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Best proximity points of implicit relation type modified {\alpha}^{3}proximal contractions
Journal of Inequalities and Applications volume 2014, Article number: 365 (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.
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
Let A and B be two nonempty subsets of metric space (X,d) and T:A\to B be a nonself mapping. We say that {x}^{\ast} is a best proximity of T if
where
We define {A}_{0} and {B}_{0} as follows:
and
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.
Let ℱ be the set of all continuous functions F:{\mathbb{R}}_{+}^{6}\to \mathbb{R} satisfying the following assertions:

(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.
Example 1 Let
where L\ge 0 and \psi \in \Psi. Then F\in \mathcal{F}.
Example 2 Let
where a+b+2c+2d<1. Then F\in \mathcal{F}.
Definition 1 Let A, B be two nonempty subsets of a metric space (X,d) and \alpha :A\times A\to [0,+\mathrm{\infty}) be a function. We say that a nonself mapping T:A\to B is {\alpha}^{3}proximal admissible if, for all {x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A,
Definition 2 Let A and B be nonempty subsets of a metric space (X,d) and \alpha :A\times A\to [0,\mathrm{\infty}) be a function. Then T:A\to B is said to be an implicit relation type modified {\alpha}^{3}proximal contraction, if for all x,y,u,v\in A,
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.
Theorem 1 Let A, B be two nonempty subsets of a metric space (X,d) such that A is complete and {A}_{0} is nonempty. Assume that T:A\to B is a continuous implicit relation type modified {\alpha}^{3}proximal contraction such that the following conditions hold:

(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.
Then T has a best proximity point. Further, the best proximity point is unique if

(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.
Proof By (ii) there exist {x}_{0},{x}_{1}\in {A}_{0} such that
On the other hand, T({A}_{0})\subseteq {B}_{0}, then there exists {x}_{2}\in {A}_{0} such that
Now, since T is {\alpha}^{3}proximal admissible, we have
Hence,
Since T({A}_{0})\subseteq {B}_{0}, there exists {x}_{3}\in {A}_{0} such that
Then we have
Again, since T is {\alpha}^{3}proximal admissible, we obtain
Also, there exists {x}_{4}\in {A}_{0} such that
and hence
By continuing this process, we construct a sequence \{{x}_{n}\} such that
for all n\in \mathbb{N}. Now, from (4.2) with u={x}_{n}, v={x}_{n+1}, x={x}_{n1}, and y={x}_{n}, we get
On the other hand from (2.2) we obtain
That is, 1\alpha ({x}_{n1},{x}_{n1})\alpha ({x}_{n},{x}_{n})\le 0 for all n\in \mathbb{N}. Therefore,
Now, since F is decreasing in {t}_{5}
and so from (F1) we get
By induction, we have
Fix \u03f5>0, there exists N\in \mathbb{N} such that
Let m,n\in \mathbb{N} with m>n\ge N. Then by the triangular inequality, we get
Consequently {lim}_{m,n,\to +\mathrm{\infty}}d({x}_{n},{x}_{m})=0. Hence \{{x}_{n}\} is a Cauchy sequence. Since A is complete, there is z\in A such that {x}_{n}\to z. Since T is continuous, T{x}_{n}\to Tz as n\to \mathrm{\infty}. Hence,
Thus z is the desired best proximity point of T.
Let x,y\in A be two best proximity point of T such that x\ne y. That is, d(x,Tx)=d(A,B)=d(y,Ty). From (iii), we get \alpha (x,y)\ge 1, \alpha (x,x)\ge 1, and \alpha (y,y)\ge 1. So by (4.2) we derive
which implies
which is a contradiction to (F4). Hence, T has a unique best proximity point. □
Theorem 2 Let A, B be two nonempty subsets of a metric space (X,d) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B is an implicit relation type modified {\alpha}^{3}proximal contraction such that the following conditions hold:

(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.
Then T has a best proximity point. Further, the best proximity point is unique if

(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.
Proof Following the proof of Theorem 1, there exist a Cauchy sequence \{{x}_{n}\}\subseteq A and z\in A such that (4.2) holds and {x}_{n}\to z as n\to +\mathrm{\infty}. On the other hand, for all n\in \mathbb{N}, we can write
Taking the limit as n\to +\mathrm{\infty} in the above inequality, we get
Since B is approximatively compact with respect to A, the sequence \{T{x}_{n}\} has a subsequence \{T{x}_{{n}_{k}}\} that converges to some {y}^{\ast}\in B. Hence,
and so z\in {A}_{0}. Now, since T({A}_{0})\subseteq {B}_{0}, we have d(w,Tz)=d(A,B) for some w\in A. By (iii) and (2.2), we have \alpha ({x}_{n},z)\ge 1, \alpha (z,z)\ge 1, and d({x}_{n+1},T{x}_{n})=d(A,B) for all n\in \mathbb{N}\cup \{0\}. Also, since T is an implicit relation type {\alpha}^{3}proximal contraction, we get
Taking the limit as n\to +\mathrm{\infty} in the above inequality and applying continuity of F, we have
Now, if we take u=d(z,w) and v=0, then we have
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.
Corollary 1 Let A, B be two nonempty subsets of a metric space (X,d) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B is a nonself mapping satisfying the following conditions:

(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.
Then T has a best proximity point. Further, the best proximity point is unique if

(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.
Corollary 2 Let A, B be two nonempty subsets of a metric space (X,d) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B is a nonself mapping satisfying the following conditions:

(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.
Then T has a best proximity point. Further, the best proximity point is unique if

(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.
Example 3 Let X=\mathbb{R} be endowed with the usual metric d(x,y)=xy, for all x,y\in X. Consider A=(\mathrm{\infty},1], B=[1,+\mathrm{\infty}) and define T:A\to B by
Define \alpha :X\times X\to [0,+\mathrm{\infty}) by
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.
Assume
then
Therefore, {u}_{1}={u}_{2}=1, that is, \alpha ({u}_{1},{u}_{2})\ge 1, \alpha ({u}_{1},{u}_{1})\ge 1, and \alpha ({u}_{2},{u}_{2})\ge 1. Further,
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])
Let A and B be nonempty closed subsets of a complete metric space (X,d) such that B is approximatively compact with respect to A. Assume that a+b+2c+2d<1. Let {A}_{0} and {B}_{0} be nonempty and T:A\to B be a nonself mapping satisfying the following assertions:

(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}
Then there exists z\in A such that
By taking \alpha (x,y)=1 in Theorem 2, we deduce the following corollary.
Corollary 4 Let A, B be two nonempty subsets of a metric space (X,d) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B is a nonself mapping such that T{A}_{0}\subseteq {B}_{0} and for all x,y,u,v\in A,
where F\in \mathcal{F}. Then T has a unique best proximity point.
Using Example 1 and Corollary 4, we deduce the following result.
Corollary 5 Let A, B be two nonempty subsets of a metric space (X,d) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B is a nonself mapping such that T{A}_{0}\subseteq {B}_{0} and, for all x,y,u,v\in A,
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.
Definition 4 Let A, B be two nonempty closed subsets of a metric space (X,d) endowed with a graph G. Then T:A\to B is said to be an implicit relation type Gproximal contraction, if, for all x,y,u,v\in A,
and
where F\in \mathcal{F}.
Theorem 3 Let A, B be two nonempty closed subsets of a metric space (X,d) endowed with a graph G. Assume that A is complete, {A}_{0} is nonempty, and T:A\to B is a continuous implicit relation type Gproximal contraction such that the following conditions hold:

(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).
Proof Define \alpha :X\times X\to [0,+\mathrm{\infty}) by
Firstly, we prove that T is an {\alpha}^{3}proximal admissible mapping. To this aim, assume
Therefore, we have
Since T is an implicit relation type Gproximal contraction, we get (u,v)\in E(G). Also, since \Delta \subseteq E(G), (u,u),(v,v)\in E(G). That is, \alpha (u,v)\ge 1, \alpha (u,u)\ge 1, \alpha (v,v)\ge 1, and
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.
Theorem 4 Let A, B be two nonempty closed subsets of a metric space (X,d) endowed with a graph G. Assume that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Also suppose that T:A\to B is an implicit relation type Gproximal contraction mapping such that the following conditions hold:

(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).
Corollary 6 Let A, B be two nonempty closed subsets of a metric space (X,d) endowed with a graph G. Assume that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume a+b+2c+2d<1. Also, suppose that T:A\to B satisfies the following conditions:

(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).
Corollary 7 Let A, B be two nonempty closed subsets of a metric space (X,d) endowed with a graph G. Assume that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Also, suppose that T:A\to B satisfies the following conditions:

(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]
Let (X,d,\u2aaf) be a partially ordered metric space. We say that a nonself mapping T:A\to B is proximally orderedpreserving if and only if, for all {x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A,
Theorem 5 Let A, B be two nonempty closed subsets of a partially ordered metric space (X,d,\u2aaf) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B satisfies the following conditions:

(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.
Proof Define \alpha :A\times A\to [0,+\mathrm{\infty}) by
Firstly, we prove that T is an {\alpha}^{3}proximal admissible mapping. To this aim, assume
Therefore, we have
Now, since T is proximally orderedpreserving, then u\u2aafv, that is, \alpha (u,v)\ge 1. Further, by (ii) we have
Moreover, from (iii) we get
Thus all the conditions of Theorem 1 hold (when L=0) and T has a best proximity point. □
Theorem 6 Let A, B be two nonempty closed subsets of a partially ordered metric space (X,d,\u2aaf) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume that T:A\to B satisfies the following conditions:

(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.
Corollary 8 Let A, B be two nonempty closed subsets of a partially ordered metric space (X,d,\u2aaf) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Assume a+b+2c+2d<1. Also, suppose that T:A\to B satisfies the following conditions:

(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.
Corollary 9 Let A, B be two nonempty closed subsets of a partially ordered metric space (X,d,\u2aaf) such that A is complete, B is approximatively compact with respect to A, and {A}_{0} is nonempty. Also, suppose that T:A\to B satisfies the following conditions:

(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]
Let T be a selfmapping on X and \alpha :X\times X\to [0,+\mathrm{\infty}) be a function. We say that T is an αadmissible mapping if
Remark 1 Note that every αadmissible mappings are {\alpha}^{3}proximal admissible mappings when A=B=X.
Definition 7 Let (X,d) be a metric space and \alpha :A\times A\to [0,\mathrm{\infty}) be a function. Then T:X\to X is said to be an implicit relation type αcontraction, if for all x,y\in X with \alpha (x,y)\ge 1, we have
where L\ge 0 and F\in \mathcal{F}.
Theorem 7 Let (X,d) be a complete metric space. Assume that T:X\to X is a continuous selfmapping satisfying the following conditions:

(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.
Theorem 8 Let (X,d) be a complete metric space. Assume that T:X\to X is a selfmapping and the following conditions hold:

(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.
Corollary 10 Let (X,d) be a complete metric space. Assume that T:X\to X is a selfmapping and the following conditions hold:

(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}
where a+b+2c+2d<1 and L\ge 0,

(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.
Corollary 11 Let (X,d) be a complete metric space. Assume that T:X\to X is a selfmapping and the following conditions hold:

(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,
where 0\le a<1 and L\ge 0,

(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]
We say that a mapping T:X\to X is a Banach Gcontraction or simply Gcontraction if T preserves edges of G, i.e.,
and T decreases weights of edges of G in the following way:
Definition 9 [15]
A mapping T:X\to X is called Gcontinuous, if for given x\in X and sequence \{{x}_{n}\}
Definition 10 Let (X,d) be a metric space endowed with a graph G. Then T:X\to X is said to be an implicit relation type Gcontraction, if, for all x,y\in X,
and
where F\in \mathcal{F}.
Theorem 9 Let (X,d) be a complete metric space endowed with a graph G. Assume that T:X\to X is a continuous selfmapping satisfying the following conditions:

(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.
Theorem 10 Let (X,d) be a complete metric space endowed with a graph G. Assume that T:X\to X is a selfmapping satisfying the following conditions:

(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)
Let (X,d,\u2aaf) be a partially ordered complete metric space. Assume that T:X\to X is a selfmapping that satisfies the following conditions:

(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,
where F\in \mathcal{F},

(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.
Corollary 12 Let (X,d,\u2aaf) be complete metric space. Assume a+b+2c+2d<1. Also, suppose that T:X\to X is a selfmapping that satisfies the following conditions:

(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.
Corollary 13 Let (X,d,\u2aaf) be complete metric space. Assume that T:X\to X is a selfmapping that satisfies the following conditions:

(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.
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Zabihi, F., Razani, A. Best proximity points of implicit relation type modified {\alpha}^{3}proximal contractions. J Inequal Appl 2014, 365 (2014). https://doi.org/10.1186/1029242X2014365
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DOI: https://doi.org/10.1186/1029242X2014365
Keywords
 fixed point
 best proximity point
 {\alpha}^{3}proximal admissible mapping
 implicit relation type {\alpha}^{3}proximal contractions
 metric space endowed with graph