Our first main result is the following best proximity point theorem for a rational proximal contraction of the first kind.
Theorem 3.1 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that ℬ is approximatively compact with respect to \mathcal{A}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a rational proximal contraction of the first kind;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in \mathcal{A} such that {B}_{\mathrm{est}}(\mathcal{T})=\{x\}. Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x.
Proof Let {x}_{0}\in {\mathcal{A}}_{0} (such a point there exists since {\mathcal{A}}_{0}\ne \mathrm{\varnothing}). Since \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}, then by the definition of {\mathcal{B}}_{0}, there exists {x}_{1}\in {\mathcal{A}}_{0} such that d({x}_{1},\mathcal{T}{x}_{0})=d(\mathcal{A},\mathcal{B}). Again, since \mathcal{T}{x}_{1}\in {\mathcal{B}}_{0}, it follows that there is {x}_{2}\in {\mathcal{A}}_{0} such that d({x}_{2},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B}). Continuing this process, we can construct a sequence \{{x}_{n}\} in {\mathcal{A}}_{0}, such that
d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}),
for every nonnegative integer n. Using the fact that \mathcal{T} is a rational proximal contraction of the first kind, we have
\begin{array}{rcl}d({x}_{n},{x}_{n+1})& \le & \alpha d({x}_{n1},{x}_{n})+\frac{\beta [1+d({x}_{n1},{x}_{n})]d({x}_{n},{x}_{n+1})}{1+d({x}_{n1},{x}_{n})}+\gamma [d({x}_{n1},{x}_{n})+d({x}_{n},{x}_{n+1})]\\ +\delta d({x}_{n1},{x}_{n+1})\\ \le & \alpha d({x}_{n1},{x}_{n})+\beta d({x}_{n},{x}_{n+1})+\gamma [d({x}_{n1},{x}_{n})+d({x}_{n},{x}_{n+1})]\\ +\delta [d({x}_{n1},{x}_{n})+d({x}_{n},{x}_{n+1})].\end{array}
It follows that
d({x}_{n},{x}_{n+1})\le kd({x}_{n1},{x}_{n}),
where k=\frac{\alpha +\gamma +\delta}{1\beta \gamma \delta}<1. Therefore, \{{x}_{n}\} is a Cauchy sequence and, since (\mathcal{X},d) is complete and \mathcal{A} is closed, the sequence \{{x}_{n}\} converges to some x\in \mathcal{A}.
Moreover, we have
\begin{array}{rcl}d(x,\mathcal{B})& \le & d(x,\mathcal{T}{x}_{n})\\ \le & d(x,{x}_{n+1})+d({x}_{n+1},\mathcal{T}{x}_{n})\\ =& d(x,{x}_{n+1})+d(\mathcal{A},\mathcal{B})\\ \le & d(x,{x}_{n+1})+d(x,\mathcal{B}).\end{array}
Taking the limit as n\to +\mathrm{\infty}, we get d(x,\mathcal{T}{x}_{n})\to d(x,\mathcal{B}). Since ℬ is approximatively compact with respect to \mathcal{A}, then the sequence \{\mathcal{T}{x}_{n}\} has a subsequence \{\mathcal{T}{x}_{{n}_{k}}\} that converges to some y\in \mathcal{B}. Therefore,
d(x,y)=\underset{k\to +\mathrm{\infty}}{lim}d({x}_{{n}_{k}+1},\mathcal{T}{x}_{{n}_{k}})=d(\mathcal{A},\mathcal{B}),
and hence x must be in {\mathcal{A}}_{0}. Since \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}, then d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}) for some u\in \mathcal{A}. Again, using the fact that \mathcal{T} is a rational proximal contraction of the first kind, we get
\begin{array}{rcl}d(u,{x}_{n+1})& \le & \alpha d(x,{x}_{n})+\frac{\beta [1+d(x,u)]d({x}_{n},{x}_{n+1})}{1+d(x,{x}_{n})}+\gamma [d(x,u)+d({x}_{n},{x}_{n+1})]\\ +\delta [d(x,{x}_{n+1})+d({x}_{n},u)].\end{array}
Taking the limit as n\to +\mathrm{\infty}, we have
d(u,x)\le (\gamma +\delta )d(u,x),
which implies x=u, since \gamma +\delta <1. Thus, it follows that
d(x,\mathcal{T}x)=d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}),
that is, x\in {B}_{\mathrm{est}}(\mathcal{T}). Now, to prove the uniqueness of the best proximity point (i.e., {B}_{\mathrm{est}}(\mathcal{T}) is singleton), assume that z is another best proximity point of \mathcal{T} so that
d(z,\mathcal{T}z)=d(\mathcal{A},\mathcal{B}).
Since \mathcal{T} is a rational proximal contraction of the first kind, we have
d(x,z)\le \alpha d(x,z)+\frac{\beta [1+d(x,x)]d(z,z)}{1+d(x,z)}+\gamma [d(x,x)+d(z,z)]+\delta [d(x,z)+d(z,x)]
which implies
d(x,z)\le (\alpha +2\delta )d(x,z).
It follows immediately that x=z, since \alpha +2\delta <1. Hence, \mathcal{T} has a unique best proximity point. □
As consequences of the Theorem 3.1, we state the following corollaries.
Corollary 3.1 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that ℬ is approximatively compact with respect to \mathcal{A}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a generalized proximal contraction of the first kind, with \alpha +2\gamma +2\delta <1;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then, there exists x\in \mathcal{A} such that {B}_{\mathrm{est}}(\mathcal{T})=\{x\}. Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to the best proximity point x.
Corollary 3.2 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that ℬ is approximatively compact with respect to \mathcal{A}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
There exists a nonnegative real number \alpha <1 such that, for all {u}_{1}, {u}_{2}, {x}_{1}, {x}_{2} in \mathcal{A}, the conditions d({u}_{1},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B}) and d({u}_{2},\mathcal{T}{x}_{2})=d(\mathcal{A},\mathcal{B}) imply that d({u}_{1},{u}_{2})\le \alpha d({x}_{1},{x}_{2});

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in \mathcal{A} such that {B}_{\mathrm{est}}(\mathcal{T})=\{x\}. Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to the best proximity point x.
The following fixedpoint result can be considered as a special case of the Theorem 3.1, when \mathcal{T} is a selfmapping.
Corollary 3.3 Let (\mathcal{X},d) be a complete metric space and \mathcal{T} be a selfmapping on \mathcal{X}. Assume that there exist nonnegative real numbers α, β, γ, δ with \alpha +\beta +2\gamma +2\delta <1 such that
\begin{array}{rcl}d(\mathcal{T}{x}_{1},\mathcal{T}{x}_{2})& \le & \alpha d({x}_{1},{x}_{2})+\frac{\beta [1+d({x}_{1},\mathcal{T}{x}_{2})]d({x}_{2},\mathcal{T}{x}_{2})}{1+d({x}_{1},{x}_{2})}+\gamma [d({x}_{1},\mathcal{T}{x}_{1})+d({x}_{2},\mathcal{T}{x}_{2})]\\ +\delta [d({x}_{1},\mathcal{T}{x}_{2})+d({x}_{2},\mathcal{T}{x}_{1})]\end{array}
for all {x}_{1},{x}_{2}\in \mathcal{X}. Then the mapping \mathcal{T} has a unique fixed point.
Remark 3.1 Note that the Corollary 3.3 is a proper extension of the contraction mapping principle [13] because the continuity of the mapping \mathcal{T} is not required. It is well known that a contraction mapping must be continuous.
Now, we state and prove a best proximity point theorem for a rational proximal contraction of the second kind.
Theorem 3.2 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that \mathcal{A} is approximatively compact with respect to ℬ. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a continuous rational proximal contraction of the second kind;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in {B}_{\mathrm{est}}(\mathcal{T}) and for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x, and \mathcal{T}x=\mathcal{T}z for all x,z\in {B}_{\mathrm{est}}(\mathcal{T}).
Proof Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence \{{x}_{n}\} in {\mathcal{A}}_{0} such that
d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}),
for every nonnegative integer n. Using the fact that \mathcal{T} is a rational proximal contraction of the second kind, we have
It follows that
d(\mathcal{T}{x}_{n},\mathcal{T}{x}_{n+1})\le kd(\mathcal{T}{x}_{n1},\mathcal{T}{x}_{n}),
where k=\frac{\alpha +\gamma +\delta}{1\beta \gamma \delta}<1. Therefore, \{\mathcal{T}{x}_{n}\} is a Cauchy sequence and, since (\mathcal{X},d) is complete, then the sequence \{\mathcal{T}{x}_{n}\} converges to some y\in \mathcal{B}.
Moreover, we have
\begin{array}{rcl}d(y,\mathcal{A})& \le & d(y,{x}_{n+1})\le d(y,\mathcal{T}{x}_{n})+d(\mathcal{T}{x}_{n},{x}_{n+1})\\ =& d(y,\mathcal{T}{x}_{n})+d(\mathcal{A},\mathcal{B})\le d(y,\mathcal{T}{x}_{n})+d(y,\mathcal{A}).\end{array}
Taking the limit as n\to +\mathrm{\infty}, we get d(y,{x}_{n})\to d(y,\mathcal{A}). Since \mathcal{A} is approximatively compact with respect to ℬ, then the sequence \{{x}_{n}\} has a subsequence \{{x}_{{n}_{k}}\} converging to some x\in \mathcal{A}. Now, using the continuity of \mathcal{T}, we obtain that
d(x,\mathcal{T}x)=\underset{k\to +\mathrm{\infty}}{lim}d({x}_{{n}_{k}+1},\mathcal{T}{x}_{{n}_{k}})=d(\mathcal{A},\mathcal{B}),
that is, x\in {B}_{\mathrm{est}}(\mathcal{T}). Finally, to prove the last assertion of the present theorem, assume that z is another best proximity point of \mathcal{T} so that
d(z,\mathcal{T}z)=d(\mathcal{A},\mathcal{B}).
Since \mathcal{T} is a rational proximal contraction of the second kind, we have
\begin{array}{rcl}d(\mathcal{T}x,\mathcal{T}z)& \le & \alpha d(\mathcal{T}x,\mathcal{T}z)+\frac{\beta [1+d(\mathcal{T}x,\mathcal{T}x)]d(\mathcal{T}z,\mathcal{T}z)}{1+d(\mathcal{T}x,\mathcal{T}z)}+\gamma [d(\mathcal{T}x,\mathcal{T}x)+d(\mathcal{T}z,\mathcal{T}z)]\\ +\delta [d(\mathcal{T}x,\mathcal{T}z)+d(\mathcal{T}z,\mathcal{T}x)]\end{array}
which implies
d(\mathcal{T}x,\mathcal{T}z)\le (\alpha +2\delta )d(\mathcal{T}x,\mathcal{T}z).
It follows immediately that \mathcal{T}x=\mathcal{T}z, since \alpha +2\delta <1. □
As consequences of the Theorem 3.2, we state the following corollaries.
Corollary 3.4 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that \mathcal{A} is approximatively compact with respect to ℬ. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a continuous generalized proximal contraction of the second kind, with \alpha +2\gamma +2\delta <1;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then, there exists x\in {B}_{\mathrm{est}}(\mathcal{T}) and for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x. Further, \mathcal{T}x=\mathcal{T}z for all x,z\in {B}_{\mathrm{est}}(\mathcal{T}).
Corollary 3.5 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that \mathcal{A} is approximatively compact with respect to ℬ. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
There exists a nonnegative real number \alpha <1 such that, for all {u}_{1}, {u}_{2}, {x}_{1}, {x}_{2} in \mathcal{A}, the conditions d({u}_{1},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B}) and d({u}_{2},\mathcal{T}{x}_{2})=d(\mathcal{A},\mathcal{B}) imply that d(\mathcal{T}{u}_{1},\mathcal{T}{u}_{2})\le \alpha d(\mathcal{T}{x}_{1},\mathcal{T}{x}_{2});

(b)
\mathcal{T} is continuous;

(c)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in {B}_{\mathrm{est}}(\mathcal{T}) and for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x. Further, \mathcal{T}x=\mathcal{T}z for all x,z\in {B}_{\mathrm{est}}(\mathcal{T}).
Remark 3.2 Note that in the Theorem 3.1 is not required the continuity of the mapping \mathcal{T}. On the contrary, the continuity of \mathcal{T} is an hypothesis of the Theorem 3.2.
Our next theorem concerns a nonselfmapping that is a rational proximal contraction of the first kind as well as a rational proximal contraction of the second kind. In this theorem, we consider only a completeness hypothesis without assuming the continuity of the nonselfmapping.
Theorem 3.3 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a rational proximal contraction of the first and second kinds;

(b)
{\mathcal{T}(\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists a unique x\in {B}_{\mathrm{est}}(\mathcal{T}). Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x.
Proof Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence \{{x}_{n}\} in {\mathcal{A}}_{0} such that
d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}),
for every nonnegative integer n. Also, using the same arguments in the proof of the Theorem 3.1, we deduce that the sequence \{{x}_{n}\} is a Cauchy sequence, and hence converges to some x\in \mathcal{A}. Moreover, on the lines of the proof of the Theorem 3.2, we obtain that the sequence \{\mathcal{T}{x}_{n}\} is a Cauchy sequence and hence converges to some y\in \mathcal{B}. Therefore, we have
d(x,y)=\underset{n\to +\mathrm{\infty}}{lim}d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}),
and hence x must be in {\mathcal{A}}_{0}. Since \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}, then d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}) for some u\in \mathcal{A}. Using the fact that \mathcal{T} is a rational proximal contraction of the first kind, we get
\begin{array}{rcl}d(u,{x}_{n+1})& \le & \alpha d(x,{x}_{n})+\frac{\beta [1+d(x,u)]d({x}_{n},{x}_{n+1})}{1+d(x,{x}_{n})}\\ +\gamma [d(x,u)+d({x}_{n},{x}_{n+1})]+\delta [d(x,{x}_{n+1})+d({x}_{n},u)].\end{array}
Taking the limit as n\to +\mathrm{\infty}, we have
d(u,x)\le (\gamma +\delta )d(u,x),
which implies that x=u, since \gamma +\delta <1. Thus, it follows that
d(x,\mathcal{T}x)=d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}),
that is, x\in {B}_{\mathrm{est}}(\mathcal{T}). Again, following the same lines of the proof of the Theorem 3.1, we prove the uniqueness of the best proximity point of the mapping \mathcal{T}. To avoid repetitions, we omit the details. □
Example 3.1 Let \mathcal{X}=\mathbb{R} endowed with the usual metric d(x,y)=xy, for all x,y\in \mathcal{X}. Define \mathcal{A}=[1,1] and \mathcal{B}=[3,2]\cup [2,3]. Then, d(\mathcal{A},\mathcal{B})=1, {\mathcal{A}}_{0}=\{1,1\} and {\mathcal{B}}_{0}=\{2,2\}. Also define \mathcal{T}:\mathcal{A}\to \mathcal{B} by
\mathcal{T}x=\{\begin{array}{cc}2\hfill & \text{if}x\text{is rational},\hfill \\ 3\hfill & \text{otherwise}.\hfill \end{array}
It is easy to show that \mathcal{T} is a rational proximal contraction of the first and second kinds and \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}. Then all the hypotheses of the Theorem 3.3 are satisfied and d(1,\mathcal{T}(1))=d(\mathcal{A},\mathcal{B}). Clearly, the Theorem 3.2 is not applicable in this case.