Abstract
Let and be nonempty subsets of a metric space and also and , . We are going to consider element such that for some . We call pair an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, -minimizing a sequence are given in a metric space.
1. Introduction
Let be a metric space and and nonempty subsets of , and is distance of and . If , then the pair is called a best proximity pair for and and put as the set of all best proximity pair . Best proximity pair evolves as a generalization of the concept of best approximation. That reader can find some important result of it in [1–4].
Now, as in [5] (see also [4, 6–11]), we can find the best proximity points of the sets and , by considering a map such that and . Best proximity pair also evolves as a generalization of the concept of fixed point of mappings. Because if , every best proximity point is a fixed point of .
We say that the point is an approximate best proximity point of the pair , if , for some .
In the following, we introduce a concept of approximate proximity pair that is stronger than proximity pair.
Definition 1.1. Let and be nonempty subsets of a metric space and a map such that , . put We say that the pair is an approximate best proximity pair if .
Example 1.2. Suppose that , , and with for . Then for some . Hence .
2. Approximate Best Proximity
In this section, we will consider the existence of approximate best proximity points for the map , such that , , and its diameter.
Theorem 2.1. Let and be nonempty subsets of a metric space . Suppose that the mapping is satisfying , , and Then the pair is an approximate best proximity pair.
Proof. Let be given and such that ; then there exists such that If , then , and and .
Theorem 2.2. Let and be nonempty subsets of a metric space . Suppose that the mapping is satisfying , and for all , where and . Then the pair is an approximate best proximity pair.
Proof. If , then Therefore, Now if , then also Therefore, and so Therefore, by Theorem 2.1, ; then pair is an approximate best proximity pair.
Definition 2.3. Let and be nonempty subsets of a metric space . Suppose that the mapping is satisfying , . We say that the sequence is T-minimizing if
Theorem 2.4. Let and be nonempty subsets of a metric space , suppose that the mapping is satisfying , . If is a T-minimizing for some , then is an approximate best pair proximity.
Proof. Since therefore, by Theorem 2.1, ; then pair is an approximate best proximity pair.
Theorem 2.5. Let and be nonempty subsets of a normed space such that is compact. Suppose that the mapping is satisfying , , is continuous and where . Then is nonempty and compact.
Proof. Since compact, there exists a such that
If , then which contradict to the definition of , ( and by (*) ). Therefore, for some and . Therefore, is nonempty.
Also, if , then , for some , and by compactness of , there exists a subsequence and a such that and so
for some , hence is compact.
Example 2.6. If , and such that then is compact, and we have That is compact.
In the following, by for a set , we will understand the diameter of the set .
Definition 2.7. Let be a continuous map such that , and . We define diameter by
Theorem 2.8. Let , such that , and . If there exists an such that for all then
Proof. If , then Put , therefore, . Hence .
3. Approximate Best Proximity for Two Maps
In this section, we will consider the existence of approximate best proximity points for two maps and , and its diameter.
Definition 3.1. Let and be nonempty subsets of a metric space and let two maps such that , . A point in is said to be an approximate-pair fixed point for in if there exists We say that the pair has the approximate-pair fixed property in if , where
Theorem 3.2. Let and be nonempty subsets of a metric space and let and be two maps such that , . If, for every , then has the approximate-pair fixed property.
Proof. For , Suppose . Since then for every . Put and . Hence and .
Theorem 3.3. Let and be nonempty subsets of a metric space and let and be two maps such that , and, for every , where and . Then if is an approximate fixed point for , or is an approximate fixed point for , then .
Proof. If , then Therefore, Now if , then also If is an approximate fixed point for , then there exists a and by (*) And ; also if is an approximate fixed point for , then there exists a and by (**) And . Therefore, .
Theorem 3.4. Let and be nonempty subsets of a metric space and let and be two continuous maps such that , . If, for every , where and , also let and be as follows: If has a convergent subsequence in , then there exists a such that .
Proof. We have If converges to , that is, , then Since is continuous, then Therefore, .
Definition 3.5. Let and be continues maps such that and . We define diameter by
Example 3.6. Suppose , , , and . Then and .
Theorem 3.7. Let and be continuous maps such that , . If there exists , then
Proof. If , then Therefore, . Then .