- Open Access
Common fixed point and invariant approximation results
Fixed Point Theory and Applications volume 2013, Article number: 135 (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
We first review needed definitions. Let X be a metric space with metric d, and . The space X is called
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 .
Let be two mappings and . Then T is called
I-nonexpansive on D if for all ;
I-contraction on D if there exists such that for all .
A point is a coincidence point (common fixed point) of I and T if (). The set of coincidence points of I and T is denoted by . The mappings I and T are called
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
Lemma 2.1 Let S be a nonempty subset of a metric space , and let T, f be self-maps of S. If is nonempty, , is complete, and T and f satisfy for all and ,
then is a singleton.
Theorem 2.2 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
for all , then .
Proof Define by for all and a fixed sequence of real numbers () converging to 1. Since is q-starshaped and , therefore for each . Also, by (2.2),
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 .
Corollary 2.8 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 satisfies, for all ,
Proof Let . Then by Lemma 1.3, and so since . As T satisfies (2.3) on and , we have
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 .
Takahashi W: A convexity in metric spaces and non-expansive mappings I. Kodai Math. Semin. Rep. 1970, 22: 142–149. 10.2996/kmj/1138846111
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/S0161171295000779
Naz A: Best approximation in strongly M -starshaped metric spaces. Rad. Mat. 2001, 10: 203–207.
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.2005.11.025
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.
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.2007.01.064
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.2007.06.008
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.2009.01.007
Meinardus G: Invarianze bei linearen approximationen. Arch. Ration. Mech. Anal. 1963, 14: 301–303.
Singh SP: An application of fixed point theorem to approximation theory. J. Approx. Theory 1979, 25: 89–90. 10.1016/0021-9045(79)90036-4
Hicks TL, Humphries MD: A note on fixed point theorems. J. Approx. Theory 1982, 34: 221–225. 10.1016/0021-9045(82)90012-0
Sahab SA, Khan MS, Sessa S: A result in best approximation theory. J. Approx. Theory 1988, 55: 349–351. 10.1016/0021-9045(88)90101-3
Al-Thagafi MA: Common fixed points and best approximation. J. Approx. Theory 1996, 85: 318–323. 10.1006/jath.1996.0045
Habiniak L: Fixed point theorems and invariant approximations. J. Approx. Theory 1989, 56: 241–244. 10.1016/0021-9045(89)90113-5
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.
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 207503
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.
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.
Hussain N: Asymptotically pseudo-contractions, Banach operator pairs and best simultaneous approximations. Fixed Point Theory Appl. 2011., 2011: Article ID 812813
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.2011.05.072
Hussain N, Pathak HK: Subweakly biased pairs and Jungck contractions with applications. Numer. Funct. Anal. Optim. 2011, 32(10):1067–1082. 10.1080/01630563.2011.587627
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.
Hussain N, Rhoades BE:-commuting maps and invariant approximations. Fixed Point Theory Appl. 2006., 2006: Article ID 24543
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/s10114-007-0935-7
Khan AR, Akbar F: Common fixed points from best simultaneous approximations. Taiwan. J. Math. 2009, 13(5):1379–1386.
Pathak HK, Hussain N: Common fixed points for Banach operator pairs with applications. Nonlinear Anal. 2008, 69: 2788–2802. 10.1016/j.na.2007.08.051
Dotson WJ Jr.: Fixed point theorems for nonexpansive mappings on star-shaped subsets of Banach spaces. J. Lond. Math. Soc. 1972, 4: 408–410.
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 23582
Latif A: A result on best approximation in p -normed spaces. Arch. Math. 2001, 37: 71–75.
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.2007.08.047
Moosaei M: Fixed point theorems in convex metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 164
Espínola R, Hussain N: Common fixed points for multimaps in metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 204981
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.
The author declares that he has no competing interests.
The author has read and approved the final manuscript.
About this article
Cite this article
Kutbi, M.A. Common fixed point and invariant approximation results. Fixed Point Theory Appl 2013, 135 (2013). https://doi.org/10.1186/1687-1812-2013-135
- common fixed point
- Banach operator pair
- strongly M-starshaped metric space
- invariant approximation