 Research
 Open Access
 Published:
Common best proximity points theorem for four mappings in metrictype spaces
Fixed Point Theory and Applications volume 2015, Article number: 47 (2015)
Abstract
In this article, we first give an existence and uniqueness common best proximity points theorem for four mappings in a metrictype space \((X, D, K)\) such that D is not necessarily continuous. An example is also given to support our main result. We also discuss the unique common fixed point existence result of four mappings defined on such a metric space.
Introduction and preliminary
Fixed point theory is essential for solving various equations of the form \(T x = x\) for selfmappings T defined on subsets of metric spaces or normed linear spaces. Given nonvoid subsets A and B of a metric space and a nonselfmapping \(T : A \rightarrow B\), the equation \(T x = x\) does not necessarily have a solution, which is known as a fixed point of the mapping T. However, in such conditions, it may be considered to determine an element x for which the error \(d(x,T x)\) is minimum, in which case x and Tx are in close proximity to each other. It is remarked that best proximity point theorems are relevant to this end. A best proximity point theorem provides sufficient conditions that confirm the existence of an optimal solution to the problem of globally minimizing the error \(d(x,T x)\), and hence the existence of a complete approximate solution to the equation \(T x = x\). In fact, with respect to the fact that \(d(x,T x) \geq d(A,B)\) for all x, a best proximity point theorem requires the global minimum of the error \(d(x,T x)\) to be the least possible value \(d(A,B)\). Eventually, a best proximity point theorem offers sufficient conditions for the existence of an element x, called a best proximity point of the mapping T, satisfying the condition that \(d(x,T x) = d(A,B)\). Moreover, it is interesting to observe that best proximity theorems also appear as a natural generalization of fixed point theorems, for a best proximity point reduces to a fixed point if the mapping under consideration is a selfmapping.
A study of several variants of contractions for the existence of a best proximity point can be found in [1–7]. Best proximity point theorems for multivalued mappings are available in [8–14]. Eldred et al. [15] have established a best proximity point theorem for relatively nonexpansive mappings. Further, Anuradha and Veeramani have investigated best proximity point theorems for proximal pointwise contraction mappings [16].
On the other hand, Khamsi and Hussain [17] generalized the definition of a metric and defined the metrictype as follows.
Definition 1.1
[17]
Let X be a nonempty set, \(K \geq1\) be a real number, and let the function \(D : X \times X \rightarrow\mathbb{R} \) satisfy the following properties:

(i)
\(D(x,y) = 0 \) if and only if \(x = y\);

(ii)
\(D(x, y)= D(y, x)\) for all \(x, y \in X\);

(iii)
\(D(x,z)\leq K(D(x,y) + D(y, z))\) for all \(x, y \in X\).
Then \((X, D,K)\) is called a metrictype space.
Obviously, for \(K = 1\), a metrictype space is simply a metric space.
Afterward, other authors proved fixed point theorems in metrictype space [18–20].
Given two nonempty subsets A and B of a metrictype space \((X,D,K)\), the following notions and notations are used in the sequel.
This study focuses upon resolving a more general problem as regards the existence of common best proximity points for pairs of nonselfmappings in metrictype space. As a result, the finding of this study verifies a common global minimal solution to the problem of minimizing the real valued multiobjective functions \(x \rightarrow d(x, Sx)\) and \(x \rightarrow d(x, Tx)\), which in turn gives rise to a common optimal approximate solution of the fixed point equations \(Sx = x\) and \(Tx = x\), where D is a metrictype space and the nonselfmappings \(S : A \rightarrow B\) and \(T : A \rightarrow B\) satisfy a contractionlike condition. Our best proximity point theorem generalizes a result due to Sadiq Basha [21]. Further, a common fixed point theorem for commuting selfmappings is a special case of our common best proximity point theorem. Now, we review some definitions used throughout the paper.
Definition 1.2
An element \(x\in A\) is said to be a common best proximity point of the nonselfmappings \(f_{1}, f_{2},\ldots,f_{n} : A \rightarrow B\) if it satisfies the condition that
Definition 1.3
The mappings \(S:A\rightarrow B\) and \(T:A\rightarrow B\) are said to be commute proximally if they satisfy the condition that
Definition 1.4
If \(A_{0}\ne\varnothing\) then the pair \((A,B)\) is said to have Pproperty if and only if for any \(x_{1}, x_{2}\in A_{0}\) and \(y_{2}, y_{2}\in B_{0}\)
Main result
We begin our study with the following definition.
Definition 2.1
Let A and B be two nonempty subsets of a metrictype space \((X,D,K)\). Nonselfmappings \(f,g,S,T: A\rightarrow B\) are said to satisfy a Kcontractive condition if there exists a nonnegative number \(\alpha< {\frac{1}{K}}\) such that for each \(x,y \in A\)
Theorem 2.2
Let A and B be nonempty subsets of a complete metrictype space \((X,D,K)\). Moreover, assume that \(A_{0}\) and \(B_{0}\) are nonempty and \(A_{0}\) is closed. Let the nonselfmappings \(f,g,S,T: A\rightarrow B\) satisfy the following conditions:

(i)
\(\{f,S\}\) and \(\{g,T\}\) commute proximally;

(ii)
the pair \((A,B)\) has the Pproperty;

(iii)
f, g, S and T are continuous;

(iv)
f, g, S, and T satisfy the Kcontractive condition;

(v)
\(f(A_{0})\subseteq T(A_{0})\), \(g(A_{0})\subseteq S(A_{0})\) and \(g(A_{0})\subseteq B_{0}\), \(f(A_{0})\subseteq B_{0}\).
Then f, g, S, and T have a unique common best proximity point.
Proof
Fix \(x_{0}\) in \(A_{0}\), since \(f(A_{0})\subseteq T(A_{0})\), then there exists an element \(x_{1}\) in \(A_{0}\) such that \(f(x_{0})=T(x_{1})\). Similarly, a point \(x_{2} \in A_{0} \) can be chosen such that \(g(x_{1}) = S(x_{2})\). Continuing this process, we obtain a sequence \(\{x_{n}\}\in A_{0}\) such that \(f(x_{2n}) = T(x_{2n+1})\) and \(g(x_{2n+1}) = S(x_{2n+2})\).
Since \(f(A_{0})\subseteq B_{0}\) and \(g(A_{0})\subseteq B_{0}\), there exists \(\{u_{n}\} \in A_{0}\) such that
Since the pair \((A,B)\) has the Pproperty, by (1) we have
thus (note that \(\frac{1}{2K}D(u_{2n1}, u_{2n+1}) \leq\frac {1}{2}[D(u_{2n1},u_{2n})+D(u_{2n},u_{2n+1})] \) and \(\alpha< 1\))
Similarly
thus (note that \(\frac{1}{2K}D(u_{2n+2}, u_{2n}) \leq\frac {1}{2}[D(u_{2n+2},u_{2n+1})+D(u_{2n+1},u_{2n})] \) and \(\alpha< 1\))
Therefore, by (2) and (3) we have
and then
Let \(m, n \in\mathbb{N}\) and \(m < n\); we have
Now (4) and \(K \alpha< 1\) imply that
then \(\{u_{n}\}\) is a Cauchy sequence.
Since \(\{u_{n}\}\subset A_{0}\) and \(A_{0}\) is a closed subset of the complete metrictype space \((X,D,K)\), we can find \(u \in A_{0}\) such that \(\lim_{n \rightarrow\infty }u_{n}=u\).
By (1) and because of the fact \(\{f,S\}\) and \(\{g,T\}\) commute proximally, \(fu_{2n1} = Su_{2n}\) and \(gu_{2n} = Tu_{2n+1}\). Therefore, the continuity of f, g, S, and T and \(n \rightarrow\infty \) ascertain that \(fu=gu=Tu=Su\).
Since \(f(A_{0})\subseteq B_{0}\), there exists \(x \in A_{0}\) such that
As \(\{f,S\}\) and \(\{g,T\}\) commute proximally, \(fx=gx=Sx=Tx\). Since \(f(A_{0})\subseteq B_{0}\), there exists \(z \in A_{0}\) such that
Because the pair \((A,B)\) has the Pproperty
which implies that \(x=z\). Thus, it follows that
then x is a common best proximity point of the mappings f, g, S, and T.
Suppose that y is another common best proximity point of the mappings f, g, S, and T, so that
As the pair \((A,B)\) has the Pproperty, from (5) and (6), we have
which implies that \(x = y\). □
Now we illustrate our common best proximity point theorem by the following example.
Example 2.3
Let \(X = [0,1]\times[0,1]\). Suppose that \(D(x,y) = d^{2}(x,y)\) for all \(x,y \in X\), where d is the Euclidean metric. Then \((X,D,K)\) is a complete metrictype space with \(K=2\). Let
Then \(D( A,B)=1\), \(A_{0} = A\), and \(B_{0} =B\). Let f, g, S, and T be defined as \(f(0,y)=(1,\frac{y}{8})\), \(g(0,y) = (1,\frac{y}{32})\), \(S(0,y) = (1,y)\), and \(T(0,y) = (1,\frac{y}{4})\). Then for all x and \(y \in X\) we have
Now, all the required hypotheses of Theorem 2.2 are satisfied. Clearly \((0,0)\) is unique common best proximity point of f, g, S, and T.
By Theorem 2.2 we also obtain the following common fixed point theorem in metrictype space.
Theorem 2.4
Let \((X,D,K)\) be a complete metrictype space. Let \(f,g,S,T : X \rightarrow X\) be given continuous mappings satisfying the Kcontractive condition such that S and T commute with f and g, respectively. Further let \(f(X)\subseteq T(X)\), \(g(X)\subseteq S(x)\). Then f, g, S, and T have a unique common fixed point.
Proof
We take the same sequence \(\{u_{n}\}\) and u as in the proof of Theorem 2.2. Due to the fact that S and T commute with f and g, respectively, we have
By continuity of f, g, S, T, and \(n \rightarrow\infty\) we have
Since \(f,g,S,T : X \rightarrow X\) satisfy the Kcontractive condition, and by (7),
we have \(D(fu,gu) \leq\alpha D(fu,gu)\). Therefore \(fu = gu\), and by (7), \(fu = gu = Su = Tu\).
We set \(w = fu = gu = Su = Tu\). Because of the fact that T commutes with g we obtain
and
Therefore, \(D(w,gw) \leq\alpha D(w,gw)\) and consequently
Similarly, we can show that
Hence, by (8) and (9) we deduce that \(w = fw = gw = Sw = Tw\). Therefore, w is a common fixed point of f, g, S, and T.
Assume to the contrary that \(p = fp = gp = Sp = Tp\) and \(q = fq = gq = Sq = Tq\) but \(p \neq q\).
We have
Consequently \(D(p,q) \leq\alpha D(p,q)\) and \(\alpha< 1\); then \(D(p,q) = 0\), a contradiction. Therefore, f, g, S, and T have a unique fixed point. □
References
AlThagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 36653671 (2009)
AminiHarandi, A: Common best proximity points theorems in metric spaces. Optim. Lett. (2014). doi:10.1007/s1159001206007
Di Bari, C, Suzuki, T, Vetro, C: Best proximity points for cyclic MeirKeeler contractions. Nonlinear Anal. 69(11), 37903794 (2008)
Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 10011006 (2006)
Karpagam, S, Agrawal, S: Best proximity point theorems for pcyclic MeirKeeler contractions. Fixed Point Theory Appl. 2009, 197308 (2009)
Sadiq Basha, S: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151, 210216 (2011)
Sadiq Basha, S: Best proximity points: global optimal approximate solution. J. Glob. Optim. 49, 1521 (2011)
AlThagafi, MA, Shahzad, N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 12091216 (2009)
AlThagafi, MA, Shahzad, N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008, 457069 (2008)
Kim, WK, Kum, S, Lee, KH: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 22162227 (2008)
Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851862 (2003)
Sadiq Basha, S, Veeramani, P: Best approximations and best proximity pairs. Acta Sci. Math. 63, 289300 (1997)
Srinivasan, PS: Best proximity pair theorems. Acta Sci. Math. 67, 421429 (2001)
Wlodarczyk, K, Plebaniak, R, Banach, A: Best proximity points for cyclic and noncyclic setvalued relatively quasiasymptotic contractions in uniform spaces. Nonlinear Anal. 70(9), 33323341 (2009)
Eldred, AA, Kirk, WA, Veeramani, P: Proximal normal structure and relatively nonexpansive mappings. Stud. Math. 171(3), 283293 (2005)
Anuradha, J, Veeramani, P: Proximal pointwise contraction. Topol. Appl. 156(18), 29422948 (2009)
Khamsi, MA, Hussain, N: KKM mappings in metric type spaces. Nonlinear Anal., Theory Methods Appl. 73(9), 31233129 (2010)
Cvetković, S, Stanić, MP, Dimitrijević, S, Simić, S: Common fixed point theorems for four mappings on cone metric type space. Fixed Point Theory Appl. 2011, 89725 (2011)
Jovanović, M, Kadelburg, Z, Radenović, S: Common fixed point results in metrictype spaces. Fixed Point Theory Appl. 2010, 978121 (2010)
Rahimi, H, Rad, GS: Some fixed point results in metric type space. J. Basic Appl. Sci. Res. 2(9), 93019308 (2012)
Sadiq Basha, S: Common best proximity points: global minimal solutions. Top (2011). doi:10.1007/s1175001101712
Acknowledgements
The authors are grateful to the referee for useful comments, which improved the manuscript, and for pointing out a number of misprints.
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 is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Lo’lo’, P., Vaezpour, S.M. & Esmaily, J. Common best proximity points theorem for four mappings in metrictype spaces. Fixed Point Theory Appl 2015, 47 (2015). https://doi.org/10.1186/s1366301502981
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366301502981
Keywords
 common best proximity point
 metrictype space
 common fixed point