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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 Mstarshaped 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, M\subset X and J=[0,1]. The space X is called

(1)
Mstarshaped [1] if there exists a continuous mapping W:X\times M\times J\to X satisfying
d(x,W(y,q,\lambda ))\le \lambda d(x,y)+(1\lambda )d(x,q)for all x,y\in X, q\in M and all \lambda \in J;

(2)
strongly Mstarshaped [2, 3] if it is Mstarshaped and satisfies the property (I), that is,
d(W(x,q,\lambda ),W(y,q,\lambda ))\le \lambda d(x,y)for all x,y\in X, q\in M and all \lambda \in J;

(3)
(strongly) convex if it is (strongly) Xstarshaped;

(4)
starshaped if it is \{q\}starshaped for some q\in 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,\lambda )=\lambda x+(1\lambda )q for all x,q\in X and all \lambda \in J. More generally, if X is a linear space with a translation invariant metric satisfying d(\lambda x+(1\lambda )y,0)\le \lambda d(x,0)+(1\lambda )d(y,0), then X is a strongly convex metric space. A subset D of an Mstarshaped metric space X is called qstarshaped if there exists q\in D\cap M such that W(x,q,\lambda )\in D for all x\in D and all \lambda \in J. For details, we refer the reader to AlThagafi [2], Guay et al. [5] and Takahashi [1].
Let I,T:X\to X be two mappings and D\subset X. Then T is called

(5)
Inonexpansive on D if d(Tx,Ty)\le d(Ix,Iy) for all x,y\in D;

(6)
Icontraction on D if there exists k\in [0,1) such that d(Tx,Ty)\le kd(Ix,Iy) for all x,y\in D.
A point x\in 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\in 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 selfmaps of a metric space X is called a Banach operator pair if the set Fix(T) is Iinvariant, namely I(Fix(T))\subseteq Fix(T). Obviously, a commuting pair (I,T) is a Banach operator pair but not conversely in general, see [6–8].
Let S\subset X and \stackrel{\u02c6}{x}\in X. Then {P}_{S}(\stackrel{\u02c6}{x})=\{x\in S:d(x,\stackrel{\u02c6}{x})=d(\stackrel{\u02c6}{x},S)\} is called the set of best Sapproximants to \stackrel{\u02c6}{x}, where d(\stackrel{\u02c6}{x},S)=inf\{d(\stackrel{\u02c6}{x},y):y\in S\} and {C}_{S}^{I}(\stackrel{\u02c6}{x})=\{x\in S:Ix\in {P}_{S}(\stackrel{\u02c6}{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 normed space X, let M be a nonempty subset of X, T(M)\subset M and u\in F(T). If {P}_{M}(u) is nonempty compact and starshaped, then {P}_{M}(u)\cap F(T)\ne \mathrm{\varnothing}.
Hicks and Humphries [11] found that Singh’s results remain true if T(M)\subset M is replaced by T(\partial M)\subset 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 selfmaps of a normed space X with u\in F(I)\cap F(T), M\subset X with T(\partial M)\subset M, and q\in F(I). If D={P}_{M}(u) is compact and qstarshaped, I(D)=D, I is continuous and linear on D, I and T are commuting on D and T is Inonexpansive on D\cup \{u\}, then {P}_{M}(u)\cap F(T)\cap F(I)\ne \mathrm{\varnothing}.
Invariant approximation results for commuting maps due to AlThagafi [13] extended and generalized Theorems 1.11.2 and the works of [11, 14, 15]. AlThagafi results were further extended by [7, 8, 16–26] to Rsubweakly commuting, pointwise Rsubweakly 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 Mstarshaped metric spaces. As application, invariant approximation results for this class of maps are derived. Our results extend and unify the work of AlThagafi [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 Mstarshaped metric space (X,d) and \stackrel{\u02c6}{x}\in X. Then {P}_{D}(\stackrel{\u02c6}{x})\subset \partial D\cap 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 selfmaps of S. If F(f) is nonempty, \mathit{clT}(F(f))\subseteq F(f), \mathit{cl}(T(M)) is complete, and T and f satisfy for all x,y\in S and 0\le h<1,
then S\cap F(T)\cap F(f) is a singleton.
Theorem 2.2 Let S be a nonempty subset of a strongly Mstarshaped metric space X and let T, f be selfmaps of S. Suppose that F(f) is qstarshaped, \mathit{clT}(F(f))\subseteq F(f), \mathit{cl}(T(S)) is compact, T is continuous on S and
for all x,y\in S, then S\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Proof Define {T}_{n}:F(f)\to F(f) by {T}_{n}x=W(Tx,q,{k}_{n}) for all x\in F(f) and a fixed sequence of real numbers {k}_{n} (0<{k}_{n}<1) converging to 1. Since F(f) is qstarshaped and \mathit{clT}(F(f))\subseteq F(f), therefore {\mathit{clT}}_{n}(F(f))\subseteq F(f) for each n\ge 1. Also, by (2.2),
for each x,y\in F(f) and 0<{k}_{n}<1. If \mathit{cl}(T(S)) is compact for each n\ge 1, then \mathit{cl}({T}_{n}(S)) is compact and hence complete. By Lemma 2.1, for each n\ge 1, there exists {x}_{n}\in F(f) such that {x}_{n}=f{x}_{n}={T}_{n}{x}_{n}. The compactness of \mathit{cl}(T(M)) implies that there exists a subsequence \{T{x}_{m}\} of \{T{x}_{n}\} such that T{x}_{m}\to z\in \mathit{cl}(T(M)) as m\to \mathrm{\infty}. Since \{T{x}_{m}\} is a sequence in T(F(f)) and \mathit{clT}(F(f))\subseteq F(f), therefore z\in F(f). Further, {x}_{m}={T}_{m}{x}_{m}=W(T{x}_{m},q,{k}_{m})\to z. By the continuity of T, we obtain Tz=z=fz. Thus, S\cap F(T)\cap F(f)\ne \mathrm{\varnothing}. □
Corollary 2.3 Let S be a nonempty subset of a strongly Mstarshaped metric space X and let T, f be selfmaps of S. Suppose that F(f) is qstarshaped, \mathit{clT}(F(f))\subseteq F(f), \mathit{cl}(T(S)) is compact, T is continuous on S and T is fnonexpansive on S, then S\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Corollary 2.4 Let S be a nonempty subset of a strongly Mstarshaped metric space X and let T, f be selfmaps of S. Suppose that F(f) is closed and qstarshaped, (T,f) is a Banach operator pair, \mathit{cl}(T(S)) is compact, T is continuous on S and T satisfies (2.2) or T is fnonexpansive on S, then S\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Corollary 2.5 ([13], Theorem 2.1)
Let M be a nonempty closed and qstarshaped subset of a normed space X and let T and f be selfmaps of M such that T(M)\subseteq f(M). Suppose that T commutes with f and q\in F(f). If \mathit{cl}(T(M)) is compact, f is continuous and linear and T is fnonexpansive on M, then M\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Corollary 2.6 (([30], Theorem 3.3))
Let M be a nonempty subset of a normed space X and let T and f be selfmaps of M. Suppose that F(f) is qstarshaped, \mathit{clT}(F(f))\subseteq F(f), \mathit{cl}(T(M)) is compact, T is continuous on M and (2.2) holds for all x,y\in M. Then M\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Corollary 2.7 ([7], Theorem 2.11)
Let M be a nonempty subset of a normed space X and let T, f be selfmaps of M. Suppose that F(f) is qstarshaped and closed \mathit{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\in M. Then M\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Corollary 2.8 Let X be a strongly Mstarshaped metric space, let f,T:X\to X be two mappings, S be a subset of X such that T(\partial S\cap S)\subset S and \stackrel{\u02c6}{x}\in F(T)\cap F(f). Suppose that {P}_{S}(\stackrel{\u02c6}{x}) is nonempty closed and qstarshaped with q\in F(f)\cap M and \mathit{cl}(T({P}_{S}(\stackrel{\u02c6}{x}))) is compact and f({P}_{S}(\stackrel{\u02c6}{x}))={P}_{S}(\stackrel{\u02c6}{x}). If T is continuous, \mathit{clT}(F(f))\subseteq F(f) and satisfies, for all x\in {P}_{S}(\stackrel{\u02c6}{x})\cup \{\stackrel{\u02c6}{x}\},
then {P}_{S}(\stackrel{\u02c6}{x})\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Proof Let x\in {P}_{S}(\stackrel{\u02c6}{x}). Then by Lemma 1.3, x\in \partial S\cap S and so Tx\in S since T(\partial S\cap S)\subset S. As T satisfies (2.3) on {P}_{S}(\stackrel{\u02c6}{x})\cup \{\stackrel{\u02c6}{x}\} and I({P}_{S}(\stackrel{\u02c6}{x}))={P}_{S}(\stackrel{\u02c6}{x}), we have
This implies that Tx\in {P}_{S}(\stackrel{\u02c6}{x}). Thus T({P}_{S}(\stackrel{\u02c6}{x}))\subset {P}_{S}(\stackrel{\u02c6}{x})=f({P}_{S}(\stackrel{\u02c6}{x})). Now Theorem 2.2 implies that {P}_{S}(\stackrel{\u02c6}{x})\cap F(T)\cap F(f)\ne \mathrm{\varnothing}. □
Theorem 2.9 Let X be a strongly Mstarshaped metric space, letf,T:X\to X be two mappings, S be a subset of X such that T(\partial S\cap S)\subset S and \stackrel{\u02c6}{x}\in F(T)\cap F(f). Suppose that {P}_{S}(\stackrel{\u02c6}{x}) is nonempty closed and qstarshaped with q\in F(f)\cap M and \mathit{cl}(T({P}_{S}(\stackrel{\u02c6}{x}))) is compact and f({P}_{S}(\stackrel{\u02c6}{x}))={P}_{S}(\stackrel{\u02c6}{x}). If T is continuous, \mathit{clT}(F(f))\subseteq F(f) and T is fnonexpansive on {P}_{S}(\stackrel{\u02c6}{x})\cup \{\stackrel{\u02c6}{x}\}, then {P}_{S}(\stackrel{\u02c6}{x})\cap F(T)\cap F(f)\ne \mathrm{\varnothing}.
Remark 2.10 A subset S of a strongly Mstarshaped metric space X is said to have the property (N) w.r.t. T [22, 28] if

(i)
T:S\to S,

(ii)
W(Tx,q,{k}_{n})\in S for some q\in S\cap M and a fixed sequence of real numbers {k}_{n} (0<{k}_{n}<1) converging to 1 and for each x\in S.
All results of the paper (Theorem 2.2Theorem 2.9) remain valid provided f is assumed to be surjective and qstarshapedness 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 Mstarshaped 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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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/10.1186/168718122013135
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DOI: https://doi.org/10.1186/168718122013135
Keywords
 common fixed point
 Banach operator pair
 strongly Mstarshaped metric space
 invariant approximation