0 Fixed Point Theory and Applications

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Common fixed point and invariant approximation results

Marwan A Kutbi*

Correspondence: mkutbi@yahoo.com Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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 c X and J = [0,1]. The space X is called

(1) M-starshaped [1] if there exists a continuous mapping W: X x M x J ^ X satisfying

d(x, W(y, q, X)) < Xd(x,y) + (1 - X)d(x, q)

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

(2) strongly M-starshaped [2, 3] if it is M-starshaped and satisfies the property (I), that is,

d(W(x, q, X), W(y, q, X)) < Xd(x,y)

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

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

(4) starshaped if it is {q}-starshaped for some q e 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, X)= Xx + (1- X)q for all x, q e X and all X e J. More generally, if X is a linear space with a translation invariant metric satisfying d(Xx + (1 - X)y,0) < Xd(x,0) + (1 - X)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 e D n M such that W(x, q, X) e D for all x e D and all X e J. For details, we refer the reader to Al-Thagafi [2], Guay etal. [5] and Takahashi [1]. Let I, T: X ^ X be two mappings and D c X. Then T is called

(5) I-nonexpansive on D if d(Tx, Ty) < d(Ix,Iy) for all x,y e D;

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

© 2013 Kutbi; licensee Springer. 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 originalworkis properly cited.

ringer

A point x e D 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

(7) commuting on D if ITx = TIx for all x e D;

(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)) c Fix(T). Obviously, a commuting pair (I, T) is a Banach operator pair but not conversely in general, see [6-8].

Let S c X and x e X. Then PS(x) = {x e S: d(x,X) = d(x, S)} is called the set of best S-approximants to x, where d(x,S) = inf{d(x,y) :y e S} and CS(x) = {x e S:Ix e PS(x)}.

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 normedspace X, letM be a nonempty subset ofX, T(M) c M and u e F(T). IfPM(u) is nonempty compact and starshaped, then Pm(u) n F(T) = 0.

Hicks and Humphries [11] found that Singh's results remain true if T(M) c M is replaced by T(dM) c 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 e F (I) n F(T), M c X with T(dM) c M, and q e F(I). IfD = PM(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 {u}, then Pm(u) n F(T) n F(I) = 0.

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, 8,16-26] to ^-subweakly commuting, pointwise ^-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], Hus-sain and Berinde [28], Hussain etal. [22], Naz [3], Latif [29], Sahab etal. [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) andx e X. Then PD(x) c dD n 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 ofS. IfF (f) is nonempty, clT (F (f)) c Ff), cl(T (M)) is complete, and T andf satisfy for all x, y e S and 0 < h <1,

then S П F(T) П Ff) 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 ofS. Suppose that F f) is q-starshaped, clT (F f)) ç F f), cl(T (S)) is compact, T is continuous on S and

for all x, y e S, then S n F (T) n F f) = 0.

Proof Define Tn : F (f) ^ F (f ) by Tnx = W (Tx, q, kn) for all x e F (f) and a fixed sequence of real numbers kn (0 < kn < 1) converging to 1. Since F f) is q-starshaped and clT (F f)) ç F (f), therefore clTn(F (f )) ç F (f ) for each n > 1. Also, by (2.2),

d(Tnx, Tny) = d(W(Tx,q,kn), W(Ty,q,kn)) = knd(Tx, Ty)

< kn max{d(fx,fy), distfx,[q, Tx]), distfy,[q, Ty]), distfx,[q, Ty]), distfy,[q, Tx])}

< kn ma^j d(fx,fy), d(fx, Tnx), d(fy, Tny), d(fy, Tnx), d(fx, Tny)}

for each x,y e Ff) and 0 < kn < 1. If cl(T(S)) is compact for each n > 1, then cl(Tn(S)) is compact and hence complete. By Lemma 2.1, for each n > 1, there exists xn e F(f ) such that xn = fxn = Tn xn . The compactness of cl(T(M)) implies that there exists a subsequence {Txm} of {Txn} such that Txm ^ z e cl(T(M)) as m ^œ. Since {Txm} is a sequence in T (F f)) and clT (F f)) ç F (f), therefore z e F f). Further, xm = Tmxm = W (Txm, q, km) ^ z. By the continuity of T, we obtain Tz = z = fz. Thus, S n F (T) n F (f) = 0. □

Corollary 2.3 Let S be a nonempty subset of a strongly M-starshaped metric space X and let T, f be self-maps ofS. Suppose that F (f) is q-starshaped, clT (F (f)) ç F (f), cl(T (S)) is compact, T is continuous on S and T isf-nonexpansive on S, then S n F(T) n Ff) = 0.

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 n F (T) n F (f) = 0.

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 andf be self-maps of M such that T (M) ç f (M). Suppose that

d(Tx, Ty) < hmax{d(fx,fy),d(Tx,fx),d(Ty,fy),d(Tx,fy),d(Ty,fx)},

\\Tx - Ty\\ < maxj \fx -fy\\,distfx,[q, Tx]),distfy,[q, Ty]), distfy,[q, Tx]), dist(fx,[q, Ty])},

T commutes withf and q e Ff). Ifcl(T (M)) is compact, f is continuous and linear and T isf-nonexpansive on M, then M П F(T) П F(f) = 0.

Corollary 2.6 (([30], Theorem 3.3)) Let M be a nonempty subset of a normed space X and let T andf be self-maps ofM. Suppose thatF(f) is q-starshaped, clT(Ff)) с Ff), cl(T(M)) iscompact, T is continuous on M and (2.2) holdsfor all x, y e M. ThenM П F(T) П F(f) = 0.

Corollary 2.7 ([7], Theorem 2.11) LetM be a nonempty subset of a normed space X and let T, f be self-maps ofM. Suppose thatF 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 e M. ThenMПF(T) ПF(f) = 0.

Corollary 2.8 Let X be a strongly M-starshaped metric space, letf, T : X ^ X be two mappings, S be a subset ofX such that T(dS П S) с S andx e F(T) П F(f ). Suppose that PS(x) is nonempty closed and q-starshaped with q e F(f ) П M and cl(T(PS(x))) is compact andf (PS(x)) = PS(x). IfT is continuous, clT(Ff)) с F(f ) and satisfies, for allx e PS(x) U {x},

d(Tx, Ty) <

d(fx,fu) if y = u,

max{d(fx,fy), distfx,[q, Tx]), dist(/y,[q, Ty]), (2.3)

distfx,[q, Ty]), distfy,[q, Tx])} if y e Ps(x),

then Ps(x) П F(T) П F(f) = 0.

Proof Let x e PS(x). Then by Lemma 1.3, x e дS П S and so Tx e S since T(dS П S) с S. As T satisfies (2.3) on PS(x) U {x} and I(PS(x)) = PS(x), we have

d(Tx,x) = d(Tx, Tx) < d(Ix,Ix) = d(Ix,x) = d(x, S).

This implies that Tx e PS(x). Thus T(PS(x)) с PS(x) =f (PS(x)). Now Theorem 2.2 implies that Ps(x) П F(T) П F(f)= 0. □

Theorem 2.9 LetX be a strongly M-starshaped metric space, letf, T: X ^ Xbe two mappings, S be a subset ofX such that T (д S П S) с S andx e F (T) П F (f). Suppose that PS(x) is nonempty closed and q-starshaped with q e F(f ) П M and cl(T(PS(x))) is compact and f (PS(x)) = PS (x). IfT is continuous, clT (F f)) с F (f) and T isf-nonexpansive onPS (x) U {x}, thenPS(x) П F(T) П F(f) = 0.

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

(i) T: S ^ S,

(ii) W(Tx, q, kn) e S for some q e S П M and a fixed sequence of real numbers kn (0 < kn < 1) converging to 1 and for each x e 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].

Competing interests

The author declares that he has no competing Interests.

Authors' contributions

The author has read and approved the finalmanuscript.

Acknowledgements

This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author

acknowledges with thanks DSR, KAU for financialsupport.

Received: 25 November 2012 Accepted: 7 May 2013 Published: 27 May 2013

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doi:10.1186/1687-1812-2013-135

Cite this article as: Kutbi: Common fixed point and invariant approximation results. Fixed Point Theory and Applications 2013 2013:135.

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