Abstract
We improve some existence theorem of best proximity points with the weak P-property, which has been recently proved by Zhang et al.
MSC:54H25, 54E50.
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1 Introduction
Let be a pair of nonempty subsets of a metric space , and let T be a mapping from A into B. Then is called a best proximity point if , where . We have proved many existence theorems of best proximity points. See, for example, [1–6]. Very recently, Caballero et al. [7] proved a new type of existence theorem, and Zhang et al. [8] generalized the theorem. The theorem proved in [8] is Theorem 8 with an additional assumption of the completeness of B. The essence of the result in [7] becomes very clear in [8], however, we have not learned the essence completely.
Motivated by the fact above, in this paper, we improve the result in [8]. Also, in order to consider the discontinuous case, we give a Kannan version.
2 Preliminaries
In this section, we give some preliminaries.
Definition 1 Let be a pair of nonempty subsets of a metric space , and define and by
and
Then
-
(Sankar Raj [9]) is said to have the P-property if and the following holds:
-
(Zhang et al.[8]) is said to have the weak P-property if and the following holds:
Proposition 2 Let be a pair of nonempty subsets of a metric space , and define and by (1) and (2). Assume that . Then the following are equivalent:
-
(i)
has the weak P-property.
-
(ii)
The conjunction of the following holds:
(ii-1) For every , there exists a unique with .
(ii-2) There exists a nonexpansive mapping Q from into such that for every .
Proof We note that because . First, we assume (i). Let and satisfy . Then from (i), we have
thus, . So (ii-1) holds. We put . Then from the definition of the weak P-property, we have for , that is, Q is nonexpansive. Conversely, we assume (ii). Let and satisfy . Then from (ii-1), we have and . Therefore,
holds. □
Lemma 3 Let be a pair of subsets of a metric space , and define and by (1) and (2). Assume that . Let T be a mapping from A into B, and let Q be a mapping from into such that for every . Then the following holds:
Proof Let be a sequence in such that converges to , and . We put . Since , we have and . Since
we have and . □
Lemma 4 Let be a metric space, let A, , be nonempty subsets such that A is complete and . Let T be a mapping from A into X such that , and let Q be a nonexpansive mapping from into . Let be the mapping whose graph is the completion of . Assume (3). Then the following hold:
-
(i)
is well-defined and nonexpansive.
-
(ii)
is equivalent to that there exists a sequence in such that and .
-
(iii)
The domain of is , where is the completion of .
-
(iv)
The range of is a subset of , where is the completion of .
-
(v)
implies .
-
(vi)
implies .
-
(vii)
The range of is a subset of A.
Proof We consider that the whole space is the completion of X. Since Q is Lipschitz continuous, is well-defined. The rest of (i) and (ii)-(iv) are obvious. By using (3), we can easily prove (v) and (vi). From the completeness of A, we obtain (vii). □
3 Fixed point theorems
In this section, we give fixed point theorems, which are used in the proofs of the main results.
Theorem 5 Let be a metric space, let A, , be nonempty subsets such that A is complete and . Let T be a contraction from A into X such that , and let Q be a nonexpansive mapping from into . Assume (3). Then has a unique fixed point in .
Proof We consider that the whole space is the completion of X. Define a nonexpansive mapping as in Lemma 4. Since T is continuous, is a subset of . Let S be the restriction of T to . Then is a contraction on . So the Banach contraction principle yields that there exists a unique fixed point z of in . Since , by Lemma 4(vi), z is a fixed point of . □
Remark
-
If and Q is the identity mapping on , then Theorem 5 becomes the Banach contraction principle [10].
-
We can prove Theorem 5 with the mapping as in the proof of Theorem 7.
We prove generalizations of Kannan’s fixed point theorem [11].
Theorem 6 Let be a metric space, let Y be a complete subset of X, and let T be a mapping from Y into X. Assume that the following hold:
-
(i)
There exists such that for all .
-
(ii)
There exists a nonempty subset Z of Y such that .
Then there exists a unique fixed point z, and for every , converges to z.
Proof Fix . Then from the proof in Kannan [11], we obtain that converges to a fixed point, and the fixed point is unique. □
Remark If , then Theorem 6 becomes Kannan’s fixed point theorem [11].
Using Theorem 6, we obtain the following.
Theorem 7 Let be a metric space, let A, , be nonempty subsets such that A is complete and . Let T be a mapping from A into X such that , and let Q be a nonexpansive mapping from into . Assume that (3) and the following hold:
-
There exist and such that
for and for all .
Then has a unique fixed point in .
Proof We consider that the whole space is the completion of X. Define a nonexpansive mapping as in Lemma 4. From the continuity of d, for . For , we have
Hence is a Kannan mapping from into X. is obvious. So by Theorem 6, there exists a unique fixed point w of in . By Lemma 4(v), and w is a fixed point of . □
Remark
-
Since T is not necessarily continuous, the range of is not necessarily included by . Because of the same reason, we cannot prove Theorem 7 with the mapping .
-
It is interesting that we do not need the completeness of any set related to directly. Of course, we need the completeness of A.
4 Main results
In this section, we give the main results.
Theorem 8 (Zhang et al. [8])
Let be a pair of subsets of a metric space , and define and by (1) and (2). Let T be a contraction from A into B. Assume that the following hold:
-
(i)
has the weak P-property.
-
(ii)
A is complete.
-
(iii)
.
Then there exists a unique such that .
Proof By Proposition 2(ii-2), there exists a nonexpansive mapping Q from into such that for every . Then by Lemma 3, all the assumptions in Theorem 5 hold. So there exists a unique fixed point z of in . This implies that . Let satisfy . Then from Proposition 2(ii-1), , and hold. Since has a unique fixed point, we obtain . Hence z is unique. □
Remark
-
If we weaken (i) to the conjunction of and (ii-2) in Proposition 2, we obtain only the existence of best proximity points.
-
In [8], we assume the completeness of B.
-
Exactly speaking, in [8], we obtained a theorem connected with Geraghty’s fixed point theorem [12]. However, in this paper, the difference between the two fixed point theorems is not essential. This means that we can easily modify Theorem 8 to be connected with Geraghty’s theorem.
Theorem 9 Let be a pair of subsets of a metric space , and define and by (1) and (2). Let T be a mapping from A into B. Assume that (i)-(iii) in Theorem 8 and the following hold:
-
(iv)
There exists such that
for .
Then there exists a unique such that .
Proof By Proposition 2(ii-2), there exists a nonexpansive mapping Q from into such that for every . Then by Theorem 7, there exists a unique fixed point w of in . This implies that , where . Let satisfy . Then from Proposition 2(ii-1), , and hold. Since , we have , and hence . Therefore, z is unique. □
Remark If we weaken (i) to the conjunction of and (ii-2) in Proposition 2, we obtain only the existence of best proximity points.
5 Additional result
In this section, we give a proposition similar to Proposition 2.
Proposition 10 Let be a pair of nonempty subsets of a metric space , and define and by (1) and (2). Assume that . Then the following are equivalent:
-
(i)
has the P-property.
-
(ii)
The conjunction of the following holds:
(ii-1) For every , there exists a unique with .
(ii-2) There exists an isometry Q from onto such that for every .
Proof We note . First, we assume (i). Let and satisfy . Then from (i), we have , thus, . So (ii-1) holds. We put . Then it is obvious that Q is isometric. For every , there exists with . From (ii-1), obviously holds, and hence Q is surjective. Conversely, we assume (ii). Let and satisfy . Then we have and . Therefore, holds. □
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The author is supported in part by the Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.
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Suzuki, T. The existence of best proximity points with the weak P-property. Fixed Point Theory Appl 2013, 259 (2013). https://doi.org/10.1186/1687-1812-2013-259
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DOI: https://doi.org/10.1186/1687-1812-2013-259