Common fixed point and invariant approximation results
© Kutbi; licensee Springer 2013
Received: 25 November 2012
Accepted: 7 May 2013
Published: 27 May 2013
Some common fixed point results for Banach operator pairs in strongly M-starshaped metric spaces are obtained. As application, invariant approximation theorems are derived.
1 Introduction and preliminaries
- (1)M-starshaped  if there exists a continuous mapping satisfying
for all , and all ;
for all , and all ;
(strongly) convex if it is (strongly) X-starshaped;
starshaped if it is -starshaped for some .
A strongly convex metric space is also said to be a metric space of hyperbolic type (see Ciric ). Obviously, every normed space X is a strongly convex metric space with W defined by for all and all . More generally, if X is a linear space with a translation invariant metric satisfying , then X is a strongly convex metric space. A subset D of an M-starshaped metric space X is called q-starshaped if there exists such that for all and all . For details, we refer the reader to Al-Thagafi , Guay et al.  and Takahashi .
I-nonexpansive on D if for all ;
I-contraction on D if there exists such that for all .
commuting on D if for all ;
weakly compatible if they commute at their coincidence points, i.e., if whenever .
The ordered pair of two self-maps of a metric space X is called a Banach operator pair if the set is I-invariant, namely . Obviously, a commuting pair is a Banach operator pair but not conversely in general, see [6–8].
Let and . Then is called the set of best S-approximants to , where and .
Theorem 1.1 Let T be a nonexpansive operator on a normed space X, let M be a nonempty subset of X, and . If is nonempty compact and starshaped, then .
Hicks and Humphries  found that Singh’s results remain true if is replaced by . In 1988, Sahab et al.  established the following result which contains the result of Hicks and Humphries and Theorem 1.1.
Theorem 1.2 Let I and T be self-maps of a normed space X with , with , and . If is compact and q-starshaped, , I is continuous and linear on D, I and T are commuting on D and T is I-nonexpansive on , then .
Invariant approximation results for commuting maps due to Al-Thagafi  extended and generalized Theorems 1.1-1.2 and the works of [11, 14, 15]. Al-Thagafi results were further extended by [7, 8, 16–26] to R-subweakly commuting, pointwise R-subweakly commuting and a Banach operator pair.
The aim of this paper is to establish certain common fixed point theorem for a Banach operator pair in the setup of strongly M-starshaped metric spaces. As application, invariant approximation results for this class of maps are derived. Our results extend and unify the work of Al-Thagafi [2, 13], Dotson , Habiniak , Hicks and Humphries , Hussain and Berinde , Hussain et al. , Naz , Latif , Sahab et al.  and Singh [10, 15].
The following result will be needed.
Lemma 1.3 
Let D be a subset of an M-starshaped metric space and . Then .
2 Main results
then is a singleton.
for all , then .
for each and . If is compact for each , then is compact and hence complete. By Lemma 2.1, for each , there exists such that . The compactness of implies that there exists a subsequence of such that as . Since is a sequence in and , therefore . Further, . By the continuity of T, we obtain . Thus, . □
Corollary 2.3 Let S be a nonempty subset of a strongly M-starshaped metric space X and let T, f be self-maps of S. Suppose that is q-starshaped, , is compact, T is continuous on S and T is f-nonexpansive on S, then .
Corollary 2.4 Let S be a nonempty subset of a strongly M-starshaped metric space X and let T, f be self-maps of S. Suppose that is closed and q-starshaped, is a Banach operator pair, is compact, T is continuous on S and T satisfies (2.2) or T is f-nonexpansive on S, then .
Corollary 2.5 (, Theorem 2.1)
Let M be a nonempty closed and q-starshaped subset of a normed space X and let T and f be self-maps of M such that . Suppose that T commutes with f and . If is compact, f is continuous and linear and T is f-nonexpansive on M, then .
Corollary 2.6 ((, Theorem 3.3))
Let M be a nonempty subset of a normed space X and let T and f be self-maps of M. Suppose that is q-starshaped, , is compact, T is continuous on M and (2.2) holds for all . Then .
Corollary 2.7 (, Theorem 2.11)
Let M be a nonempty subset of a normed space X and let T, f be self-maps of M. Suppose that is q-starshaped and closed is compact, T is continuous on M, is a Banach operator pair and satisfies (2.2) for all . Then .
This implies that . Thus . Now Theorem 2.2 implies that . □
Theorem 2.9 Let X be a strongly M-starshaped metric space, let be two mappings, S be a subset of X such that and . Suppose that is nonempty closed and q-starshaped with and is compact and . If T is continuous, and T is f-nonexpansive on , then .
for some and a fixed sequence of real numbers () converging to 1 and for each .
All results of the paper (Theorem 2.2-Theorem 2.9) remain valid provided f is assumed to be surjective and q-starshapedness of the set is replaced by the property respectively. Consequently, recent results due to Hussain and Berinde  and Hussain et al.  are improved and extended.
Remark 2.11 Recently, in , the author obtained certain fixed point theorems in convex metric spaces. Using Theorems 3.2 and 3.4  and the technique in , we can prove more common fixed point and approximation results for Banach pairs satisfying generalized nonexpansive conditions in a strongly M-starshaped metric space X.
Remark 2.12 All results of the paper can be proved for multivalued Banach operator pairs defined and studied in .
This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author acknowledges with thanks DSR, KAU for financial support.
- Takahashi W: A convexity in metric spaces and non-expansive mappings I. Kodai Math. Semin. Rep. 1970, 22: 142–149. 10.2996/kmj/1138846111MATHView ArticleGoogle Scholar
- Al-Thagafi MA: Best approximation and fixed points in strong M -starshaped metric spaces. Int. J. Math. Math. Sci. 1995, 18: 613–616. 10.1155/S0161171295000779MATHMathSciNetView ArticleGoogle Scholar
- Naz A: Best approximation in strongly M -starshaped metric spaces. Rad. Mat. 2001, 10: 203–207.MATHMathSciNetGoogle Scholar
- Ciric LB: Contractive type non-self mappings on metric spaces of hyperbolic type. J. Math. Anal. Appl. 2006, 317: 28–42. 10.1016/j.jmaa.2005.11.025MATHMathSciNetView ArticleGoogle Scholar
- Guay MD, Singh KL, Whitfield JHM: Fixed point theorems for nonexpansive mappings in convex metric spaces. 80. In Proc. Conf. on Nonlinear Analysis. Edited by: Singh SP, Burry JH. Dekker, New York; 1992:179–189.Google Scholar
- Chen J, Li Z: Common fixed points for Banach operator pairs in best approximation. J. Math. Anal. Appl. 2007, 336: 1466–1475. 10.1016/j.jmaa.2007.01.064MATHMathSciNetView ArticleGoogle Scholar
- Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćirić type I -contractions. J. Math. Anal. Appl. 2008, 338: 1351–1363. 10.1016/j.jmaa.2007.06.008MATHMathSciNetView ArticleGoogle Scholar
- Khan AR, Akbar F: Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces. J. Math. Anal. Appl. 2009, 354: 469–477. 10.1016/j.jmaa.2009.01.007MATHMathSciNetView ArticleGoogle Scholar
- Meinardus G: Invarianze bei linearen approximationen. Arch. Ration. Mech. Anal. 1963, 14: 301–303.MATHMathSciNetGoogle Scholar
- Singh SP: An application of fixed point theorem to approximation theory. J. Approx. Theory 1979, 25: 89–90. 10.1016/0021-9045(79)90036-4MATHMathSciNetView ArticleGoogle Scholar
- Hicks TL, Humphries MD: A note on fixed point theorems. J. Approx. Theory 1982, 34: 221–225. 10.1016/0021-9045(82)90012-0MATHMathSciNetView ArticleGoogle Scholar
- Sahab SA, Khan MS, Sessa S: A result in best approximation theory. J. Approx. Theory 1988, 55: 349–351. 10.1016/0021-9045(88)90101-3MATHMathSciNetView ArticleGoogle Scholar
- Al-Thagafi MA: Common fixed points and best approximation. J. Approx. Theory 1996, 85: 318–323. 10.1006/jath.1996.0045MATHMathSciNetView ArticleGoogle Scholar
- Habiniak L: Fixed point theorems and invariant approximations. J. Approx. Theory 1989, 56: 241–244. 10.1016/0021-9045(89)90113-5MATHMathSciNetView ArticleGoogle Scholar
- Singh SP: Applications of fixed point theorems in approximation theory. In Applied Nonlinear Analysis. Edited by: Lakshmikantham V. Academic Press, New York; 1979:389–394.Google Scholar
- Akbar F, Khan AR: Common fixed point and approximation results for noncommuting maps on locally convex spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 207503Google Scholar
- Ciric LB, Hussain N, Akbar F, Ume JS: Common fixed points for Banach operator pairs from the set of best approximations. Bull. Belg. Math. Soc. Simon Stevin 2009, 16: 319–336.MATHMathSciNetGoogle Scholar
- Ćirić LB, Hussain N, Cakic N: Common fixed points for Ciric type f -weak contraction with applications. Publ. Math. (Debr.) 2010, 76(1–2):31–49.MATHGoogle Scholar
- Hussain N: Asymptotically pseudo-contractions, Banach operator pairs and best simultaneous approximations. Fixed Point Theory Appl. 2011., 2011: Article ID 812813Google Scholar
- Hussain N, Khamsi MA, Latif A: Banach operator pairs and common fixed points in hyperconvex metric spaces. Nonlinear Anal. 2011, 74: 5956–5961. 10.1016/j.na.2011.05.072MATHMathSciNetView ArticleGoogle Scholar
- Hussain N, Pathak HK: Subweakly biased pairs and Jungck contractions with applications. Numer. Funct. Anal. Optim. 2011, 32(10):1067–1082. 10.1080/01630563.2011.587627MATHMathSciNetView ArticleGoogle Scholar
- Hussain N, O’Regan D, Agarwal RP: Common fixed point and invariant approximation results on non-starshaped domains. Georgian Math. J. 2005, 12: 659–669.MATHMathSciNetGoogle Scholar
- Hussain N, Rhoades BE:-commuting maps and invariant approximations. Fixed Point Theory Appl. 2006., 2006: Article ID 24543Google Scholar
- O’Regan D, Hussain N: Generalized I -contractions and pointwise R -subweakly commuting maps. Acta Math. Sin. Engl. Ser. 2007, 23: 1505–1508. 10.1007/s10114-007-0935-7MATHMathSciNetView ArticleGoogle Scholar
- Khan AR, Akbar F: Common fixed points from best simultaneous approximations. Taiwan. J. Math. 2009, 13(5):1379–1386.MATHMathSciNetGoogle Scholar
- Pathak HK, Hussain N: Common fixed points for Banach operator pairs with applications. Nonlinear Anal. 2008, 69: 2788–2802. 10.1016/j.na.2007.08.051MATHMathSciNetView ArticleGoogle Scholar
- Dotson WJ Jr.: Fixed point theorems for nonexpansive mappings on star-shaped subsets of Banach spaces. J. Lond. Math. Soc. 1972, 4: 408–410.MATHMathSciNetView ArticleGoogle Scholar
- Hussain N, Berinde V: Common fixed point and invariant approximation results in certain metrizable topological vector spaces. Fixed Point Theory Appl. 2006., 2006: Article ID 23582Google Scholar
- Latif A: A result on best approximation in p -normed spaces. Arch. Math. 2001, 37: 71–75.MATHMathSciNetGoogle Scholar
- Al-Thagafi MA, Shahzad N: Banach operator pairs, common fixed points, invariant approximations and ∗-nonexpansive multimaps. Nonlinear Anal. 2008, 69: 2733–2739. 10.1016/j.na.2007.08.047MATHMathSciNetView ArticleGoogle Scholar
- Moosaei M: Fixed point theorems in convex metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 164Google Scholar
- Espínola R, Hussain N: Common fixed points for multimaps in metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 204981Google Scholar
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