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Common fixed point and invariant approximation results
Fixed Point Theory and Applications volume 2013, Article number: 135 (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.
1 Introduction and preliminaries
We first review needed definitions. Let X be a metric space with metric d, and . The space X is called
-
(1)
M-starshaped [1] if there exists a continuous mapping satisfying
for all , and all ;
-
(2)
strongly M-starshaped [2, 3] if it is M-starshaped and satisfies the property , that is,
for all , and all ;
-
(3)
(strongly) convex if it is (strongly) X-starshaped;
-
(4)
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 [4]). 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 [2], Guay et al. [5] and Takahashi [1].
Let be two mappings and . Then T is called
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(5)
I-nonexpansive on D if for all ;
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(6)
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
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(7)
commuting on D if for all ;
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(8)
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 .
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, and . If is nonempty compact and starshaped, then .
Hicks and Humphries [11] found that Singh’s results remain true if is replaced by . 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 , 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 [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 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 and . Then .
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 , 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 ([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 . 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 (([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 is q-starshaped, , is compact, T is continuous on M and (2.2) holds for all . Then .
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 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 ,
then .
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 .
Remark 2.10 A subset S of a strongly M-starshaped metric space X is said to have the property w.r.t. T [22, 28] if
-
(i)
,
-
(ii)
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 [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].
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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.
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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
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DOI: https://doi.org/10.1186/1687-1812-2013-135