Skip to main content

Common fixed point and invariant approximation results

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.

1 Introduction and preliminaries

We first review needed definitions. Let X be a metric space with metric d, MX and J=[0,1]. The space X is called

  1. (1)

    M-starshaped [1] if there exists a continuous mapping W:X×M×JX satisfying

    d ( x , W ( y , q , λ ) ) λd(x,y)+(1λ)d(x,q)

    for all x,yX, qM and all λJ;

  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 ) ) λd(x,y)

    for all x,yX, qM and all λJ;

  3. (3)

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

  4. (4)

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

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)=λx+(1λ)q for all x,qX and all λJ. More generally, if X is a linear space with a translation invariant metric satisfying d(λx+(1λ)y,0)λd(x,0)+(1λ)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 qDM such that W(x,q,λ)D for all xD and all λJ. For details, we refer the reader to Al-Thagafi [2], Guay et al. [5] and Takahashi [1].

Let I,T:XX be two mappings and DX. Then T is called

  1. (5)

    I-nonexpansive on D if d(Tx,Ty)d(Ix,Iy) for all x,yD;

  2. (6)

    I-contraction on D if there exists k[0,1) such that d(Tx,Ty)kd(Ix,Iy) for all x,yD.

A point xD is a coincidence point (common fixed point) of I and T if Ix=Tx (x=Ix=Tx). 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 ITx=TIx for all xD;

  2. (8)

    weakly compatible if they commute at their coincidence points, i.e., if ITx=TIx whenever Ix=Tx.

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 SX and x ˆ X. Then P S ( x ˆ )={xS: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):yS} and C S I ( x ˆ )={xS:Ix 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 uF(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 uF(I)F(T), MX with T(M)M, and qF(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 )DD.

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,yS and 0h<1,

d(Tx,Ty)hmax { 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 SF(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,yS, then SF(T)F(f).

Proof Define T n :F(f)F(f) by T n x=W(Tx,q, k n ) for all xF(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 n1. 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,yF(f) and 0< k n <1. If cl(T(S)) is compact for each n1, then cl( T n (S)) is compact and hence complete. By Lemma 2.1, for each n1, 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 zcl(T(M)) as m. Since {T x m } is a sequence in T(F(f)) and clT(F(f))F(f), therefore zF(f). Further, x m = T m x m =W(T x m ,q, k m )z. By the continuity of T, we obtain Tz=z=fz. Thus, SF(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 SF(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 SF(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 qF(f). If cl(T(M)) is compact, f is continuous and linear and T is f-nonexpansive on M, then MF(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,yM. Then MF(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,yM. Then MF(T)F(f).

Corollary 2.8 Let X be a strongly M-starshaped metric space, let f,T:XX be two mappings, S be a subset of X such that T(SS)S and x ˆ F(T)F(f). Suppose that P S ( x ˆ ) is nonempty closed and q-starshaped with qF(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(Tx,Ty){ 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, xSS and so TxS since T(SS)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(Tx, x ˆ )=d(Tx,T x ˆ )d(Ix,I x ˆ )=d(Ix, x ˆ )=d( x ˆ ,S).

This implies that Tx 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, letf,T:XX be two mappings, S be a subset of X such that T(SS)S and x ˆ F(T)F(f). Suppose that P S ( x ˆ ) is nonempty closed and q-starshaped with qF(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:SS,

  2. (ii)

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

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].

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/1138846111

    MATH  Article  Google 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/S0161171295000779

    MATH  MathSciNet  Article  Google Scholar 

  3. Naz A: Best approximation in strongly M -starshaped metric spaces. Rad. Mat. 2001, 10: 203–207.

    MATH  MathSciNet  Google 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.2005.11.025

    MATH  MathSciNet  Article  Google 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.2007.01.064

    MATH  MathSciNet  Article  Google 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.2007.06.008

    MATH  MathSciNet  Article  Google 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.2009.01.007

    MATH  MathSciNet  Article  Google Scholar 

  9. Meinardus G: Invarianze bei linearen approximationen. Arch. Ration. Mech. Anal. 1963, 14: 301–303.

    MATH  MathSciNet  Google 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)90036-4

    MATH  MathSciNet  Article  Google Scholar 

  11. Hicks TL, Humphries MD: A note on fixed point theorems. J. Approx. Theory 1982, 34: 221–225. 10.1016/0021-9045(82)90012-0

    MATH  MathSciNet  Article  Google 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)90101-3

    MATH  MathSciNet  Article  Google Scholar 

  13. Al-Thagafi MA: Common fixed points and best approximation. J. Approx. Theory 1996, 85: 318–323. 10.1006/jath.1996.0045

    MATH  MathSciNet  Article  Google Scholar 

  14. Habiniak L: Fixed point theorems and invariant approximations. J. Approx. Theory 1989, 56: 241–244. 10.1016/0021-9045(89)90113-5

    MATH  MathSciNet  Article  Google 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 207503

    Google 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.

    MATH  MathSciNet  Google 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.

    MATH  Google Scholar 

  19. Hussain N: Asymptotically pseudo-contractions, Banach operator pairs and best simultaneous approximations. Fixed Point Theory Appl. 2011., 2011: Article ID 812813

    Google 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.2011.05.072

    MATH  MathSciNet  Article  Google Scholar 

  21. Hussain N, Pathak HK: Subweakly biased pairs and Jungck contractions with applications. Numer. Funct. Anal. Optim. 2011, 32(10):1067–1082. 10.1080/01630563.2011.587627

    MATH  MathSciNet  Article  Google 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.

    MATH  MathSciNet  Google Scholar 

  23. Hussain N, Rhoades BE: C q -commuting maps and invariant approximations. Fixed Point Theory Appl. 2006., 2006: Article ID 24543

    Google 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/s10114-007-0935-7

    MATH  MathSciNet  Article  Google Scholar 

  25. Khan AR, Akbar F: Common fixed points from best simultaneous approximations. Taiwan. J. Math. 2009, 13(5):1379–1386.

    MATH  MathSciNet  Google 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.2007.08.051

    MATH  MathSciNet  Article  Google 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.

    MATH  MathSciNet  Article  Google 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 23582

    Google Scholar 

  29. Latif A: A result on best approximation in p -normed spaces. Arch. Math. 2001, 37: 71–75.

    MATH  MathSciNet  Google 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.2007.08.047

    MATH  MathSciNet  Article  Google Scholar 

  31. Moosaei M: Fixed point theorems in convex metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 164

    Google Scholar 

  32. Espínola R, Hussain N: Common fixed points for multimaps in metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 204981

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marwan A Kutbi.

Additional information

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The author has read and approved the final manuscript.

Rights and permissions

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

Reprints and Permissions

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

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

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