Common best proximity points theorem for four mappings in metric-type spaces
- Parvaneh Lo’lo’^{1},
- Seyyed Mansour Vaezpour^{2}Email author and
- Jafar Esmaily^{3}
https://doi.org/10.1186/s13663-015-0298-1
© Lo’lo’ et al.; licensee Springer. 2015
Received: 7 November 2014
Accepted: 18 March 2015
Published: 9 April 2015
Abstract
In this article, we first give an existence and uniqueness common best proximity points theorem for four mappings in a metric-type 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.
Keywords
1 Introduction and preliminary
Fixed point theory is essential for solving various equations of the form \(T x = x\) for self-mappings T defined on subsets of metric spaces or normed linear spaces. Given non-void subsets A and B of a metric space and a non-self-mapping \(T : A \rightarrow B\), 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 self-mapping.
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 non-expansive 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 metric-type as follows.
Definition 1.1
[17]
- (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 \(x, y \in X\).
Obviously, for \(K = 1\), a metric-type space is simply a metric space.
Afterward, other authors proved fixed point theorems in metric-type space [18–20].
Definition 1.2
Definition 1.3
Definition 1.4
2 Main result
We begin our study with the following definition.
Definition 2.1
Theorem 2.2
- (i)
\(\{f,S\}\) and \(\{g,T\}\) commute proximally;
- (ii)
the pair \((A,B)\) has the P-property;
- (iii)
f, g, S and T are continuous;
- (iv)
f, g, S, and T satisfy the K-contractive condition;
- (v)
\(f(A_{0})\subseteq T(A_{0})\), \(g(A_{0})\subseteq S(A_{0})\) and \(g(A_{0})\subseteq B_{0}\), \(f(A_{0})\subseteq B_{0}\).
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 \(\{u_{n}\}\subset A_{0}\) and \(A_{0}\) is a closed subset of the complete metric-type 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, \(fu_{2n-1} = Su_{2n}\) and \(gu_{2n} = Tu_{2n+1}\). Therefore, the continuity of f, g, S, and T and \(n \rightarrow\infty \) ascertain that \(fu=gu=Tu=Su\).
Now we illustrate our common best proximity point theorem by the following example.
Example 2.3
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 metric-type space.
Theorem 2.4
Let \((X,D,K)\) be a complete metric-type space. Let \(f,g,S,T : X \rightarrow X\) be given continuous mappings satisfying the K-contractive 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
Assume to the contrary that \(p = fp = gp = Sp = Tp\) and \(q = fq = gq = Sq = Tq\) but \(p \neq q\).
Declarations
Acknowledgements
The authors are grateful to the referee for useful comments, which improved the manuscript, and for pointing out a number of misprints.
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.
Authors’ Affiliations
References
- Al-Thagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665-3671 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Amini-Harandi, A: Common best proximity points theorems in metric spaces. Optim. Lett. (2014). doi:10.1007/s11590-012-0600-7 MathSciNetGoogle Scholar
- Di Bari, C, Suzuki, T, Vetro, C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69(11), 3790-3794 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001-1006 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Karpagam, S, Agrawal, S: Best proximity point theorems for p-cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009, 197308 (2009) View ArticleMathSciNetGoogle Scholar
- Sadiq Basha, S: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151, 210-216 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Sadiq Basha, S: Best proximity points: global optimal approximate solution. J. Glob. Optim. 49, 15-21 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Al-Thagafi, MA, Shahzad, N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209-1216 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Al-Thagafi, MA, Shahzad, N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008, 457069 (2008) View ArticleMathSciNetGoogle Scholar
- Kim, WK, Kum, S, Lee, KH: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 2216-2227 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851-862 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Sadiq Basha, S, Veeramani, P: Best approximations and best proximity pairs. Acta Sci. Math. 63, 289-300 (1997) MathSciNetGoogle Scholar
- Srinivasan, PS: Best proximity pair theorems. Acta Sci. Math. 67, 421-429 (2001) MATHGoogle Scholar
- Wlodarczyk, K, Plebaniak, R, Banach, A: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 70(9), 3332-3341 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Eldred, AA, Kirk, WA, Veeramani, P: Proximal normal structure and relatively nonexpansive mappings. Stud. Math. 171(3), 283-293 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Anuradha, J, Veeramani, P: Proximal pointwise contraction. Topol. Appl. 156(18), 2942-2948 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Khamsi, MA, Hussain, N: KKM mappings in metric type spaces. Nonlinear Anal., Theory Methods Appl. 73(9), 3123-3129 (2010) View ArticleMATHMathSciNetGoogle Scholar
- 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) Google Scholar
- Jovanović, M, Kadelburg, Z, Radenović, S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010, 978121 (2010) View ArticleGoogle Scholar
- Rahimi, H, Rad, GS: Some fixed point results in metric type space. J. Basic Appl. Sci. Res. 2(9), 9301-9308 (2012) Google Scholar
- Sadiq Basha, S: Common best proximity points: global minimal solutions. Top (2011). doi:10.1007/s11750-011-0171-2 Google Scholar