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# Common fixed point and invariant approximation results

Fixed Point Theory and Applications20132013:135

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

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

## 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, $M\subset X$ and $J=\left[0,1\right]$. The space X is called
1. (1)
M-starshaped  if there exists a continuous mapping $W:X×M×J\to X$ satisfying
$d\left(x,W\left(y,q,\lambda \right)\right)\le \lambda d\left(x,y\right)+\left(1-\lambda \right)d\left(x,q\right)$

for all $x,y\in X$, $q\in M$ and all $\lambda \in J$;

2. (2)
strongly M-starshaped [2, 3] if it is M-starshaped and satisfies the property $\left(I\right)$, that is,
$d\left(W\left(x,q,\lambda \right),W\left(y,q,\lambda \right)\right)\le \lambda d\left(x,y\right)$

for all $x,y\in X$, $q\in M$ and all $\lambda \in J$;

3. (3)

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

4. (4)

starshaped if it is $\left\{q\right\}$-starshaped for some $q\in X$.

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 $W\left(x,q,\lambda \right)=\lambda x+\left(1-\lambda \right)q$ 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 $d\left(\lambda x+\left(1-\lambda \right)y,0\right)\le \lambda d\left(x,0\right)+\left(1-\lambda \right)d\left(y,0\right)$, 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 $W\left(x,q,\lambda \right)\in D$ for all $x\in D$ and all $\lambda \in J$. For details, we refer the reader to Al-Thagafi , Guay et al.  and Takahashi .

Let $I,T:X\to X$ be two mappings and $D\subset X$. Then T is called
1. (5)

I-nonexpansive on D if $d\left(Tx,Ty\right)\le d\left(Ix,Iy\right)$ for all $x,y\in D$;

2. (6)

I-contraction on D if there exists $k\in \left[0,1\right)$ such that $d\left(Tx,Ty\right)\le kd\left(Ix,Iy\right)$ 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 $C\left(I,T\right)$. The mappings I and T are called
1. (7)

commuting on D if $ITx=TIx$ for all $x\in D$;

2. (8)

weakly compatible if they commute at their coincidence points, i.e., if $ITx=TIx$ whenever $Ix=Tx$.

The ordered pair $\left(I,T\right)$ of two self-maps of a metric space X is called a Banach operator pair if the set $Fix\left(T\right)$ is I-invariant, namely $I\left(Fix\left(T\right)\right)\subseteq Fix\left(T\right)$. Obviously, a commuting pair $\left(I,T\right)$ is a Banach operator pair but not conversely in general, see .

Let $S\subset X$ and $\stackrel{ˆ}{x}\in X$. Then ${P}_{S}\left(\stackrel{ˆ}{x}\right)=\left\{x\in S:d\left(x,\stackrel{ˆ}{x}\right)=d\left(\stackrel{ˆ}{x},S\right)\right\}$ is called the set of best S-approximants to $\stackrel{ˆ}{x}$, where $d\left(\stackrel{ˆ}{x},S\right)=inf\left\{d\left(\stackrel{ˆ}{x},y\right):y\in S\right\}$ and ${C}_{S}^{I}\left(\stackrel{ˆ}{x}\right)=\left\{x\in S:Ix\in {P}_{S}\left(\stackrel{ˆ}{x}\right)\right\}$.

In 1963, Meinardus  employed the Schauder fixed point theorem to prove a result regarding invariant approximation. In 1979, Singh  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, $T\left(M\right)\subset M$ and $u\in F\left(T\right)$. If ${P}_{M}\left(u\right)$ is nonempty compact and starshaped, then ${P}_{M}\left(u\right)\cap F\left(T\right)\ne \mathrm{\varnothing }$.

Hicks and Humphries  found that Singh’s results remain true if $T\left(M\right)\subset M$ is replaced by $T\left(\partial M\right)\subset M$. 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 $u\in F\left(I\right)\cap F\left(T\right)$, $M\subset X$ with $T\left(\partial M\right)\subset M$, and $q\in F\left(I\right)$. If $D={P}_{M}\left(u\right)$ is compact and q-starshaped, $I\left(D\right)=D$, I is continuous and linear on D, I and T are commuting on D and T is I-nonexpansive on $D\cup \left\{u\right\}$, then ${P}_{M}\left(u\right)\cap F\left(T\right)\cap F\left(I\right)\ne \mathrm{\varnothing }$.

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, 1626] 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 $\left(X,d\right)$ and $\stackrel{ˆ}{x}\in X$. Then ${P}_{D}\left(\stackrel{ˆ}{x}\right)\subset \partial D\cap D$.

## 2 Main results

The following result will be needed (see Lemma 2.10  and Lemma 2.2 ).

Lemma 2.1 Let S be a nonempty subset of a metric space $\left(X,d\right)$, and let T, f be self-maps of S. If $F\left(f\right)$ is nonempty, $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$, $\mathit{cl}\left(T\left(M\right)\right)$ is complete, and T and f satisfy for all $x,y\in S$ and $0\le h<1$,
$d\left(Tx,Ty\right)\le hmax\left\{d\left(fx,fy\right),d\left(Tx,fx\right),d\left(Ty,fy\right),d\left(Tx,fy\right),d\left(Ty,fx\right)\right\},$
(2.1)

then $S\cap F\left(T\right)\cap F\left(f\right)$ 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 $F\left(f\right)$ is q-starshaped, $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$, $\mathit{cl}\left(T\left(S\right)\right)$ is compact, T is continuous on S and
$\begin{array}{rl}\parallel Tx-Ty\parallel \le & max\left\{\parallel fx-fy\parallel ,dist\left(fx,\left[q,Tx\right]\right),dist\left(fy,\left[q,Ty\right]\right),\\ dist\left(fy,\left[q,Tx\right]\right),dist\left(fx,\left[q,Ty\right]\right)\right\},\end{array}$
(2.2)

for all $x,y\in S$, then $S\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

Proof Define ${T}_{n}:F\left(f\right)\to F\left(f\right)$ by ${T}_{n}x=W\left(Tx,q,{k}_{n}\right)$ for all $x\in F\left(f\right)$ and a fixed sequence of real numbers ${k}_{n}$ ($0<{k}_{n}<1$) converging to 1. Since $F\left(f\right)$ is q-starshaped and $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$, therefore ${\mathit{clT}}_{n}\left(F\left(f\right)\right)\subseteq F\left(f\right)$ for each $n\ge 1$. Also, by (2.2),
$\begin{array}{rcl}d\left({T}_{n}x,{T}_{n}y\right)& =& d\left(W\left(Tx,q,{k}_{n}\right),W\left(Ty,q,{k}_{n}\right)\right)\\ =& {k}_{n}d\left(Tx,Ty\right)\\ \le & {k}_{n}max\left\{d\left(fx,fy\right),dist\left(fx,\left[q,Tx\right]\right),dist\left(fy,\left[q,Ty\right]\right),\\ dist\left(fx,\left[q,Ty\right]\right),dist\left(fy,\left[q,Tx\right]\right)\right\}\\ \le & {k}_{n}max\left\{d\left(fx,fy\right),d\left(fx,{T}_{n}x\right),d\left(fy,{T}_{n}y\right),d\left(fy,{T}_{n}x\right),d\left(fx,{T}_{n}y\right)\right\}\end{array}$

for each $x,y\in F\left(f\right)$ and $0<{k}_{n}<1$. If $\mathit{cl}\left(T\left(S\right)\right)$ is compact for each $n\ge 1$, then $\mathit{cl}\left({T}_{n}\left(S\right)\right)$ is compact and hence complete. By Lemma 2.1, for each $n\ge 1$, there exists ${x}_{n}\in F\left(f\right)$ such that ${x}_{n}=f{x}_{n}={T}_{n}{x}_{n}$. The compactness of $\mathit{cl}\left(T\left(M\right)\right)$ implies that there exists a subsequence $\left\{T{x}_{m}\right\}$ of $\left\{T{x}_{n}\right\}$ such that $T{x}_{m}\to z\in \mathit{cl}\left(T\left(M\right)\right)$ as $m\to \mathrm{\infty }$. Since $\left\{T{x}_{m}\right\}$ is a sequence in $T\left(F\left(f\right)\right)$ and $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$, therefore $z\in F\left(f\right)$. Further, ${x}_{m}={T}_{m}{x}_{m}=W\left(T{x}_{m},q,{k}_{m}\right)\to z$. By the continuity of T, we obtain $Tz=z=fz$. Thus, $S\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$. □

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 $F\left(f\right)$ is q-starshaped, $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$, $\mathit{cl}\left(T\left(S\right)\right)$ is compact, T is continuous on S and T is f-nonexpansive on S, then $S\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

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 $F\left(f\right)$ is closed and q-starshaped, $\left(T,f\right)$ is a Banach operator pair, $\mathit{cl}\left(T\left(S\right)\right)$ is compact, T is continuous on S and T satisfies (2.2) or T is f-nonexpansive on S, then $S\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

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 $T\left(M\right)\subseteq f\left(M\right)$. Suppose that T commutes with f and $q\in F\left(f\right)$. If $\mathit{cl}\left(T\left(M\right)\right)$ is compact, f is continuous and linear and T is f-nonexpansive on M, then $M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

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 $F\left(f\right)$ is q-starshaped, $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$, $\mathit{cl}\left(T\left(M\right)\right)$ is compact, T is continuous on M and (2.2) holds for all $x,y\in M$. Then $M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

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 $F\left(f\right)$ is q-starshaped and closed $\mathit{cl}\left(T\left(M\right)\right)$ is compact, T is continuous on M, $\left(T,f\right)$ is a Banach operator pair and satisfies (2.2) for all $x,y\in M$. Then $M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

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 $T\left(\partial S\cap S\right)\subset S$ and $\stackrel{ˆ}{x}\in F\left(T\right)\cap F\left(f\right)$. Suppose that ${P}_{S}\left(\stackrel{ˆ}{x}\right)$ is nonempty closed and q-starshaped with $q\in F\left(f\right)\cap M$ and $\mathit{cl}\left(T\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)\right)$ is compact and $f\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)={P}_{S}\left(\stackrel{ˆ}{x}\right)$. If T is continuous, $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$ and satisfies, for all $x\in {P}_{S}\left(\stackrel{ˆ}{x}\right)\cup \left\{\stackrel{ˆ}{x}\right\}$,
$d\left(Tx,Ty\right)\le \left\{\begin{array}{cc}d\left(fx,fu\right)\hfill & \mathit{\text{if}}y=u,\hfill \\ max\left\{d\left(fx,fy\right),dist\left(fx,\left[q,Tx\right]\right),dist\left(fy,\left[q,Ty\right]\right),\hfill \\ \phantom{\rule{1em}{0ex}}dist\left(fx,\left[q,Ty\right]\right),dist\left(fy,\left[q,Tx\right]\right)\right\}\hfill & \mathit{\text{if}}y\in {P}_{S}\left(\stackrel{ˆ}{x}\right),\hfill \end{array}$
(2.3)

then ${P}_{S}\left(\stackrel{ˆ}{x}\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

Proof Let $x\in {P}_{S}\left(\stackrel{ˆ}{x}\right)$. Then by Lemma 1.3, $x\in \partial S\cap S$ and so $Tx\in S$ since $T\left(\partial S\cap S\right)\subset S$. As T satisfies (2.3) on ${P}_{S}\left(\stackrel{ˆ}{x}\right)\cup \left\{\stackrel{ˆ}{x}\right\}$ and $I\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)={P}_{S}\left(\stackrel{ˆ}{x}\right)$, we have
$d\left(Tx,\stackrel{ˆ}{x}\right)=d\left(Tx,T\stackrel{ˆ}{x}\right)\le d\left(Ix,I\stackrel{ˆ}{x}\right)=d\left(Ix,\stackrel{ˆ}{x}\right)=d\left(\stackrel{ˆ}{x},S\right).$

This implies that $Tx\in {P}_{S}\left(\stackrel{ˆ}{x}\right)$. Thus $T\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)\subset {P}_{S}\left(\stackrel{ˆ}{x}\right)=f\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)$. Now Theorem 2.2 implies that ${P}_{S}\left(\stackrel{ˆ}{x}\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$. □

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 $T\left(\partial S\cap S\right)\subset S$ and $\stackrel{ˆ}{x}\in F\left(T\right)\cap F\left(f\right)$. Suppose that ${P}_{S}\left(\stackrel{ˆ}{x}\right)$ is nonempty closed and q-starshaped with $q\in F\left(f\right)\cap M$ and $\mathit{cl}\left(T\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)\right)$ is compact and $f\left({P}_{S}\left(\stackrel{ˆ}{x}\right)\right)={P}_{S}\left(\stackrel{ˆ}{x}\right)$. If T is continuous, $\mathit{clT}\left(F\left(f\right)\right)\subseteq F\left(f\right)$ and T is f-nonexpansive on ${P}_{S}\left(\stackrel{ˆ}{x}\right)\cup \left\{\stackrel{ˆ}{x}\right\}$, then ${P}_{S}\left(\stackrel{ˆ}{x}\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{\varnothing }$.

Remark 2.10 A subset S of a strongly M-starshaped metric space X is said to have the property $\left(N\right)$ w.r.t. T [22, 28] if
1. (i)

$T:S\to S$,

2. (ii)

$W\left(Tx,q,{k}_{n}\right)\in S$ 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 $F\left(f\right)$ is replaced by the property $\left(N\right)$ 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 .

## 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

(1)
Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

## References 