- Research
- Open Access

# Some common fixed-point and invariant approximation results with generalized almost contractions

- Savita Rathee
^{1}and - Anil Kumar
^{1}Email author

**2014**:23

https://doi.org/10.1186/1687-1812-2014-23

© Rathee and Kumar; licensee Springer. 2014

**Received:**7 July 2013**Accepted:**5 January 2014**Published:**24 January 2014

## Abstract

In this paper, the concept of a generalized almost $(f,g)$-contraction is introduced and we establish some common fixed-point results for the noncommuting generalized almost $(f,g)$-contraction in the setup of metric spaces and normed linear spaces, where the set of fixed points of *f* and *g* need not be starshaped. As applications, invariant approximation results are proved. Supporting examples are also given.

## Keywords

- best approximation
- Banach operator pair
- generalized almost contraction
- property (N)
- jointly continuous contractive family

## 1 Introduction

The classical Banach contraction principle is a very popular tool for solving problems in nonlinear analysis. It has various applications to operator theory, variational analysis, and approximation theory, so it has been extended in many ways (see, *e.g.*, [1–30]).

In 2004, Berinde [1] defined the notion of a weak contraction mapping, which is more general than a contraction mapping. However, in [2] Berinde renamed it as an almost contraction, which is more appropriate.

**Definition 1.1**Let ($X,d$) be a complete metric space. A map $T:X\to X$ is called an almost contraction if there exist a constant $\delta \in (0,1)$ and some $L\ge 0$ such that

Berinde [1] proved some fixed-point theorems for almost contractions in a complete metric space which generalized the results of Kannan [3], Chatterjea [4], and Zamfirescu [5].

In 2008, Babu *et al*. [6] defined the class of mappings satisfying ‘condition (B)’ as follows.

**Definition 1.2**Let $(X,d)$ be a metric space. A map $T:X\to X$ is said to satisfy ‘condition (B)’ if there exist a constant $\delta \in (0,1)$ and some $L\ge 0$ such that

for all $x,y\in X$.

They prove that any map *T* satisfying ‘condition (B)’ has a unique fixed point in complete metric spaces. They also discuss quasi-contraction, almost contraction, and the class of mappings that satisfy ‘condition (B)’ in detail.

Afterwards Berinde [7] generalized the above definition and proved the following fixed-point result.

**Theorem 1.3**

*Let*$(X,d)$

*be a complete metric space and let*$T:X\to X$

*be a mapping for which there exist*$\delta \in (0,1)$

*and some*$L\ge 0$

*such that for all*$x,y\in X$

*where*

*Then* *T* *has a unique fixed point*.

The contractive condition (1.3) is termed as generalized almost contraction.

Recently, Abbas and Ilić in [15] introduced the following definition.

**Definition 1.4**Let

*T*and

*f*be two self-maps of a metric space $(X,d)$. A map

*T*is called a generalized almost

*f*-contraction if there exist $\delta \in (0,1)$ and some $L\ge 0$ such that

If $f=\text{identity map}$, then condition (1.3) can be obtained as particular case of condition (1.4). However, in [15] Abbas and Ilić obtained various common fixed-point and invariant approximation results for such mappings under the assumption of weak compatibility of maps.

Recently, Chen and Li [10] introduced the class of Banach operator pairs, as a new class of noncommuting mappings and obtained some common fixed-point and invariant approximation results for this class of maps. This class of noncommuting maps is different from the class of noncommuting maps (*viz.* *R*-subcommuting, *R*-sub-weakly commuting, ${C}_{q}$-commuting, compatible, weakly compatible *etc.*) studied in [11–13, 15, 17–19, 27–29]. So, it has been further studied by various authors (see, *e.g.*, [16, 21, 22, 24]).

In this article, we introduce the class of generalized almost $(f,g)$-contraction and consequently establish some common fixed-point results for the noncommuting generalized almost $(f,g)$-contraction in the framework of metric spaces and normed linear spaces, where the set of fixed points of *f* and *g* need not be starshaped. As an application, invariant approximation results are proved. The proved results generalize and extend the corresponding results of Chen and Li [10], Al-Thagafi and Shahzad [16], Akbar *et al*. [22], Chandok and Narang [24], Al-Thagafi [25] and Jungck and Sessa [26], Shahzad [28] to the class of generalized almost $(f,g)$-contractions.

## 2 Preliminaries

First, we introduce some well-known notations and definitions that will be needed in the sequel.

*M*be a subset of

*X*and

*f*,

*T*be self-maps of

*M*. A point $x\in M$ is a coincidence point (common fixed point) of

*f*and

*T*if $fx=Tx$ ($fx=Tx=x$). The set of coincidence points of

*f*and

*T*is denoted by $C(f,T)$ and the set of fixed points of

*f*is denoted by $F(f)$. The pair $\{f,T\}$ is called

- (1)
commuting if $Tfx=fTx$ for all $x\in M$,

- (2)
compatible [8] if ${lim}_{n\to \mathrm{\infty}}d(Tf{x}_{n},fT{x}_{n})=0$ whenever $\{{x}_{n}\}$ is a sequence in

*M*such that ${lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}T{x}_{n}=t$ for some $t\in M$, - (3)
weakly compatible [9] if $Tfx=fTx$ for all $x\in C(f,T)$,

- (4)
a Banach operator pair [10] if the set $F(f)$ is

*T*-invariant, namely $T(F(f))\subseteq F(f)$.

Obviously, a commuting pair $(T,f)$ is a Banach operator pair but not conversely. If $(T,f)$ is a Banach operator pair, then $(f,T)$ need not be Banach operator pair (see [10]).

Let *M* be a subset of a normed space $(X,\parallel \cdot \parallel )$. The set ${B}_{M}(p)=\{x\in M:\parallel x-p\parallel =dist(p,M)\}$ is called the set of best approximants to $p\in X$ out of *M*, where $dist(p,M)=inf\{\parallel y-p\parallel :p\in M\}$. We denote by ℕ and $cl(M)$ ($wcl(M)$) the set of positive integers and the closure (weak closure) of a set *M* in *X*, respectively.

*M*is said to be (a)

*q*-starshaped if there exists $q\in M$ such that the line segment $[q,x]=\{(1-k)q+kx:0\le k\le 1\}$ joining

*q*to

*x*is contained in

*M*for all $x\in M$; (b) convex if $kx+(1-k)y\in M$ for all $x,y\in M$. The map

*f*defined on a set

*M*is called

*M*is

*q*-starshaped with $q\in F(f)$ and is both

*T*- and

*f*-invariant. Then

*T*and

*f*are called

- (1)
${C}_{q}$-commuting [11] if $fTx=Tfx$ for all $x\in {C}_{q}(f,T)$, where ${C}_{q}(f,T)=\bigcup \{C(f,{T}_{k}):0\le k\le 1\}$ where ${T}_{k}(x)=(1-k)q+kTx$,

- (2)
*R*-subcommuting on*M*[12] if, for all $x\in M$, there exists a real number $R>0$ such that $\parallel Tfx-fTx\parallel \le \frac{R}{k}\parallel kTx+(1-k)q-fx\parallel $, $0<k\le 1$, - (3)
*R*-sub-weakly commuting on*M*[13] if, for all $x\in M$, there exists a real number $R>0$ such that $\parallel Tfx-fTx\parallel \le Rdist(fx,[q,Tx])$.

*X*is said to satisfy Opial’s condition if, whenever $\{{x}_{n}\}$ is a sequence in

*X*such that $\{{x}_{n}\}$ converges weakly to $x\in X$, the inequality

holds for all $y\ne x$. A Hilbert space and the space ${l}_{p}$ ($1<p<\mathrm{\infty}$) satisfy Opial’s condition. The map $T:M\to X$ is said to be demiclosed at zero if, whenever $\{{x}_{n}\}$ is a sequence in *M* such that $\{{x}_{n}\}$ converges weakly to $x\in M$ and $\{T{x}_{n}\}$ converges to 0, then $Tx=0$.

The following important extension of the concept of starshapedness was defined by Naimpally *et al*. [14] and has been studied by many authors.

**Definition 2.1**A subset

*M*of a linear space

*X*is said to have property (N) with respect to

*T*if

- (1)
$T:M\to M$,

- (2)
$(1-{k}_{n})q+{k}_{n}Tx\in M$, for some $q\in M$ and a fixed sequence of real numbers ${k}_{n}$ ($0<{k}_{n}<1$) converging to 1 and for each $x\in M$.

It is to be noted that each *T*-invariant *q*-starshaped set has property (N) but converse does not hold in general. This is shown by the following example.

**Example 2.2** Let $X=\mathfrak{R}$ be the set of real numbers and $M=\{1/n,\text{where}n\text{is a natural}\phantom{\rule{0.25em}{0ex}}\text{number}\}$ be endowed with the usual norm. Define $Tx=1$ for each $x\in M$. Then clearly *M* is not *q*-starshaped but has property (N) with respect to *T*, for $q=1$, ${k}_{n}=1-1/n$.

## 3 Main results

First we introduce the notion of a generalized almost $(f,g)$-contraction.

**Definition 3.1**Let $(X,d)$ be a metric space and

*f*,

*g*be self-maps of

*X*. A mapping $T:X\to X$ is said to be a generalized almost $(f,g)$-contraction if there exist $\delta \in (0,1)$ and some $L\ge 0$ such that

If $g=f$, then Definition 1.4 is a particular case of Definition 3.1. If $g=f=I$ (identity operator), then equation (1.3) can be obtained as a special case of equation (3.1).

Here we observe that if *T* satisfies ‘condition (B)’ then *T* is a generalized almost contraction but its converse need not be true. This is shown by the following example.

**Example 3.2**Let $X=[0,\mathrm{\infty})$ be endowed with the Euclidean metric $d(x,y)=|x-y|$. We define a mapping $T:X\to X$ by

Then *T* is a generalized almost contraction with $\delta =\frac{2}{3}$ and $L=0$. But *T* does not satisfy ‘condition (B)’ at $x=\frac{3}{4}$, $y=1$ for any $\delta \in (0,1)$ and $L\ge 0$.

In (3.1) if $L=0$, then *T* is called a generalized $(f,g)$-contraction. Obviously, a generalized $(f,g)$-contraction implies a generalized almost $(f,g)$-contraction, but the converse is not true in general.

**Example 3.3**Let $X=\{0,1,2\}$ with the usual metric and $f,g:X\to X$ be given by $f(x)=g(x)=1$ for all $x\in X$. Also define a mapping $T:X\to X$ as

Then *T* is a generalized almost $(f,g)$-contraction with any $\delta \in (0,1)$ and $L\ge 2$. But *T* is not a generalized $(f,g)$-contraction at $x=0$, $y=1$ or $x=1$, $y=2$ for any $\delta \in (0,1)$.

The following lemma is a particular case of the main theorem of Abbas and Ilić [15].

**Lemma 3.4** *Let* *M* *be a nonempty subset of a metric space* $(X,d)$, *and* *T* *be a self*-*map of* *M*. *Assume that* $cl(T(M))\subseteq M$, $cl(T(M))$ *is complete*, *and* *T* *is a generalized almost contraction*. *Then* $M\cap F(T)$ *is singleton*.

Now, we start with the following common fixed-point result, which will be used in sequel.

**Theorem 3.5** *Let* *M* *be a nonempty subset of a metric space* $(X,d)$, *and* *T*, *f* *and* *g* *be self*-*maps of* *M*. *Assume that* $F(f)\cap F(g)$ *is nonempty*, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$, $cl(T(M))$ *is complete*, *and* *T* *is a generalized almost* $(f,g)$-*contraction*. *Then* $M\cap F(T)\cap F(f)\cap F(g)$ *is singleton*.

*Proof*The completeness of $cl(T(M))$ implies that of $cl(T(F(f)\cap F(g)))$. Further, by a generalized almost $(f,g)$-contraction of

*T*, for all $x,y\in F(f)\cap F(g)$, we have

Hence *T* is a generalized almost contraction mapping on $F(f)\cap F(g)$ and $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$. By Lemma 3.4, *T* has a unique fixed point *z* in $F(f)\cap F(g)$ and consequently $M\cap F(T)\cap F(f)\cap F(g)$ is singleton. □

**Corollary 3.6** *Let* *M* *be a nonempty subset of a metric space* $(X,d)$, *and* *T*, *f* *and* *g* *be self*-*maps of* *M* *such that* $(T,f)$ *and* $(T,g)$ *are Banach operator pairs on* *M*. *Assume that* $cl(T(M))$ *is complete*, *T* *is a generalized almost* $(f,g)$-*contraction and* $F(f)\cap F(g)$ *is nonempty and closed*. *Then* $M\cap F(T)\cap F(f)\cap F(g)$ *is singleton*.

In Theorem 3.5 if we take $L=0$, then we easily obtain the following result, which improves and extends Lemma 3.1 of Chen and Li [10] and Theorem 2.2 of Al-Thagafi and Shahzad [16].

**Corollary 3.7** *Let* *M* *be a nonempty subset of a metric space* $(X,d)$, *and* *T*, *f*, *and* *g* *be self*-*maps on* *M*. *Assume that* $F(f)\cap F(g)$ *is nonempty*, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$, $cl(T(M))$ *is complete*, *and* *T* *is a generalized* $(f,g)$-*contraction*. *Then* $M\cap F(T)\cap F(f)\cap F(g)$ *is singleton*.

**Remark 3.8** By comparing Theorem 2.1 of Shahzad [17] with Corollary 3.7 (when $g=f$), their assumptions that *M* is closed, $T(M)\subseteq f(M)$, *T* is continuous and $(T,f)$ is *R*-weakly commuting pair on *M* are replaced with ‘$F(f)$ is nonempty, $cl(T(F(f)))\subseteq F(f)$’.

**Theorem 3.9**

*Let*

*M*

*be a nonempty subset of a normed*(

*respectively*,

*Banach*)

*space*

*X*

*and*

*T*,

*f*,

*and*

*g*

*be self*-

*maps of*

*M*.

*If*$F(f)\cap F(g)$

*has the property*(N)

*with respect to*

*T*, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (

*respectively*, $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$),

*and there exists a constant*$L\ge 0$

*such that*

*where*

*and*

*then* $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $, *provided* $cl(T(M))$ *is compact* (*respectively*, $wcl(T(M))$ *is weakly compact*) *and* *T* *is continuous* (*respectively*, $I-T$ *is demiclosed at* 0, *where* *I* *stands for identity map*).

*Proof*As $T(F(f)\cap F(g))\subseteq F(f)\cap F(g)$ and $F(f)\cap F(g)$ has the property (N) with respect to

*T*, for each $n\in \mathbb{N}$, we can define ${T}_{n}:F(f)\cap F(g)\to F(f)\cap F(g)$ by ${T}_{n}x=(1-{k}_{n})q+{k}_{n}Tx$ for all $x\in F(f)\cap F(g)$ and a fixed sequence of real numbers ${k}_{n}$ ($0<{k}_{n}<1$) converging to 1. Since $F(f)\cap F(g)$ has the property (N) with respect to

*T*, and $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (respectively, $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$), we have $cl({T}_{n}(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (respectively, $wcl({T}_{n}(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$) for each $n\in \mathbb{N}$. Also, by the inequality (3.2),

for all $x,y\in F(f)\cap F(g)$, ${L}_{n}:={k}_{n}L$, and $0<{k}_{n}<1$. Thus, for each $n\in \mathbb{N}$, ${T}_{n}$ is a generalized $(f,g)$-almost contraction.

*M*such that ${x}_{n}=f({x}_{n})=g({x}_{n})={T}_{n}({x}_{n})$. The compactness of $cl(T(M))$ implies that there exists a subsequence $\{T{x}_{m}\}$ of $\{T{x}_{n}\}$ such that $T{x}_{m}\to z\in cl(T(M))$. Since $\{T{x}_{m}\}$ is a sequence in $T(F(f)\cap F(g))$ and $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$, we have $z\in F(f)\cap F(g)$. Moreover,

As *T* is continuous on *M*, we have $Tz=z$. Thus $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

Next, the weak compactness of $wcl(T(M))$ implies that $wcl({T}_{n}(M))$ is weakly compact and hence complete due to completeness of *X*. From Theorem 3.5, for each $n\ge 1$, there is a unique ${x}_{n}$ in *M* such that ${x}_{n}=f({x}_{n})=g({x}_{n})={T}_{n}({x}_{n})$. The weak compactness of $wcl(T(M))$ implies that there is a subsequence $\{T{x}_{m}\}$ of $\{T{x}_{n}\}$ such that $T{x}_{m}$ converges weakly to $z\in wcl(T(M))$. Since $\{T{x}_{m}\}$ is a sequence in $T(F(f)\cap F(g))$ and $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$, therefore $z\in F(f)\cap F(g)$. Also we have $(I-T){x}_{m}\to 0$ as $m\to \mathrm{\infty}$. Further, demiclosedness of $I-T$ at 0 implies $z=Tz$, thus $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $. □

**Corollary 3.10** *Let* *M* *be a nonempty subset of a normed* (*respectively*, *Banach*) *space* *X* *and* *T*, *f*, *and* *g* *be self*-*maps of* *M*. *If* $F(f)\cap F(g)$ *has the property* (N) *with respect to* *T* *and is closed* (*respectively*, *weakly closed*), $(T,f)$ *and* $(T,g)$ *are Banach operator pairs and satisfy* (3.2) *for all* $x,y\in M$. *Then* $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $, *provided* $cl(T(M))$ *is compact* (*respectively*, $wcl(T(M))$ *is weakly compact*) *and* *T* *is continuous* (*respectively*, $I-T$ *is demiclosed at* 0, *where* *I* *stands for the identity map*).

**Remark 3.11** (1) By comparing Theorem 2.2 of Shahzad [17] with the first case of Theorem 3.9 (when $g=f$, $L=0$), their assumptions ‘$q\in F(f)$, *M* is closed and *q*-starshaped, *f* is linear and continuous on *M*, $T(M)\subseteq f(M)$ and $(T,f)$ is *R*-sub-weakly commuting pair on *M*’ are replaced with ‘*M* is a nonempty subset, $F(f)$ has the property (N) with respect to *T*, $cl(T(F(f)))\subseteq F(f)$’.

(2) By comparing Theorem 2.2(i) of Hussain and Jungck [18] with the first case of Theorem 3.9 (when $L=0$), their assumptions ‘*M* is complete and *q*-starshaped, *f* and *g* are continuous and affine on *M*, $T(M)\subseteq f(M)\cap g(M)$, $q\in F(f)\cap F(g)$, and $(T,f)$ and $(T,g)$ are *R*-sub-weakly commuting pair on *M*’ are replaced with ‘$F(f)\cap F(g)$ has the property (N) with respect to *T*, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$’.

(3) By comparing Theorem 2.2(ii) of Hussain and Jungck [18] with the second case of Theorem 3.9 (when $L=0$), their assumptions ‘*M* is weakly compact and *q*-starshaped, *f* and *g* are affine and continuous on *M*, $T(M)\subseteq f(M)\cap g(M)$, $q\in F(f)\cap F(g)$, and $(T,f)$ and $(T,g)$ are *R*-sub-weakly commuting pair on *M*, and $f-T$ is demiclosed at 0’ are replaced with ‘$wcl(T(M))$ is weakly compact, $F(f)\cap F(g)$ has the property (N) with respect to *T*, $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ and $I-T$ is demiclosed at 0’.

**Remark 3.12**If the contractive condition (3.2) in Theorem 3.9 is replaced with the stronger contractive condition

then continuity of *T* can be relaxed in the first case of Theorem 3.9.

*Proof*The proof will be similar to the first case of Theorem 3.9. To prove $Tz=z$, instead of continuity of

*T*, using (3.3) we have

This is possible only if $T{x}_{m}\to Tz$ as $m\to \mathrm{\infty}$, which implies $Tz=z$. □

Let $C={B}_{M}(p)\cap {C}_{M}^{f,g}(p)$, where ${C}_{M}^{f,g}(p)=\{x\in M:fx\in {B}_{M}(p),gx\in {B}_{M}(p)\}$.

**Corollary 3.13** *Let* *X* *be a normed* (*respectively*, *Banach*) *space and let* *T*, *f*, *and* *g* *be self*-*maps of* *X*. *If* $p\in X$ *and* $D\subseteq C$, ${D}_{0}:=D\cap F(f)\cap F(g)$ *has the property* (N) *with respect to* *T*, $cl(T({D}_{0}))\subseteq {D}_{0}$ (*respectively*, $wcl(T({D}_{0}))\subseteq {D}_{0}$), $cl(T(D))$ *is compact* (*respectively*, $wcl(T(D))$ *is weakly compact*), *T* *is continuous on* *D* (*respectively*, $I-T$ *is demiclosed at* 0, *where* *I* *stands for identity map*) *and* (3.2) *holds for all* $x,y\in D$, *then* ${B}_{M}(p)\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

**Corollary 3.14** *Let* *X* *be a normed* (*respectively*, *Banach*) *space and let* *T*, *f*, *and* *g* *be self*-*maps of* *X*. *If* $p\in X$ *and* $D\subseteq {B}_{M}(p)$, ${D}_{0}:=D\cap F(f)\cap F(g)$ *has the property* (N) *with respect to* *T*, $cl(T({D}_{0}))\subseteq {D}_{0}$ (*respectively*, $wcl(T({D}_{0}))\subseteq {D}_{0}$), $cl(T(D))$ *is compact* (*respectively*, $wcl(T(D))$ *is weakly compact*), *T* *is continuous on* *D* (*respectively*, $I-T$ *is demiclosed at* 0, *where* *I* *stands for the identity map*) *and* (3.2) *holds for all* $x,y\in D$, *then* ${B}_{M}(p)\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

**Remark 3.15** Corollaries 3.13 and 3.14 improve and develop Theorems 2.8-2.11 of Hussain and Jungck [18] and Theorems 3.1-3.4 of Song [19] to the non-starshaped domain.

Denote by ${\mathcal{L}}_{0}$ the class of closed convex subsets of *X* containing 0. For $M\in {\mathcal{L}}_{0}$, we define ${M}_{p}=\{x\in M:\parallel x\parallel \le 2\parallel p\parallel \}$. Clearly ${B}_{M}(p)\subseteq {M}_{p}\in {\mathcal{L}}_{0}$.

The following invariant approximation result constitutes an extension of Theorem 2.6 of Al-Thagafi and Shahzad [16] and Corollary 2.10 of [29] to a non-starshaped domain.

**Theorem 3.16** *Let* *X* *be a normed* (*respectively*, *Banach*) *space and* $T,f,g:X\to X$. *If* $p\in X$ *and* $M\in {\mathcal{L}}_{0}$ *such that* $T({M}_{p})\subseteq M$, $cl(T({M}_{p}))$ *is compact* (*respectively*, $wcl(T({M}_{p}))$ *is weakly compact*), *and* $\parallel Tx-p\parallel \le \parallel x-p\parallel $ *for all* $x\in {M}_{p}$, *then* ${B}_{M}(p)$ *is nonempty*, *closed*, *and convex with* $T({B}_{M}(p))\subseteq {B}_{M}(p)$. *If*, *in addition*, *D* *is a subset of* ${B}_{M}(p)$, ${D}_{0}:=D\cap F(f)\cap F(g)$ *has the property* (N) *with respect to* *T*, $cl(T({D}_{0}))\subseteq {D}_{0}$ (*respectively*, $wcl(T({D}_{0}))\subseteq {D}_{0}$), *T* *is continuous on* *D* (*respectively*, $I-T$ *is demiclosed at* 0, *where* *I* *stands for the identity map*) *and* (3.2) *holds for all* $x,y\in D$, *then* ${B}_{M}(p)\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

*Proof*We may assume that $p\notin M$. If $y\in M\mathrm{\setminus}{M}_{p}$, then $\parallel y\parallel >2\parallel p\parallel $ and, so

Thus $dist(p,{M}_{p})=dist(p,M)$. Assume that $cl(T({M}_{p}))$ is compact, then by the continuity of the norm there exists $z\in cl(T({M}_{p}))$ such that $\parallel z-p\parallel =dist(p,clT({M}_{p}))$.

for all $x\in {M}_{p}$. It follows that $\parallel z-p\parallel =dist(p,M)$. Thus ${B}_{M}(p)$ is nonempty, closed, and convex with $T({B}_{M}(p))\subseteq {B}_{M}(p)$. The compactness of $cl(T({M}_{p}))$ (respectively, weak compactness of $wcl(T({M}_{p}))$) implies that $cl(T(D))$ is compact (respectively, $wcl(T(D))$ is weakly compact). Then by Corollary 3.14, ${B}_{M}(p)\cap F(T)\cap F(f)\cap F(g)\ne \varphi $. □

Now, we present some non-trivial examples in support of Theorem 3.9.

**Example 3.17**Let $X=\mathfrak{R}$ be the set of real numbers with the usual norm and $M=[0,1)$. We define mappings $f,g,T:M\to M$ by

and $T(x)=\frac{2}{3}$, for $0\le x<1$.

Here we observe that $F(f)\cap F(g)=\{0,\frac{2}{3}\}$, $cl(T(F(f)\cap F(g)))=\{\frac{2}{3}\}\subseteq F(f)\cap F(g)$ and $cl(T(M))=\{\frac{2}{3}\}$ is compact. Clearly $F(f)\cap F(g)$ is not starshaped but has property (N) with respect to T, for $q=\frac{2}{3}$ and ${k}_{n}=1-1/n$. Further, the mappings *T*, *f*, and *g* satisfy the contractive condition (3.2) and also *T* is continuous. Hence all the conditions of the first case of Theorem 3.9 are satisfied and consequently *T*, *f*, and *g* have a common fixed point, $x=\frac{2}{3}$.

**Remark 3.18** In Example 3.17, it is interesting to note that Theorem 2.19 of Hussain and Cho [21], and Corollary 3.10 of Akbar *et al*. [22] cannot apply, since $F(f)\cap F(g)$ is not *q*-starshaped.

**Example 3.19**Let $X=\mathfrak{R}$ be the set of real numbers with the usual norm and $M=[0,1]$. Define $f,g,T:M\to M$ by

Clearly $F(f)\cap F(g)=\{x,x\text{is rational in}M\}$ has property (N) with respect to *T*, for $q=0$, ${k}_{n}=1-1/n$. Further, $cl(T(F(f)\cap F(g)))=\{0,1\}\subseteq F(f)\cap F(g)$, $cl(T(M))=\{0,1\}$ is compact and *T*, *f*, and *g* satisfy the contractive condition (3.2). Hence all the conditions of the first case of Theorem 3.9 are satisfied except the continuity of *T*. Note that $F(T)\cap F(f)\cap F(g)=\varphi $.

**Remark 3.20** It is to be noted that the maps *T*, *f*, and *g* given in Example 3.19 do not satisfy the contractive condition (3.3) at the point $x=\frac{1}{2}$, $y=1$.

## 4 Results with joint contractive family

Dotson [23] proved some results concerning the existence of fixed points of nonexpansive mappings on a certain class of non-convex sets. For proving these results, he extends the concept of starshapedness by introducing the following class of non-convex set.

*M*be a subset of a normed space

*X*and $\mathrm{\Gamma}=\{{h}_{x}:x\in M\}$ be a family of functions from $[0,1]$ to

*M*such that ${h}_{x}(1)=x$ for each $x\in M$. The family Γ is said to be contractive if there exists a function $\phi :(0,1)\to (0,1)$ such that for all $x,y\in M$ and all $t\in (0,1)$, we have

Such a family Γ is said to be jointly continuous (jointly weakly continuous) if $t\to {t}_{0}$ in $[0,1]$ and $x\to {x}_{0}$ ($x\to {x}_{0}$ weakly) in *M*; then ${h}_{x}(t)\to {h}_{{x}_{0}}(t)$ (${h}_{x}(t)\to {h}_{{x}_{0}}(t)$ weakly) in *M*.

We observe that if *M* is *q*-starshaped subset of a normed linear space *X* and ${h}_{x}(t)=(1-t)q+tx$, for each $x\in M$, $q\in M$ and $t\in [0,1]$, then Γ is a contractive jointly continuous and jointly weakly continuous family with $\phi (t)=t$. Thus the class of subsets of *X* with the property of contractiveness and joint continuity contains the class of starshaped sets which in turns contains the class of convex sets.

We shall denote ${Y}_{q}^{Tx}=\{{h}_{Tx}(k):0\le k\le 1\}$ where $q={h}_{Tx}(0)$.

The following results properly contain Theorems 3.2 and 3.3 of [10], Theorems 1 and 2 of [24] and improves Theorem 2.2 of [25], Theorem 6 of [26].

**Theorem 4.1**

*Let*

*M*

*be a nonempty subset of a normed*(

*respectively*,

*Banach*)

*space*

*X*

*and*

*T*,

*f*

*and*

*g*

*be self*-

*maps of*

*M*.

*Suppose*$F(f)\cap F(g)$

*is nonempty and has a contractive*,

*jointly continuous*(

*respectively*,

*jointly weakly continuous*)

*family of functions*$\mathrm{\Gamma}=\{{h}_{x}:x\in F(f)\cap F(g)\}$, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (

*respectively*, $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$),

*and there exists a constant*$L\ge 0$

*such that*

*for all*$x,y\in M$,

*where*

*and*

*Then* $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $, *provided* $cl(T(M))$ *is compact* (*respectively*, $wcl(T(M))$ *is weakly compact*) *and* *T* *is continuous* (*respectively*, *T* *is weakly continuous*).

*Proof* For each natural number *n*, let ${k}_{n}=\frac{n}{n+1}$. Define ${T}_{n}:F(f)\cap F(g)\to F(f)\cap F(g)$ by ${T}_{n}(x)={h}_{Tx}({k}_{n})$ for all $x\in F(f)\cap F(g)$. Since $F(f)\cap F(g)$ has a contractive family and $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (respectively, $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$), so for each $n\in \mathbb{N}$, $cl({T}_{n}(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (respectively, $wcl({T}_{n}(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$).

Thus, for each $n\in \mathbb{N}$, ${T}_{n}$ is a generalized almost $(f,g)$-contraction.

By the continuity of *T*, we obtain $z=T(z)$. Thus, $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

*X*. From Theorem 3.5 for each $n\ge 1$, there exists a unique ${x}_{n}\in F(f)\cap F(g)$ such that ${x}_{n}=f({x}_{n})=g({x}_{n})={T}_{n}({x}_{n})$. The weak compactness of $wcl(T(M))$ implies that there is a subsequence $\{T{x}_{m}\}$ of $\{T{x}_{n}\}$ such that $T{x}_{m}$ converges weakly to $z\in wcl(T(M))$ as $m\to \mathrm{\infty}$. Since $\{T{x}_{m}\}$ is a sequence in $T(F(f)\cap F(g))$ and $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$, we have $z\in F(f)\cap F(g)$. By the joint weak continuity of the family we obtain

Since the weak topology is Hausdorff, by weak continuity of *T*, we have $z=T(z)$. Thus, $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $. □

**Remark 4.2** By comparing Theorem 2.2(i) of Chandok and Narang [27] with the first case of Theorem 4.1 (when $L=0$), their assumptions ‘*M* is complete and has a contractive jointly continuous family Γ with $g({h}_{x}(k))={h}_{gx}(k)$ and $f({h}_{x}(k))={h}_{fx}(k)$ for $k\in (0,1)$, $cl(T(M))\subseteq f(M)\cap g(M)$, the pairs $(T,f)$ and $(T,g)$ are ${C}_{q}$-commuting and *f*, *g* are continuous on *M*’ are replaced with ‘*M* is nonempty subset, $F(f)\cap F(g)$ is nonempty and has a contractive jointly continuous family Γ, and $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$’.

**Corollary 4.3**

*Let*

*M*

*be a nonempty subset of a normed*(

*respectively*,

*Banach*)

*space*

*X*

*and*

*T*,

*f*,

*and*

*g*

*be self*-

*maps of*

*M*.

*Suppose*$F(f)\cap F(g)$

*is*

*q*-

*starshaped*, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$ (

*respectively*, $wcl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$),

*and there exists a constant*$L\ge 0$

*such that*

*for all*$x,y\in M$,

*where*

*and*

*Then* $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $, *provided* $cl(T(M))$ *is compact* (*respectively*, $wcl(T(M))$ *is weakly compact*) *and* *T* *is continuous* (*respectively*, *T* *is weakly continuous*).

**Remark 4.4** (1) By comparing Theorem 2.3(i) of Abbas and Ilić [15] with the first case of Corollary 4.3 (when $g=f$), their assumptions ‘*M* is *q*-starshaped, $cl(T(M))\subseteq f(M)$, *f* and *T* are weakly compatible on *M*’ are replaced with ‘$F(f)$ is *q*-starshaped, $cl(T(F(f)))\subseteq F(f)$’.

(2) By comparing Theorem 2.3(ii) of Abbas and Ilić [15] with the second case of Corollary 4.3 (when $g=f$), their assumptions ‘*M* is *q*-starshaped, $cl(T(M))\subseteq f(M)$, *f* and *T* are weakly compatible on *M*, *f* is weakly continuous and $f-T$ is demiclosed at 0’ are replaced with ‘$F(f)$ is *q*-starshaped, $cl(T(F(f)))\subseteq F(f)$ and *T* is weakly continuous’.

(3) By comparing Theorem 2.4 of Song [19] with the first case of Corollary 4.3 (when $L=0$), their assumptions ‘*M* is *q*-starshaped, $cl(T(M))\subseteq f(M)\cap g(M)$, the pairs $(T,f)$ and $(T,g)$ are ${C}_{q}$-commuting, *f* and *g* are *q*-affine and continuous on *M*’ are replaced with ‘$F(f)\cap F(g)$ is *q*-starshaped, $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$’.

**Corollary 4.5** *Let* *M* *be a nonempty subset of a normed* (*respectively*, *Banach*) *space* *X* *and* *T*, *f*, *and* *g* *be self*-*maps of* *M*. *If* *M* *has a contractive jointly continuous* (*respectively*, *jointly weakly continuous*) *family* $\mathrm{\Gamma}=\{{h}_{x}:x\in M\}$ *such that* $g({h}_{x}(k))={h}_{gx}(k)$ *and* $f({h}_{x}(k))={h}_{fx}(k)$ *for all* $x\in M$, $k\in [0,1]$. *Suppose* $F(f)\cap F(g)$ *is nonempty*, *closed* (*respectively*, *weakly closed*), $cl(T(M))$ *is compact* (*respectively*, $wcl(T(M))$ *is weakly compact*), *T* *is continuous* (*respectively*, *weakly continuous*), $(T,f)$ *and* $(T,g)$ *are Banach operator pair on* *M* *and satisfy* (4.1). *Then* $M\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

*Proof*For each natural number

*n*, define ${T}_{n}:M\to M$ by ${T}_{n}(x)={h}_{Tx}({k}_{n})$, for all $x\in M$. Clearly, for each $n\ge 1$, ${T}_{n}$ is a self-map on

*M*. Since $(T,f)$ is Banach operator pair on

*M*, for each $x\in F(f)$, we have $Tx\in F(f)$. Consider

This implies that ${T}_{n}x\in F(f)$ for each $x\in F(f)$. Thus for each $n\in \mathbb{N}$, $({T}_{n},f)$ is a Banach operator pair on *M*. Similarly, for each $n\in \mathbb{N}$, $({T}_{n},g)$ is a Banach operator on *M*. Now the result follows from Theorem 4.1. □

**Corollary 4.6** *Let* *X* *be a normed* (*respectively*, *Banach*) *space and let* *T*, *f*, *and* *g* *be self*-*maps of* *X*. *If* $p\in X$ *and* $D\subseteq C$, ${D}_{0}:=D\cap F(f)\cap F(g)$ *is nonempty*, *has a contractive jointly continuous* (*respectively*, *jointly weakly continuous*) *family of functions* $\mathrm{\Gamma}=\{{h}_{x}:x\in {D}_{0}\}$, $cl(T({D}_{0}))\subseteq {D}_{0}$ (*respectively*, $wcl(T({D}_{0}))\subseteq {D}_{0}$), $cl(T(D))$ *is compact* (*respectively*, $wcl(T(D))$ *is weakly compact*), *T* *is continuous on* *D* (*respectively*, *T* *is weakly continuous*) *and* (4.1) *holds for all* $x,y\in D$, *then* ${B}_{M}(p)\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

**Corollary 4.7** *Let* *X* *be a normed* (*respectively*, *Banach*) *space and* *T*, *f*, *and* *g* *be self*-*maps of* *X*. *If* $p\in X$ *and* $D\subseteq {B}_{M}(p)$, ${D}_{0}:=D\cap F(f)\cap F(g)$ *is nonempty*, *has a contractive jointly continuous* (*respectively*, *jointly weakly continuous*) *family of* $\mathrm{\Gamma}=\{{h}_{x}:x\in {D}_{0}\}$, $cl(T({D}_{0}))\subseteq {D}_{0}$ (*respectively*, $wcl(T({D}_{0}))\subseteq {D}_{0}$), $cl(T(D))$ *is compact* (*respectively*, $wcl(T(D))$ *is weakly compact*), *T* *is continuous on* *D* (*respectively*, *T* *is weakly continuous*) *and* (4.1) *holds for all* $x,y\in D$, *then* ${B}_{M}(p)\cap F(T)\cap F(f)\cap F(g)\ne \varphi $.

**Remark 4.8** (1) Theorems 4.1 and 4.2 of Chen and Li [10], Theorems 3 and 4 of Chandok and Narang [24] are particular cases of Corollaries 4.6 and 4.7.

(2) By Proposition 2.2 of Chen and Li [10], it can be concluded that Corollary 4.5 extends and generalizes Corollary 2.1 of Shahzad [28].

Now we present two examples in support of Theorem 4.1 and Theorem 3.5, respectively.

**Example 4.9**Let $X=\mathfrak{R}$ be the set of real numbers with the usual norm and $M=[0,1]$. Assume $T(x)=\frac{1}{2}$, for every

*x*in

*M*and define $f,g:M\to M$ by

We observe that the family Γ is contractive jointly continuous for $\phi (t)={t}^{2}$, $t\in (0,1)$. Thus all the conditions of Theorem 4.1 are satisfied. Consequently *T*, *f*, and *g* have a common fixed point. Here it is seen that $x=\frac{1}{2}$ is the common fixed point of *T*, *f*, and *g*.

**Remark 4.10** (1) Theorem 2.2(i) of Chandok and Narang [27] cannot apply to Example 4.9, since *f* is not continuous.

(2) It is interesting to note that the results of Akbar *et al*. [22] cannot apply to Example 4.9, since $F(f)\cap F(g)$ is not *q*-starshaped.

**Example 4.11**Let $X=M=\{\alpha ,\beta ,\gamma ,\delta \}$ and let $d:X\times X\to \mathfrak{R}$ be given as

Clearly $F(f)\cap F(g)=\{\beta \}$ and $cl(T(F(f)\cap F(g)))=\{\beta \}\subseteq F(f)\cap F(g)$. Further *T* is a generalized almost $(f,g)$-contraction for $\delta =\frac{19}{20}$ and $L=0$. Hence, all the conditions of Theorem 3.5 are satisfied. Consequently *T*, *f*, and *g* have a unique common fixed point. Here it is seen that $x=\beta $ is the unique common fixed point of *T*, *f*, and *g*.

**Remark 4.12** (1) In Example 4.11, $f(M)=\{\alpha ,\beta \}$, $g(M)=\{\alpha ,\beta ,\gamma ,\delta \}$ and $T(M)=\{\beta ,\delta \}$, therefore $clT(M)$ is not contained in $f(M)\cap g(M)$. Hence Theorem 2.1 of Song [19] cannot apply to Example 4.11.

(2) In Example 4.11, if we take $g(x)=f(x)=\{\begin{array}{ll}\beta ,& x\ne \gamma ,\\ \alpha ,& x=\gamma ,\end{array}$ then *T* and *f* does not satisfy the contractive condition of Lemma 3.1 of [10] and Theorem 2.2 of [16] at $x=\gamma $, $y=\alpha $. Hence Lemma 3.1 of [10] and Theorem 2.2 of [16] cannot apply to Example 4.11.

**Remark 4.13** (1) Example 3.3 satisfies all the conditions of Theorem 3.5 except the condition $cl(T(F(f)\cap F(g)))\subseteq F(f)\cap F(g)$. Note that $F(T)\cap F(f)\cap F(g)\ne \varphi $.

## Declarations

### Acknowledgements

The author would like to thank the referees for their valuable suggestions, which helped to improve the presentation of the paper.

## Authors’ Affiliations

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