# Common fixed point and invariant approximation results

- Marwan A Kutbi
^{1}Email author

**2013**:135

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

© Kutbi; licensee Springer 2013

**Received: **25 November 2012

**Accepted: **7 May 2013

**Published: **27 May 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.

### Keywords

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

## 1 Introduction and preliminaries

*X*be a metric space with metric

*d*, $M\subset X$ and $J=[0,1]$. The space

*X*is called

- (1)
*M*-starshaped [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
*M*-starshaped [2, 3] if it is*M*-starshaped 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)

*X*-starshaped; - (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 *M*-starshaped metric space *X* is called *q*-starshaped 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 Al-Thagafi [2], Guay *et al.* [5] and Takahashi [1].

*T*is called

- (5)
*I*-nonexpansive on*D*if $d(Tx,Ty)\le d(Ix,Iy)$ for all $x,y\in D$; - (6)
*I*-contraction on*D*if there exists $k\in [0,1)$ such that $d(Tx,Ty)\le kd(Ix,Iy)$ for all $x,y\in D$.

*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 self-maps of a metric space *X* is called a Banach operator pair if the set $Fix(T)$ is *I*-invariant, namely $I(Fix(T))\subseteq Fix(T)$. 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 *S*-approximants 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 self*-*maps 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* *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* $D\cup \{u\}$, *then* ${P}_{M}(u)\cap F(T)\cap F(I)\ne \mathrm{\varnothing}$.

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* $(X,d)$ *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 self*-

*maps 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*

*M*-

*starshaped metric space*

*X*

*and let*

*T*,

*f*

*be self*-

*maps of*

*S*.

*Suppose that*$F(f)$

*is*

*q*-

*starshaped*, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(S))$

*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

*q*-starshaped 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* *M*-*starshaped metric space* *X* *and let* *T*, *f* *be self*-*maps of* *S*. *Suppose that* $F(f)$ *is* *q*-*starshaped*, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(S))$ *is compact*, *T* *is continuous on* *S* *and* *T* *is* *f*-*nonexpansive 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* *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*, $\mathit{cl}(T(S))$ *is compact*, *T* *is continuous on* *S* *and* *T* *satisfies* (2.2) *or* *T* *is* *f*-*nonexpansive 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* *q*-*starshaped subset of a normed space* *X* *and let* *T* *and* *f* *be self*-*maps 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* *f*-*nonexpansive 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 self*-*maps of* *M*. *Suppose that* $F(f)$ *is* *q*-*starshaped*, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(M))$ *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 self*-*maps of* *M*. *Suppose that* $F(f)$ *is* *q*-*starshaped 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*

*M*-

*starshaped 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*

*q*-

*starshaped 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* *M*-*starshaped 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* *q*-*starshaped 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* *f*-*nonexpansive 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

*M*-starshaped 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 2.2-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].

## Declarations

### 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.

## Authors’ Affiliations

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