# Common best proximity points theorem for four mappings in metric-type spaces

## 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.

## 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 [17]. Best proximity point theorems for multivalued mappings are available in [814]. 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]

Let X be a non-empty set, $$K \geq1$$ be a real number, and let the function $$D : X \times X \rightarrow\mathbb{R}$$ satisfy the following properties:

1. (i)

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

2. (ii)

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

3. (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 metric-type space.

Obviously, for $$K = 1$$, a metric-type space is simply a metric space.

Afterward, other authors proved fixed point theorems in metric-type space [1820].

Given two non-empty subsets A and B of a metric-type space $$(X,D,K)$$, the following notions and notations are used in the sequel.

\begin{aligned}& D(A,B)= \inf \bigl\{ d(x,y): x \in A , y\in B\bigr\} ; \\& A_{0}=\bigl\{ x \in A : D(x,y)= D(A,B) \mbox{ for some } y \in B\bigr\} ; \\& B_{0}=\bigl\{ y\in B : D(x,y)= D(A,B) \mbox{ for some } x \in A\bigr\} . \end{aligned}

This study focuses upon resolving a more general problem as regards the existence of common best proximity points for pairs of non-self-mappings in metric-type space. As a result, the finding of this study verifies a common global minimal solution to the problem of minimizing the real valued multi-objective 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 metric-type space and the non-self-mappings $$S : A \rightarrow B$$ and $$T : A \rightarrow B$$ satisfy a contraction-like condition. Our best proximity point theorem generalizes a result due to Sadiq Basha [21]. Further, a common fixed point theorem for commuting self-mappings 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 non-self-mappings $$f_{1}, f_{2},\ldots,f_{n} : A \rightarrow B$$ if it satisfies the condition that

$$D(x,f_{1}x) = D(x,f_{2}x) = \cdots= D(x,f_{n}x) = D(A,B).$$

### 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

$$\bigl[D(u,Sx) = D(v,Tx) = D(A,B)\bigr]\quad\Rightarrow\quad Sv=Tu.$$

### Definition 1.4

If $$A_{0}\ne\varnothing$$ then the pair $$(A,B)$$ is said to have P-property if and only if for any $$x_{1}, x_{2}\in A_{0}$$ and $$y_{2}, y_{2}\in B_{0}$$

$$\left \{ \begin{array}{@{}l} D(x_{1},y_{1})=D(A,B),\\ D(x_{2},y_{2})=D(A,B) \end{array} \right . \quad\Longrightarrow\quad D(x_{1},x_{2})=D(y_{1},y_{2}).$$

## Main result

We begin our study with the following definition.

### Definition 2.1

Let A and B be two non-empty subsets of a metric-type space $$(X,D,K)$$. Non-self-mappings $$f,g,S,T: A\rightarrow B$$ are said to satisfy a K-contractive condition if there exists a non-negative number $$\alpha< {\frac{1}{K}}$$ such that for each $$x,y \in A$$

$$D(fx,gy) \leq{\alpha} \max \biggl\{ D(Sx,Ty) , D(fx,Sx) , D(Ty,gy), \frac {1}{2K}\bigl[D(Sx, gy)+D(fx, Ty)\bigr]\biggr\} .$$

### Theorem 2.2

Let A and B be non-empty subsets of a complete metric-type space $$(X,D,K)$$. Moreover, assume that $$A_{0}$$ and $$B_{0}$$ are non-empty and $$A_{0}$$ is closed. Let the non-self-mappings $$f,g,S,T: A\rightarrow B$$ satisfy the following conditions:

1. (i)

$$\{f,S\}$$ and $$\{g,T\}$$ commute proximally;

2. (ii)

the pair $$(A,B)$$ has the P-property;

3. (iii)

f, g, S and T are continuous;

4. (iv)

f, g, S, and T satisfy the K-contractive condition;

5. (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

$$D\bigl(u_{2n},f(x_{2n})\bigr) = D(A,B) \quad \mbox{and} \quad D\bigl(u_{2n+1}, g(x_{2n+1})\bigr)= D(A,B).$$
(1)

Since the pair $$(A,B)$$ has the P-property, by (1) we have

\begin{aligned} D(u_{2n},u_{2n+1}) =&D(fx_{2n},gx_{2n+1}) \\ \leq&\alpha \max\biggl\{ D(Sx_{2n},Tx_{2n+1}) , D(fx_{2n},Sx_{2n}) , D(Tx_{2n+1},gx_{2n+1}), \\ &{}\frac{1}{2K}\bigl[D(Sx_{2n}, gx_{2n+1})+D(fx_{2n}, Tx_{2n+1})\bigr]\biggr\} \\ \leq&\alpha \max\biggl\{ D(u_{2n-1},u_{2n}) , D(u_{2n},u_{2n-1}) , D(u_{2n},u_{2n+1}),\\ &{} \frac{1}{2K}\bigl[D(u_{2n-1}, u_{2n+1})+D(u_{2n}, u_{2n})\bigr]\biggr\} , \end{aligned}

thus (note that $$\frac{1}{2K}D(u_{2n-1}, u_{2n+1}) \leq\frac {1}{2}[D(u_{2n-1},u_{2n})+D(u_{2n},u_{2n+1})]$$ and $$\alpha< 1$$)

$$D(u_{2n},u_{2n+1})\leq{\alpha}D(u_{2n-1},u_{2n}).$$
(2)

Similarly

\begin{aligned} D(u_{2n+1},u_{2n+2}) =&D(fx_{2n+2},gx_{2n+1}) \\ \leq&\alpha \max\biggl\{ D(Sx_{2n+2},Tx_{2n+1}) , D(fx_{2n+2},Sx_{2n+2}) , D(Tx_{2n+1},gx_{2n+1}), \\ &{}\frac{1}{2K}\bigl[D(Sx_{2n+2}, gx_{2n+1})+D(fx_{2n+2}, Tx_{2n+1})\bigr]\biggr\} \\ \leq&\alpha \max\biggl\{ D(u_{2n+1},u_{2n}) , D(u_{2n+2},u_{2n+1}) , D(u_{2n},u_{2n+1}),\\ &{} \frac{1}{2K}\bigl[D(u_{2n+1}, u_{2n+1})+D(u_{2n+2}, u_{2n})\bigr]\biggr\} , \end{aligned}

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 $$\alpha< 1$$)

$$D(u_{2n+1},u_{2n+2})\leq{\alpha}D(u_{2n},u_{2n+1}).$$
(3)

Therefore, by (2) and (3) we have

$$D(u_{n},u_{n+1})\leq\alpha D(u_{n-1},u_{n}),$$

and then

$$D(u_{n},u_{n+1})\leq\alpha^{n} D(u_{0}, u_{1}).$$
(4)

Let $$m, n \in\mathbb{N}$$ and $$m < n$$; we have

\begin{aligned} D(u_{m}, u_{n}) \leq& K\bigl[D(u_{m}, u_{m+1})+ D(u_{m+1}, u_{n})\bigr] \\ \leq&K D(u_{m}, u_{m+1}) + K^{2} \bigl[D(u_{m+1}, u_{m+2}) + D(u_{m+2}, u_{n}) \bigr]\\ \leq& \cdots \\ \leq& K D(u_{m}, u_{m+1}) + K^{2}D(u_{m+1}, u_{m+2})+ \cdots\\ &{}+ K^{n-m-1}\bigl[D(u_{n-2}, u_{n-1}) + D(u_{n-1}, u_{n})\bigr] \\ \leq& K D(u_{m}, u_{m+1}) + K^{2}D(u_{m+1}, u_{m+2} )+\cdots\\ &{}+ K^{n-m-1}D(u_{n-2}, u_{n-1})+ K^{n-m}D(u_{n-1}, u_{n}). \end{aligned}

Now (4) and $$K \alpha< 1$$ imply that

\begin{aligned} D(u_{m},u_{n}) \leq& \bigl(K \alpha^{m} + K^{2} \alpha^{m+1}+\cdots + K^{n-m}\alpha^{n-1} \bigr)D(u_{0} , u_{1}) \\ \leq& K {\alpha}^{m} \bigl( 1+ K \alpha+ \cdots+ {(K \alpha)}^{n-m-1}\bigr)D(u_{0} , u_{1}) \\ \leq& \frac{K {\alpha}^{m}}{1-K\alpha}D(u_{0},u_{1})\rightarrow0 \quad \mbox{when } m \rightarrow\infty; \end{aligned}

then $$\{u_{n}\}$$ is a Cauchy sequence.

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 $$fu=gu=Tu=Su$$.

Since $$f(A_{0})\subseteq B_{0}$$, there exists $$x \in A_{0}$$ such that

$$D(A,B)=D(x, fu)=D(x,gu)=D(x,Su)=D(x,Tu).$$

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

$$D(A,B)=D(z,fx)=D(z,gx)=D(z,Sx)=D(z,Tx).$$

Because the pair $$(A,B)$$ has the P-property

\begin{aligned} D(x,z) =&D(fu,gx)\\ \leq&\alpha \max\biggl\{ D(Su,Tx),D(fu,Su),D(Tx,gx), \frac {1}{2K}\bigl[D(Su,gx)+D(fu,Tx)\bigr]\biggr\} \\ \leq& \alpha \max\biggl\{ D(x,z),D(x,x),D(z,z),\frac {1}{2K} \bigl[D(x,z)+d(x,z)\bigr]\biggr\} \\ \leq& \alpha D(x,z), \end{aligned}

which implies that $$x=z$$. Thus, it follows that

$$D(A,B)=D(x,fx)=(x,gx)=(x,Tx)=(x,Sx),$$
(5)

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

$$D(A,B)=D(y,fy)=(y,gy)=(y,Ty)=(y,Sy).$$
(6)

As the pair $$(A,B)$$ has the P-property, from (5) and (6), we have

$$D(x,y)\leq\alpha D(x,y),$$

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 metric-type space with $$K=2$$. Let

$$A:=\bigl\{ (0,x): 0 \leq x \leq1\bigr\} , \qquad B := \bigl\{ (1,y) : 0 \leq y \leq 1\bigr\} .$$

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

$$D(fx,gy)= \biggl(\frac{x}{8}-\frac{y}{32}\biggr)^{2} = \frac{1}{64}D(Sx , Ty).$$

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

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

$$fu_{2n-1} = Su_{2n}, \qquad gu_{2n}=Tu_{2n+1}.$$

By continuity of f, g, S, T, and $$n \rightarrow\infty$$ we have

\begin{aligned} fu = Su, \qquad gu = Tu. \end{aligned}
(7)

Since $$f,g,S,T : X \rightarrow X$$ satisfy the K-contractive condition, and by (7),

\begin{aligned} D(fu,gu) \leq& \alpha \max \biggl\{ D(Su,Tu),D(fu,Su),D(Tu,gu),\frac {1}{2K} \bigl[D(Su,gu)+D(fu,Tu)\bigr]\biggr\} \\ \leq& \alpha \max \biggl\{ D(fu,gu),D(fu,fu),D(gu,gu),\frac {1}{2K} \bigl[(fu,gu)+(fu,gu)\bigr]\biggr\} , \end{aligned}

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

$$gw = gTu = Tgu = Tw,$$

and

\begin{aligned} D(w,gw) =& D(fu ,gw)\\ \leq&\alpha \max \biggl\{ D(Su,Tw),D(fu,Su),D(Tw,gw), \frac {1}{2K}\bigl[D(Su,gw)+D(fu,Tw)\bigr]\biggr\} \\ \leq&\alpha \max \biggl\{ D(w,gw),D(w,w),D(gw,gw),\frac {1}{2K} \bigl[(w,gw)+(w,gw)\bigr]\biggr\} . \end{aligned}

Therefore, $$D(w,gw) \leq\alpha D(w,gw)$$ and consequently

\begin{aligned} w = gw = Tw. \end{aligned}
(8)

Similarly, we can show that

\begin{aligned} w = fw = Sw. \end{aligned}
(9)

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

\begin{aligned} D(p,q) &= D(fp ,gq)\\ &\leq\alpha \max \biggl\{ D(Sp,Tq),D(fp,Sp),D(Tq,gq), \frac {1}{2K}\bigl[D(Sp,gq)+D(fp,Tq)\bigr]\biggr\} \\ &\leq\alpha \max \biggl\{ D(p,q),D(p,p),D(q,q),\frac{1}{2K} \bigl[(p,q)+(p,q)\bigr]\biggr\} . \end{aligned}

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. □

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## Acknowledgements

The authors are grateful to the referee for useful comments, which improved the manuscript, and for pointing out a number of misprints.

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Correspondence to Seyyed Mansour Vaezpour.

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The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Lo’lo’, P., Vaezpour, S.M. & Esmaily, J. Common best proximity points theorem for four mappings in metric-type spaces. Fixed Point Theory Appl 2015, 47 (2015). https://doi.org/10.1186/s13663-015-0298-1