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Existence of common fixed point and best proximity point for generalized nonexpansive type maps in convex metric space
 Savita Rathee†^{1},
 Kusum Dhingra†^{1}Email author and
 Anil Kumar†^{2}Email author
 Received: 15 July 2016
 Accepted: 27 September 2016
 Published: 9 November 2016
Abstract
Here, we extend the notion of (E.A.) property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1–14, 2014) by introducing a new class of selfmaps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of selfmaps. Also, we guarantee the existence of common best proximity points for this class of maps satisfying generalized nonexpansive type condition. We furnish an example in support of the proved results.
Keywords
 Common property (E.A.)
 Common fixed point
 Best proximity point
 Compatible maps
 qaffine
 pqaffine
Mathematics Subject Classification
 46T99
 47H10
 54H25
Introduction and preliminaries
In 2002, Aamri and El Moutawakil (2002) obtained the notion of (E.A.) property for a single pair of selfmaps. In the recent past, Liu et al. (2005) introduced common property (E.A.) and extend the concept of (E.A.) property defined by Aamri and El Moutawakil (2002) to two pairs of selfmaps.
Definition 1
In 1970, Takahashi (1970) introduced the notion of convexity into metric space and proved several fixed point theorems for nonexpansive mappings in the context of convex metric space. Then after, Beg and Azam (1987), Fu and Huang (1991), Ciric (1993), and many others have obtained fixed point theorems in convex metric spaces. Very recently, Kumar and Rathee (2014) defined the concept of (E.A.) property in the setup of convex metric space and ensure the existence of common fixed point for a pair of maps satisfying this property by omitting the assumption that the range of one map is contained in other.
In the present work, we define the concept of common property (E.A.) in the context of convex metric space and extend the results of Kumar and Rathee (2014) to four selfmaps by utilizing this newly introduced concept. Further, we ensure the existence of common best proximity point for generalized nonexpansive type maps.
Before going to the main work, we recall some standard notations, known definitions and results which is required in the sequel. Throughout this paper, \({\mathbb {N}}\) and \({\mathbb {R}}\) denote the set of natural numbers and the set of real numbers, respectively.
Definition 2
A metric space (X, d) equipped with a convex structure is called a convex metric space.
Definition 3
A subset M of a convex metric space (X, d) is called a convex set (Takahashi 1970) 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 (Guay et al. 1982) if there exists \(q\in M\) such that \(W(x,q,\lambda ) \in M\) for all \(x\in M\) and \(\lambda \in [0,1]\).
Clearly, each convex set M is starshaped with respect to any \(q\in M\) but the converse assertion is not true. Thus, the class of starshaped set properly contains the class of convex set.
Definition 4
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 space, for details (see Guay et al. 1982; Takahashi 1970).
Definition 5
 (1)
affine (AlThagafi and Shahzad 2006; Huang and Li 1996), 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 (AlThagafi and Shahzad 2006; Kumar and Rathee 2014), if M is qstarshaped and \(I(W(x,q,\lambda ))= W(Ix,q,\lambda )\) for all \(x\in M\) and \(\lambda \in [0,1]\).
Definition 6
 (1)
commuting if \(ITx=TIx\) for all \(x \in M\).
 (2)
compatible (Jungck 1986) if \(\lim _{n\rightarrow \infty }d(ITx_m,TIx_m)=0,\) whenever \(\{x_n\}\) is a sequence in X such that \(\lim _{n\rightarrow \infty }Ix_n =\lim _{n\rightarrow \infty }Tx_n=t\in X\).
For more details about these classes, one can refer to (see Agarwal et al. 2014). In 1998, Pant (1998) defined the concept of reciprocal continuity as follows.
Definition 7
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 Imdad et al. 2011, 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
Obviously, compatible maps which satisfy (E.A.) property are subcompatible but the converse statement does not hold in general (see Rouzkard et al. 2012, Example 2.5)
Definition 9
Obviously, if the pair (I, T) satisfy (E.A.) property with respect to q, then I and T satisfy (E.A.) property but converse assertion is not necessarily true (see Kumar and Rathee 2014, Example 12).
Main results
We start to this section with following definition.
Definition 10
Remark 11
In Definition 10, if \(A=B\) and \(S=T\), then Definition 9 can be obtained as a particular case of Definition 10. Therefore the common property (E.A.) defined here extends the notion of (E.A.) property in convex metric space defined by Kumar and Rathee (2014).
The following Lemma is particular case of the Theorem 4.1 of Chauhan and Pant (2014).
Lemma 12
Now, we start with the following theorem.
Theorem 13
Proof
Now, we have to prove that \(y =z.\)
This implies that \(M \bigcap F(A) \bigcap F(B) \bigcap F(S) \bigcap F(T) \ne \phi\). \(\square\)
Corollary 14
Corollary 15
Now we present an example in support of our theorem.
Example 16
We have to check the following:
 (i):

A and B are qaffine with \(q =\frac{1}{3}\)
 (ii):

The pair (A, S) and (B, T) satisfying common property (E.A.) w.r.t. \(q = \frac{1}{3}\).
 (iii):

A, B, S and T are compatible.
Proof
(i) If \(x \in \left[ 1, \frac{1}{3}\right]\), then \(W\left( x, \frac{1}{3}, \lambda \right) = (\lambda )x +(1  \lambda )\frac{1}{3} \in \left[ 1, \frac{1}{3}\right]\).
That implies \(A\left( W\left( x, \frac{1}{3}, \lambda \right) \right) = W\left( Ax, \frac{1}{3}, \lambda \right) .\)
Now we shall prove that B is qaffine with \(q = \frac{1}{3}.\)
Proof
(ii) Clearly \(A\left(\frac{1}{3}\right) =B\left(\frac{1}{3}\right) =\frac{1}{3}.\)
Proof
(iii) Here, we shall prove that the pairs (A, S) and (B, T) are compatible.
 (i)
If \(x =y\), then
Subcase (i): if \(x= y \in \left[ 1, \frac{1}{3}\right]\), then \(d(Sx, Tx) = 0.\) So the inequality holds trivially.
Subcase (ii): if \(x =y \in \left[ \frac{1}{3}, \frac{2}{3}\right]\), thenThat implies \(d(Sx,Tx) \le d(Ax,Bx).\)$$\begin{aligned} d(Sx, Tx)&= \left \frac{x}{2} + \frac{1}{6}\frac{x+1}{4}\right \\&= \left \frac{6x+23x3}{12}\right \\&= \left \frac{3x1}{12}\right \\&= \frac{1}{4}\left x\frac{1}{3}\right \\ d(Ax,Bx)&= \left \frac{5}{3} 4x1+2x \right \\&= \left \frac{1}{3} 2x\right \\&= 2\left x\frac{1}{3}\right. \end{aligned}$$  (ii)
If \(x \ne y\), then
Subcase (i): if \(x \ne y \in [0, \frac{1}{3}]\), then \(d(Sx, Ty) = 0\). Inequality trivially holds.
Subcase (ii): if \(x \ne y \in \left[ \frac{1}{3}, \frac{2}{3}\right]\), thenThis implies \(d(Sx,Ty) \le d(Ax, By).\)$$\begin{aligned} d(Sx, Ty)&= \left \frac{x}{2} + \frac{1}{6}\frac{y+1}{4}\right \\&= \left \frac{6x+23y3}{12}\right \\&= \left \frac{6x3y1}{12}\right \\&= \frac{1}{2}\left x\frac{y}{2} \frac{1}{6}\right \\ d(Ax,By)&= \left \frac{5}{3} 4x1+2y \right \\&= \left \frac{2}{3} 4x +2y\right \\&= 4\left x\frac{y}{2}\frac{1}{6}\right. \end{aligned}$$Subcase (iii): if \(x \in \left[ 1,\frac{1}{3}\right]\) and \(y \in \left[ \frac{1}{3}, \frac{2}{3}\right]\), thenTherefore, we get \(d(Sx,Ty) \le d(Ax, By).\)$$\begin{aligned} d(Sx, Ty)&= \left \frac{1}{3} \frac{y+1}{4}\right \\&= \left \frac{43y3}{12}\right \\&= \left \frac{13y}{12}\right \\&= \frac{1}{4}\left \frac{1}{3}y\right \\ d(Ax,By)&= \left \frac{1}{3} 1+2y \right \\&= \left 2y\frac{2}{3} \right \\&= 2\left \frac{1}{3}y\right. \end{aligned}$$Subcase (iv): if \(x \in \left[ \frac{1}{3}, \frac{2}{3}\right]\) and \(y \in \left[ 1, \frac{1}{3}\right]\), thenSo, we have \(d(Sx,Ty) \le d(Ax, By).\)$$\begin{aligned} d(Sx, Ty)&= \left \frac{x}{2} \frac{1}{6}\right \\&= \frac{1}{2}\left x\frac{1}{3}\right \\ d(Ax,By)&= \left \frac{5}{3} 4x\frac{1}{3} \right \\&= 4\left \frac{1}{3}x\right. \end{aligned}$$
Remark 17
It is to be noted that, in Example 16, \(S(M) \not \subset A(M)\) and \(T(M)\not \subset B(M)\). Therefore all the existing common fixed point theorems which ensure the existence of common fixed point for the maps under the hypothesis that range of one set is contained in other are not applicable to Example 16 (see Chen and Li (2007), Rathee and Kumar (2014a, b), Shahzad (2001)).
Application to invariant approximation
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 \(P_M(p)\).
As an application of Theorem 13, we present an invariant approximation theorems.
Theorem 18
Proof
Define \(D= P_M(p)\cap C_M^{A,B}(p)\), where \(C_M^{A,B}(p)=\{x\in M : Ax\in P_M(p)\quad \text {and}\quad Bx\in P_M(p)\}\)
Theorem 19
Proof
Best proximity point
First we discuss the concept of best proximity. Let \(T:A \rightarrow B\) be a map where A and B are two nonempty subsets of a metric space (X, d) and let A and B are disjoint subsets of a metric space then the equation \(Tx=x\) might have no solution. Therefore in case of nonselfmaps we are not sure about the existence of fixed point. In such a case we try to minimize the distance d(x, Tx) and a point x for which d(x, Tx) is minimum is called a best proximity point. In the recent years there have been many interesting best proximity point theorems are proved, for example, see De la Sen et al. (2013), Eldred and Veeramani (2006), Prolla (1983), Reich (1978), and Sankar Raj (2011), Sehgal and Singh (1988). In the present section we prove a new best proximity theorem for four maps but before this we recall some definitions which are required in the sequel.
Definition 20
 (i):

A is pstarshaped set and B is qstarshaped set;
 (ii):

\(f(W(x,p, \lambda )) = W(fx,q,\lambda )\).
Definition 21
Definition 22
Let (X, d) be a convex metric space and A and B be two nonempty subsets of X such that B is qstarshaped set. A pair (f, S) of two nonselfmaps from A to B is said to be proximally commuting if for some \(\lambda \in [0,1]\) whenever \(d(x,W(Su,q,\lambda ))=d(y,fu)=d(A,B) \Longrightarrow W(Sy,q, \lambda ) = fx\).
Definition 23
Now we presents a best proximity point theorem:
Theorem 24
 (i)
Two pairs (f, S) and (g, T) satisfying common property (E.A.) w.r.t q and proximally commuting;
 (ii)
\(T(A) \subseteq f(A), S(A) \subseteq g(A), f(A_{0}) \subseteq B_{0}, g(A_{0}) \subseteq B_{0}\);
 (iii)
The pair (A, B) has Pproperty;
 (iv)
f, g, S and, T satisfying the condition \(d(Sx,Ty) \le \max \{d(fx,gy),dist(fx,[Sx,q]),dist(gy,[Ty,q]), \frac{1}{2}[dist(fx,[Ty,q])+d(gy,[Sx,q])]\};\)
 (v)
Two mappings S and T are pqaffine.
Proof
For each \(n \in N\), we define sequences \(T_{n}: A \rightarrow B\) and \(S_{n}: A \rightarrow B\) by \(T_{n}y = W(Ty,q, \lambda _{n})\) and \(S_{n}x =W(Sx,q, \lambda _{n})\) for all \(x,y \in A\) and \(\lambda _{n}\) is a sequence in (0, 1) such that \(\lambda _{n} \rightarrow 1\)
Conclusion
In this note, we defined the common property (E.A.) in the context of convex metric space that means here we assign the algebraic structure to the common property (E.A.) that is already exists in metric space. Due to this, we have been able to obtained a set of common fixed point theorems in which to ensure the existence of common fixed points the condition of range of one set is contained in other is not required. Thus, this newly introduced concept plays a great role in solving many kinds of physical sciences problems which can be recast in terms of common fixed point problems.
Notes
Declarations
Authors’ contributions
All the authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to express their gratitude to the editor and the refrees for their valuable suggestions to improve the presentation of this manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.
Authors’ Affiliations
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