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