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The existence of optimal approximate solution theorems for generalized αproximal contraction nonself mappings and applications
Fixed Point Theory and Applications volume 2013, Article number: 323 (2013)
Abstract
In this paper, we investigate the sufficient conditions to find a best proximity point for a certain class of nonself mappings. It is well known that optimization problems can be transformed to the problems of the existence of a best proximity point. Hence, improvement in the best proximity point theory implicitly develops the theory of optimization. Our presented results generalize, extent and improve various wellknown results on the topic in the literature. In particular, we consider some applications of our results to the best proximity point theorems on a class of metric spaces endowed with an arbitrary binary relation which involves the partially ordered metric spaces.
MSC:47H10, 54H25.
1 Introduction and preliminaries
Various nonlinear problems arising in several branches of mathematics, besides some other quantitative sciences such as economics, biology, physics and engineering, can be transformed to a fixed point problem of the form Tx=x for a selfmapping T defined on a subset of the metric space (X,d). Among them, optimization problems, differential equation problems, integral equation problems, variational problems, equilibrium problems have attracted attention of researchers. The renowned Banach contraction principle of Banach [1] is a very crucial and popular tool for solving fixed point problems in the setting of a selfmapping T. On the other hand, if T is not a selfmapping (T:A\to B where A and B are nonempty subsets of a metric space X), then T does not necessarily have a fixed point. Therefore, the equation Tx=x could have no solution. In this case, it is natural to search for a point x\in A in a way that the distance between x and its image Tx is as small as possible. In this discussion, the best approximation theorem plays an important role in studying the existence of an approximate solution that is optimal for the equation Tx=x.
Let A, B be subsets of the metric space (X,d). Under the setting of d(x,Tx)=d(A,B):=inf\{d(x,y):x\in A\text{and}y\in B\}, x is called a best proximity point. Notice that the point x\in A is the global minimum of the error involved for an approximate solution of the equation Tx=x. Therefore, we conclude that the purpose of the existence theorems for a best proximity point is to provide sufficient conditions to solve a minimization problem. One of the initial and most interesting best approximation theorems in this direction was given by Fan [2]. Following this result, several authors have reported a number of results in this field; see, e.g., [3–14] and the references therein.
The notion of approximate compactness, introduced by Efimov and Stechkin [15], plays an important role in the theory of approximation. The properties of approximately compact sets have been studied intensively. The following definition is a generalization of this concept.
Definition 1.1 Let A and B be two nonempty subsets of the metric space (X,d). Then B is said to be approximately compact with respect to A if every sequence \{{y}_{n}\} of B, satisfying the condition d(x,{y}_{n})\to d(x,B) as n\to \mathrm{\infty} for some x\in A, has a convergent subsequence.
Remark 1.1 Any nonempty subset of the metric space (X,d) is approximately compact with respect to itself. Moreover, if B is compact, then B is approximately compact with respect to A.
On the other hand, Samet et al. [16] introduced the concept of αadmissible mapping as follows.
Definition 1.2 [16]
Let X be a nonempty set, T:X\to X and \alpha :X\times X\to [0,\mathrm{\infty}). We say that T is an αadmissible if for all x,y\in X, we have
Using this concept, they proved some fixed point theorems. The authors also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces. Moreover, they applied their main results to solve certain types of ordinary differential equations. Afterward, a number of papers have reported on fixed point theory with applications to ordinary differential equations via the concept of αadmissible mapping in various directions (see [17–22] and references therein).
Recently, Jleli and Samet [23] introduced the notion of αproximal admissible mapping which is a nonself version of an αadmissible mapping.
Definition 1.3 [23]
Let A and B be two nonempty subsets of the metric space (X,d) and \alpha :A\times A\to [0,\mathrm{\infty}). A mapping T:A\to B is said to be αproximal admissible if for all u,v,x,y\in A,
Remark 1.2 If A=B, then the notion of αproximal admissibility T coincides with the concept of αadmissibility.
Inspired by [23], we introduce the class of nonself mappings, generalized αproximal contraction mappings. Also, we establish the existence theorems for a best proximity point for mappings in this class under the assumptions of approximate compactness and nonapproximate compactness of subspaces. Our presented results generalize, extent and improve various wellknown results such as the results of Banach [1], Kannan [24], Chatterjea [25], Berinde [26]. As an application, we apply our results to the existence theorems for a best proximity point on a metric space endowed with an arbitrary binary relation. Furthermore, we give the special case of these results in partially ordered metric spaces.
2 Best proximity point theorems for an αproximal contraction nonself mapping
In the sequel, unless otherwise specified, ℝ and ℕ denote the set of real numbers and the set of positive integers, respectively.
Let A and B be two nonempty subsets of a metric space (X,d). The following notations will be used in the sequel:
For a nonself mapping T:A\to B, we define the set of all best proximity points of T by {\mathcal{B}}_{est}(T), that is,
Remark 2.1 Kirk et al. [27] gave sufficient conditions to ensure that {A}_{0} and {B}_{0} are nonempty. Also, we obtain that if A and B are closed subsets of a normed linear space such that d(A,B)>0, then {A}_{0} and {B}_{0} are contained in the boundaries of A and B, respectively (see [28]).
Now, we introduce new classes of generalized proximal contraction nonself mappings.
Definition 2.1 Let A and B be two nonempty subsets of a metric space (X,d) and \alpha :A\times A\to [0,\mathrm{\infty}). A mapping T:A\to B is said to be a generalized αproximal contraction of the first kind if there exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
Definition 2.2 Let A and B be two nonempty subsets of the metric space (X,d) and \alpha :A\times A\to [0,\mathrm{\infty}). A mapping T:A\to B is said to be a generalized αproximal contraction of the second kind if there exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
Next, we give our first main result which is the best proximity point theorem for a generalized αproximal contraction.
Theorem 2.1 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X such that B is approximately compact with respect to A. Assume that \alpha :A\times A\to [0,\mathrm{\infty}), {A}_{0} and {B}_{0} are nonempty sets and T:A\to B is a generalized αproximal contraction of the first kind such that the following conditions hold:

(a)
T is αproximal admissible;

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge 1;

(d)
If \{{x}_{n}\} is a sequence in A such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then \alpha ({x}_{n},x)\ge 1 for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if \alpha ({z}_{1},{z}_{2})\ge 1 for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
Proof From (c), there exist {x}_{0},{x}_{1}\in A such that
and
Since T({A}_{0})\subseteq {B}_{0}, then by the definition of {B}_{0}, there exists {x}_{2}\in {A}_{0} such that
Since T is αproximal admissible, by (2.1), (2.2) and (2.3), we have
Again, since T({A}_{0})\subseteq {B}_{0}, there exists {x}_{3}\in {A}_{0} such that
Using (2.3), (2.4), (2.5) and the assumption that T is αproximal admissible, we get
Continuing this process, we can construct a sequence \{{x}_{n}\} in {A}_{0} such that
and
for all n\in \mathbb{N}. Since T is a generalized αproximal contraction of the first kind, for each n\in \mathbb{N}, we have
It follows that d({x}_{n},{x}_{n+1})\le Kd({x}_{n1},{x}_{n}) for all n\in \mathbb{N}, where K:=\frac{{\theta}_{1}+{\theta}_{2}+{\theta}_{3}}{1{\theta}_{2}{\theta}_{3}}\in [0,1). By induction we get
for all n\in \mathbb{N}. It easily follows that for m,n\in \mathbb{N} such that m<n,
This implies that \{{x}_{n}\} is a Cauchy sequence. Now, since X is complete and A is closed, the sequence \{{x}_{n}\} converges to some z\in A. Further, we have
for all n\in \mathbb{N}. Therefore, d(z,T{x}_{n})\to d(z,B) as n\to \mathrm{\infty}. Since B is approximately compact with respect to A, then the sequence \{T{x}_{n}\} has a subsequence \{T{x}_{{n}_{k}}\} converging to some element w\in B. Therefore, d(z,w)={lim}_{k\to \mathrm{\infty}}d({x}_{{n}_{k}+1},T{x}_{{n}_{k}})=d(A,B), and hence z must be a member of {A}_{0}. Because of the fact that T({A}_{0}) is contained in {B}_{0}, d(u,Tz)=d(A,B) for some element u in A.
From the hypothesis (d) and (2.8), we get
for all n\in \mathbb{N}. Now, we have
for all n\in \mathbb{N}. Since T is a generalized αproximal contraction of the first kind, we get
for all n\in \mathbb{N}. Taking the limit as n\to \mathrm{\infty} in the above inequality, we get z=u. Thus, it follows that d(z,Tz)=d(u,Tz)=d(A,B). Therefore, z is at least one of the best proximity points of T.
Finally, we prove the uniqueness of the best proximity point. Assume that \tilde{z} is another best proximity point of T such that \alpha (z,\tilde{z})\ge 1, then we have
Again, since T is a generalized αproximal contraction of the first kind, we obtain that
which implies that d(z,\tilde{z})=0, and then z=\tilde{z}. Hence T has a unique best proximity point. This completes the proof. □
Now we give some examples to support our result.
Example 2.2 Let X=\mathbb{R} with the usual metric d:X\times X\to [0,\mathrm{\infty}) defined by d(x,y):=xy. Clearly, (X,d) is a complete metric space. Also, let A=(\mathrm{\infty},2] and B=[\frac{1}{3},200] be two closed subsets of X. It is obtained that B is compact and so B is approximately compact with respect to A. Define a nonself mapping T:A\to B by
Clearly, d(A,B)=\frac{7}{3} and
It is easy to see that T({A}_{0})\subseteq {B}_{0}. Define \alpha :A\times A\to [0,\mathrm{\infty}) by
Now we show that T is a generalized αproximal contraction of the first kind. Assume that u,v,x,y\in A such that
Then
Since Tz\in [\frac{1}{3},\frac{8}{15}] for all z\in [3,2], we get u=v=2. Therefore,
where {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L are nonnegative real numbers with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1. This shows that T is a generalized αproximal contraction of the first kind.
Next, we prove that T is αproximal admissible. Assume that u,v,x,y\in A such that
Then we have x,y\in [3,2] and u=v=2. So \alpha (u,v)=\alpha (2,2)=1 and thus T is αproximal admissible.
It is easy to see that there exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge 1.
Assume that \{{x}_{n}\} is a sequence in A such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty}. Therefore, {x}_{n}\in [3,2] for all n\in \mathbb{N}. By the closeness of [3,2], we get x\in [3,2] and hence \alpha ({x}_{n},x)\ge 1 for all n\in \mathbb{N}.
Consequently, all the hypotheses of Theorem 2.1 are satisfied and so T has at least one best proximity point, that is, a point −2 such that
Example 2.3 Let X=\mathbb{R} with the usual metric d:X\times X\to [0,\mathrm{\infty}) defined by d(x,y):=xy. Clearly, (X,d) is a complete metric space. Also, let A=[1,1]\cup \{5,10\} and B=[\frac{1}{3},\frac{1}{3}]\cup \{5,10\} be two closed subsets of X. It is obtained that B is compact and so B is approximately compact with respect to A. Define a nonself mapping T:A\to B by
Clearly, d(A,B)=0 and
It is easy to see that T({A}_{0})\subseteq {B}_{0}. Define \alpha :A\times A\to [0,\mathrm{\infty}) by
Now we show that T is a generalized αproximal contraction of the first kind with {\theta}_{1}=\frac{1}{2}, {\theta}_{2}=\frac{1}{10}, {\theta}_{3}=\frac{1}{10} and L=2. Assume that u,v,x,y\in A such that
Then
Hence, u=Tx=\frac{1}{3}x and v=Ty=\frac{1}{3}y. Therefore,
This shows that T is a generalized αproximal contraction of the first kind.
Next, we prove that T is αproximal admissible. Assume that u,v,x,y\in A such that
Then we have x,y\in [1,1] and so
and
Thus \alpha (u,v)=\alpha (Tx,Ty)\ge 1. This implies that T is αproximal admissible.
It is easy to see that there exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge 1.
Suppose that \{{x}_{n}\} is a sequence in A such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty}. Therefore, {x}_{n}\in [1,1] for all n\in \mathbb{N}. Since [1,1] is closed, we get x\in [1,1] and hence \alpha ({x}_{n},x)\ge 1 for all n\in \mathbb{N}.
Consequently, all the hypotheses of Theorem 2.1 are satisfied and so T has at least one best proximity point, that is, a point 0 such that
It is easy to see that Theorem 2.1 yields the following corollary.
Corollary 2.4 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X such that B is approximately compact with respect to A. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B satisfies the following conditions:
(a′) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B).
Then there exists unique z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has a unique best proximity point.
Proof By taking \alpha (x,y)=1 for all x,y\in A in Theorem 2.1, we get this result. □
In Corollary 2.4, if T is a selfmapping, then we get the following fixed point theorem.
Corollary 2.5 Let (X,d) be a complete metric space, A be a nonempty, closed subset of X and T:A\to A. If there exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that
for all x,y\in A, then T has a unique fixed point.
Remark 2.2

1.
If {\theta}_{2}={\theta}_{3}=L=0 in Corollary 2.5, we get the generalized and improved result of Banach [1].

2.
If {\theta}_{1}={\theta}_{3}=L=0 in Corollary 2.5, we get the generalized and improved result of Kannan [24].

3.
If {\theta}_{1}={\theta}_{2}=L=0 in Corollary 2.5, we get the generalized and improved result of Chatterjea [25].

4.
If {\theta}_{2}={\theta}_{3}=0 in Corollary 2.5, we get the generalized and improved result of Berinde [26].
Next, we give the existence theorem of a best proximity point for a generalized αproximal contraction of the second kind.
Theorem 2.6 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X such that A is approximately compact with respect to B. Assume that \alpha :A\times A\to [0,\mathrm{\infty}), {A}_{0} and {B}_{0} are nonempty sets and T:A\to B is a continuous generalized αproximal contraction of the second kind such that the following conditions hold:

(a)
T is αproximal admissible;

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge 1.
Then there exists an element z\in {\mathcal{B}}_{est}(T). Further, T{z}_{1}=T{z}_{2} whenever {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T) and \alpha ({z}_{1},{z}_{2})\ge 1.
Proof Following the arguments in Theorem 2.1, we can construct a sequence \{{x}_{n}\} in {A}_{0} such that
and
for all n\in \mathbb{N}. Since T is a generalized αproximal contraction of the second kind, we have
for all n\in \mathbb{N}. It follows that d(T{x}_{n},T{x}_{n+1})\le Kd(T{x}_{n1},T{x}_{n}) for all n\in \mathbb{N}, where K:=\frac{{\theta}_{1}+{\theta}_{2}+{\theta}_{3}}{1{\theta}_{2}{\theta}_{3}}<1. Similar to the proof in Theorem 2.1, we obtain that the sequence \{T{x}_{n}\} is a Cauchy sequence in B. Since B is a closed subspace of the complete metric space X, B is complete. Then the sequence \{T{x}_{n}\} converges to some \stackrel{\u02c6}{y}\in B. Further, we have
for all n\in \mathbb{N}. Therefore, d(\stackrel{\u02c6}{y},{x}_{n})\to d(\stackrel{\u02c6}{y},A) as n\to \mathrm{\infty}. Since A is approximately compact with respect to B, then the sequence \{{x}_{n}\} has a subsequence \{{x}_{{n}_{k}}\} converging to some element z\in A. Now, using the continuity of T, we obtain that d(z,Tz)={lim}_{k\to \mathrm{\infty}}d({x}_{{n}_{k}+1},T{x}_{{n}_{k}})=d(A,B). Hence z\in {\mathcal{B}}_{est}(T).
Finally, we may assume that {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T) and \alpha ({z}_{1},{z}_{2})\ge 1. Therefore, we get
Since T is a generalized αproximal contraction of the second kind, we have
which implies that
It follows from {\theta}_{1}+2{\theta}_{3}\in [0,1) that d(T{z}_{1},T{z}_{2})=0 and hence T{z}_{1}=T{z}_{2}. This completes the proof. □
As a consequence of Theorem 2.6, we state the following corollary.
Corollary 2.7 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X such that A is approximately compact with respect to B. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B is continuous such that the following conditions hold:
(a″) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B).
Then there exists an element z\in {\mathcal{B}}_{est}(T). Further, T{z}_{1}=T{z}_{2} for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T).
Proof By taking \alpha (x,y)=1 for all x,y\in A in Theorem 2.6, we get this result. □
In Corollary 2.7, if T is a selfmapping, then we get the following fixed point theorem.
Corollary 2.8 Let (X,d) be a complete metric space, A be a nonempty, closed subset of X, and let T:A\to A be a continuous mapping. If there exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that
for all x,y\in A, then T has a unique fixed point.
In the next theorem, we give conditions for the existence of a best proximity point for a nonself mapping that is a generalized αproximal contraction of the first and second kinds. In this theorem, we consider only a completeness hypothesis without assuming the continuity of the nonself mapping and the approximate compactness of the subspace.
Theorem 2.9 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X. Assume that \alpha :A\times A\to [0,\mathrm{\infty}), {A}_{0} and {B}_{0} are nonempty sets and T:A\to B is a generalized αproximal contraction of the first and second kinds such that the following conditions hold:

(a)
T is αproximal admissible;

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge 1;

(d)
If \{{x}_{n}\} is a sequence in A such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then \alpha ({x}_{n},x)\ge 1 for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if \alpha ({z}_{1},{z}_{2})\ge 1 for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
Proof Following the arguments in Theorem 2.1, we can construct a sequence \{{x}_{n}\} in {A}_{0} such that
and
for all n\in \mathbb{N}.
Also, using similar arguments as in the proof of Theorem 2.1, we conclude that the sequence \{{x}_{n}\} is a Cauchy sequence, and hence converges to some z\in A. Moreover, on the lines of Theorem 2.6, we obtain that the sequence \{T{x}_{n}\} is a Cauchy sequence and hence converges to some \stackrel{\u02c6}{y}\in B. Therefore, we have d(z,\stackrel{\u02c6}{y})={lim}_{n\to \mathrm{\infty}}d({x}_{n+1},T{x}_{n})=d(A,B), hence z must be in {A}_{0}. Since T({A}_{0})\subseteq {B}_{0}, then d(\stackrel{\u02c6}{u},Tz)=d(A,B) for some \stackrel{\u02c6}{u}\in A.
From the assumption (d) and (2.12), we have
for all n\in \mathbb{N}.
Now we have
for all n\in \mathbb{N}. Using the fact that T is a generalized αproximal contraction of the first kind, we have
for all n\in \mathbb{N}. Taking the limit as n\to \mathrm{\infty} in the previous inequality, we have
Since {\theta}_{2}+{\theta}_{3}\in [0,1), we get d(z,\stackrel{\u02c6}{u})=0 and then z=\stackrel{\u02c6}{u}. Thus, it follows that d(z,Tz)=d(\stackrel{\u02c6}{u},Tz)=d(A,B). Therefore, z\in {\mathcal{B}}_{est}(T).
For the uniqueness of a best proximity point of T, we proceed similarly to the proof of Theorem 2.1. Then, in order to avoid repetition, the details are omitted. □
Corollary 2.10 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B satisfies the following conditions:
(a′) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
(a″) There exist nonnegative real numbers {\overline{\theta}}_{1}, {\overline{\theta}}_{2}, {\overline{\theta}}_{3} and \overline{L} with {\overline{\theta}}_{1}+2{\overline{\theta}}_{2}+2{\overline{\theta}}_{3}<1 such that for all u,v,x,y\in A,

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B).
Then there exists a unique z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has a unique best proximity point.
Proof By taking \alpha (x,y)=1 for all x,y\in A in Theorem 2.9, we get this result. □
3 Best proximity point theorems on a metric space endowed with an arbitrary binary relation
In this section, we apply our results in the previous section (Theorems 2.1, 2.6 and 2.9) to the best proximity point theorems on a metric space endowed with an arbitrary binary relation. Moreover, we obtain a special case of these results as corollaries in partially ordered metric spaces.
Before presenting our results, we need a few preliminaries. Let (X,d) be a metric space and ℛ be an arbitrary binary relation over X. Denote
this is the symmetric relation attached to R. Clearly,
Definition 3.1 Let (X,d) be a metric space, let ℛ be a binary relation over X, and let A and B be two nonempty subsets of X. We say that T:A\to B is a proximal comparative mapping if
for all u,v,x,y\in A.
We have the following best proximity point result.
Theorem 3.1 Let (X,d) be a complete metric space, let ℛ be a binary relation over X, and let A and B be two nonempty, closed subsets of X such that B is approximately compact with respect to A. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B such that the following conditions hold:

(A)
T is proximal comparative mapping;
(A′) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,

(B)
T({A}_{0})\subseteq {B}_{0};

(C)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and {x}_{0}\mathcal{S}{x}_{1};

(D)
If \{{x}_{n}\} is a sequence in A such that {x}_{n}\mathcal{S}{x}_{n+1} for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then {x}_{n}\mathcal{S}x for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if {z}_{1}\mathcal{S}{z}_{2} for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
Proof Define the mapping \alpha :A\times A\to [0,\mathrm{\infty}) by
Since T is a proximal comparative mapping, we have
for all u,v,x,y\in A. By the definition of α, we obtain that
for all u,v,x,y\in A. This shows that T is αproximal admissible.
From ({A}^{\prime}) and the construction of mapping α, we get
for all u,v,x,y\in A, that is, T is a generalized αproximal contraction of the first kind.
Since condition (C) implies that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge 1, condition (c) in Theorem 2.1 holds. It is easy to see that condition (D) implies condition (d) in Theorem 2.1.
Now, all the hypotheses of Theorem 2.1 are satisfied, and the desired result follows immediately from this theorem. □
Theorem 3.2 Let (X,d) be a complete metric space, let ℛ be a binary relation over X, and let A and B be two nonempty, closed subsets of X such that A is approximately compact with respect to B. Assume that {A}_{0} and {B}_{0} are nonempty and T:A\to B is continuous such that the following conditions hold:

(A)
T is a proximal comparative mapping;
(A″) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and {x}_{0}\mathcal{S}{x}_{1}.
Then there exists an element z\in {\mathcal{B}}_{est}(T). Further, T{z}_{1}=T{z}_{2} whenever {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T) and {z}_{1}\mathcal{S}{z}_{2}.
Proof The result follows from Theorem 2.6 by considering the mapping α as in the proof of Theorem 3.1. □
Theorem 3.3 Let (X,d) be a complete metric space, let ℛ be a binary relation over X, and let A and B be two nonempty, closed subsets of X. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B such that the following conditions hold:

(A)
T is a proximal comparative mapping;
(A′) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
(A″) There exist nonnegative real numbers {\overline{\theta}}_{1}, {\overline{\theta}}_{2}, {\overline{\theta}}_{3} and L with {\overline{\theta}}_{1}+2{\overline{\theta}}_{2}+2{\overline{\theta}}_{3}<1 such that for all u,v,x,y\in A,

(B)
T({A}_{0})\subseteq {B}_{0};

(C)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and {x}_{0}\mathcal{S}{x}_{1};

(D)
If \{{x}_{n}\} is a sequence in A such that {x}_{n}\mathcal{S}{x}_{n+1} for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then {x}_{n}\mathcal{S}x for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if {z}_{1}\mathcal{S}{z}_{2} for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
Proof This result follows from Theorem 2.9 by considering the mapping α given in the proof of Theorem 3.1. □
Next, we deduce Theorems 3.1, 3.2 and 3.3 to the special case in the context of partially ordered metric spaces. Before studying the next results, we give the following definitions.
Definition 3.2 Let X be a nonempty set. Then (X,d,\u2aaf) is called a partially ordered metric space if (X,d) is a metric space and (X,\u2aaf) is a partially ordered set.
For the partially ordered set (X,\u2aaf), we define
Definition 3.3 Let (X,d,\u2aaf) be a partially ordered metric space, and let A and B be two nonempty subsets of X. We say that T:A\to B is a proximal comparative mapping with respect to ⪯ if
for all u,v,x,y\in A.
It is obtained that ⪯ is a binary operation on X and \asymp \phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\u2aaf\cup {\u2aaf}^{1}. Therefore, we get the following best proximity point results in a partially ordered metric space.
Corollary 3.4 Let (X,d,\u2aaf) be a complete partially ordered metric space, and let A and B be two nonempty, closed subsets of X such that B is approximately compact with respect to A. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B such that the following conditions hold:

(A)
T is proximal comparative with respect to ⪯;
(A′) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,

(B)
T({A}_{0})\subseteq {B}_{0};

(C)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and {x}_{0}\asymp {x}_{1};

(D)
If \{{x}_{n}\} is a sequence in A such that {x}_{n}\asymp {x}_{n+1} for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then {x}_{n}\asymp x for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if {z}_{1}\asymp {z}_{2} for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
Corollary 3.5 Let (X,d,\u2aaf) be a complete partially ordered metric space, and let A and B be two nonempty, closed subsets of X such that A is approximately compact with respect to B. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B is continuous such that the following conditions hold:

(A)
T is a proximal comparative mapping with respect to ⪯;
(A″) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,

(B)
T({A}_{0})\subseteq {B}_{0};

(C)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and {x}_{0}\asymp {x}_{1}.
Then there exists an element z\in {\mathcal{B}}_{est}(T). Further, T{z}_{1}=T{z}_{2} whenever {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T) and {z}_{1}\asymp {z}_{2}.
Corollary 3.6 Let (X,d,\u2aaf) be a complete partially ordered metric space, and let A and B be two nonempty, closed subsets of X. Assume that {A}_{0} and {B}_{0} are nonempty sets and T:A\to B such that the following conditions hold:

(A)
T is a proximal comparative mapping with respect to ⪯;
(A′) There exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
(A″) There exist nonnegative real numbers {\overline{\theta}}_{1}, {\overline{\theta}}_{2}, {\overline{\theta}}_{3} and L with {\overline{\theta}}_{1}+2{\overline{\theta}}_{2}+2{\overline{\theta}}_{3}<1 such that for all u,v,x,y\in A,

(B)
T({A}_{0})\subseteq {B}_{0};

(C)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and {x}_{0}\asymp {x}_{1};

(D)
If \{{x}_{n}\} is a sequence in A such that {x}_{n}\asymp {x}_{n+1} for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then {x}_{n}\asymp x for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if {z}_{1}\asymp {z}_{2} for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
4 Remarks and conclusions
Recently, Salimi et al. [21] presented some fixed point results by modifying \alpha \psi contraction types in [16] and [29].
Definition 4.1 [21]
Let G:X\to X be a mapping on a metric space (X,d), and let \alpha ,\eta :X\times X\to [0,\mathrm{\infty}) be two functions. We say that G is an αadmissible mapping with respect to η if for x,y\in X, we have
Inspired by Salimi et al. [21] one can suggest the following definitions.
Definition 4.2 Let A and B be two nonempty subsets of a metric space (X,d) and \alpha ,\beta :A\times A\to [0,\mathrm{\infty}). A mapping T:A\to B is said to be an (\alpha ,\beta )proximal contraction of the first kind if there exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
Definition 4.3 Let A and B be two nonempty subsets of a metric space (X,d) and \alpha ,\beta :A\times A\to [0,\mathrm{\infty}). A mapping T:A\to B is said to be an (\alpha ,\beta )proximal contraction of the second kind if there exist nonnegative real numbers {\theta}_{1}, {\theta}_{2}, {\theta}_{3} and L with {\theta}_{1}+2{\theta}_{2}+2{\theta}_{3}<1 such that for all u,v,x,y\in A,
By replacing an (\alpha ,\beta )proximal contraction of the first kind with an αproximal contraction of the first kind (that is, by replacing Definition 4.2 with Definition 2.1) in the statements of Theorem 2.1, one can get the following theorem.
Theorem 4.1 Let (X,d) be a complete metric space, and let A and B be two nonempty, closed subsets of X such that B is approximately compact with respect to A. Assume that \alpha ,\beta :A\times A\to [0,\mathrm{\infty}), {A}_{0} and {B}_{0} are nonempty sets and T:A\to B is an (\alpha ,\beta )proximal contraction of the first kind such that the following conditions hold:

(a)
T is αproximal admissible with respect to β;

(b)
T({A}_{0})\subseteq {B}_{0};

(c)
There exist {x}_{0},{x}_{1}\in A such that d({x}_{1},T{x}_{0})=d(A,B) and \alpha ({x}_{0},{x}_{1})\ge \beta ({x}_{0},{x}_{1});

(d)
If \{{x}_{n}\} is a sequence in A such that \alpha ({x}_{n},{x}_{n+1})\ge \beta ({x}_{n},{x}_{n+1}) for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty} for some x\in A, then \alpha ({x}_{n},x)\ge \beta ({x}_{n},x) for all n\in \mathbb{N}.
Then there exists an element z\in {A}_{0} such that d(z,Tz)=d(A,B), that is, T has at least one best proximity point. Moreover, if \alpha ({z}_{1},{z}_{2})\ge \beta ({z}_{1},{z}_{2}) for all {z}_{1},{z}_{2}\in {\mathcal{B}}_{est}(T), then T has a unique best proximity point.
In this case, the proof is the analog of the proof of Theorem 2.1. In the same setting, the analogues of Theorem 2.6, Theorem 2.9, Theorem 3.1, Theorem 3.2, Theorem 3.3 can be obtained easily.
Very recently, Berzig and Karapınar [30] proved that the first main result of Salimi et al. [21] follows from a result of Karapınar and Samet [29]. Thus, Definition 4.2 and Definition 4.3 can be considered as a consequence of Definition 2.1 and Definition 2.2.
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Karapınar, E., Sintunavarat, W. The existence of optimal approximate solution theorems for generalized αproximal contraction nonself mappings and applications. Fixed Point Theory Appl 2013, 323 (2013). https://doi.org/10.1186/168718122013323
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DOI: https://doi.org/10.1186/168718122013323
Keywords
 approximately compact
 best proximity point
 binary relation
 generalized αproximal contraction of the first kind
 generalized αproximal contraction of the second kind