 Research
 Open access
 Published:
Some common fixed point and invariant approximation results for nonexpansive mappings in convex metric space
Fixed Point Theory and Applications volume 2014, Article number: 182 (2014)
Abstract
In this work, we introduce a new class of selfmaps which satisfy the (E.A.) property with respect to some q\in M, where M is qstarshaped subset of a convex metric space and common fixed point results are established for this new class of selfmaps. After that we obtain some invariant approximation results as an application. Our results represent a very strong variant of the several recent results existing in the literature. We also provide some illustrative examples in the support of proved results.
MSC:46T99, 47H10, 54H25.
1 Introduction
In 1976, Jungck [1] established some common fixed point results for a pair of commuting selfmaps in the setting of complete metric space. The first ever attempt to relax the commutativity of mappings was initiated by Sessa [2] who introduced a class of noncommuting maps called ‘namely’ weak commutativity. Further, in order to enlarge the domain of noncommuting mappings, Jungck [3] in 1986 introduced the concept of ‘compatible maps’ as a generalization of weakly commuting maps.
Definition 1 Two selfmaps I and T of a metric space (X,d) are called compatible if and only if
whenever \{{x}_{n}\} is a sequence in X such that {lim}_{n\to \mathrm{\infty}}I{x}_{n}={lim}_{n\to \mathrm{\infty}}T{x}_{n}=t for some t\in X.
In 2002, Aamri and Moutawakil [4] obtained the notion of (E.A.) property which enables us to study the existence of a common fixed points of selfmaps satisfying nonexpansive or Lipschitz type condition in the setting of noncomplete metric space.
Definition 2 Two selfmaps I and T of a metric space (X,d) are said to satisfy the (E.A.) property if there exists a sequence \{{x}_{n}\} in X such that
On the other side, in 1970 Takahashi [5] introduced the notion of convexity into the metric space, studied properties of such spaces and proved several fixed point theorems for nonexpansive mappings. Afterward Guay et al. [6], Beg and Azam [7], Fu and Huang [8], Ding [9], Ćirić et al. [10], and many others have studied fixed point theorem in convex metric spaces. In the recent past, fixed point theorems have been extensively applied to best approximation theory. Meinardus [11] was the first who employed the Schauder’s fixed point theorem to prove a result regarding invariant approximation. Later on, Brosowski [12] generalized the result of Meinardus under different settings. Further significant contribution to this area was made by a number of authors (see [13–35]). Many of them considered the pair of commuting or noncommuting mappings in the setting of normed or Banach spaces. In 1992, Beg et al. [36] proved some results on the existence of a common fixed point in the setting of a convex metric space and utilized the same to prove the best approximation results. After that, several authors studied common fixed point and invariant approximation results in the setting of convex metric space (see [36–40] and references therein).
In this work, we introduce a new class of selfmaps which satisfy the (E.A.) property with respect to some q\in M, where M is qstarshaped subset of a convex metric space and establish some common fixed point results for this class of selfmaps. After that we obtain some invariant approximation results as application. Our results represent a very strong variant of the several recent results existing in the literature.
2 Preliminaries
Firstly, we recall some useful definitions and auxiliary results that will be needed in the sequel. Throughout this paper, ℕ and ℝ denote the set of natural numbers and the set of real numbers, respectively.
Definition 3 [5]
Let (X,d) be a metric space. A continuous mapping W:X\times X\times [0,1]\to X is called a convex structure on X if, for all x,y\in X and \lambda \in [0,1], we have
for all u\in X.
A metric space (X,d) equipped with a convex structure is called a convex metric space.
Definition 4 A subset M of a convex metric space (X,d) is called a convex set if W(x,y,\lambda )\in M for all x,y\in M and \lambda \in [0,1]. The set M is said to be qstarshaped if there exists q\in M such that W(x,q,\lambda )\in M for all x\in M and \lambda \in [0,1]. A set M is called starshaped if it is qstarshaped with respect to any q\in M.
Clearly, each convex set M is starshaped but the converse assertion is not true. Thus, the class of starshaped sets properly contains the class of convex sets.
Definition 5 A convex metric space (X,d) is said to satisfy the Property (I), if for all x,y,z\in X and \lambda \in [0,1],
A normed linear space X and each of its convex subset are simple examples of convex metric spaces with W given by W(x,y,\lambda )=\lambda x+(1\lambda )y for all x,y\in X and 0\le \lambda \le 1. Also, Property (I) is always satisfied in a normed linear space. There are many convex metric spaces which are not normed linear spaces (see [5, 6]). For further information on a convex metric space, refer to [5–10, 36–42].
Definition 6 Let (X,d) be a convex metric space and M a subset of X. A mapping I:M\to M is said to be

(1)
affine, if M is convex and I(W(x,y,\lambda ))=W(Ix,Iy,\lambda ) for all x,y\in M and \lambda \in [0,1];

(2)
qaffine, if M is qstarshaped and I(W(x,q,\lambda ))=W(Ix,q,\lambda ) for all x\in M and \lambda \in [0,1].
In [43] Pant define the concept of reciprocal continuity as follows.
Definition 7 Let (X,d) be a metric space and I,T:X\to X. Then the pair (I,T) is said to be reciprocally continuous if and only if
whenever \{{x}_{n}\} is a sequence in X such that {lim}_{n\to \mathrm{\infty}}I{x}_{n}={lim}_{n\to \mathrm{\infty}}T{x}_{n}=t for some t\in X.
It is easy to see that if I and T are continuous, then the pair (I,T) is reciprocally continuous but the converse is not true in general (see [[44], Example 2.3]). Moreover, in the setting of common fixed point theorems for compatible pairs of selfmappings satisfying some contractive conditions, continuity of one of the mappings implies their reciprocal continuity.
Definition 8 [45]
A pair (I,T) of selfmaps of a metric space (X,d) is said to be subcompatible if there exists a sequence \{{x}_{n}\} such that
Obviously, compatible maps which satisfy the (E.A.) property are subcompatible but the converse statement does not hold in general (see [40], Example 2.5).
Definition 9 Let (X,d) be a metric space, M a nonempty subset of X, and I and T be selfmaps of M. A point x\in M is a coincidence point (common fixed point) of I and T if Ix=Tx (Ix=Tx=x). The set of coincidence points of I and T is denoted by C(I,T) and the set of fixed points of I and T is denoted by F(I) and F(T), respectively. The pair \{I,T\} is called:

(1)
Commuting if ITx=TIx for all x\in M.

(2)
Weakly compatible [46] if ITx=TIx for all x\in C(I,T).

(3)
Banach operator pair [24] if the set F(I) is Tinvariant, i.e. T(F(I))\subseteq F(I).
For more details about these classes, one can refer [27, 47].
Definition 10 [19]
Let M be a qstarshaped subset of convex metric space (X,d) such that q\in F(I) and is both I and Tinvariant. Then the selfmaps I and T are called Rsubweakly commuting on M if for all x\in M, there exists a real number R>0 such that d(ITx,TIx)\le Rdist(Ix,[q,Tx]), where [q,x]=\{W(x,q,\lambda ):0\le \lambda \le 1\}.
Clearly, Rsubweakly commuting maps are compatible but the converse assertion is not necessarily true (see [31], Example 15).
For a nonempty subset M of a metric space (X,d) and p\in X, an element y\in M is called a best approximation to p if d(p,y)=dist(p,M), where dist(p,M)=inf\{d(p,z):z\in M\}. The set of all best approximations to p is denoted by {B}_{M}(p).
3 Main results
We start to this section with following definition.
Definition 11 Let M be a qstarshaped subset of a convex metric space (X,d) and let I,T:M\to M with q\in F(I). The pair (I,T) is said to satisfy the (E.A.) property with respect to q if there exists a sequence \{{x}_{n}\} in M such that for all \lambda \in [0,1]
where {T}_{\lambda}x=W(Tx,q,\lambda ).
Obviously, if the pair (I,T) satisfies the (E.A.) property with respect to q, then I and T satisfy the (E.A.) property but the converse assertion is not necessarily true. This can be seen by the following simple example.
Example 12 Let X={\mathbb{R}}^{2} be equipped with the metric d:{\mathbb{R}}^{2}\times {\mathbb{R}}^{2}\to \mathbb{R} defined as
and let M=\{(x,y):x\ge 1,y\ge 1\}. Then (X,d) is a convex metric space with W(x,y,\lambda )=\lambda x+(1\lambda )y, 0\le \lambda \le 1, and M is qstarshaped with q=(2,3). Define I,T:M\to M by
Firstly, we show I and T satisfy the (E.A.) property. Take a sequence \{{z}_{n}\}=\{({x}_{n},{y}_{n})\} in M such that {z}_{n}\to (1,1), then {x}_{n}\to 1 and {y}_{n}\to 1. Thus
Now we will show that the pair (I,T) does not satisfy the (E.A.) property with respect to q=(2,3). Take \{{z}_{n}\}=\{({x}_{n},{y}_{n})\} be any sequence in M, then {lim}_{n\to \mathrm{\infty}}{T}_{0}{z}_{n}=(2,3) and hence
But, for {x}_{n}\to 2, {y}_{n}\to 3, we get
Thus, it is not possible to find a sequence \{{z}_{n}\} in M such that for each \lambda \in [0,1]
So, the pair (I,T) does not satisfy the (E.A.) property with respect to q=(2,3).
Remark 13 If M is convex subset of a convex metric space X and p is common fixed point of the selfmaps I and T of M, then the pair (I,T) satisfies the (E.A.) property with respect to p but converse is not true in general. This can be seen by the following example.
Example 14 Let X=\mathbb{R} be endowed with the usual metric and let M=[0,1]. Define I,T:M\to M by
Clearly, X is a convex metric space with W(x,y,\lambda )=\lambda x+(1\lambda )y and M is \frac{1}{2}starshaped. Take {x}_{n}=\frac{1}{4}\frac{1}{2(n+1)} for each n\ge 1.Then for each \lambda \in [0,1], we have
Hence the pair (I,T) satisfies the (E.A.) property with respect to q=\frac{1}{2}, but I and T do not have a common fixed point.
The following lemma is a particular case of Theorem 4.1 of Chauhan and Pant [48].
Lemma 15 Let I and T be selfmaps of a metric space (X,d). If the pair (I,T) is subcompatible, reciprocally continuous and satisfy
for some \lambda \in (0,1) and all x,y\in X. Then I and T have a unique common fixed point in X.
Now we prove our first result.
Theorem 16 Let M be a nonempty qstarshaped subset of a convex metric space (X,d) with Property (I) and let I and T be continuous selfmaps on M such that the pair (I,T) satisfies the (E.A.) property with respect to q. Assume that I is qaffine, cl(T(M)) is compact. If I and T are compatible and satisfy the inequality
for all x,y\in M, then M\cap F(T)\cap F(I)\ne \varphi.
Proof For each n\in \mathbb{N}, we define {T}_{n}:M\to M by
where {\lambda}_{n} is a sequence in (0,1) such that {\lambda}_{n}\to 1.
Now, we have to show that for each n\in \mathbb{N}, the pair ({T}_{n},I) is subcompatible. Since I and T satisfy the (E.A.)property with respect to q, there exists a sequence \{{x}_{m}\} in M such that for all \lambda \in [0,1]
where {T}_{\lambda}{x}_{m}=W(T{x}_{m},q,\lambda ).
Since {\lambda}_{n}\in (0,1), in the light of (3.5) and (3.6), for each n\in \mathbb{N}, we have
Thus, we have
Now, using the fact that I is qaffine and Property (I) is satisfied, we get
Since I and T are compatible, in view of (3.6), we have
Now, letting m\to \mathrm{\infty} in (3.8), we get
Hence, on account of (3.7) and (3.9), it follows that the pair ({T}_{n},I) is subcompatible for each n\in \mathbb{N}. Since I and T are continuous, for each n\in \mathbb{N}, the pair ({T}_{n},I) is reciprocally continuous. Also, by (3.4),
for each x,y\in M and 0<{\lambda}_{n}<1. By Lemma 15, for each n\in \mathbb{N}, there exists {x}_{n}\in M such that {x}_{n}=I{x}_{n}={T}_{n}{x}_{n}.
Now 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 as m\to \mathrm{\infty}. Further, it follows that
By the continuity of I and T, we obtain Iz=z=Tz. Thus, M\cap F(T)\cap F(I)\ne \varphi. □
Now we present a nontrivial example in support of Theorem 16.
Example 17 Let X=\mathbb{R} endowed with the usual metric and let M=[0,1]. Define I,T:M\to M by
Then (X,d) is a convex metric space with the convex structure W(x,y,\lambda )=\lambda x+(1\lambda )y. Firstly, we check the following:

(a)
I is q affine with q=\frac{1}{2}.

(b)
The pair (I,T) satisfies the (E.A.) property with respect to q=\frac{1}{2}.

(c)
I and T are compatible.
Proof (a) Let x\in [0,\frac{1}{2}]. Then W(x,\frac{1}{2},\lambda )=\lambda x+(1\lambda )\frac{1}{2}\in [0,\frac{1}{2}] and hence
Again, if x\in [\frac{1}{2},1], then W(x,\frac{1}{2},\lambda )\in [\frac{1}{2},1], therefore we have
So, I(W(x,\frac{1}{2},\lambda ))=W(Ix,\frac{1}{2},\lambda ) for all x\in M and hence I is qaffine with q=\frac{1}{2}. □
Proof (b) Clearly, I(q)=q for q=\frac{1}{2}. Take {x}_{n}=\frac{1}{2}\frac{1}{n+1}, n\ge 1, then for each n, {x}_{n}\in [0,\frac{1}{2}). So for each \lambda \in [0,1], we have
Thus, the pair (I,T) satisfies the (E.A.) property with respect to q=\frac{1}{2}. □
Proof (c) If \{{x}_{n}\} be a sequence in M such that {lim}_{n\to \mathrm{\infty}}I{x}_{n}={lim}_{n\to \mathrm{\infty}}T{x}_{n}=t for some t\in M, then t lies in the closure of both I(M)=[0,\frac{1}{2}] and T(M)=[\frac{1}{2},\frac{3}{4}], so t=\frac{1}{2}. Using the continuity of I and T we obtain
Hence, I and T are compatible. □
Now we will show that the inequality (3.4) holds for each x,y\in M. If x=y, then d(Tx,Ty)=0 and hence (3.4) obviously holds. Let x,y\in M with x\ne y, then we have the following cases.

(1)
If x,y\in [0,\frac{1}{2}], then d(Tx,Ty)=0 and so inequality (3.4) trivially holds.

(2)
If x,y\in [\frac{1}{2},1], then
d(Tx,Ty)=\frac{1}{2}xy<xy=d(Ix,Iy).
Thus, the inequality (3.4) holds.

(3)
If x\in [0,\frac{1}{2}] and y\in [\frac{1}{2},1], then Tx=\frac{1}{2}, Ty=\frac{y}{2}+\frac{1}{4}, and Ix=\frac{1}{2}, Iy=1y. Therefore
d(Tx,Ty)=\frac{1}{2}y\frac{1}{2}<y\frac{1}{2}=d(Ix,Iy).
Hence, the inequality (3.4) holds.

(3)
If x\in [\frac{1}{2},1] and y\in [0,\frac{1}{2}], then, due to the symmetric property of metric d, the inequality (3.4) follows from case 3.
So, for each x,y\in M, the maps I and T satisfy the inequality (3.4). Also, cl(T(M))=[\frac{1}{2},\frac{3}{4}] is compact, I and T are continuous. Thus, from the above discussion we conclude that I and T satisfy all the hypotheses of Theorem 16 and consequently, M\cap F(T)\cap F(I)\ne \varphi. Here \frac{1}{2}\in M is a common fixed point of I and T.
Remark 18 Note that, in Example 17, T(M)=[\frac{1}{2},\frac{3}{4}]\u2288[0,\frac{1}{2}]=I(M). Also, most of the common fixed point results in which the pair of maps is taken to be commuting, weakly commuting, Rsubweakly commuting, compatible, and weakly compatible guarantee the existence of a common fixed point under the hypothesis T(M)\subseteq I(M) (for example see [17–22, 25, 28–30, 36–40]). Thus, all these results are not applicable to finding the common fixed point of the maps I and T defined in Example 17.
Now, we present an example that will show if the condition ‘The pair (I,T) satisfy the (E.A.) property with respect to q’ of Theorem 16 fails to hold, then I and T may not have a common fixed point.
Example 19 Let X=[0,\mathrm{\infty}) be endowed with the usual metric and M=[0,2]. Define I,T:M\to M by
Then (X,d) is a convex metric space with W(x,y,\lambda )=\lambda x+(1\lambda )y and M is qstarshaped with q=\frac{1}{3}. Clearly, I and T are continuous and cl(T(M))=[\frac{1}{6},1] is compact. Using a routine calculation as is done in Example 17, it can easily be shown that I is qaffine with q=\frac{1}{3} and also the maps I and T satisfy inequality (3.4) for each x,y\in M. Now we show that the map I and T do not satisfy the (E.A.) property. On the contrary, assume \{{x}_{n}\} is a sequence in M such that
Then t lies in the closure of both I(M)=[\frac{1}{3},2] and T(M)=[\frac{1}{6},1], so t\in [\frac{1}{3},1]. Further, employing the definition of maps I and T with 3.14, we have
This is not true for any t\in [\frac{1}{3},1] and hence our assumption is wrong, so there does not exist any sequence \{{x}_{n}\} in M such that 3.14 holds. Thus, the maps I and T do not satisfy the (E.A.) property and consequently the pair (I,T) does not satisfy (E.A.)property with respect to q=\frac{1}{3}. Moreover, I and T are vacuously compatible and we observe I and T have no common fixed point.
Thus, if we relax the condition ‘The pair (I,T) satisfy the (E.A.) property with respect to q’ of Theorem 16, then I and T may not have a common fixed point.
The following corollaries immediately follow from Theorem 16.
Corollary 20 Let M be a nonempty qstarshaped subset of a convex metric space (X,d) with Property (I) and let I and T be continuous selfmaps on M such that the pair (I,T) satisfies the (E.A.) property with respect to q. Assume that I is qaffine, cl(T(M)) is compact. If I and T are compatible and satisfy the inequality
for all x,y\in M, then M\cap F(T)\cap F(I)\ne \varphi.
Corollary 21 Let M be a nonempty qstarshaped subset of a convex metric space (X,d) with Property (I) and let I and T be continuous selfmaps on M such that the pair (I,T) satisfies the (E.A.) property with respect to q. Assume that I is qaffine, cl(T(M)) is compact. If I and T are Rsubweakly commuting and satisfy the inequality
for all x,y\in M, then M\cap F(T)\cap F(I)\ne \varphi.
4 Invariant approximation
Now, we present some invariant approximation results as an application of Theorem 16.
Theorem 22 Let I and T be selfmaps of a convex metric space (X,d) with Property (I), p\in F(I)\cap F(T), and M be a subset of X such that T(\delta M\cap M)\subseteq M, where δM denotes the boundary of M. Suppose that {B}_{M}(p) is nonempty, qstarshaped with I({B}_{M}(p))\subset {B}_{M}(p) and I is qaffine and continuous on {B}_{M}(p). If the maps I and T are compatible, satisfy the (E.A.) property with respect to q on {B}_{M}(p), and also satisfy for all x,y\in {B}_{M}(p)\cup \{p\}
then I and T have a common fixed point in {B}_{M}(p), provided cl(T({B}_{M}(p))) is compact and T is continuous on {B}_{M}(p).
Proof Let x\in {B}_{M}(p). Then for all \lambda \in (0,1), we have
Thus, it follows that \{W(x,p,\lambda ):\lambda \in (0,1)\}\cap M=\varphi and so x\in \delta M\cap M. As T(\delta M\cap M)\subseteq M, therefore Tx\in M. Since Ix\in {B}_{M}(p) and p\in F(I)\cap F(T), on account of (4.1), we have
which shows that Tx\in {B}_{M}(p), and in all I and T are selfmaps on {B}_{M}(p). In view of Theorem 16 there exists a z\in {B}_{M}(p) such that z is a common fixed point of I and T. □
Example 23 Consider X=[0,1] equipped with the usual metric, M=(0,\frac{1}{2}] and define a mapping W:X\times X\times [0,1]\to X
Then (X,d) is a convex metric space with Property (I). Define I,T:X\to X by
Clearly, F(I)=\{\frac{1}{2},\frac{2}{3}\}, F(T)=[\frac{1}{2},\frac{2}{3}] and T(\delta M\cap M)=T(\frac{1}{2})=\frac{1}{2}\subseteq (0,\frac{1}{2}]=M. Take p=\frac{2}{3}\in F(I)\cap F(T)=\{\frac{1}{2},\frac{2}{3}\}, then {B}_{M}(p)=\{\frac{1}{2}\}. Here, we observe that {B}_{M}(p) is nonempty, q=\frac{1}{2}starshaped with I({B}_{M}(p))=\{\frac{1}{2}\}\subset {B}_{M}(p), and also I is qaffine and continuous on {B}_{M}(p).
Further, I and T are commuting on {B}_{M}(p) and hence compatible. Also, cl(T({B}_{M}(p)))=\{\frac{1}{2}\} is compact, T is continuous on {B}_{M}(p) and on account of Remark 13, the pair (I,T) satisfies the (E.A.) property with respect to q=\frac{1}{2}. Moreover, it can easily be checked that I and T satisfy inequality 4.1 for all x,y\in {B}_{M}(p)\cup \{p\}. Thus all the conditions of Theorem 22 are satisfied and consequently I and T have a common fixed point in {B}_{M}(p). Here x=\frac{1}{2} is such point.
Corollary 24 Let I and T be selfmaps of a convex metric space (X,d) with Property (I), p\in F(I)\cap F(T), and M be a subset of X such that T(\delta M\cap M)\subseteq M, where δM denotes the boundary of M. Suppose that {B}_{M}(p) is nonempty, qstarshaped with I({B}_{M}(p))\subset {B}_{M}(p) and I is qaffine and continuous on {B}_{M}(p). If the maps I and T are Rsubweakly commuting, satisfy the (E.A.) property with respect to q on {B}_{M}(p) and also satisfy for all x,y\in {B}_{M}(p)\cup \{p\}
then I and T have a common fixed point in {B}_{M}(p), provided cl(T({B}_{M}(p))) is compact and T is continuous on {B}_{M}(p).
We define D={B}_{M}(p)\cap {C}_{M}^{I}(p), where {C}_{M}^{I}(p)=\{x\in M:Ix\in {B}_{M}(p)\}.
Theorem 25 Let I and T be selfmaps of a convex metric space (X,d) with Property (I), p\in F(I)\cap F(T), and M be a subset of X such that T(\delta M\cap M)\subseteq M, where δM denotes the boundary of M. Suppose that D is nonempty, qstarshaped with I(D)\subset D and I is qaffine and nonexpansive on D. If the maps I and T are compatible, satisfy the (E.A.) property with respect to q on D, and also satisfy for all x,y\in D\cup \{p\}
then I and T have a common fixed point in {B}_{M}(p), provided cl(T(D)) is compact and T is continuous on D.
Proof Let x\in D. Then x\in {B}_{M}(p), and therefore, proceeding as in the proof of Theorem 22, we have Tx\in {B}_{M}(p). Since I is nonexpansive and p\in F(I)\cap F(T), it follows from (4.4) that
Thus ITx\in {B}_{M}(p) and so Tx\in {C}_{M}^{I}(p), which gives Tx\in D. Hence I and T are selfmaps on D. Now, in the light of Theorem 16, there exists z\in {B}_{M}(p) such that z is a common fixed point of I and T. □
Corollary 26 Let I and T be selfmaps of a convex metric space (X,d) with Property (I), p\in F(I)\cap F(T), and M be a subset of X such that T(\delta M\cap M)\subseteq M, where δM denotes the boundary of M. Suppose that D is nonempty, qstarshaped with I(D)\subset D, and I is qaffine and nonexpansive on D. If the maps I and T are Rsubweakly commuting, satisfy the (E.A.) property with respect to q on D, and also satisfy for all x,y\in D\cup \{p\}
then I and T have a common fixed point in {B}_{M}(p), provided cl(T(D)) is compact and T is continuous on D.
Let {D}_{M}^{R,I}(p)={B}_{M}(p)\cap {G}_{R,I}^{M}(p), where {G}_{M}^{R,I}(p)=\{x\in M:d(Ix,p)\le (2R+1)dist(p,M)\}.
Theorem 27 Let I and T be selfmaps of a convex metric space (X,d) with Property (I), p\in F(I)\cap F(T), and M be a subset of X such that T(\delta M\cap M)\subseteq M, where δM denotes the boundary of M. Suppose that {D}_{M}^{R,I}(p) is nonempty, qstarshaped with I({D}_{M}^{R,I}(p))\subset {D}_{M}^{R,I}(p), and I is qaffine and continuous on {D}_{M}^{R,I}(p). If the maps I and T are Rsubweakly commuting, satisfy the (E.A.) property with respect to q on {D}_{M}^{R,I}(p), and also satisfy for all x,y\in {D}_{M}^{R,I}(p)\cup \{p\}
then I and T have a common fixed point in {B}_{M}(p), provided cl(T({D}_{M}^{R,I}(p))) is compact and T is continuous on {D}_{M}^{R,I}(p).
Proof Let x\in {D}_{M}^{R,I}(p), then using an argument similar to that in Theorem 22, we have Tx\in {B}_{M}(p). Since I and T are Rsubweakly commuting and p\in F(I)\cap F(T), on account of (4.6) it follows that
Thus Tx\in {G}_{R,I}^{M}(p). Hence I and T are selfmaps on {D}_{M}^{R,I}(p). Regarding, Theorem 16, there exists z\in {B}_{M}(p) such that z is a common fixed point of I and T. □
Remark 28

(1)
The class of compatible and Rsubweakly commuting are different from the class of Banach operator pairs and so our results are different from the results of [41].

(2)
In the existing literature, common fixed point results are proved with the assumption T(M)\subseteq I(M) but here we replace it with the assumption of the (E.A.) property with respect to some q\in M.
References
Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83(41):261–263.
Sessa S: On a weak commutativity condition of mappings in fixed point consideration. Publ. Inst. Math. (Belgr.) 1982, 32: 149–153.
Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 4: 771–779.
Aamri M, El Moutawakil D: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 2002, 270: 181–188. 10.1016/S0022247X(02)000598
Takahashi WA: A convexity in metric spaces and nonexpansive mapping. I. Kodai Math. Semin. Rep. 1970, 22: 142–149. 10.2996/kmj/1138846111
Guay MD, Singh KL, Whitfield JHM: Fixed point theorems for nonexpansive mappings in convex metric spaces. Lecture Notes in Pure and Applied Mathematics 80. In Proceedings of Conference on Nonlinear Analysis. Dekker, New York; 1982:179–189.
Beg I, Azam A: Fixed point on starshaped subset of convex metric spaces. Indian J. Pure Appl. Math. 1987, 18: 594–596.
Fu JY, Huang NJ: Common fixed point theorems for weakly commuting mappings in convex metric spaces. J. Jiangxi Univ. 1991, 3: 39–43.
Ding DP: Iteration processes for nonlinear mappings in convex metric spaces. J. Math. Anal. Appl. 1998, 132: 114–122.
Ćirić L, Rakočevic V, Radenović S, Rajović M, Lazović R: Common fixed point theorems for nonself mappings in metric spaces of hyperbolic type. J. Comput. Appl. Math. 2010, 233: 2966–2974. 10.1016/j.cam.2009.11.042
Meinardus G: Invarianz bei linearen approximationen. Arch. Ration. Mech. Anal. 1963, 14: 301–303.
Brosowski B: Fixpunksatze in der approximations theorie. Mathematica 1969, 11: 195–220.
Singh SP: An application of a fixed point theorem to approximation theory. J. Approx. Theory 1979, 25: 89–90. 10.1016/00219045(79)900364
Subrahmanyan PV: An application of a fixed point theorem to best approximation. J. Approx. Theory 1977, 20: 165–172. 10.1016/00219045(77)900703
Habiniak L: Fixed point theorems and invariant approximations. J. Approx. Theory 1989, 56: 241–244. 10.1016/00219045(89)901135
Sahab SA, Khan MS, Sessa S: A result in best approximation theory. J. Approx. Theory 1988, 55: 349–351. 10.1016/00219045(88)901013
AlThagafi MA: Common fixed points and best approximation. J. Approx. Theory 1996, 85: 318–323. 10.1006/jath.1996.0045
Shahzad N: Noncommuting maps and best approximation. Rad. Mat. 2000/01, 10: 77–83.
Shahzad N: Invariant approximations and R subweakly commuting maps. J. Math. Anal. Appl. 2001, 1: 39–45.
Pathak HK, Cho YJ, Kang SM: An application of fixed point theorems in best approximation theory. Int. J. Math. Math. Sci. 1998, 21(3):467–470. 10.1155/S0161171298000659
Jungck G, Hussain N: Compatible maps and invariant approximations. J. Math. Anal. Appl. 2007, 325: 1003–1012. 10.1016/j.jmaa.2006.02.058
Shahzad N: Invariant approximation, generalized I contractions and R subweakly commuting maps. Fixed Point Theory Appl. 2005, 1: 79–86.
AlThagafi MA, Shahzad N: Banach operator pair, common fixed points, invariant approximation, and *nonexpansive multimaps. Nonlinear Anal. 2008, 69: 2733–2739. 10.1016/j.na.2007.08.047
Chen J, Li Z: Common fixed points for Banach operators in best approximations. J. Math. Anal. Appl. 2007, 336: 1466–1475. 10.1016/j.jmaa.2007.01.064
Hussain N, Jungck G:Common fixed point and invariant approximation result for noncommuting generalized(f,g)nonexpansive maps. J. Math. Anal. Appl. 2006, 2: 851–861.
Rathee S, Kumar A: Some common fixed point and invariant approximation results with generalized almost contractions. Fixed Point Theory Appl. 2014., 2014: Article ID 23
Agarwal RP, Bisht RK, Shahzad N: A comparison of various noncommuting conditions in metric fixed point theory and their applications. Fixed Point Theory Appl. 2014., 2014: Article ID 38
O’Regan D, Shahzad N: Invariant approximation for generalized I contraction. Numer. Funct. Anal. Optim. 2005, 26: 565–575. 10.1080/NFA200067306
Shahzad N: On R subweakly commuting maps and invariant approximations in Banach spaces. Georgian Math. J. 2005, 12: 157–162.
Jungck G, Sessa S: Fixed point theorems in best approximation theory. Math. Jpn. 1995, 42: 249–252.
Rathee S, Kumar A: Some common fixed point results for modified subcompatible maps and related invariant approximation results. Abstr. Appl. Anal. 2014., 2014: Article ID 505067
Rathee S, Kumar A, Tas K: Invariant approximation results via common fixed point theorems for generalized weak contractions maps. Abstr. Appl. Anal. 2014., 2014: Article ID 752107
AlThagafi MA, Shahzad N: Generalized I nonexpansive selfmaps and invariant approximations. Acta Math. Sin. Engl. Ser. 2008, 24(5):867–876. 10.1007/s101140075598x
Shahzad N, Bassindowa G: Fixed point theorems for Suzukigeneralized nonexpansive mappings with application. J. Nonlinear Convex Anal. 2012, 13(4):657–666.
Alghamdi MA, Dhompongsa S, Shahzad N: Common fixed points for Suzukigeneralized nonexpansive maps. J. Appl. Math. 2013., 2013: Article ID 206759
Beg I, Shahzad N, Iqbal M: Fixed point theorems and best approximation in convex metric space. Approx. Theory Appl 1992, 8(4):97–105.
Nashine HK, Imdad M: Common fixed point and invariant approximations for subcompatible mappings in convex metric space. Math. Commun. 2011, 16: 1–12.
Beg I, Abbas M: Common fixed points and invariant approximations in convex metric space. Soochow J. Math. 2007, 33(4):1–10.
Narang TD, Chandok S: Common fixed points and invariant approximations of R subweakly commuting maps in convex metric spaces. Ukr. Math. J. 2011, 62(10):1585–1596. 10.1007/s112530110451x
Rouzkard F, Imdad M, Nashine HK: New common fixed point theorems and invariant approximation in convex metric space. Bull. Belg. Math. Soc. Simon Stevin 2012, 19: 311–328.
Kutbi MA: Common fixed point and invariant approximation result. Fixed Point Theory Appl. 2013., 2013: Article ID 135
Abbas M, Jovanović M, Radenović S, Sretenović A, Simić S: Abstract metric spaces and approximating fixed points of a pair of contractive type mappings. J. Comput. Anal. Appl. 2011, 13(2):243–253.
Pant RP: Common fixed points for four mappings. Bull. Calcutta Math. Soc. 1998, 90: 281–286.
Imdad M, Ali J, Tanveer M: Remarks on some recent metrical common fixed point theorems. Appl. Math. Lett. 2011, 24: 1165–1169. 10.1016/j.aml.2011.01.045
Jungck G: Common fixed points for noncontinuous nonselfmaps on nonmetric spaces. Far East J. Math. Sci. 1996, 4(2):199–215.
Kadelburg Z, Radenović S, Shahzad N: A note on various classes of compatibletype pairs of mappings and common fixed point theorems. Abstr. Appl. Anal. 2013., 2013: Article ID 697151
Chauhan S, Pant BD: Fixed point theorems for compatible and subsequentially continuous mappings in Menger spaces. J. Nonlinear Sci. Appl. 2014, 7: 78–89.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kumar, A., Rathee, S. Some common fixed point and invariant approximation results for nonexpansive mappings in convex metric space. Fixed Point Theory Appl 2014, 182 (2014). https://doi.org/10.1186/168718122014182
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122014182