Common fixed points of a family of strictly pseudocontractive mappings
© Qin et al.; licensee Springer. 2013
Received: 15 July 2013
Accepted: 14 October 2013
Published: 11 November 2013
In this article, fixed point problems of a family of strictly pseudocontractive mappings are investigated based on a viscosity iterative process. Strong convergence theorems are established in a real q-uniformly Banach space.
MSC:47H09, 47J05, 47J25.
Keywordsaccretive operator iterative process fixed point nonexpansive mapping zero point
Fixed point problems of nonlinear mappings as an important branch of nonlinear analysis theory have been applied in many disciplines, including economics, optimization, image recovery, mechanics, quantum physics, transportation and control theory; for more details, see [1–31] and the references therein.
Strictly pseudocontractive mappings, which act as a link between nonexpansive mappings and pseudocontractive mappings, have been extensively studied by many authors; see [20–31] and the references therein. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively.
Recently, many authors studied the following convex feasibility problem (CFP): , where is an integer, and each is assumed to be the fixed point set of a nonlinear mapping , . There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration , computer tomography  and radiation therapy treatment planning .
In this paper, we investigate the problem of finding a common fixed point of a finite family of strictly pseudocontractive mappings based on a viscosity approximation iterative process. Strong convergence theorems of common fixed points are established in a real q-uniformly Banach space.
for all and ;
for all .
The Banach space E is uniformly smooth if and only if . Let . The Banach space E is said to be q-uniformly smooth if there exists a constant such that . Indeed, there is no Banach space which is q-uniformly smooth with . Hilbert spaces, (or ) spaces and Sobolev spaces , where , are 2-uniformly smooth.
The class of κ-strictly pseudocontractive mappings was first introduced by Browder and Petryshyn  in Hilbert spaces.
where is a fixed point. Banach’s contraction mapping principle guarantees that has a unique fixed point in C. In the case of T having a fixed point, Browder  proved that converges strongly to a fixed point of T in the framework of Hilbert spaces. Reich  extended Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth Banach space, then converges strongly to a fixed point of T, and the limit defines the (unique) sunny nonexpansive retraction from C onto ; for more details, see  and the reference therein.
It is not hard to see that also enjoys a unique fixed point. Xu proved that converges to a fixed point of T as , and defines the unique sunny nonexpansive retraction from C onto .
where the sequence is in the interval .
In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm has only weak convergence. In many disciplines, including economics, image recovery and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence is often much more desirable than weak convergence, for it translates the physically tangible property that the energy of the error between the iterate and the solution x eventually becomes arbitrarily small. We also remark here that many authors have been instigating the problem of modifying the normal Mann iteration process to have strong convergence for κ-strictly pseudocontractive mappings; see [24–27] and the references therein.
Let D be a nonempty subset of C. Let . Q is said to be a contraction iff ; sunny iff for each and , we have ; sunny nonexpansive retraction iff Q is sunny, nonexpansive and a 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 , describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Q is sunny and nonexpansive;
, , .
In this paper, we investigate the problem of modifying the normal Mann iteration process for a family of κ-strictly pseudocontractive mappings. Strong convergence of the purposed iterative process is obtained in a real q-uniformly Banach space. In order to prove our main results, we need the following tools.
Lemma 2.1 
Let C be a nonempty subset of a real q-uniformly smooth Banach space E, and let be a κ-strict pseudocontraction. For , we define for every . Then, as , where , is nonexpansive such that .
Lemma 2.2 
where D is some fixed positive constant.
Lemma 2.3 
Lemma 2.4 
Lemma 2.5 
Let E be a smooth Banach space, and let C be a nonempty convex subset of E. Given an integer , assume that is a finite family of -strict pseudocontractions such that . Assume that is a positive sequence such that . Then .
Lemma 2.6 
for arbitrary positive real numbers a and b.
3 Main results
In view of Lemma 2.3, we find the desired conclusion immediately. This completes the proof. □
Remark 3.2 Theorem 3.1 mainly improves the corresponding results in Yuan et al.  from 2-uniformly smooth Banach spaces to q-uniformly smooth Banach spaces. Theorem 3.1 is applicable to the spaces and for all .
From Theorem 3.1, we have the following result immediately.
Remark 3.4 Corollary 3.3 improves the corresponding results in Zhou  from 2-uniformly smooth Banach spaces to q-uniformly smooth Banach spaces and relaxes the restrictions imposed on the parameter in Zhang and Su .
The first author was supported by the Natural Science Foundation of Zhejiang Province (Q12A010097) and the National Natural Science Foundation of China (11126334). The second author was supported by the Fundamental Research Funds for the Central Universities of China (2011YJS075) and the Scientific Research Fund of Hebei Provincial Education Department (QN20132030). The authors are grateful to the editor and two anonymous reviewers’ suggestions which improved the contents of the article.
- Park S:A review of the KKM theory on -space or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355–382.Google Scholar
- Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
- Dhage BC, Nashine HK, Patil VS: Common fixed points for some variants of weakly contraction mappings in partially ordered metric spaces. Adv. Fixed Point Theory 2013, 3: 29–48.Google Scholar
- Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar
- Dhage BC, Jadhav NS: Differential inequalities and comparison theorems for first order hybrid integro-differential equations. Adv. Inequal. Appl. 2013, 2: 61–80.Google Scholar
- Qin X, Cho SY, Kang SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 2011, 49: 679–693. 10.1007/s10898-010-9556-2MathSciNetView ArticleGoogle Scholar
- Chen JH: Fixed point iterations of semigroups of nonexpansive mappings. J. Semigroup Theory Appl. 2013., 2013: Article ID 9Google Scholar
- Wang ZM, Lou W: A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.Google Scholar
- Lv S, Wu C: Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping. Eng. Math. Lett. 2012, 1: 44–57.Google Scholar
- Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.Google Scholar
- Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.View ArticleGoogle Scholar
- Shen J, Pang LP: An approximate bundle method for solving variational inequalities. Commun. Optim. Theory 2012, 1: 1–18.Google Scholar
- Fattorini HO: Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge; 1999.View ArticleGoogle Scholar
- Dhage BC, Kamble GP, Metkar RG: On generalized Mellin-Hardy integral transformations. Eng. Math. Lett. 2013, 2: 67–80.Google Scholar
- Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 1979, 16: 964–979. 10.1137/0716071MathSciNetView ArticleGoogle Scholar
- Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.MathSciNetGoogle Scholar
- Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013. 10.1007/s10898-012-0017-yGoogle Scholar
- Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4Google Scholar
- Osu BO, Solomon OU: A stochastic algorithm for the valuation of financial derivatives using the hyperbolic distributional variates. Math. Finance Lett. 2012, 1: 43–56.Google Scholar
- Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8MathSciNetView ArticleGoogle Scholar
- Kangtunyakarn A, Suantai S: Strong convergence of a new iterative scheme for a finite family of strict pseudo-contractions. Comput. Math. Appl. 2010, 60: 680–694. 10.1016/j.camwa.2010.05.016MathSciNetView ArticleGoogle Scholar
- Yuan Q, Cho SY, Shang M: Strong convergence of an iterative process for a family of strictly pseudocontractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 117Google Scholar
- Qin X, Cho YJ, Kang SM, Shang MJ: A hybrid iterative scheme for asymptotically κ -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2009, 70: 1902–1911. 10.1016/j.na.2008.02.090MathSciNetView ArticleGoogle Scholar
- Qin X, Shang M, Kang SM: Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2009, 70: 1257–1264. 10.1016/j.na.2008.02.009MathSciNetView ArticleGoogle Scholar
- Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039–4046. 10.1016/j.na.2008.08.012MathSciNetView ArticleGoogle Scholar
- Zhou H: Convergence theorems for λ -strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Anal. 2008, 69: 3160–3173. 10.1016/j.na.2007.09.009MathSciNetView ArticleGoogle Scholar
- Zhang H, Su Y: Strong convergence theorems for strict pseudo-contractions in q -uniformly smooth Banach spaces. Nonlinear Anal. 2009, 70: 3236–3242. 10.1016/j.na.2008.04.030MathSciNetView ArticleGoogle Scholar
- Wang ZM: Convergence theorem on total asymptotically pseudocontractive mapping. J. Math. Comput. Sci. 2013, 3: 788–798.Google Scholar
- Ceng LC, Yao JC: Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense. Acta Appl. Math. 2011, 115: 167–191. 10.1007/s10440-011-9614-xMathSciNetView ArticleGoogle Scholar
- Takahshi W, Wong NC, Yao JC: Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems. Fixed Point Theory Appl. 2012., 2012: Article ID 181Google Scholar
- Ceng LC, Ansari QH, Yao JC: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. J. Nonlinear Convex Anal. 2010, 11: 283–308.MathSciNetGoogle Scholar
- Kotzer T, Cohen N, Shamir J: Image restoration by a novel method of parallel projection onto constraint sets. Opt. Lett. 1995, 20: 1172–1174. 10.1364/OL.20.001172View ArticleGoogle Scholar
- Sezan MI, Stark H: Application of convex projection theory to image recovery in tomograph and related areas. In Image Recovery: Theory and Application. Edited by: Stark H. Academic Press, Orlando; 1987:155–270.Google Scholar
- Censor Y, Zenios SA Numerical Mathematics and Scientific Computation. In Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York; 1997.Google Scholar
- Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleGoogle Scholar
- Browder FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach space. Arch. Ration. Mech. Anal. 1967, 24: 82–90.MathSciNetView ArticleGoogle Scholar
- Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleGoogle Scholar
- Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleGoogle Scholar
- Bruck RE: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 1973, 47: 341–355. 10.2140/pjm.1973.47.341MathSciNetView ArticleGoogle Scholar
- Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleGoogle Scholar
- Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289MathSciNetView ArticleGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleGoogle Scholar
- Mitrinovic DS: Analytic Inequalities. Springer, New York; 1970.View ArticleGoogle Scholar
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