Our first main result is the following best proximity point theorem for a rational proximal contraction of the first kind.
Theorem 3.1 Let be a complete metric space and and ℬ be two nonempty, closed subsets of such that ℬ is approximatively compact with respect to . Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
is a rational proximal contraction of the first kind;
-
(b)
.
Then there exists such that . Further, for any fixed , the sequence , defined by , converges to x.
Proof Let (such a point there exists since ). Since , then by the definition of , there exists such that . Again, since , it follows that there is such that . Continuing this process, we can construct a sequence in , such that
for every nonnegative integer n. Using the fact that is a rational proximal contraction of the first kind, we have
It follows that
where . Therefore, is a Cauchy sequence and, since is complete and is closed, the sequence converges to some .
Moreover, we have
Taking the limit as , we get . Since ℬ is approximatively compact with respect to , then the sequence has a subsequence that converges to some . Therefore,
and hence x must be in . Since , then for some . Again, using the fact that is a rational proximal contraction of the first kind, we get
Taking the limit as , we have
which implies , since . Thus, it follows that
that is, . Now, to prove the uniqueness of the best proximity point (i.e., is singleton), assume that z is another best proximity point of so that
Since is a rational proximal contraction of the first kind, we have
which implies
It follows immediately that , since . Hence, has a unique best proximity point. □
As consequences of the Theorem 3.1, we state the following corollaries.
Corollary 3.1 Let be a complete metric space and and ℬ be two nonempty, closed subsets of such that ℬ is approximatively compact with respect to . Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
is a generalized proximal contraction of the first kind, with ;
-
(b)
.
Then, there exists such that . Further, for any fixed , the sequence , defined by , converges to the best proximity point x.
Corollary 3.2 Let be a complete metric space and and ℬ be two nonempty, closed subsets of such that ℬ is approximatively compact with respect to . Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
There exists a nonnegative real number such that, for all , , , in , the conditions and imply that ;
-
(b)
.
Then there exists such that . Further, for any fixed , the sequence , defined by , converges to the best proximity point x.
The following fixed-point result can be considered as a special case of the Theorem 3.1, when is a self-mapping.
Corollary 3.3 Let be a complete metric space and be a self-mapping on . Assume that there exist nonnegative real numbers α, β, γ, δ with such that
for all . Then the mapping 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 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 be a complete metric space and and ℬ be two nonempty, closed subsets of such that is approximatively compact with respect to ℬ. Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
is a continuous rational proximal contraction of the second kind;
-
(b)
.
Then there exists and for any fixed , the sequence , defined by , converges to x, and for all .
Proof Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence in such that
for every nonnegative integer n. Using the fact that is a rational proximal contraction of the second kind, we have
It follows that
where . Therefore, is a Cauchy sequence and, since is complete, then the sequence converges to some .
Moreover, we have
Taking the limit as , we get . Since is approximatively compact with respect to ℬ, then the sequence has a subsequence converging to some . Now, using the continuity of , we obtain that
that is, . Finally, to prove the last assertion of the present theorem, assume that z is another best proximity point of so that
Since is a rational proximal contraction of the second kind, we have
which implies
It follows immediately that , since . □
As consequences of the Theorem 3.2, we state the following corollaries.
Corollary 3.4 Let be a complete metric space and and ℬ be two nonempty, closed subsets of such that is approximatively compact with respect to ℬ. Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
is a continuous generalized proximal contraction of the second kind, with ;
-
(b)
.
Then, there exists and for any fixed , the sequence , defined by , converges to x. Further, for all .
Corollary 3.5 Let be a complete metric space and and ℬ be two nonempty, closed subsets of such that is approximatively compact with respect to ℬ. Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
There exists a nonnegative real number such that, for all , , , in , the conditions and imply that ;
-
(b)
is continuous;
-
(c)
.
Then there exists and for any fixed , the sequence , defined by , converges to x. Further, for all .
Remark 3.2 Note that in the Theorem 3.1 is not required the continuity of the mapping . On the contrary, the continuity of is an hypothesis of the Theorem 3.2.
Our next theorem concerns a nonself-mapping 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 nonself-mapping.
Theorem 3.3 Let be a complete metric space and and ℬ be two nonempty, closed subsets of . Assume that and are nonempty and is a nonself-mapping such that:
-
(a)
is a rational proximal contraction of the first and second kinds;
-
(b)
.
Then there exists a unique . Further, for any fixed , the sequence , defined by , converges to x.
Proof Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence in such that
for every nonnegative integer n. Also, using the same arguments in the proof of the Theorem 3.1, we deduce that the sequence is a Cauchy sequence, and hence converges to some . Moreover, on the lines of the proof of the Theorem 3.2, we obtain that the sequence is a Cauchy sequence and hence converges to some . Therefore, we have
and hence x must be in . Since , then for some . Using the fact that is a rational proximal contraction of the first kind, we get
Taking the limit as , we have
which implies that , since . Thus, it follows that
that is, . 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 . To avoid repetitions, we omit the details. □
Example 3.1 Let endowed with the usual metric , for all . Define and . Then, , and . Also define by
It is easy to show that is a rational proximal contraction of the first and second kinds and . Then all the hypotheses of the Theorem 3.3 are satisfied and . Clearly, the Theorem 3.2 is not applicable in this case.