Abstract
The purpose of this paper is to present a new method for the research of best proximity point theorems of nonlinear mappings in metric spaces. In this paper, the P-operator technique, which changes non-self-mapping to self-mapping, provides a new and simple method of proof. Best proximity point theorems for weakly contractive and weakly Kannan mappings, generalized best proximity point theorems for generalized contractions, and best proximity points for proximal cyclic contraction mappings have been proved by using this new method. Meanwhile, many recent results in this area have been improved.
MSC:47H05, 47H09, 47H10.
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1 Introduction and preliminaries
Several problems can be changed to equations of the form , where T is a given self-mapping defined on a subset of a metric space, a normed linear space, a topological vector space or some suitable space. However, if T is a non-self-mapping from A to B, then the aforementioned equation does not necessarily admit a solution. In this case, one would contemplate finding an approximate solution x in A such that the error is minimum, where d is the distance function. In view of the fact that is at least , a best proximity point theorem (for short BPPT) guarantees the global minimization of by the requirement that an approximate solution x satisfies the condition . Such optimal approximate solutions are called best proximity points of the mapping T. Interestingly, best proximity point theorems also serve as a natural generalization of fixed point theorems, for a best proximity point becomes a fixed point if the mapping under consideration is a self-mapping. Research on the best proximity point is an important topic in the nonlinear functional analysis and applications (see [1–18]).
Let A, B be two nonempty subsets of a complete metric space and consider a mapping . The best proximity point problem is whether we can find an element such that . Since for any , in fact, the optimal solution to this problem is the one for which the value attained.
Let A, B be two nonempty subsets of a metric space . We denote by and the following sets:
where .
It is interesting to notice that and are contained in the boundaries of A and B, respectively, provided A and B are closed subsets of a normed linear space such that (see [1]).
2 BPPT for weakly contractive and weakly Kannan mappings
Let A and B be nonempty subsets of a metric space . An operator is said to be contractive if there exists such that for any . The well-known Banach contraction principle says: Let be a complete metric space, and be a contraction of X into itself. Then T has a unique fixed point in X.
In the last 50 years, the Banach contraction principle has been extensively studied and generalized in many settings. One of the generalizations is the weakly contractive mapping.
Definition 2.1 [3]
Let be a metric space. A mapping is said to be weakly contractive provided that
for all , where the function , holds, for every , that
The fixed point theorem for weakly contractive mapping was presented in [3].
Theorem 2.2 Let be a complete metric space. If is a weakly contractive mapping, then f has a unique fixed point and the Picard sequence of iterates converges, for every , to .
One type of contraction which is different from the Banach contraction is Kannan mappings. In [11], Kannan obtained the following fixed point theorem.
Theorem 2.3 [11]
Let be a complete metric space and let be a mapping such that
for all and some , then f has a unique fixed point . Moreover, the Picard sequence of iterates converges, for every , to .
In [12], the authors introduce a more general weakly Kannan mapping and obtain its fixed point theorem.
Definition 2.4 [12]
Let be a metric space. A mapping is said to be weakly Kannan if there exists , which satisfies for every and for all
and
Theorem 2.5 [12]
Let be a complete metric space. If is a weakly Kannan mapping, then f has a unique fixed point and the Picard sequence of iterates converges, for every , to .
In this section, we first obtain best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in metric spaces. Further, we extend the results to partial metric spaces. The P-operator technique, which changes non-self-mapping to self-mapping, provides a new and simple proof. Many recent results in this area have been improved.
Before giving the main results, we need the following notations and basic facts.
Definition 2.6 [13]
Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the P-property if and only if for any and ,
In [13], the author proves that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.
In [4], the P-property has been weakened to the weak P-property. An example that satisfies the P-property but not the weak P-property can be found there.
Definition 2.7 [4]
Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the weak P-property if and only if for any and ,
Example [4]
Consider , where d is the Euclidean distance and the subsets and .
Obviously, , and . Furthermore,
however,
We can see that the pair satisfies the weak P-property but not the P-property.
Firstly, we present the following definitions.
Definition 2.8 Let be a pair of nonempty closed subsets of a complete metric space . A mapping is said to be weakly contractive provided that
for all , where the function holds, for every , and
Definition 2.9 Let be a pair of nonempty closed subsets of a complete metric space. A mapping is said to be weakly Kannan if there exists which satisfies for every and for all
and
Next we prove the best proximity point theorems for weakly contractive and weakly Kannan mappings in metric spaces.
Theorem 2.10 Let be a pair of nonempty closed subsets of a complete metric space such that . Let be a weakly contractive mapping defined as Definition 2.8. Suppose that and the pair has the weak P-property. Then T has a unique best proximity point and the iteration sequence defined by
converges, for every , to .
Proof We first prove that is closed. Let be a sequence such that . It follows from the weak P-property that
as , where and , . Then is a Cauchy sequence so that converges strongly to a point . By the continuity of metric d we have , that is, and hence is closed.
Let be the closure of ; we claim that . In fact, if , then there exists a sequence such that . By the continuity of T and the closedness of we have . That is .
Define an operator , by . Since the pair has the weak P-property, we have
for any . This shows that is a weak contraction from complete metric subspace into itself. Using Theorem 2.2, we can see that has a unique fixed point . That is, , which implies that
Therefore, is the unique one in such that . It is easy to see that is also the unique one in A such that . The Picard iteration sequence
converges, for every , to . The iteration sequence defined by
is exactly the subsequence of , so that it converges, for every , to . This completes the proof. □
Theorem 2.11 Let be a pair of nonempty closed subsets of a complete metric space such that . Let be a continuous weakly Kannan mapping defined as Definition 2.9. Suppose that and the pair has the weak P-property. Then T has a unique best proximity point and the iteration sequence defined by
converges, for every , to .
Proof The closedness of and have been proved in Theorem 2.10. Now we define an operator , by . Since the pair has weak P-property, we have
for any . This shows that is a weakly Kannan mapping from complete metric subspace into itself. Using Theorem 2.5, we can see that a unique fixed point . That is, , which implies that
Therefore, is the unique one in such that . It is easy to see that is also the unique one in A such that . The Picard iteration sequence
converges, for every , to . Since the iteration sequence defined by
is exactly the subsequence of , it converges, for every , to . This completes the proof. □
Example 2.12 Let , , , and define as follows:
We have , , . It is obvious that satisfy the P-property so it must satisfy the weakly P-property. Meanwhile
where
That is, f is a weakly contractive mapping. All conditions of Theorem 2.10 hold, the conclusion of Theorem 2.10 is also correct, that is, f has a unique best proximity point such that . On the other hand, it is obvious that the iteration sequence defined by
converges, for every , to , since
In fact, from we know that , so there exists a number such that . Furthermore, and hence .
Example 2.13 Let , , . For , , we have the following equivalence relations:
We define a function as follows:
From the above equivalence relations, we get
Therefore, we define a mapping as follows:
We have , , . It is obvious that satisfy the P-property and so must satisfy the weakly P-property. Meanwhile the following inequality holds:
where . That is, T is a continuous weakly Kannan mapping. All conditions of Theorem 2.11 hold, the conclusion of Theorem 2.11 is also correct, that is, T has a unique best proximity point such that . On the other hand, it is obvious that the iteration sequence defined by
converges, for every , to , since .
3 Generalized BPPT for generalized contractions
Definition 3.1 [3]
A mapping is said to be a proximal contraction of the first kind if there exists a non-negative number such that
for all .
Recently in [9], Amini-Harandi et al. introduced the following new class of proximal contractions and proved the following result.
Definition 3.2 [9]
A mapping is said to be a -contraction if
for all , where is a function obeying the following conditions:
and is a mapping. If g is the identity operator, is said to be a φ-contraction.
Definition 3.3 An element x in A is said to be a best proximity point of the mapping if it satisfies the condition that .
Theorem 3.4 [9]
Let A and B be nonempty closed subsets of a complete metric space such that B is approximately compact with respect to A. Moreover, assume that and are nonempty. Let and satisfy the following conditions.
-
(a)
T is a -proximal contraction,
-
(b)
,
-
(c)
g is a one-to-one continuous map such that is uniformly continuous,
-
(d)
.
Then there exists a unique element such that . Further, for any fixed element , the sequence defined by converges to x.
The purpose of this section is to improve the result of Amini-Harandi et al. by using a new simple method of proof without the hypothesis of approximate compactness to B.
The following lemma is important for our results, which is actually a generalized Banach’s fixed point theorem.
Lemma 3.5 Let A be a subset of a complete metric space , and let a continuous mapping with conditions
and , where is a function obeying the following conditions:
Then for any fixed element , the sequence defined by converges to a point . Further, x is a fixed point of T.
Proof We claim that is a Cauchy sequence. Suppose, to the contrary, that is not a Cauchy sequence. Then there exist and two subsequences of integers , such that
Since () is obvious, we may also assume
by choosing to be the smallest number exceeding for which (3.1) holds. From (3.1) and (3.2) we have
Taking the limit as , we get
By the triangle inequality
Taking the sup-limit as , we get
a contradiction. Therefore is a Cauchy sequence. Since X is complete, there exists such that . It is obvious from the continuity of T and that x is a fixed point of T. This completes the proof. □
Now, we are ready to state our main result in this section.
Theorem 3.6 Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let and satisfy the following conditions.
-
(a)
g is a one-to-one continuous map such that is uniformly continuous;
-
(b)
T is a -contraction with .
Then there exists a unique element such that . Further, for any fixed element , the sequence defined by , converges to .
Proof Let
It is obvious that is a metric on the A. Now we prove is a complete metric space. Let be a Cauchy sequence, we have
Since is uniformly continuous, we have
and hence is a Cauchy sequence. Since is a complete metric space, there exists an element such that as . Since g is continuous, we have as . This completes the proof of the completeness of .
For any , from (b) we know . Since T is a -contraction, there exists a unique such that . We denote . Then is a mapping. Further, we define a composite mapping from into itself. Since T is a -contraction, we have
for any ,
for any , and
for any . From the above inequality, we also know that the mapping is continuous on the , so we can expand the definition of onto such that it is still continuous on the . By using Lemma 3.5, we know for any fixed element , that the sequence defined by
which is equivalent to , converges to a point . Further, is a fixed point of . That is, which is equivalent to . Since T is a -contraction, this is unique. This completes the proof. □
Corollary 3.7 Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let and satisfy the following conditions.
-
(a)
g is a one-to-one continuous map such that is uniformly continuous;
-
(b)
T is a proximal contraction of the first kind with .
Then there exists a unique element such that . Further, for any fixed element , the sequence defined by converges to .
Corollary 3.8 Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let be is a φ-contraction with . Then there exists a unique best proximity point of T. Further, for any fixed element , the sequence defined by converges to .
Remark 3.9 In Theorem 3.6, we do not need the hypothesis of approximate compactness to B. Therefore, Theorem 3.6 improved substantially the results of Theorem 3.4. On the other hand, the method of proof is also different.
4 BPPT for proximal cyclic contraction mappings
Definition 4.1 [1]
Given non-self-mappings and , the pair is said to form a proximal cyclic contraction if there exists a non-negative number such that
for all and .
Definition 4.2 [1]
A mapping is said to be a proximal contraction of the first kind if there exists a non-negative number such that
for all .
Definition 4.3 An element x in A is said to be a best proximity point of the mapping if it satisfies the condition that .
In [1], the author proved the following result, a generalized best proximity point theorem for non-self-proximal contractions of the first kind.
Theorem 4.4 [1]
Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let , and satisfy the following conditions.
-
(a)
S and T are proximal contractions of the first kind.
-
(b)
and .
-
(c)
The pair forms a proximal cyclic contraction.
-
(d)
g is an isometry.
-
(e)
and .
Then there exist a unique element x in A and a unique element y in B satisfying the conditions that
Further, for any fixed element in , the sequence , defined by
converges to the element x. For any fixed element in , the sequence , defined by
converges to the element y.
On the other hand, a sequence of elements in A converges to x if there is a sequence of positive numbers for which , , where satisfies the condition that .
In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 4.6.
Definition 4.5 [14]
Let be a metric space. A mapping is said to be a Geraghty-contraction if there exists such that for any
where the class Γ denotes those functions satisfying the following condition:
Theorem 4.6 [14]
Let be a complete metric space and be a Geraghty-contraction. Then T has a unique fixed point and, for any , the iterative sequence converges to .
Definition 4.7 [2]
A mapping is called Geraghty’s proximal contraction of the first kind if there exists such that
for all .
In [2], the authors proved the following result.
Theorem 4.8 [2]
Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let , , and satisfy the following conditions.
-
(a)
S, T are Geraghty’s proximal contractions of the first kind.
-
(b)
and .
-
(c)
The pair forms a proximal cyclic contraction.
-
(d)
g is an isometry.
-
(e)
and .
Then there exist a unique element in A and a unique element in B satisfying the conditions that
Further, for any fixed element in , the sequence , defined by
converges to the element . For any fixed element in , the sequence , defined by
converges to the element .
On the other hand, a sequence of elements in A converges to x if there is a sequence of positive numbers for which , , where satisfies the condition that .
The purpose of this section is to prove best proximity point theorems for proximal cyclic contractions and weakly proximal contractions by using the new method of proof. Our results improve and extend the recent results of some others. Meanwhile, we point out a mistake in Theorem 4.8.
Theorem 4.9 Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let , , and satisfy the following conditions.
-
(a)
S, T are Geraghty’s proximal contractions of the first kind.
-
(b)
and .
-
(c)
The pair forms a proximal cyclic contraction.
-
(d)
g is an isometry.
-
(e)
and .
Then there exist a unique element in A and a unique element in B satisfying the conditions that
Further, for any fixed element in , the sequence , defined by
converges to the element . For any fixed element in , the sequence , defined by
converges to the element .
On the other hand, assume . Then a sequence of elements in A converges to if there is a sequence of positive numbers for which , , where satisfies the condition that .
Proof For any , from (b) we know . Since S is a Geraghty-contraction, there exists a unique such that . We denote . Then is a mapping. Further, we define a composite mapping from into itself. Since S is a Geraghty-contraction, we have
for any . From the above inequality, we also know that the mapping is continuous, so we can expand the definition of onto . Because we do not need the continuity of function , we define another function as follows:
It is easy to see . From (4.1) we get
for any . From (4.2) we know is a Geraghty-contraction. By using Theorem 4.6, we claim has a unique fixed point in , that is, , which implies and hence . By using the same method, we can prove that there exists a unique element in such that . On the other hand, from (c) we have
which implies and hence .
Since is a Geraghty-contraction, for any fixed element in , the sequence , defined by converges to the element . This sequence also is defined by . For the same reason, for any fixed element in , the sequence , defined by , converges to the element . This sequence also is defined by .
Finally, , which gives us
It is easy to prove which implies . This completes the proof. □
Remark 4.10 If , then Theorem 4.9 yields Theorem 4.4.
Remark 4.11 In the reference [2], from line 15 to line 20 on page 7, the following argument is wrong, so the final conclusion of Theorem 1.8 is not correct.
The wrong argument For any , choose a positive integer N such that for all . Observe that
Since ε is arbitrary, we can conclude that for all the sequence is nonincreasing and bounded below and hence converges to some non-negative real number r.
Counter-example Let
then
and, for any , we can choose a positive integer N such that for all , and hence
But is not nonincreasing and as .
Next we prove the best proximal point theorem for a weakly proximal contractive mapping.
Definition 4.12 Let A and B be nonempty subsets of a complete metric space. A mapping is called weak proximal contraction if
for all , where for the function we have, for every ,
Theorem 4.13 Let A and B be nonempty closed subsets of a complete metric space such that and are nonempty. Let , and satisfy the following conditions.
-
(a)
S, T are weakly proximal contractions.
-
(b)
and .
-
(c)
The pair forms a proximal cyclic contraction.
-
(d)
g is an isometry.
-
(e)
and .
Then there exist a unique element in A and a unique element in B satisfying the conditions that
Further, for any fixed element in , the sequence , defined by
converges to the element . For any fixed element in , the sequence , defined by
converges to the element .
On the other hand, assume . Then a sequence of elements in A converges to if there is a sequence of positive numbers for which , , where satisfies the condition that .
Proof For any , from (b) we know . Since S is a weakly contractive mapping, there exists a unique such that . We denote . Then is a mapping. Further, we define a composite mapping from into itself. Since S is a weakly contractive mapping, then we have
for any . From above inequality, we also know the mapping is continuous, so we can expand the definition of onto . From the above inequality we know that is a weak contractive mapping. By using Theorem 2.2, we claim that has a unique fixed point in , that is, , which implies and hence . By using the same method, we can prove that there exists a unique element in such that . On the other hand, from (c) we have
which implies and hence .
Since is a weak contractive mapping, for any fixed element in , the sequence , defined by converges to the element . This sequence also is defined by . By the same reason, for any fixed element in , the sequence , defined by converges to the element . This sequence also is defined by .
Finally, , which gives us
It is easy to prove , which implies , This completes the proof. □
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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Sun, Y., Su, Y. & Zhang, J. A new method for the research of best proximity point theorems of nonlinear mappings. Fixed Point Theory Appl 2014, 116 (2014). https://doi.org/10.1186/1687-1812-2014-116
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DOI: https://doi.org/10.1186/1687-1812-2014-116