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Best proximity points for generalized αϕGeraghty proximal contraction mappings and its applications
Fixed Point Theory and Applications volume 2016, Article number: 72 (2016)
Abstract
In this paper, we introduce the new notion of generalized αϕGeraghty proximal contraction mappings and investigate the existence of the best proximity point for such mappings in complete metric spaces. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature.
Introduction
Let A and B be two nonempty subsets of a metric space \((X, d)\). An element \(x\in A\) is said to be a fixed point of a given map \(T : A\rightarrow B\) if \(T x = x\). Clearly, \(T (A)\cap A\neq\emptyset\) is a necessary (but not sufficient) condition for the existence of a fixed point of T. If \(T (A) \cap A=\emptyset\), then the set of fixed points of T is empty. In such a situation, one often attempts to find an element x which is in some sense closest to Tx. Best approximation theory and best proximity point analysis have been developed in this direction. Let \(A\cap B=\emptyset\) and \(T:A\rightarrow B\) be a nonselfmapping. A best proximity point of the mapping T is a point \(x^{*}\in A\) satisfying the equality \(d(x^{*},Tx^{*})=d(A,B)\), where \(d(A,B)=\inf\{ d(x,y): x\in A, y\in B\}\). The goal of best proximity point theory is to furnish sufficient conditions that ensure the existence of best proximity points. An operator \(T:A\cup B \rightarrow A\cup B\) is said to be a cyclic contraction if \(T(A)\subseteq B\) and \(T(B)\subseteq A\) and there exists \(k\in (0,1)\) such that
A best proximity point theorem for cyclic contraction mappings has been detailed by Anthony and Veeramani [1]. A great number of generalizations of this theorem appear in the literature. For more details of this approach, we refer the reader to [2–7]. We introduce the class \(\mathcal{F}\) of those functions \(\beta: [0,\infty)\rightarrow[0, 1)\) satisfying the following condition:
Recently, Karapinar [8] introduced a new class of contraction mappings called generalized αϕGeraghty contraction type mappings. Let Φ denote the class of all functions \(\phi: [0,\infty )\rightarrow[0,\infty)\) which satisfy the following conditions:

(a)
ϕ is nondecreasing;

(b)
ϕ is continuous;

(c)
\(\phi(t) = 0 \Leftrightarrow t = 0\).
Definition 1.1
([8])
Let \((X, d)\) be a metric space, and let \(\alpha:X\times X\rightarrow [0,\infty)\) be a function. A mapping \(T:X\rightarrow X\) is said to be a generalized αϕGeraghty contraction if there exists \(\beta\in\mathcal{F}\) such that for all \(x,y\in X\)
where \(\phi\in\Phi\) and
Let \(T:X \rightarrow X\) be a mapping and \(\alpha:X\times X\rightarrow [0,\infty)\) be a function. Then T is said to be αadmissible [9] if
Definition 1.2
([10])
Let \((X, d)\) be a complete metric space, \(\alpha: X\times X\rightarrow[0,\infty)\) be a function, and let \(T:X\rightarrow X\) be a mapping. The sequence \(\{x_{n}\}\) is said to be αregular if \(\alpha(x_{n}, x_{n+1}) \geq1\) for all \(n\in \mathbb{N}\) and \(x_{n}\rightarrow x\in X\) as \(n\rightarrow\infty\), implies that there exists a subsequence \(\{x_{n_{k}}\} \text{ of }\{x_{n}\} \) such that \(\alpha(x_{n_{k}}, x)\geq1 \text{ for all } k\).
The main result obtained in [8] is the following fixed point theorem.
Theorem 1.3
Let \((X, d)\) be a complete metric space, \(\alpha:X\times X\rightarrow \mathbb{R}\) be a function, and let \(T : X \rightarrow X\) be a map. Suppose that the following conditions are satisfied:

(i)
T is generalized αϕGeraghty contraction type map;

(ii)
T is triangular αadmissible;

(iii)
there exists \(x_{1} \in X\) such that \(\alpha(x_{1}, Tx_{1}) \geq1\);

(iv)
either, T is continuous, or any sequence \(\{x_{n}\}\) is αregular;
where an αadmissible map T is said to be triangular αadmissible [9] if
Then T has a fixed point \(x^{*}\in X\), and \(\{T^{n}x_{1}\}\) converges to \(x^{*}\).
We refer the reader to [11–13] for further examples.
In this work, we extend the concept of generalized αϕGeraghty contraction type mappings to generalized αϕGeraghty proximal contraction mappings to the case of nonself mappings. More precisely, we study the existence and uniqueness of best proximity points for generalized αϕGeraghty proximal contraction nonselfmappings. Several applications and interesting consequences of our obtained results are presented.
Let A and B be two nonempty subsets of a metric space \((X,d)\). We denote by \(A_{0}\) and \(B_{0}\) the following sets:
Definition 1.4
An element \(x^{*}\in A\) is said to be a best proximity point of the nonselfmapping \(T : A\rightarrow B\) if it satisfies the following condition:
We denote the set of all best proximity points of T by \(P_{T} (A)\), that is,
Let A and B be two nonempty closed subsets of metric space \((X,d)\). B is said to be approximatively compact with respect to A if every sequence \(\{y_{n}\}\) in B, satisfying the condition \(\lim_{n\rightarrow \infty}d(x,y_{n})=d(x,B)\) for some \(x\in A\), has a convergent subsequence.
Definition 1.5
([10]) Let \(T:A\rightarrow B\) be a map and \(\alpha: X\times X\rightarrow [0,\infty)\) be a function. The mapping T is said to be αproximal admissible if
for all \(x,y,u,v\in A\).
Main results
We start this section with the following definition.
Definition 2.1
Let A and B be two nonempty subsets of metric space \((X,d)\) and \(T:A\rightarrow B\) be a mapping. We say that T has the RJproperty if for any sequence \(\{x_{n}\} \subseteq A\),
In order to illustrate RJproperty, we present some examples.
Example 2.2
Let A and B be two nonempty closed subsets of metric space \((X,d)\) and \(T:A\rightarrow B\) be a continuous mapping. Let \(\lim_{n\rightarrow \infty}d (x_{n+1},Tx_{n})=d(A,B)\) and \(\lim_{n\rightarrow\infty}x_{n}=x\). Since T is continuous, \(\lim_{n\rightarrow\infty}Tx_{n}=Tx\). This implies that
Therefore \(x\in A_{0}\), which implies that T has the RJproperty.
Example 2.3
Let A and B be two nonempty closed subsets of metric space \((X,d)\) such that B is approximatively compact with respect to A and \(T:A\rightarrow B\) be a mapping. Let \(\lim_{n\rightarrow\infty}x_{n}=x\) and \(\lim_{n\rightarrow\infty}d (x_{n+1},Tx_{n})=d(A,B)\). For any \(n\in \mathbb{N}\), we have
Thus \(\lim_{n\rightarrow\infty}d (x,Tx_{n})=d(A,B)\). Also for any \(n\in \mathbb{N}\), we have \(d(x,B)\leq d(x,Tx_{n})\). Thus
which implies that \(\lim_{n\rightarrow\infty}d(x,Tx_{n})=d(x,B)\). Since B is approximatively compact with respect to A, there exist a subsequence \(\{x_{n_{k}}\}\subseteq\{x_{n}\}\) and \(y\in B\) such that \(\lim_{n\rightarrow\infty}Tx_{n_{k}}=y\). Hence
which implies that \(x\in A_{0}\). Therefore T has the RJproperty.
Lemma 2.4
Let \(T: A \rightarrow B\) be a triangular αproximal admissible mapping. Assume that \(\{x_{n}\}\) is a sequence in A such that \(\alpha(x_{n+1}, x_{n})\geq1\), for all \(n\in\mathbb{N}\). Then we have \(\alpha(x_{n}, x_{m})\geq1\) for all \(m, n \in\mathbb{ N}\) with \(n < m\).
Proof
Let \(n< m\) and \(m=n+k\). For \(k=1\), obviously we have \(\alpha (x_{n},x_{m})\geq1\). Let \(k>1\). Then
Since T is a triangular αadmissible mapping, \(\alpha (x_{n},x_{n+2})\geq1\). If \(k=2\), the proof is complete. Otherwise for \(k>2\) by continuing in this process we can complete the proof. □
Definition 2.5
Let A and B be two nonempty subsets of metric space \((X, d)\), and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. A mapping \(T:A\rightarrow B\) is said to be a generalized αϕGeraghty proximal contraction if there exists \(\beta\in\mathcal{F}\) such that for all \(x,y,u,v\in A\),
where
and \(\phi\in\Phi\).
Now we prove the following theorem, which extends, improves, and generalizes some earlier results in the literature on best proximity point theorems.
Theorem 2.6
Let A and B be two nonempty subsets of the complete metric space \((X, d)\), \(\alpha:X\times X\rightarrow\mathbb{R}\) be a function, and let \(T :A \rightarrow B\) be a mapping. Suppose that the following conditions are satisfied:

(i)
T is a generalized αϕGeraghty proximal contraction type mapping;

(ii)
\(T(A_{0})\subseteq B_{0}\) and T is triangular αproximal admissible;

(iii)
T has the RJproperty;

(iv)
if \(\{x_{n}\}\) is a sequence in A such that \(\alpha (x_{n}, x_{n+1}) \geq1\) for all n and \(x_{n}\rightarrow x\in A\) as \(n\rightarrow\infty\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n_{k}}, x)\geq1\) for all k;

(v)
there exist \(x_{0},x_{1}\in A\) such that
$$d(x_{1},Tx_{0})=d(A,B)\quad \textit{and} \quad \alpha(x_{0},x_{1})\geq1. $$
Then there exists an element \(x^{*}\in A_{0}\) such that
Moreover, if \(\alpha(x,y)\geq1\) for all \(x,y\in P_{T}(A)\), then \(x^{*}\) is the unique best proximity point of T.
Proof
Let \(x_{1}, x_{0} \in A\) be such that
Therefore \(x_{1}\in A_{0}\). Since \(T(A_{0})\subseteq B_{0}\), there exists \(x_{2}\in A_{0}\) such that \(d(x_{2},Tx_{1})=d(A,B)\). Now, we have
Since T is αproximal admissible, \(\alpha(x_{2},x_{1})\geq1\). Thus, we have
Continuing this process, by induction, we can construct a sequence \(\{ x_{n}\}\subseteq A_{0}\) such that
Therefore for any \(n\in\mathbb{N}\), we have
Since T is a generalized αϕGeraghty proximal contraction type mapping, we have
Also we have
If \(M(x_{n1},x_{n},x_{n},x_{n+1})=d(x_{n},x_{n+1})\), applying (2.2), we deduce that
which is a contradiction. Thus, we conclude that
Now, from (2.2) and (2.3), we get
Regarding the properties of ϕ, implies that
Hence, we deduce that the sequence \(\{d(x_{n}, x_{n+1})\}\) is nonnegative and decreasing. Consequently, there exists \(r \geq0\) such that \(\lim_{n\rightarrow\infty }d(x_{n},x_{n+1})=r\).
Suppose that there exists \(n_{0}\in\mathbb{N}\) such that \(d(x_{n_{0}},x_{n_{0}+1})=0\). This implies that \(x_{n_{0}} = x_{n_{0}+1}\). Applying (2.1), we deduce that
This is the desired result. Now let, for any \(n\in\mathbb{N}\), \(d(x_{n},x_{n+1})\neq0\). In the sequel, we prove that \(r = 0\). In the contrary case suppose that \(r > 0\). Then from (2.2) and (2.3), we have
which implies that \(\lim_{n\rightarrow\infty}\beta(\phi (d(x_{n1},x_{n})))=1\). Since \(\beta\in\mathcal{F}\),
This implies that \(r=0\), which is a contradiction. Therefore \(\lim_{n\rightarrow\infty}d(x_{n},x_{n+1})=0\). Now, we shall prove that \(\{ x_{n}\}\) is a Cauchy sequence in the metric space \((X,d)\). Note that for any \(m, n\in\mathbb{N}\), we have
Then, for any \(m,n\in\mathbb{N}\), we have
Also for any \(m,n\in\mathbb{N}\), we have
Since \(\lim_{n\rightarrow\infty}d(x_{n},x_{n+1})=0\),
In the following, we show that \(\{x_{n}\}\) is a Cauchy sequence. In the contrary case, we have
By using the triangular inequality and taking the limit as \(n\rightarrow\infty\), we derive
Combining (2.4), (2.5), and (2.6) with the continuity of ϕ, we get
Since \(\limsup_{m,n\rightarrow\infty}d(x_{n},x_{m})=r>0\), we deduce that
By taking the fact \(\beta\in\mathcal{F}\), we get
which is a contradiction. Therefore, \(\{x_{n}\}\) is a Cauchy sequence. Since \(\{x_{n}\}\) is a sequence in complete metric space \((X,d)\), there exists \(x^{*}\in X\) such that \(\lim_{n\rightarrow\infty} x_{n}=x^{*}\). RJproperty of T, implies that \(x^{*}\in A_{0}\). Since \(T(A_{0})\subseteq B_{0}\), there exists \(w\in A_{0}\) such that \(d(w,Tx^{*})=d(A,B)\). We shall prove that \(w=x^{*}\). In the contrary case let \(w\neq x^{*}\).
Property (iv) implies that there exists a subsequence \(\{x_{n_{k}}\} \text{ of }\{x_{n}\}\) such that \(\alpha(x_{n_{k}}, x)\geq1 \text{ for all } k\in\mathbb{N}\). Without loss of generality, we assume that
For any \(n\in\mathbb{N}\), we have \(d(x_{n+1},Tx_{n})=d(A,B)\text{ and }d(w,Tx^{*})=d(A,B)\). Using the fact that T is a generalized αϕGeraghty proximal contraction type mapping, for any \(n\in \mathbb{N}\), we have
Also for any \(n\in\mathbb{N}\), we have
Let there exist a subsequence \(\{x_{n_{k}}\}\subseteq\{x_{n}\}\) such that
Thus for any \(k\in\mathbb{N}\), we have
Taking the limit of both sides as \(n\rightarrow\infty\), implies that \(\beta(\phi(d(x^{*},w)))=1\), which is a contradiction. Thus there exists \(k\in\mathbb{N}\) such that
From this, together with (2.7), (2.8), and by taking the limit as \(n\rightarrow\infty\), we deduce that \(d(x^{*},w)=0\). This is a contradiction. Therefore \(x^{*}=w\), which implies that
Hence \(x^{*}\) is the best proximity point of T.
For the uniqueness, let \(\alpha(x,y)\geq1\) for all \(x,y\in P_{T}(A)\). Suppose that \(x_{1}\) and \(x_{2}\) are two best proximity points of T with \(x_{1}\neq x_{2}\). Therefore
Also, we have
Since \(\alpha(x_{1},x_{2})\geq1\) and T is a generalized αϕGeraghty proximal contraction type mapping, we get
which is a contradiction. Hence the best proximity point is unique. □
If in Theorem 2.6 we take \(\phi(t) = t\text{ for all }t \geq0\), then we deduce the following corollary.
Corollary 2.7
Let A and B be two nonempty closed subsets of the complete metric space \((X, d)\), \(\alpha:X\times X\rightarrow\mathbb{R}\) be a function, and let \(T :A \rightarrow B\) be a mapping. Suppose that the following conditions are satisfied:

(i)
T is a generalized αGeraghty proximal contraction type mapping, that is,
$$ \textstyle\begin{array}{lcl} \left . \textstyle\begin{array}{l} d (u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B) \end{array}\displaystyle \right \} \quad\Longrightarrow\quad \alpha(x,y)d(u,v) \leq\beta\bigl(M(x,y,u,v)\bigr)M(x,y,u,v), \end{array} $$where \(M(x,y,u,v)=\max\{d(x,y),d(x,u),d(y,v)\}\), for any \(x,y,u,v\in A\).

(ii)
The conditions (ii), (iii), (iv) and (v) of Theorem 2.6 are satisfied.
Then there exists an element \(x^{*}\in A_{0}\) such that
Moreover, if \(\alpha(x,y)\geq1\) for all \(x,y\in P_{T}(A)\), then \(x^{*}\) is the unique best proximity point of T.
By Example 2.2 a continuous map has the RJproperty and if all conditions of Theorem 2.6 are satisfied, then T has a best proximity point. In the next theorem, we prove that in Theorem 2.6, if mapping T is continuous, then condition (iv) is not needed.
Theorem 2.8
Let A, B be two nonempty subsets of the complete metric space \((X, d)\), \(\alpha:X\times X\rightarrow\mathbb{R}\) be a function, and let \(T :A \rightarrow B\) be a mapping. Suppose that the following conditions are satisfied:

(i)
The conditions (i), (ii) and (v) of Theorem 2.6 are satisfied;

(ii)
T is continuous.
Then there exists an element \(x^{*}\in A_{0}\) such that
Moreover, if \(\alpha(x,y)\geq1\) for all \(x,y\in P_{T}(A)\), then \(x^{*}\) is the unique best proximity point of T.
Proof
Let \(x_{0},x_{1} \in A\) be such that
Therefore \(x_{1}\in A_{0}\). Since \(T(A_{0})\subseteq B_{0}\), there exists \(x_{2}\in A_{0}\) such that \(d(x_{2},Tx_{1})=d(A,B)\). Now, we have
Since T is αproximal admissible, \(\alpha(x_{1},x_{2})\geq1\). Thus, we have
Continuing this process, by induction, we can construct a sequence \(\{ x_{n}\}\subseteq A_{0}\) such that
Following the lines in the proof of Theorem 2.6, there exists a sequence \(\{x_{n}\}\) such that \(d(x_{n+1},Tx_{n})=d(A,B)\) for all n, and the sequence \(\{x_{n}\}\) converges to some \(x^{*}\in A\). Since T is continuous, obviously
Therefore \(x^{*}\) is the best proximity point of T. If \(\alpha (x,y)\geq1\) for all \(x,y\in P_{T}(A)\), following the lines in the proof of Theorem 2.6, we see that the best proximity point is unique. □
To illustrate our results given in Theorem 2.6, we present the following example, which shows that Theorem 2.6 is a proper generalization of Corollary 2.7.
Example 2.9
Consider \(X=\mathbb{R}^{2}\) with the usual metric. Let A and B be the subsets of X defined by
Obviously, \(d(A,B) = 1\). Moreover, it is easily seen that \(A_{0} = A\). Let \(T : A\rightarrow B\) be the mapping defined as
Also define \(\alpha: X\times X\rightarrow\mathbb{R}\) by
In the sequel, we check that T is a generalized αϕGeraghty proximal contraction type mapping. Define \(\beta: [0,\infty )\rightarrow[0,1)\) and \(\phi: [0,\infty)\rightarrow[0,\infty)\) by
Then \(\beta\in\mathcal{F}\), \(\phi\in\Phi\). Let \(x,y\in A\). Then \(t=d(x,y)\in[0,5]\). Also, it is easy to show that
Let \(x=(0,x_{0})\), \(y=(0,y_{0})\), \(u=(0,u_{0})\), \(v=(0,v_{0})\in A\) satisfied the following conditions:
Then \(u_{0}=\frac{4}{5}\ln(x_{0}+1)\text{ and }v_{0}=\frac{4}{5}\ln(y_{0}+1)\). Since \(f(t)=\arctan(\frac{1}{2}t^{3})\) is a nondecreasing function, from (2.9), we have
Hence T is a generalized αϕGeraghty proximal contraction type mapping. Obviously, the other conditions of Theorem 2.6 are satisfied. Therefore T has an unique best proximity point.
Note that \(x^{*}=(0,0)\) is the best proximity point of T.
Applying Example 2.3 and Theorem 2.6 we have the following corollary.
Corollary 2.10
Let A, B be two nonempty subsets of the complete metric space \((X, d)\), \(\alpha:X\times X\rightarrow\mathbb{R}\) be a function, and let \(T :A \rightarrow B\) be a mapping. Suppose that the following conditions are satisfied:

(i)
The conditions (i), (ii), (iv) and (v) of Theorem 2.6 are satisfied;

(ii)
B is approximatively compact with respect to A.
Then there exists an element \(x^{*}\in A_{0}\) such that
Moreover, if \(\alpha(x,y)\geq1\) for all \(x,y\in P_{T}(A)\), then \(x^{*}\) is the unique best proximity point of T.
Applications in fixed point theory
As applications of our results, we prove some new fixed point theorems as follows. We start with the following fixed point theorem which is proved by Karapinar in [8].
Theorem 3.1
Let \((X, d)\) be a complete metric space, \(\alpha:X\times X\rightarrow \mathbb{R}\) be a function, and let \(T : X \rightarrow X\) be a map. Suppose that the following conditions are satisfied:

(i)
T is generalized αϕGeraghty contraction type map;

(ii)
T is triangular αadmissible;

(iii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}) \geq1\);

(iv)
either, T is continuous, or \(\{x_{n}\}\) is αregular.
Then T has a fixed point \(x^{*}\in X\), and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\).
Proof
Let \(A=B=X\). First, we prove that T is a generalized αϕGeraghty proximal contraction type map. Let \(x,y,u,v\in X\), satisfy the following conditions:
Since \(d(A,B)=0\), we have \(u=Tx\) and \(v=Ty\). T is a generalized αϕGeraghty contraction mapping, which implies that
Also,
Therefore
which implies that T is a generalized αϕGeraghty proximal contraction type map. Let
Then αadmissible property of T implies that \(\alpha (u,v)=\alpha(Tx,Ty)\geq1\). Therefore T is a triangular αproximal admissible mapping. Applying condition (iii), there exists \(x_{0} \in X\) such that \(\alpha (x_{0}, Tx_{0}) \geq1\).
If \(x_{1}=Tx_{0}\), then
Since the pair \((A,B)\) has the RJproperty, the conditions of Theorem 2.6 are satisfied, and so there exists \(x^{*}\in X\) such that \(d(x^{*},Tx^{*})=0\), which implies that \(Tx^{*}=x^{*}\). □
Remark 3.2
The fixed point of a generalized αϕGeraghty contraction mapping is unique if it satisfies the following condition:
(\(H_{1}\)) For all \(x, y\in \mathrm{Fix}(T)\), there exists \(z\in X\) such that \(\alpha(x, z) \geq1\) and \(\alpha(z, y)\geq1\).
Note that triangular αadmissible property of T and condition (\(H_{1}\)) imply that
Now, applying Theorems 2.6 and 2.8, the fixed point is unique.
Let \(\phi(t)=t\). Then we have the following definition and corollary.
Definition 3.3
([14]) Let \((X, d)\) be a metric space and \(\alpha:X\times X\rightarrow \mathbb {R}\) be a function. A map \(T:X\rightarrow X\) is said to be generalized αGeraghty contraction type map if there exists \(\beta\in\mathcal{F}\) such that
where
Corollary 3.4
([14]) Let \((X, d)\) be a complete metric space, \(\alpha:X\times X\rightarrow \mathbb{R}\) be a function, and let \(T : X \rightarrow X\) be a map. Suppose that the following conditions are satisfied:

(i)
T is generalized αGeraghty contraction map;

(ii)
T is triangular αadmissible;

(iii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}) \geq1\);

(iv)
either, T is continuous, or \(\{x_{n}\}\) is αregular.
Then T has a fixed point \(x^{*}\in X\), and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\).
Further, if for all \(x, y \in \mathrm{Fix}(T)\), there exists \(z\in X\) such that \(\alpha(x, z) \geq1\) and \(\alpha(z,y) \geq1\), and so fixed point of T is unique.
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Hamzehnejadi, J., Lashkaripour, R. Best proximity points for generalized αϕGeraghty proximal contraction mappings and its applications. Fixed Point Theory Appl 2016, 72 (2016). https://doi.org/10.1186/s1366301605610
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DOI: https://doi.org/10.1186/s1366301605610
MSC
 47H10
 54H25
 11J83
Keywords
 best proximity point
 generalized αϕGeraghty proximal contraction mapping
 Geraghty contraction mapping
 fixed point