1 Introduction

In 1969, Fan in [2] proposed the concept best proximity point result for non-self continuous mappings \(T:A\longrightarrow X\) where A is a non-empty compact convex subset of a Hausdorff locally convex topological vector space X. He showed that there exists a such that \(d(a,Ta)=d(Ta,A)\). Many extensions of Fan’s theorems were established in the literature, such as in work by Reich [3], Sehgal and Singh [4] and Prolla [5].

In 2010, [1], Basha introduce the concept of best proximity point of a non-self mapping. Furthermore he introduced an extension of the Banach contraction principle by a best proximity theorem. Later on, several best proximity points results were derived (see e.g. [6,7,8,9,10,11,12,13,14,15,16,17,18,19]). Best proximity point theorems for non-self set valued mappings have been obtained in [20] by Jleli and Samet, in the context of proximal orbital completeness condition which is weaker than the compactness condition.

The aim of this article is to generalize the results of Basha [21] by introducing proximal β-quasi-contractive mappings which involve suitable comparison functions. As a consequence of our theorem, we obtain the result of Basha in [21] and an analogous result on proximal quasi-contractions is obtained which was first introduced by Jleli and Samet in [20].

2 Preliminaries and definitions

Let \((M,N)\) be a pair of non-empty subsets of a metric space \((X,d)\). The following notations will be used throughout this paper: \(d(M,N):=\inf\{d(m,n):m\in M, n\in N\}\); \(d(x,N):=\inf\{d(x,n):n\in N\}\).

Definition 2.1

([1])

Let \(T:M\rightarrow N\) be a non-self-mapping. An element \(a_{\ast}\in M\) is said to be a best proximity point of T if \(d(a_{\ast },Ta_{\ast})=d(M,N)\).

Note that in the case of self-mapping, a best proximal point is the normal fixed point, see [22, 23].

Definition 2.2

([21])

Given non-self-mappings \(S:M\rightarrow N\) and \(T:N\rightarrow M \). The pair \((S,T)\) is said to form a proximal cyclic contraction if there exists a non-negative number \(k<1\) such that

$$ d(u,Sa)=d(M,N)\quad\mbox{and}\quad d(v,Tb)=d(M,N)\Longrightarrow d(u,v)\leq kd(a,b)+(1-k)d(M,N) $$

for all \(u,a\in M\) and \(v,b\in N\).

Definition 2.3

([21])

A non-self-mapping \(S: M\rightarrow N\) is said to be a proximal contraction of the first kind if there exists a non-negative number \(\alpha<1 \) such that

$$ d(u_{1},Sa_{1})=d(M,N) \quad\mbox{and}\quad d(u_{2},Sa_{2})=d(M,N) \Longrightarrow d(u_{1},u_{2})\le\alpha d(a_{1},a_{2}) $$

for all \(u_{1},u_{2},a_{1},a_{2} \in M\).

Definition 2.4

([24])

Let \(\beta\in(0,+\infty)\). A β-comparison function is a map \(\varphi:[0,+\infty)\rightarrow{}[0,+\infty)\) satisfying the following properties:

\((P_{1})\) :

φ is nondecreasing.

\((P_{2})\) :

\(\lim_{n\rightarrow\infty}\varphi _{\beta }^{n}(t)=0\) for all \(t>0\), where \(\varphi_{\beta}^{n}\) denote the nth iteration of \(\varphi_{\beta}\) and \(\varphi_{\beta}(t)=\varphi (\beta t)\).

\((P_{3})\) :

There exists \(s\in(0,+\infty)\) such that \(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(s)<\infty\).

\((P_{4})\) :

\((\mathrm{id}-\varphi_{\beta} ) \circ\varphi _{\beta}(t) \leq\varphi_{\beta} \circ(\mathrm{id}-\varphi_{\beta})(t) \mbox{ for all } t \geq0\), where \(\mathrm{id}: [0,\infty) \longrightarrow[0,\infty) \) is the identity function.

Throughout this work, the set of all functions φ satisfying \((P_{1}), (P_{2})\) and \((P_{3})\) will be denoted by \(\varPhi_{\beta}\).

Remark 2.1

Let \(\alpha,\beta\in(0,+\infty)\). If \(\alpha<\beta\), then \(\varPhi_{\beta}\subset\varPhi_{\alpha}\).

We recall the following useful lemma concerning the comparison functions \(\varPhi_{\beta}\).

Lemma 2.1

([24])

Let \(\beta\in(0,+\infty)\) and \(\varphi\in \varPhi_{\beta}\). Then

  1. (i)

    \(\varphi_{\beta}\) is nondecreasing;

  2. (ii)

    \(\varphi_{\beta} (t) < t\) for all \(t > 0\);

  3. (iii)

    \(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(t) < \infty\) for all \(t > 0 \).

Definition 2.5

([20])

A non-self-mapping \(T:M\rightarrow N\) is said to be a proximal quasi-contraction if there exists a number \(q\in {}[ 0,1)\) such that

$$ d(u,v)\leq q\max\bigl\{ d(a,b),d(a,u),d(b,v),d(a,v),d(b,u)\bigr\} $$

whenever \(a,b,u,v\in M\) satisfy the condition that \(d(u,Ta)=d(M,N)\) and \(d(v,Tb)=d(M,N)\).

3 Main results and theorems

Now, we start this section by introducing the following concept.

Definition 3.1

Let \(\beta\in(0,+\infty)\). A non-self mapping \(T:M\rightarrow N\) is said to be a proximal β-quasi-contraction if and only if there exist \(\varphi\in\varPhi_{\beta }\) and positive numbers \(\alpha_{0},\ldots,\alpha_{4}\) such that

$$ d(u,v)\leq\varphi\bigl(\max \bigl\{ \alpha_{0}d(a,b),\alpha _{1}d(a,u),\alpha _{2}d(b,v),\alpha_{3}d(a,v), \alpha_{4}d(b,u) \bigr\} \bigr). $$

For all \(a,b,u,v\in M\) satisfying, \(d(u,Ta)=d(M,N)\) and \(d(v,Tb)=d(M,N)\).

Let \((M,N)\) be a pair of non-empty subsets of a metric space \((X,d)\). The following notations will be used throughout this paper: \(M_{0}:=\{u\in M:\text{ there exists }v\in N\text{ with }d(u,v)=d(M,N)\} \);\(N_{0}:=\{v\in N:\text{ there exists }u\in M\text{ with }d(u,v)=d(M,N)\} \).

Our main result is giving by the following best proximity point theorems.

Theorem 3.1

Let \((M,N)\) be a pair of non-empty closed subsets of a complete metric space \((X,d)\) such that \(M_{0}\) and \(N_{0}\) are non-empty. Let \(S:M\longrightarrow N\) and \(T:N\longrightarrow M\) be two mappings satisfying the following conditions:

\((C_{1})\) :

\(S(M_{0})\subset N_{0}\) and \(T(N_{0})\subset M_{0}\);

\((C_{2})\) :

there exist \(\beta_{1}, \beta_{2}\geq\max\{ \alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}, 2\alpha_{4}\}\) such that S is a proximal \(\beta_{1}\)-quasi-contraction mapping (say, \(\psi\in\varPhi_{\beta_{1}}\)) and T is a proximal \(\beta_{2} \)-quasi-contraction mapping (say, \(\phi\in\varPhi_{\beta_{2}}\)).

\((C_{3})\) :

The pair \((S,T)\) forms a proximal cyclic contraction.

\((C_{4})\) :

Moreover, one of the following two assertions holds:

  1. (i)

    ψ and ϕ are continuous;

  2. (ii)

    \(\beta_{1},\beta_{2}>\max\{\alpha_{2},\alpha _{3}\} \).

Then S has a unique best proximity point \(a_{\ast}\in M\) and T has a unique best proximity point \(b_{\ast}\in N\). Also these best proximity points satisfy \(d(a_{\ast},b_{\ast})=d(M,N)\).

Proof

Since \(M_{0}\) is a non-empty set, \(M_{0}\) contains at least one element, say \(a_{0}\in M_{0}\). Using the first hypothesis of the theorem, there exists \(a_{1}\in M_{0}\) such that \(d(a_{1},Sa_{0})=d(M,N)\). Again, since \(S(M_{0})\subset N_{0}\), there exists \(a_{2}\in M_{0}\) such that \(d(a_{2},Sa_{1})=d(M,N)\). Continuing this process in a similar fashion to find \(a_{n+1}\in M_{0}\) such that \(d(a_{n+1},Sa_{n})=d(M,N)\). Since S is a proximal \(\beta_{1}\)-quasi-contraction mapping for \(\psi\in\varPhi_{\beta_{1}}\) and since

$$ d(a_{n+1},Sa_{n})=d(a_{n},Sa_{n-1})=d(M,N) \text{,} $$
(1)

then by Definition 3.1 we have

d ( a n + 1 , a n ) ≤ ψ ( max { α 0 d ( a n , a n − 1 ) , α 1 d ( a n , a n + 1 ) , α 2 d ( a n , a n − 1 ) , α 4 d ( a n + 1 , a n − 1 ) } ) ≤ ψ ( max { α 0 d ( a n , a n − 1 ) , α 1 d ( a n , a n + 1 ) , α 2 d ( a n , a n − 1 ) α 4 d ( a n − 1 , a n ) + α 4 d ( a n , a n + 1 ) } ) ≤ ψ ( max { α 0 d ( a n , a n − 1 ) , α 1 d ( a n , a n + 1 ) , α 2 d ( a n , a n − 1 ) 2 α 4 max { d ( a n − 1 , a n ) , d ( a n , a n + 1 ) } } ) ≤ ψ ( β 1 max { d ( a n , a n − 1 ) , d ( a n , a n + 1 ) } ) = ψ β 1 ( max { d ( a n , a n − 1 ) , d ( a n , a n + 1 ) } ) .
(2)

Now, if \(\max\{ d(a_{n},a_{n-1}), d(a_{n},a_{n+1})\}= d(a_{n},a_{n+1})\), then by Lemma 2.1 the above inequality becomes

$$d(a_{n+1},a_{n})\leq\psi_{\beta_{1}}\bigl(d(a_{n+1},a_{n}) \bigr)< d(a_{n+1},a_{n}), $$

which is a contradiction. Thus, \(\max\{ d(a_{n},a_{n-1}), d(a_{n},a_{n+1}) \}= d(a_{n},a_{n-1})\), then the above inequality (2) becomes

$$ d(a_{n+1},a_{n})\leq \psi_{\beta_{1}}\bigl(d(a_{n-1},a_{n}) \bigr)). $$

By applying induction on n, the above inequality gives

$$ d(a_{n+1},a_{n})\leq \psi_{\beta_{1}}^{n} \bigl(d(a_{0},a_{1})\bigr)\quad \forall n\geq1. $$
(3)

Now, from the axioms of metric and Eq. (3), for positive integers \(n< m\), we get

$$ d(a_{n},a_{m})\leq\sum_{k=n}^{m-1}d(a_{k},a_{k+1}) \leq \sum_{k=n}^{m-1}\psi_{\beta_{1}}^{k} \bigl(d(a_{1},a_{0})\bigr)\leq \sum _{k=1}^{\infty}\psi_{\beta_{1}}^{k} \bigl(d(a_{1},a_{0})\bigr)< \infty. $$

Hence, for every \(\epsilon>0\) there exists \(N>0\) such that

$$ d(a_{n},a_{m})\leq\sum_{k=n}^{m-1}d(a_{k},a_{k+1})< \epsilon \quad\text{for all }m>n>N. $$

Therefore, \(d(a_{n},a_{m})<\epsilon\) for all \(m>n>N\). That is \(\{ a_{n}\}\) is a Cauchy sequence in M. But M is a closed subset of the complete metric space X, then \(\{a_{n}\}\) converges to some element \(a_{\ast}\in M\).

Since \(T(N_{0})\subset M_{0}\), by using a similar argument as above, there exists a sequence \(\{b_{n}\}\subset N_{0}\) such that \(d(b_{n+1},Tb_{n})=d(M,N)\) for each n. Since T is a proximal \(\beta _{2}\)-quasi-contraction mapping (say \(\phi\in\varPhi_{\beta_{2}}\)) and since \(d(b_{n+1},Tb_{n})=d(b_{n},Tb_{n-1})=d(M,N)\), we deduce from Definition 3.1 that

d ( b n + 1 , b n ) ≤ ϕ ( max { α 0 d ( b n , b n − 1 ) , α 1 d ( b n , b n + 1 ) , α 2 d ( b n , b n − 1 ) , α 4 d ( b n − 1 , b n + 1 ) } ) ≤ ϕ ( max { α 0 d ( b n , b n − 1 ) , α 1 d ( b n , b n + 1 ) , α 2 d ( b n , b n − 1 ) , α 4 d ( b n − 1 , b n ) + α 4 d ( b n , b n + 1 ) } ) ≤ ϕ ( max { α 0 d ( b n , b n − 1 ) , α 1 d ( b n , b n + 1 ) , α 2 d ( b n , b n − 1 ) , 2 α 4 max { d ( b n − 1 , b n ) , d ( b n , b n + 1 ) } } ) ≤ ϕ ( β 2 max { d ( b n , b n − 1 ) , d ( b n , b n + 1 ) } ) = ϕ β 2 ( max { d ( b n , b n − 1 ) , d ( b n , b n + 1 ) } ) .

Using a similar argument as in the case of \(\{a_{n}\}\), one can show that \(\{b_{n}\}\) is a Cauchy sequence in the closed subset N of the complete space X. Thus \(\{b_{n}\}\) converges to \(b_{\ast}\in N\). Now we shall show that \(a_{\ast}\) and \(b_{\ast}\) are best proximal points of S and T, respectively. As the pair \((S,T)\) forms a proximal cyclic contraction, it follows that

$$ d(a_{n+1},b_{n+1})\leq kd(a_{n},b_{n})+(1-k)d(M,N). $$
(4)

Taking the limit as \(n\longrightarrow+\infty\), in Eq. (4) we get \(d(a_{\ast},b_{\ast})\leq kd(a_{\ast},b_{\ast})+(1-k)d(M,N)\), and so, \((1-k) d(a_{\ast},b_{\ast})\leq (1-k)d(M,N)\). This implies

$$ d(a_{\ast},a_{\ast})\leq d(M,N). $$
(5)

Using the fact that \(d(M,N)\leq d(a_{\ast},b_{\ast})\) and (5), we get \(d(a_{\ast},b_{\ast})=d(M,N)\). Therefore, we conclude that \(a_{\ast }\in M_{0}\) and \(b_{\ast}\in N_{0}\).

From one hand, since \(S(M_{0})\subset N_{0}\) and \(T(N_{0})\subset M_{0}\), there exist \(u\in M\) and \(v\in N\) such that

$$ d(u,Sa_{\ast})=d(v,Tb_{\ast})=d(M,N). $$
(6)

On the other hand, by (1), (6) and using the hypothesis of the theorem that S is a proximal \(\beta_{1}\)-quasi-contraction mapping, we deduce that

$$\begin{aligned} &d(a_{n+1},u) \\ &\quad\leq\psi\bigl(\max\bigl\{ \alpha_{0}d(a_{n},a_{\ast}), \alpha _{1}d(a_{n},a_{n+1}),\alpha_{2}d(a_{\ast},u), \alpha _{3}d(a_{n},u),\alpha _{4}d(a_{\ast},a_{n+1}) \bigr\} \bigr). \end{aligned}$$
(7)

For simplicity, we denote

$$\rho=d(a_{\ast},u) $$

and

$$ A_{n}=\max\bigl\{ \alpha _{0}d(a_{n},a_{\ast}), \alpha_{1}d(a_{n},a_{n+1}),\alpha_{2}d(a_{\ast },u), \alpha_{3}d(a_{n},u),\alpha_{4}d(a_{\ast},a_{n+1}) \bigr\} . $$

Thus,

$$ {\lim_{n\longrightarrow+\infty}A_{n}=\max\{\alpha_{2}, \alpha_{3}\} \rho}. $$
(8)

Now, we show by contradiction that \(\rho=0\). Suppose that \(\rho>0\). First, we consider the case where the assertion (i) of \((C_{4})\) is satisfied, that is, ψ is continuous. Then, taking the limit as \(n\rightarrow\infty\) in (7) and using (8) and Lemma 2.1, we obtain

$$ \rho\leq\psi\bigl(\max\{\alpha_{2},\alpha_{3}\}\rho\bigr) \leq\psi(\beta _{1}\rho)=\psi_{\beta_{1}} (\rho) < \rho, $$

which is a contradiction. Now, we assume the case where the assertion (ii) of \((C_{4})\) is satisfied, that is, \(\beta_{1}>\max\{\alpha_{2},\alpha _{3}\}\). Then there exist \(\epsilon>0\) and integer \(N>0\) such that, for all \(n>N\), we have

$$ A_{n}< \bigl(\max\{\alpha_{2},\alpha_{3}\}+ \epsilon\bigr)\rho\quad\text{and}\quad \beta_{1}>\max\{ \alpha_{2},\alpha_{3}\}+\epsilon. $$

Therefore, the inequality (7) turns into the following inequality:

$$\begin{aligned} d(a_{n+1},u) &\leq\psi(A_{n}) \\ &\leq\psi\bigl(\bigl(\max\{\alpha_{2},\alpha_{3}\}+ \epsilon\bigr)\rho\bigr)=\psi _{\beta_{1}}\biggl(\frac{\max\{\alpha_{2},\alpha_{3}\}+\epsilon}{\beta _{1}}\rho\biggr). \end{aligned}$$

Since \(\psi\in\varPhi_{\beta_{1}}\), by Lemma 2.1 we have

$$ d(a_{n+1},u)< \frac{\max\{\alpha_{2},\alpha_{3}\}+\epsilon}{\beta _{1}}\rho< \rho. $$

By letting \(n\rightarrow\infty\), the above inequality yields

$$ \rho\leq\frac{\max\{\alpha_{2},\alpha_{3}\}+\epsilon}{\beta _{1}}\rho < \rho, $$

which is a contradiction as well. Thus, in both two cases we get \(0=\rho =d(a_{\ast},u)\), which means that \(u=a_{\ast}\) and so from equation (6) we get \(d(a_{\ast},Sa_{\ast})=d(M,N)\). That is \(a_{\ast}\) is a best proximity point for S.

Similarly, by using word by word the above argument after replacing u by v, S by T, \(\beta_{1}\) by \(\beta_{2}\) and ψ by ϕ, we get that \(v=b_{\ast}\) and hence by (6) \(b_{\ast}\) is a best proximity point for the non-self mapping T.

Now, we shall prove that the obtained best proximity points \(a_{\ast}\) of S is unique. Assume to the contrary that there exists \(x\in M \) such that \(d(x,Sx)=d(M,N)\) and \(x\neq a_{\ast}\). Since S is a proximal \(\beta_{1}\)-quasi-contractive mapping, we obtain

$$\begin{aligned} d(a_{\ast},x)& \leq\psi\bigl(\max \bigl\{ \alpha_{0}d(a_{\ast},x), \alpha _{1}d(x,x),\alpha _{2}d(a_{\ast},a_{\ast}), \alpha_{3}d(a_{\ast},x),\alpha _{4}d(a_{\ast},x) \bigr\} \bigr) \\ & \leq\psi \bigl(\max\{\alpha_{0},\alpha_{3}, \alpha_{4}\} d(a_{\ast },x) \bigr) \\ & \leq\psi\bigl(\beta_{1}d(a_{\ast},x) \bigr)= \psi_{\beta_{1}} \bigl(d(a_{\ast },x) \bigr) \\ & < d(a_{\ast},x), \end{aligned}$$

which is a contradiction. Similarly, using the same as above and the fact that T is a proximal \(\beta_{2}\)-quasi-contractive mapping, we see that the best proximity point \(b_{\ast}\) of T is unique. □

In Theorem 3.1 by taking \(\alpha_{0}=\alpha_{1}=\alpha_{2}=\alpha_{3}=0 , \alpha _{4}=1,\beta_{1}=\beta_{2}=1\) and \(\psi(t)=\phi(t)=qt\) which is a continuous function and belongs to \(\varPhi_{1}\), we obtain Corollary 3.3 in [21].

Corollary 3.1

Let \((M,N)\) be a pair of non-empty closed subsets of a complete metric space \((X,d)\) such that \(M_{0}\) and \(M_{0}\) are non-empty. Let \(S:M\longrightarrow N\) and \(T:N\longrightarrow M\) be mappings satisfy the following conditions:

\((d_{1})\) :

\(S(A_{0})\subset M_{0}\) and \(T(M_{0})\subset N_{0}\).

\((d_{2})\) :

S and T are proximal quasi-contractions.

\((d_{3})\) :

The pair \((S,T)\) form a proximal cyclic contraction.

Then S has a unique best proximity point \(a_{\ast }\in M\) such that \(d(a_{\ast},Sa_{\ast})=d(M,N)\) and T has a unique best proximity point \(b_{\ast}\in N\) such that \(d(b_{\ast},Tb_{\ast })=d(M,N)\). Also, these best proximity points satisfies \(d(a_{\ast },b_{\ast })=d(M,N)\).

Proof

The result follows immediately from Theorem 3.1 by taking \(\alpha _{0} = \alpha_{1} =\alpha_{2} =\alpha_{3} = 1 \) and \(\alpha_{4} = \frac {1}{2} \), \(\beta_{1}=\beta_{2} = 1\) and \(\psi(t)=\phi(t)=qt\). □

The following definition, which was introduced in [24], is needed to derive a fixed point result as a consequence of our main theorem.

Definition 3.2

([24])

Let X be a non-empty set. A mapping \(T:X\longrightarrow X\) is called β-quasi-contractive, if there exist \(\beta>0\) and \(\varphi \in\varPhi_{\beta}\) such that

$$ d(Ta,Tb)\leq\varphi\bigl(H_{T}(a,b)\bigr), $$

where

$$ H_{T}(a,b)=\max\bigl\{ \alpha_{0}d(a,b), \alpha_{1}d(a,Ta),\alpha _{2}d(b,Tb),\alpha_{3}d(a,Tb), \alpha_{4}d(b,Ta)\bigr\} , $$

with \(\alpha_{i}\geq0\) for \(i=0, 1,2,3,4\).

Corollary 3.2

Let \((X,d)\) be a complete metric space. Let \(S,T:X\longrightarrow X\) be two self-mappings satisfying the following conditions:

\((E_{1})\) :

S is \(\beta_{1}\)-quasi-contractive ( say, \(\psi\in\varPhi_{\beta_{1}}\)) and T is \(\beta_{2}\)-quasi-contractive (say, \(\phi\in\varPhi_{\beta_{2}}\)).

\((E_{2})\) :

For all \(a,b\in X,d(Sa,Tb)\leq kd(a,b)\) for some \(k\in(0,1)\).

\((E_{3})\) :

Moreover, one of the following assertions holds:

  1. (i)

    ψ and ϕ are continuous;

  2. (ii)

    \(\beta_{1},\beta_{2}>\max\{\alpha _{2},\alpha_{3}\}\).

Then S and T have a common unique fixed point.

Proof

This result follows from Theorem 3.1 by taking \(M=N=X\) and noticing that the hypotheses \((E_{1})\) and \((E_{2})\) of the corollary coincide with the first, second and the third conditions of Theorem 3.1. □

Example 3.1

Let \(X=\mathbb{R}\) with the metric \(d(x,y)=|x-y|\), then \((X,d)\) is complete metric space. Let \(M=[0,1]\) and \(N=[2,3]\). Also, let \(S:M\longrightarrow N\) and \(T:N\longrightarrow M\) be defined by \(S(x)=3-x\) and \(T(y)=3-y\). Then it is easy to see that \(d(M,N)=1\), \(M_{0}=\{1\}\) and \(N_{0}=\{2\}\). Thus, \(S(M_{0}) = S(\{1\}) = \{2\} = N_{0}\) and \(T(M_{0}) = T(\{2\}) = \{1\} = M_{0}\).

Now we show that the pair \((S,T)\) forms a proximal cyclic contraction. \(d(u,Sa) = d(M,N) =1\) implies that \(u=a=1 \in M\) and \(d(v,Tb = d(M,N) =1\) implies that \(v=b=2 \in N\).

Now, since \(d(u,Sa)=d(1,S(1))= d(1,2)=1=d(M,N)\) and \(d(v,Tb)=d(2,T(2))= d(2,1)=1=d(M,N)\). Therefore,

$$\begin{aligned} 1&= d(u,v) = d(1,2) \\ &\leq k \bigl(d(1,2)\bigr) + (1-k) d(M,N) \\ &= k + (1-k) = 1. \end{aligned}$$

So, \((S,T)\) are proximal cyclic contraction for any \(0\leq k<1\). Now we shall show that S is proximal \(\beta_{1}\)-quasi-contraction mapping with \(\psi(t)=\frac{1}{7}t,\beta_{1}=2\) and \(\alpha_{i}=\frac{1}{5}\) for\(i=0,1,2,3\) and \(\alpha_{4} = \frac{1}{100}\). Note that \(\psi(t)= \frac{1}{7}t \in\varPhi_{2} \) since \(\psi_{\beta_{1}}t= \psi_{2}t= \frac{2}{7} t \). As above the only \(a,b,u,v\in M\) such that \(d(u,Sa)=d(M,N)=1=d(v,Sb)\) is \(a=b=u=v =1 \in M\). But

$$\begin{aligned} 0&=d(u,v) =d(1,1) \\ &\leq\frac{1}{7}\max\biggl\{ \frac{1}{6}d(a,b), \frac{1}{6}d(a,u),\frac{1}{6} d(b,v), \frac{1}{6}d(a,v),\frac{1}{100}d(b,u)\biggr\} \\ &= \psi\biggl(\max\biggl\{ \frac{1}{6}d(1,1),\frac{1}{100}d(1,1) \biggr\} \biggr) \\ &= \psi\bigl(\max\{0,0,0,0,0\}\bigr) \\ &= 0. \end{aligned}$$

So, S is a proximal \(\beta_{1}\)-quasi-contraction mapping. We deduce using our Theorem 3.1, that S has a unique best proximity point which is \(a_{\ast} =1\) in this example.

Similarly, by using the same argument as above, we can show that T is proximal \(\beta_{2}\)-quasi-contraction mapping with \(\phi(t)=\frac{1}{8}t,\beta_{2}=3\) and \(\alpha_{i}=\frac{1}{6}\) for\(i=0,1,2,3\) and \(\alpha_{4} = \frac{1}{100}\). Note that \(\phi(t)= \frac{1}{8}t \in\varPhi_{3} \) since \(\phi_{\beta_{2}}t= \phi_{3}(t)= \frac{3}{8} t \). As above the only \(a,b,u,v\in N\) such that \(d(u,Ta)=d(M,N)=1=d(v,Tb)\) is \(a=b=u=v =2 \in M\). But

$$\begin{aligned} 0&=d(u,v) =d((2,2) \\ &\leq\frac{1}{8}\max\biggl\{ \frac{1}{6}d(a,b), \frac{1}{6}d(a,u),\frac {1}{6}d(b,v),\frac{1}{6}d(a,v), \frac{1}{100}d(b,u)\biggr\} \\ &= \phi\biggl(\max\biggl\{ \frac{1}{6}d(2,2), \frac{1}{100}d(2,2) \biggr\} \biggr) \\ &= \phi\bigl(\max\{0,0,0,0,0\}\bigr) \\ &= 0. \end{aligned}$$

So, T is a proximal \(\beta_{2}\)-quasi-contraction mapping. We deduce, using Theorem 3.1, that T has a unique best proximity point which is \(b_{\ast} =2\).

Finally, \(\psi(t)\) and \(\phi(t)\) are continuous mappings as well as \(\beta_{1}, \beta_{2} > \max_{0\leq i \leq3}\{\alpha_{i} \} \). Therefore

$$ d(a_{\ast},b_{\ast})=d(1,2)=1=d(M,N). $$

4 Conclusion

Improvements to some best proximity point theorems are proposed. In particular, the result due to Basha [21] for proximal contractions of first kind is generalized. Furthermore, we propose a similar result on existence and uniqueness of best proximity point of proximal quasi-contractions introduced by Jleli and Samet in [20]. This has been achieved by introducing β-quasi-contractions involving β-comparison functions introduced in [24].