307–314. 10.1090/S0002-9939-1970-0264195-6View ArticleMATHGoogle Scholar
- Krasnosel’skiĭ MA: Two observations about the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123–127.Google Scholar
- Schaefer H: Über die methode sukzessiver approximationen. Jahresber. Dtsch. Math.-Ver. 1957, 59: 131–140.MathSciNetGoogle Scholar
- Opial Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–5597. 10.1090/S0002-9904-1967-11761-0View ArticleMathSciNetMATHGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleMATHGoogle Scholar
- Chidume CE: An approximation method for monotone Lipschitzian operators in Hilbert space. J. Aust. Math. Soc. 1986, 41: 59–63. 10.1017/S144678870002807XView ArticleMathSciNetMATHGoogle Scholar
- Chidume CE: Iterative solution of nonlinear equations of the monotone and dissipative types. Appl. Anal. 1989, 33: 79–86. 10.1080/00036818908839862View ArticleMathSciNetMATHGoogle Scholar
- Ding XP, Deng L: Iterative solution of nonlinear equations of the monotone and dissipative types in uniformly smooth Banach spaces. J. Sichuan Normal Univ. 1994, 17: 43–48.MathSciNetMATHGoogle Scholar
- Zhu L: Iterative solution of nonlinear equations involving m -accretive operators in Banach spaces. J. Math. Anal. Appl. 1994, 188: 410–415. 10.1006/jmaa.1994.1434View ArticleMathSciNetMATHGoogle Scholar
- Chidume CE, Osilike MO: Approximation methods for nonlinear operator equations of the m -accretive type. J. Math. Anal. Appl. 1995, 189: 225–239. 10.1006/jmaa.1995.1014View ArticleMathSciNetMATHGoogle Scholar
- Barbu V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden; 1976.View ArticleMATHGoogle Scholar
- Ding XP: Iterative process with errors of nonlinear equations involving m -accretive operators. J. Math. Anal. Appl. 1997, 209: 191–201. 10.1006/jmaa.1997.5356View ArticleMathSciNetMATHGoogle Scholar
- Chang SS, Cho Y, Zhou H: Iterative Methods for Nonlinear Operator Equations in Banach Spaces. Nova Science Publishers, Hauppauge; 2002.MATHGoogle Scholar
- Reich S: Constructive techniques for accretive and monotone operators. In Applied Nonlinear Analysis. Academic Press, New York; 1979:335–345.Google Scholar
- Liu LW: Strong convergence of iteration methods for equations involving accretive operators in Banach spaces. Nonlinear Anal. 2000, 42: 271–276. 10.1016/S0362-546X(98)00345-9View ArticleMathSciNetMATHGoogle Scholar
- Zeng LC: On the convergence rate estimates of iteration methods for equations involving accretive operators in Banach spaces. Math. Appl. 2002, 15(2):80–84.View ArticleMathSciNetMATHGoogle Scholar
- Liu ZQ, Xu YG, Kang SM: Almost stable iteration schemes for local strongly pseudocontractive and local strongly accretive operators in real uniformly smooth Banach spaces. Acta Math. Univ. Comen. 2008, 77: 285–298.MathSciNetMATHGoogle Scholar
- Liu ZQ, Xu YG, Cho YJ: Iteration solution of nonlinear equations with ϕ -strongly accretive operators. Arch. Math. 2001, 77: 508–516. 10.1007/PL00000524View ArticleMathSciNetMATHGoogle Scholar
- Liu ZQ, Kang SM: Convergence theorems for ϕ -strongly accretive and ϕ -hemicontractive operators. J. Math. Anal. Appl. 2001, 253: 35–49. 10.1006/jmaa.2000.6973View ArticleMathSciNetMATHGoogle Scholar
- Osilike MO: Iterative solutions of nonlinear ϕ -strongly accretive operator equations in arbitrary Banach spaces. Nonlinear Anal. 1999, 36: 1–9. 10.1016/S0362-546X(97)00566-XView ArticleMathSciNetMATHGoogle Scholar
- Osilike MO: Iterative solution of nonlinear equations of the ϕ -strongly accretive type. J. Math. Anal. Appl. 1996, 200: 259–271. 10.1006/jmaa.1996.0203View ArticleMathSciNetMATHGoogle Scholar
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