Open Access

Common fixed point and invariant approximation results

Fixed Point Theory and Applications20132013:135

https://doi.org/10.1186/1687-1812-2013-135

Received: 25 November 2012

Accepted: 7 May 2013

Published: 27 May 2013

Abstract

Some common fixed point results for Banach operator pairs in strongly M-starshaped metric spaces are obtained. As application, invariant approximation theorems are derived.

MSC:47H10, 54H25.

Keywords

common fixed point Banach operator pair strongly M-starshaped metric space invariant approximation

1 Introduction and preliminaries

We first review needed definitions. Let X be a metric space with metric d, M X and J = [ 0 , 1 ] . The space X is called
  1. (1)
    M-starshaped [1] if there exists a continuous mapping W : X × M × J X satisfying
    d ( x , W ( y , q , λ ) ) λ d ( x , y ) + ( 1 λ ) d ( x , q )

    for all x , y X , q M and all λ J ;

     
  2. (2)
    strongly M-starshaped [2, 3] if it is M-starshaped and satisfies the property ( I ) , that is,
    d ( W ( x , q , λ ) , W ( y , q , λ ) ) λ d ( x , y )

    for all x , y X , q M and all λ J ;

     
  3. (3)

    (strongly) convex if it is (strongly) X-starshaped;

     
  4. (4)

    starshaped if it is { q } -starshaped for some q X .

     

A strongly convex metric space is also said to be a metric space of hyperbolic type (see Ciric [4]). Obviously, every normed space X is a strongly convex metric space with W defined by W ( x , q , λ ) = λ x + ( 1 λ ) q for all x , q X and all λ J . More generally, if X is a linear space with a translation invariant metric satisfying d ( λ x + ( 1 λ ) y , 0 ) λ d ( x , 0 ) + ( 1 λ ) d ( y , 0 ) , 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 q D M such that W ( x , q , λ ) D for all x D and all λ J . For details, we refer the reader to Al-Thagafi [2], Guay et al. [5] and Takahashi [1].

Let I , T : X X be two mappings and D X . Then T is called
  1. (5)

    I-nonexpansive on D if d ( T x , T y ) d ( I x , I y ) for all x , y D ;

     
  2. (6)

    I-contraction on D if there exists k [ 0 , 1 ) such that d ( T x , T y ) k d ( I x , I y ) for all x , y D .

     
A point x D is a coincidence point (common fixed point) of I and T if I x = T x ( x = I x = T x ). The set of coincidence points of I and T is denoted by C ( I , T ) . The mappings I and T are called
  1. (7)

    commuting on D if I T x = T I x for all x D ;

     
  2. (8)

    weakly compatible if they commute at their coincidence points, i.e., if I T x = T I x whenever I x = T x .

     

The ordered pair ( I , T ) of two self-maps of a metric space X is called a Banach operator pair if the set Fix ( T ) is I-invariant, namely I ( Fix ( T ) ) Fix ( T ) . Obviously, a commuting pair ( I , T ) is a Banach operator pair but not conversely in general, see [68].

Let S X and x ˆ X . Then P S ( x ˆ ) = { x S : d ( x , x ˆ ) = d ( x ˆ , S ) } is called the set of best S-approximants to x ˆ , where d ( x ˆ , S ) = inf { d ( x ˆ , y ) : y S } and C S I ( x ˆ ) = { x S : I x P S ( x ˆ ) } .

In 1963, Meinardus [9] employed the Schauder fixed point theorem to prove a result regarding invariant approximation. In 1979, Singh [10] proved the following extension of the result of Meinardus.

Theorem 1.1 Let T be a nonexpansive operator on a normed space X, let M be a nonempty subset of X, T ( M ) M and u F ( T ) . If P M ( u ) is nonempty compact and starshaped, then P M ( u ) F ( T ) .

Hicks and Humphries [11] found that Singh’s results remain true if T ( M ) M is replaced by T ( M ) M . In 1988, Sahab et al. [12] 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 u F ( I ) F ( T ) , M X with T ( M ) M , and q F ( I ) . If D = P M ( u ) is compact and q-starshaped, I ( D ) = D , I is continuous and linear on D, I and T are commuting on D and T is I-nonexpansive on D { u } , then P M ( u ) F ( T ) F ( I ) .

Invariant approximation results for commuting maps due to Al-Thagafi [13] extended and generalized Theorems 1.1-1.2 and the works of [11, 14, 15]. Al-Thagafi results were further extended by [7, 8, 1626] 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 [27], Habiniak [14], Hicks and Humphries [11], Hussain and Berinde [28], Hussain et al. [22], Naz [3], Latif [29], Sahab et al. [12] and Singh [10, 15].

The following result will be needed.

Lemma 1.3 [2]

Let D be a subset of an M-starshaped metric space ( X , d ) and x ˆ X . Then P D ( x ˆ ) D D .

2 Main results

The following result will be needed (see Lemma 2.10 [7] and Lemma 2.2 [8]).

Lemma 2.1 Let S be a nonempty subset of a metric space ( X , d ) , and let T, f be self-maps of S. If F ( f ) is nonempty, clT ( F ( f ) ) F ( f ) , cl ( T ( M ) ) is complete, and T and f satisfy for all x , y S and 0 h < 1 ,
d ( T x , T y ) h max { d ( f x , f y ) , d ( T x , f x ) , d ( T y , f y ) , d ( T x , f y ) , d ( T y , f x ) } ,
(2.1)

then S F ( T ) F ( f ) 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 F ( f ) is q-starshaped, clT ( F ( f ) ) F ( f ) , cl ( T ( S ) ) is compact, T is continuous on S and
T x T y max { f x f y , dist ( f x , [ q , T x ] ) , dist ( f y , [ q , T y ] ) , dist ( f y , [ q , T x ] ) , dist ( f x , [ q , T y ] ) } ,
(2.2)

for all x , y S , then S F ( T ) F ( f ) .

Proof Define T n : F ( f ) F ( f ) by T n x = W ( T x , q , k n ) for all x F ( f ) and a fixed sequence of real numbers k n ( 0 < k n < 1 ) converging to 1. Since F ( f ) is q-starshaped and clT ( F ( f ) ) F ( f ) , therefore clT n ( F ( f ) ) F ( f ) for each n 1 . Also, by (2.2),
d ( T n x , T n y ) = d ( W ( T x , q , k n ) , W ( T y , q , k n ) ) = k n d ( T x , T y ) k n max { d ( f x , f y ) , dist ( f x , [ q , T x ] ) , dist ( f y , [ q , T y ] ) , dist ( f x , [ q , T y ] ) , dist ( f y , [ q , T x ] ) } k n max { d ( f x , f y ) , d ( f x , T n x ) , d ( f y , T n y ) , d ( f y , T n x ) , d ( f x , T n y ) }

for each x , y F ( f ) and 0 < k n < 1 . If cl ( T ( S ) ) is compact for each n 1 , then cl ( T n ( S ) ) is compact and hence complete. By Lemma 2.1, for each n 1 , there exists x n F ( f ) such that x n = f x n = T n x n . The compactness of cl ( T ( M ) ) implies that there exists a subsequence { T x m } of { T x n } such that T x m z cl ( T ( M ) ) as m . Since { T x m } is a sequence in T ( F ( f ) ) and clT ( F ( f ) ) F ( f ) , therefore z F ( f ) . Further, x m = T m x m = W ( T x m , q , k m ) z . By the continuity of T, we obtain T z = z = f z . Thus, S F ( T ) F ( f ) . □

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 F ( f ) is q-starshaped, clT ( F ( f ) ) F ( f ) , cl ( T ( S ) ) is compact, T is continuous on S and T is f-nonexpansive on S, then S F ( T ) F ( f ) .

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 F ( f ) is closed and q-starshaped, ( T , f ) is a Banach operator pair, cl ( T ( S ) ) is compact, T is continuous on S and T satisfies (2.2) or T is f-nonexpansive on S, then S F ( T ) F ( f ) .

Corollary 2.5 ([13], 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 T ( M ) f ( M ) . Suppose that T commutes with f and q F ( f ) . If cl ( T ( M ) ) is compact, f is continuous and linear and T is f-nonexpansive on M, then M F ( T ) F ( f ) .

Corollary 2.6 (([30], 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 F ( f ) is q-starshaped, clT ( F ( f ) ) F ( f ) , cl ( T ( M ) ) is compact, T is continuous on M and (2.2) holds for all x , y M . Then M F ( T ) F ( f ) .

Corollary 2.7 ([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 F ( f ) is q-starshaped and closed cl ( T ( M ) ) is compact, T is continuous on M, ( T , f ) is a Banach operator pair and satisfies (2.2) for all x , y M . Then M F ( T ) F ( f ) .

Corollary 2.8 Let X be a strongly M-starshaped metric space, let f , T : X X be two mappings, S be a subset of X such that T ( S S ) S and x ˆ F ( T ) F ( f ) . Suppose that P S ( x ˆ ) is nonempty closed and q-starshaped with q F ( f ) M and cl ( T ( P S ( x ˆ ) ) ) is compact and f ( P S ( x ˆ ) ) = P S ( x ˆ ) . If T is continuous, clT ( F ( f ) ) F ( f ) and satisfies, for all x P S ( x ˆ ) { x ˆ } ,
d ( T x , T y ) { d ( f x , f u ) if y = u , max { d ( f x , f y ) , dist ( f x , [ q , T x ] ) , dist ( f y , [ q , T y ] ) , dist ( f x , [ q , T y ] ) , dist ( f y , [ q , T x ] ) } if y P S ( x ˆ ) ,
(2.3)

then P S ( x ˆ ) F ( T ) F ( f ) .

Proof Let x P S ( x ˆ ) . Then by Lemma 1.3, x S S and so T x S since T ( S S ) S . As T satisfies (2.3) on P S ( x ˆ ) { x ˆ } and I ( P S ( x ˆ ) ) = P S ( x ˆ ) , we have
d ( T x , x ˆ ) = d ( T x , T x ˆ ) d ( I x , I x ˆ ) = d ( I x , x ˆ ) = d ( x ˆ , S ) .

This implies that T x P S ( x ˆ ) . Thus T ( P S ( x ˆ ) ) P S ( x ˆ ) = f ( P S ( x ˆ ) ) . Now Theorem 2.2 implies that P S ( x ˆ ) F ( T ) F ( f ) . □

Theorem 2.9 Let X be a strongly M-starshaped metric space, let f , T : X X be two mappings, S be a subset of X such that T ( S S ) S and x ˆ F ( T ) F ( f ) . Suppose that P S ( x ˆ ) is nonempty closed and q-starshaped with q F ( f ) M and cl ( T ( P S ( x ˆ ) ) ) is compact and f ( P S ( x ˆ ) ) = P S ( x ˆ ) . If T is continuous, clT ( F ( f ) ) F ( f ) and T is f-nonexpansive on P S ( x ˆ ) { x ˆ } , then P S ( x ˆ ) F ( T ) F ( f ) .

Remark 2.10 A subset S of a strongly M-starshaped metric space X is said to have the property ( N ) w.r.t. T [22, 28] if
  1. (i)

    T : S S ,

     
  2. (ii)

    W ( T x , q , k n ) S for some q S M and a fixed sequence of real numbers k n ( 0 < k n < 1 ) converging to 1 and for each x S .

     

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 F ( f ) is replaced by the property ( N ) respectively. Consequently, recent results due to Hussain and Berinde [28] and Hussain et al. [22] are improved and extended.

Remark 2.11 Recently, in [31], the author obtained certain fixed point theorems in convex metric spaces. Using Theorems 3.2 and 3.4 [31] and the technique in [7], 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 [32].

Declarations

Acknowledgements

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.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdul Aziz University

References

  1. 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
  2. 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
  3. Naz A: Best approximation in strongly M -starshaped metric spaces. Rad. Mat. 2001, 10: 203–207.MATHMathSciNetGoogle Scholar
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. Meinardus G: Invarianze bei linearen approximationen. Arch. Ration. Mech. Anal. 1963, 14: 301–303.MATHMathSciNetGoogle Scholar
  10. 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
  11. 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
  12. 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
  13. Al-Thagafi MA: Common fixed points and best approximation. J. Approx. Theory 1996, 85: 318–323. 10.1006/jath.1996.0045MATHMathSciNetView ArticleGoogle Scholar
  14. Habiniak L: Fixed point theorems and invariant approximations. J. Approx. Theory 1989, 56: 241–244. 10.1016/0021-9045(89)90113-5MATHMathSciNetView ArticleGoogle Scholar
  15. 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
  16. 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
  17. 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
  18. 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
  19. Hussain N: Asymptotically pseudo-contractions, Banach operator pairs and best simultaneous approximations. Fixed Point Theory Appl. 2011., 2011: Article ID 812813Google Scholar
  20. 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
  21. 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
  22. 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
  23. Hussain N, Rhoades BE: C q -commuting maps and invariant approximations. Fixed Point Theory Appl. 2006., 2006: Article ID 24543Google Scholar
  24. 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
  25. Khan AR, Akbar F: Common fixed points from best simultaneous approximations. Taiwan. J. Math. 2009, 13(5):1379–1386.MATHMathSciNetGoogle Scholar
  26. 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
  27. 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
  28. 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
  29. Latif A: A result on best approximation in p -normed spaces. Arch. Math. 2001, 37: 71–75.MATHMathSciNetGoogle Scholar
  30. 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
  31. Moosaei M: Fixed point theorems in convex metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 164Google Scholar
  32. Espínola R, Hussain N: Common fixed points for multimaps in metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 204981Google Scholar

Copyright

© Kutbi; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.