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 Open Access
Best proximity points of implicit relation type modified ${\alpha}^{3}$proximal contractions
 Farzaneh Zabihi^{1} and
 Abdolrahman Razani^{1}Email author
https://doi.org/10.1186/1029242X2014365
© Zabihi and Razani; licensee Springer. 2014
 Received: 29 June 2014
 Accepted: 9 September 2014
 Published: 24 September 2014
Abstract
In this paper, we introduce the concept of an ${\alpha}^{3}$proximal admissible mappings and establish the existence of best proximity point theorems for implicit relation type modified ${\alpha}^{3}$proximal contractions. As applications of our theorems, we derive some new best proximity point results for implicit relation type contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some best proximity point results in the literature. Several interesting consequences of our theorems are also provided.
MSC:46N40, 47H10, 54H25, 46T99.
Keywords
 fixed point
 best proximity point
 ${\alpha}^{3}$proximal admissible mapping
 implicit relation type ${\alpha}^{3}$proximal contractions
 metric space endowed with graph
1 Introduction
In nonlinear functional analysis, one of the most significant research areas is fixed point theory. On the other hand, fixed point theory has an application in distinct branches of mathematics and also in different sciences, such as engineering, computer science, economics, etc. In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this celebrated result, many authors have generalized, improved, and extended this result in the context of different abstract spaces for various operators.
On the other hand, several classical fixed point theorems and common fixed point theorems have been recently unified by considering general contractive conditions expressed by an implicit relation (see Popa [1, 2]). Following Popa’s approach, many results on fixed point, common fixed points, and coincidence points have been obtained, in various ambient spaces (see [3–8], and references therein). On the other hand, Samet et al. [9] introduced and studied αψcontractive mappings in complete metric spaces and provided applications of the results to ordinary differential equations. More recently, Salimi et al. [10] modified the notions of αψcontractive and αadmissible mappings and established fixed point theorems to modify the results in [9]. For more details and applications of this line of research, we refer the reader to some related papers [11–13] and references therein. In this paper, we introduce the concept of an ${\alpha}^{3}$proximal admissible mappings and establish the existence of best proximity point theorems for implicit relation type modified ${\alpha}^{3}$proximal contractions. As applications of our theorems, we derive some new best proximity point results for implicit relation type contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some best proximity point results in the literature.
2 Main results
We denote by Ψ the set of all nondecreasing functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ such that ${\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the n th iterate of ψ.

(F1) if $F(u,v,v,u,u+v,0)\le 0$, where $u,v>0$, then $u\le \psi (v)$;

(F2) $F({t}_{1},\dots ,{t}_{6})$ is decreasing in ${t}_{5}$;

(F3) if $F(u,v,0,u+v,u,v)\le 0$, where $u,v\ge 0$, then $u\le \psi (v)$;

(F4) $F(u,u,0,0,u,u)>0$ for all $u>0$.
where $L\ge 0$ and $\psi \in \Psi $. Then $F\in \mathcal{F}$.
where $a+b+2c+2d<1$. Then $F\in \mathcal{F}$.
where $L\ge 0$ and $F\in \mathcal{F}$.
Definition 3 Let $(X,d)$ be a metric space and A and B be two nonempty subsets of X. Then B is said to be approximatively compact with respect to A if every sequence $\{{y}_{n}\}$ in B, satisfying the condition $d(x,{y}_{n})\to d(x,B)$ for some x in A, has a convergent subsequence.
 (i)T is an ${\alpha}^{3}$proximal admissible mapping and$T({A}_{0})\subseteq {B}_{0},$
 (ii)there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B),\phantom{\rule{2em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1,\phantom{\rule{2em}{0ex}}\alpha ({x}_{0},{x}_{0})\ge 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{1},{x}_{1})\ge 1.$
 (iii)
for every $x,y\in A$ with $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $\alpha (x,y)\ge 1$, $\alpha (x,x)\ge 1$, and $\alpha (y,y)\ge 1$.
Thus z is the desired best proximity point of T.
which is a contradiction to (F4). Hence, T has a unique best proximity point. □
 (i)
T is an ${\alpha}^{3}$proximal admissible mapping and $T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B),\phantom{\rule{2em}{0ex}}\alpha ({x}_{0},{x}_{0})\ge 1,\phantom{\rule{2em}{0ex}}\alpha ({x}_{1},{x}_{1})\ge 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1,$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},x)\ge 1$ and $\alpha (x,x)\ge 1$.
 (iv)
for every $x,y\in A$, where $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $\alpha (x,y)\ge 1$, $\alpha (x,x)\ge 1$, and $\alpha (y,y)\ge 1$.
and so from (F3) we get $u\le \psi (v)$. That is, $d(z,w)\le \psi (0)=0$. Thus, $z=w$. Hence z is a best proximity point of T. Uniqueness follows similarly to the proof of Theorem 1. □
Using Example 2 and Theorem 2 we obtain the following corollary.
 (i)
T is an ${\alpha}^{3}$proximal admissible mapping and $T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B),\phantom{\rule{2em}{0ex}}\alpha ({x}_{0},{x}_{0})\ge 1,\phantom{\rule{2em}{0ex}}\alpha ({x}_{1},{x}_{1})\ge 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1,$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},x)\ge 1$ and $\alpha (x,x)\ge 1$,
 (iv)there exist nonnegative real numbers a, b, c, d with $a+b+2c+2d<1$, such that for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$,$\begin{array}{r}\{\begin{array}{l}\alpha ({x}_{1},{x}_{2})\ge 1,\\ d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})+L\alpha ({x}_{1},{x}_{1})\alpha ({x}_{2},{x}_{2})\le ad({x}_{1},{x}_{2})+b\frac{[1+d({x}_{1},{u}_{1})]d({x}_{2},{u}_{2})}{1+d({x}_{1},{x}_{2})}\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})+L\alpha ({x}_{1},{x}_{1})\alpha ({x}_{2},{x}_{2})\le}+c[d({x}_{1},{u}_{1})+d({x}_{2},{u}_{2})]\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})+L\alpha ({x}_{1},{x}_{1})\alpha ({x}_{2},{x}_{2})\le}+d[d({x}_{1},{u}_{2})+d({x}_{2},{u}_{1})]+L,\end{array}$
where $L\ge 0$.
 (v)
for every $x,y\in A$, where $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $\alpha (x,y)\ge 1$, $\alpha (x,x)\ge 1$, and $\alpha (y,y)\ge 1$.
If in Corollary 1 we take $b=c=d=0$, then we have the following corollary.
 (i)
T is an ${\alpha}^{3}$proximal admissible mapping and $T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B),\phantom{\rule{2em}{0ex}}\alpha ({x}_{0},{x}_{0})\ge 1,\phantom{\rule{2em}{0ex}}\alpha ({x}_{1},{x}_{1})\ge 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1,$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},x)\ge 1$ and $\alpha (x,x)\ge 1$,
 (iv)there exists a nonnegative real number a with $a<1$, such that for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$,$\{\begin{array}{l}\alpha ({x}_{1},{x}_{2})\ge 1,\\ d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})+L\alpha ({x}_{1},{x}_{1})\alpha ({x}_{2},{x}_{2})\le ad({x}_{1},{x}_{2})+L,$
where $L\ge 0$.
 (v)
for every $x,y\in A$, where $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $\alpha (x,y)\ge 1$, $\alpha (x,x)\ge 1$, and $\alpha (y,y)\ge 1$.
Clearly, B is approximatively compact with respect to A and $d(A,B)=2$. Then ${A}_{0}=\{1\}$ and ${B}_{0}=\{1\}$. Clearly, $T({A}_{0})\subseteq {B}_{0}$, $d(1,T(1))=d(A,B)=2$, and $\alpha (1,1)\ge 1$.
that is, T is an ${\alpha}^{3}$proximal admissible mapping and condition (iv) of Corollary 1 holds true. Moreover, if $\{{x}_{n}\}$ is a sequence such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\{{x}_{n}\}\subseteq [2,1]$ and hence $x\in [2,1]$. Consequently, $\alpha (x,x)\ge 1$ and $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Therefore all the conditions of Corollary 1 hold for this example and T has a best proximity point. Here $z=1$ is the best proximity point of T.
If in Corollary 1 we take $\alpha (x,y)=1$, then we have the following corollary.
Corollary 3 (Theorem 3.1 of [14])
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)$\begin{array}{r}\{\begin{array}{l}d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le ad({x}_{1},{x}_{2})+b\frac{[1+d({x}_{1},{u}_{1})]d({x}_{2},{u}_{2})}{1+d({x}_{1},{x}_{2})}\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}+c[d({x}_{1},{u}_{1})+d({x}_{2},{u}_{2})]+d[d({x}_{1},{u}_{2})+d({x}_{2},{u}_{1})].\end{array}$
By taking $\alpha (x,y)=1$ in Theorem 2, we deduce the following corollary.
where $F\in \mathcal{F}$. Then T has a unique best proximity point.
Using Example 1 and Corollary 4, we deduce the following result.
where $\psi \in \Psi $. Then T has a unique best proximity point.
3 Some results in metric spaces endowed with a graph
Consistent with Jachymski [15], let $(X,d)$ be a metric space and Δ denotes the diagonal of the Cartesian product $X\times X$. Consider a directed graph G such that the set $V(G)$ of its vertices coincides with X, and the set $E(G)$ of its edges contains all loops, i.e., $E(G)\supseteq \Delta $. We assume G has no parallel edges, so we can identify G with the pair $(V(G),E(G))$. Moreover, we may treat G as a weighted graph (see [15]) by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length N ($N\in \mathbb{N}$) is a sequence ${\{{x}_{i}\}}_{i=0}^{N}$ of $N+1$ vertices such that ${x}_{0}=x$, ${x}_{N}=y$ and $({x}_{n1},{x}_{n})\in E(G)$ for $i=1,\dots ,N$. A graph G is connected if there is a path between any two vertices. G is weakly connected if $\tilde{G}$ is connected (see for details [12, 15, 16]).
In 2006, Espínola and Kirk [17] established an important combination of fixed point theory and graph theory.
where $F\in \mathcal{F}$.
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}({x}_{0},{x}_{1})\in E(G).$
Then T has a best proximity point. Further, the best proximity point is unique if, for every $x,y\in A$ such that $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $(x,y)\in E(G)$.
when $L=0$. Thus T is an ${\alpha}^{3}$proximal admissible mapping with $T({A}_{0})\subseteq {B}_{0}$ and continuous implicit relation type Gproximal contraction. From (ii) there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that $d({x}_{1},T{x}_{0})=d(A,B)$ and $({x}_{0},{x}_{1})\in E(G)$, that is, $d({x}_{1},T{x}_{0})=d(A,B)$, $\alpha ({x}_{0},{x}_{1})\ge 1$, $\alpha ({x}_{0},{x}_{0})\ge 1$, and $\alpha ({x}_{1},{x}_{1})\ge 1$. Hence, all the conditions of Theorem 1 are satisfied and T has a best proximity point. □
Similarly, by using Theorem 2, we can prove the following theorem.
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}({x}_{0},{x}_{1})\in E(G),$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $({x}_{n},{x}_{n+1})\in E(G)$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $({x}_{n},x)\in E(G)$ for all $n\in \mathbb{N}\cup \{0\}$.
Then T has a best proximity point. Further, the best proximity point is unique if, for every $x,y\in A$ such that $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $(x,y)\in E(G)$.
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}({x}_{0},{x}_{1})\in E(G),$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $({x}_{n},{x}_{n+1})\in E(G)$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $({x}_{n},x)\in E(G)$ for all $n\in \mathbb{N}\cup \{0\}$,
 (iv)for ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in {A}_{0}$,$\begin{array}{r}\{\begin{array}{l}({x}_{1},{x}_{2})\in E(G),\\ d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le ad({x}_{1},{x}_{2})+b\frac{[1+d({x}_{1},{u}_{1})]d({x}_{2},{u}_{2})}{1+d({x}_{1},{x}_{2})}\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}+c[d({x}_{1},{u}_{1})+d({x}_{2},{u}_{2})]\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}+d[d({x}_{1},{u}_{2})+d({x}_{2},{u}_{1})].\end{array}$
Then T has a best proximity point. Further, the best proximity point is unique if, for every $x,y\in A$ such that $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $(x,y)\in E(G)$.
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}({x}_{0},{x}_{1})\in E(G),$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $({x}_{n},{x}_{n+1})\in E(G)$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $({x}_{n},x)\in E(G)$ for all $n\in \mathbb{N}\cup \{0\}$,
 (iv)for ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in {A}_{0}$,$\begin{array}{r}\{\begin{array}{l}({x}_{1},{x}_{2})\in E(G),\\ d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le \psi (max\{d({x}_{1},{x}_{2}),d({x}_{1},{u}_{1}),d({x}_{2},{u}_{2}),\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}\frac{d({x}_{2},{u}_{1})+d({x}_{1},{u}_{2})}{2}\left\}\right)\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}+Lmin\{d({x}_{1},{u}_{1}),d({x}_{2},{u}_{2}),d({x}_{2},{u}_{1}),d({x}_{1},{u}_{2})\},\end{array}$
where $\psi \in \Psi $.
Then T has a best proximity point. Further, the best proximity point is unique if, for every $x,y\in A$ such that $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $(x,y)\in E(G)$.
4 Some results in metric spaces endowed with a partially ordered
The study of existence of fixed points in partially ordered sets has been established by Ran and Reurings [18] with applications to matrix equations. Agarwal et al. [19], Ćirić et al. [20], and Hussain et al. [12, 21] obtained some new fixed point results for nonlinear contractions in partially ordered Banach and metric spaces with some applications. In this section, as an application of our results we derive some new best proximity point results whenever the range space is endowed with a partial order.
Definition 5 [22]
 (i)
T is continuous and proximally orderedpreserving such that $T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$
 (iii)for all $x,y,u,v\in A$,$\begin{array}{r}\{\begin{array}{l}x\u2aafy,\\ d(u,Tx)=d(A,B),\\ d(y,Ty)=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}F(d(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v))\le 0.\end{array}$(4.1)
Then T has a best proximity point.
Thus all the conditions of Theorem 1 hold (when $L=0$) and T has a best proximity point. □
 (i)
T is proximally orderedpreserving such that $T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$
 (iii)for all $x,y,u,v\in A$,$\begin{array}{r}\{\begin{array}{l}x\u2aafy,\\ d(u,Tx)=d(A,B),\\ d(y,Ty)=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}F(d(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v))\le 0,\end{array}$(4.2)
 (iv)
if $\{{x}_{n}\}$ is an increasing sequence in A converging to $x\in A$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
Then T has a best proximity point.
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that ${x}_{n}\u2aaf{x}_{n+1}$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}\cup \{0\}$,
 (iv)for ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in {A}_{0}$,$\begin{array}{r}\{\begin{array}{l}{x}_{1}\u2aaf{x}_{2},\\ d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le ad({x}_{1},{x}_{2})+b\frac{[1+d({x}_{1},{u}_{1})]d({x}_{2},{u}_{2})}{1+d({x}_{1},{x}_{2})}\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}+c[d({x}_{1},{u}_{1})+d({x}_{2},{u}_{2})]+d[d({x}_{1},{u}_{2})+d({x}_{2},{u}_{1})].\end{array}$
Then T has a best proximity point. Further, the best proximity point is unique if, for every $x,y\in A$ such that $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $x\u2aafy$.
 (i)
$T({A}_{0})\subseteq {B}_{0}$,
 (ii)there exist elements ${x}_{0},{x}_{1}\in {A}_{0}$ such that$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that ${x}_{n}\u2aaf{x}_{n+1}$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}\cup \{0\}$,
 (iv)for ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in {A}_{0}$,$\begin{array}{r}\{\begin{array}{l}{x}_{1}\u2aaf{x}_{2},\\ d({u}_{1},T{x}_{1})=d(A,B),\\ d({u}_{2},T{x}_{2})=d(A,B)\end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le \psi (max\{d({x}_{1},{x}_{2}),d({x}_{1},{u}_{1}),d({x}_{2},{u}_{2}),\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}\frac{d({x}_{2},{u}_{1})+d({x}_{1},{u}_{2})}{2}\left\}\right)\\ \phantom{\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d({u}_{1},{u}_{2})\le}+Lmin\{d({x}_{1},{u}_{1}),d({x}_{2},{u}_{2}),d({x}_{2},{u}_{1}),d({x}_{1},{u}_{2})\},\end{array}$
where $\psi \in \Psi $.
Then T has a best proximity point. Further, the best proximity point is unique if, for every $x,y\in A$ such that $d(x,Tx)=d(A,B)=d(y,Ty)$, we have $x\u2aafy$.
5 Application to fixed point theory
5.1 Implicit relation type modified αcontraction
Definition 6 [9]
Remark 1 Note that every αadmissible mappings are ${\alpha}^{3}$proximal admissible mappings when $A=B=X$.
where $L\ge 0$ and $F\in \mathcal{F}$.
 (i)
T is αadmissible,
 (ii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$,
 (iii)
T is an implicit relation type modified αcontraction.
Then T has a fixed point.
 (i)
T is αadmissible,
 (ii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$,
 (iii)
T is an implicit relation type modified αcontraction,
 (iv)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha (x,x)\ge 1$ and $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$.
Then T has a fixed point.
Using Example 2 and Theorem 8, we deduce the following result.
 (i)
T is αadmissible,
 (ii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$,
 (iii)for all $x,y\in X$ with $\alpha (x,y)\ge 1$ we have$\begin{array}{rcl}d(Tx,Ty)+L\alpha (x,x)\alpha (y,y)& \le & ad(x,y)+\frac{b[1+d(x,Tx)]d(y,Ty)}{1+d(x,y)}\\ +c[d(x,Tx)+d(y,Ty)]\\ +d[d(y,Tx)+d(x,Ty)]+L,\end{array}$
 (iv)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha (x,x)\ge 1$ and $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$.
Then T has a fixed point.
 (i)
T is αadmissible,
 (ii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$,
 (iii)for all $x,y\in X$ with $\alpha (x,y)\ge 1$ we have$d(Tx,Ty)+L\alpha (x,x)\alpha (y,y)\le ad(x,y)+L,$
 (iv)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha (x,x)\ge 1$ and $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$.
Then T has a fixed point.
5.2 Implicit relation type Gcontraction
Definition 8 [15]
Definition 9 [15]
where $F\in \mathcal{F}$.
 (i)
there exists ${x}_{0}$ in X such that $({x}_{0},T{x}_{0})\in E(G)$,
 (ii)
T is an implicit relation type Gcontraction.
Then T has a fixed point.
 (i)
there exists ${x}_{0}$ in X such that $({x}_{0},T{x}_{0})\in E(G)$,
 (ii)
T is an implicit relation type Gcontraction,
 (iii)
if $\{{x}_{n}\}$ is a sequence in X such that $({x}_{n},{x}_{n+1})\in E(G)$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $({x}_{n},x)\in E(G)$ for all $n\in \mathbb{N}$.
Then T has a fixed point.
5.3 Implicit relation type ordered contraction
Theorem 11 ([3], Theorem 3.2)
 (i)
there exists ${x}_{0}$ in X such that ${x}_{0}\u2aafT{x}_{0}$,
 (ii)for all $x,y\in X$ with $x\u2aafy$ we have$F(d(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(y,Tx),d(x,Ty))\le 0,$
 (iii)
either T is continuous or if $\{{x}_{n}\}$ is an increasing sequence in X such that ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
Then T has a fixed point.
 (i)
there exists an element ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$,
 (ii)
if $\{{x}_{n}\}$ is an increasing sequence in X such that ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}\cup \{0\}$,
 (iii)for $x,y\in X$ with $x\u2aafy$,$\begin{array}{rcl}d(Tx,Ty)& \le & ad(x,y)+b\frac{[1+d(x,Tx)]d(y,Ty)}{1+d(x,y)}\\ +c[d(x,Tx)+d(y,Ty)]+d[d(x,Ty)+d(y,Tx)].\end{array}$
Then T has a fixed point.
 (i)
there exist element ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$,
 (ii)
if $\{{x}_{n}\}$ is an increasing sequence in X such that ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}\cup \{0\}$,
 (iii)for $x,y\in X$ with $x\u2aafy$,$\begin{array}{rcl}d(Tx,Ty)& \le & \psi (max\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(y,Tx)+d(x,Ty)}{2}\})\\ +Lmin\{d(x,Tx),d(y,Ty),d(y,Tx),d(x,Ty)\},\end{array}$
where $\psi \in \Psi $. Then T has a fixed point.
Declarations
Authors’ Affiliations
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