- Research Article
- Open Access
© Yuan Qing et al. 2011
- Received: 21 November 2010
- Accepted: 8 February 2011
- Published: 1 March 2011
- Banach Space
- Variational Inequality
- Convex Subset
- Iterative Algorithm
- Nonexpansive Mapping
We use to denote the unique fixed point of , which yields that . In 1967, Browder  proved the following theorem.
In , Moudafi proposed a viscosity approximation method which was considered by many authors [2–8]. If is a Hilbert space, is a nonexpansive mapping and is a contraction, he proved the following theorems.
For each , we denote by the resolvent of , that is, . Note that if is -accretive, then is nonexpansive and , for all . We also denote by the Yosida approximation of , that is, . It is known that is a nonexpansive mapping from to .
where is a real sequence , is a contractive mapping, and is a nonexpansive mapping with a fixed point. Strong convergence theorems of fixed points are obtained in a uniformly smooth Banach space; see  for more details.
Very recently, Zegeye and Shahzad  studied the common zero problem of a family of -accretive mappings. To be more precise, they proved the following result.
where is a real sequence which satisfies the following conditions: ; ; or and with for for and . If every nonempty, closed, bounded convex subset of has the fixed point property for a nonexpansive mapping, then converges strongly to a common solution of the equations for .
where with for , and is a real sequence in . It is proved that the sequence generated in the iterative algorithms (1.17) and (1.18) converges strongly to a common zero point of a finite family of -accretive mappings in reflexive Banach spaces, respectively.
for , , , where , then (see ).
In order to prove our main results, we also need the following lemmas.
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 .
Lemma 2.2 (see ).
Let be a Banach space satisfying a weakly continuous duality map, let be a nonempty, closed, convex subset of , and let be a nonexpansive mapping with a fixed point. Then, is demiclosed at zero, that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .
Lemma 2.3 (see ).
Let be a nonempty, closed, convex subset of a strictly convex Banach space . Let , , be a family of -accretive mappings such that . Let be real numbers in such that and , where . Then, is nonexpansive and .
Lemma 2.4 (see ).
We need the strong convergence of the implicit algorithm (1.17) to prove the strong convergence of the explicit algorithm (3.14).
Let be a strictly convex and reflexive Banach space which has a weakly continuous duality map with the gauge . Lek be a nonempty, closed, convex subset of and a contractive mapping. Let be a family of -accretive mappings with . Let for each . For any , let be generated by the algorithm (1.18), where with for , , and is a sequence in which satisfies the following conditions: and . Assume also that
As an application of Theorems 3.1 and 3.2, we have the following results for a single mapping.
- Browder FE: Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces. Archive for Rational Mechanics and Analysis 1967, 24: 82–90.MATHMathSciNetView ArticleGoogle Scholar
- Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MATHMathSciNetView ArticleGoogle Scholar
- Cho YJ, Qin X: Viscosity approximation methods for a family of -accretive mappings in reflexive Banach spaces. Positivity 2008,12(3):483–494. 10.1007/s11117-007-2181-8MATHMathSciNetView ArticleGoogle Scholar
- Cho YJ, Kang SM, Qin X: Some results on -strictly pseudo-contractive mappings in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1956–1964. 10.1016/j.na.2008.02.094MATHMathSciNetView ArticleGoogle Scholar
- Cho YJ, Kang SM, Qin X: Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces. Computers & Mathematics with Applications 2008,56(8):2058–2064. 10.1016/j.camwa.2008.03.035MATHMathSciNetView ArticleGoogle Scholar
- Ceng L-C, Xu H-K, Yao J-C: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(4):1402–1412. 10.1016/j.na.2007.06.040MATHMathSciNetView ArticleGoogle Scholar
- Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):99–112. 10.1016/j.na.2009.06.042MATHMathSciNetView ArticleGoogle Scholar
- Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MATHMathSciNetView ArticleGoogle Scholar
- Kim T-H, Xu H-K: Strong convergence of modified Mann iterations. Nonlinear Analysis: Theory, Methods & Applications 2005,61(1–2):51–60. 10.1016/j.na.2004.11.011MATHMathSciNetView ArticleGoogle Scholar
- Xu H-K: Strong convergence of an iterative method for nonexpansive and accretive operators. Journal of Mathematical Analysis and Applications 2006,314(2):631–643. 10.1016/j.jmaa.2005.04.082MATHMathSciNetView ArticleGoogle Scholar
- Zegeye H, Shahzad N: Strong convergence theorems for a common zero for a finite family of -accretive mappings. Nonlinear Analysis: Theory, Methods & Applications 2007,66(5):1161–1169. 10.1016/j.na.2006.01.012MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Zhou H, Cho YJ, Kang SM: Zeros and mapping theorems for perturbations of -accretive operators in Banach spaces. Computers & Mathematics with Applications 2005,49(1):147–155. 10.1016/j.camwa.2005.01.012MATHMathSciNetView ArticleGoogle Scholar
- Dominguez Benavides T, Lopez Acedo G, Xu H-K: Iterative solutions for zeros of accretive operators. Mathematische Nachrichten 2003, 248/249: 62–71. 10.1002/mana.200310003MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA: On successive approximations for nonexpansive mappings in Banach spaces. Glasgow Mathematical Journal 1971, 12: 6–9. 10.1017/S0017089500001063MATHMathSciNetView ArticleGoogle Scholar
- Liu G, Lei D, Li S: Approximating fixed points of nonexpansive mappings. International Journal of Mathematics and Mathematical Sciences 2000,24(3):173–177. 10.1155/S0161171200003252MATHMathSciNetView ArticleGoogle Scholar
- Maiti M, Saha B: Approximating fixed points of nonexpansive and generalized nonexpansive mappings. International Journal of Mathematics and Mathematical Sciences 1993,16(1):81–86. 10.1155/S0161171293000092MATHMathSciNetView ArticleGoogle Scholar
- Park S: Fixed point theorems in locally -convex spaces. Nonlinear Analysis: Theory, Methods & Applications 2002,48(6):869–879. 10.1016/S0362-546X(00)00220-0MATHMathSciNetView ArticleGoogle Scholar
- Park S: Fixed point theory of multimaps in abstract convex uniform spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2468–2480. 10.1016/j.na.2009.01.081MATHView ArticleMathSciNetGoogle Scholar
- Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. Journal of Mathematical Analysis and Applications 2007,329(1):415–424. 10.1016/j.jmaa.2006.06.067MATHMathSciNetView ArticleGoogle Scholar
- Yao Y, Liou Y-C: Strong convergence to common fixed points of a finite family of asymptotically nonexpansive map. Taiwanese Journal of Mathematics 2007,11(3):849–865.MATHMathSciNetGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis, Fixed Point theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
- Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Mathematische Zeitschrift 1967, 100: 201–225. 10.1007/BF01109805MATHMathSciNetView ArticleGoogle Scholar
- Xu ZB, Roach GF: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. Journal of Mathematical Analysis and Applications 1991,157(1):189–210. 10.1016/0022-247X(91)90144-OMATHMathSciNetView ArticleGoogle Scholar
- Lim T-C, Xu HK: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 1994,22(11):1345–1355. 10.1016/0362-546X(94)90116-3MATHMathSciNetView ArticleGoogle Scholar
- Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.