Abstract
Consider a non-self-mapping , where is a pair of nonempty subsets of a metric space . In this paper, we study the existence and uniqueness of solutions to the global optimization problem , where T belongs to the class of proximal quasi-contraction mappings.
MSC:41A65, 90C30, 47H10.
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1 Introduction
Let be a metric space and be a pair of nonempty subsets of X. Consider a non-self-mapping . An element is said to be a best proximity point of the mapping T iff . Clearly, if , a best proximity point of T will be a fixed point of T. The aim of best proximity point analysis is to provide sufficient conditions assuring the existence and uniqueness of best proximity points, and build algorithms that can serve to approximate such points.
In [1], Sankar Raj introduced the concept of P-property and obtained a best proximity point result for a class of weakly non-self-contractive mappings. Later on, many authors considered different classes of contractive mappings under the P-property (see, e.g., [2–4]). Very recently, Abkar and Gabeleh [5] observed that the most best proximity point theorems obtained under the P-property can be deduced from existing fixed point results in the literature.
In [6], Sadiq Basha presented an extension of Banach’s contraction principle via a best proximity point theorem under the assumption: B is approximatively compact with respect to A. Later on, various best proximity point results are derived under this assumption (see, e.g., [7–9]). In a recent work, Sadiq Basha and Shahzad [9] established new best proximity point results for generalized proximal contractions of first and second kind. For other results on best proximity point analysis, we refer the reader to [10–19].
This paper discusses the existence and uniqueness of best proximity points for a new class of non-self-mappings. More precisely, we introduce in this work the class of proximal quasi-contractive mappings, and we establish new best proximity point results for such mappings. In our results, we consider only proximal contractions of the first kind. Moreover, the compactness assumption, used in many previous works, is not assumed. We show that the results obtained in [6, 9] are particular cases of our main result.
The paper is organized as follows. In Section 2, we introduce the class of proximal quasi-contractive mappings, and the concept of proximal orbital completeness. Section 3 presents some useful lemmas that will be used to show that our proximal orbital completeness concept is weaker than the compactness condition imposed on the pair , and the proximal condition of the second kind. In Section 4, we present and establish our main result. Finally, in Section 5, we show that many existing best proximity point results including the obtained results in [6, 9] are particular cases of our main theorem.
2 Definitions
Through this paper, ℕ denotes the set of natural numbers, and .
Let be a metric space and a pair of nonempty subsets of X. We consider the following notations:
Definition 2.1 An element is said to be a best proximity point of the non-self-mapping iff
Reference [6] introduced the following concept.
Definition 2.2 We say that B is approximatively compact with respect to A iff every sequence satisfying the condition that for some x in A, has a convergent subsequence.
Reference [9] introduced the following concepts.
Definition 2.3 A non-self-mapping is said to be a generalized proximal contraction of the first kind iff there exist non-negative numbers α, β, γ, δ with such that
where .
Definition 2.4 A non-self-mapping is said to be a generalized proximal contraction of the second kind iff there exist non-negative numbers α, β, γ, δ with such that
where .
In this paper, we introduce the following concepts.
Definition 2.5 A non-self-mapping is said to be a proximal quasi-contraction iff there exists a number such that
where .
Remark 2.1 Clearly, we have the following implication: T is a generalized proximal contraction of the first kind ⇒ T is a proximal quasi-contraction.
If T is a self-mapping on A, then the requirement in the preceding definition reduces to the condition that
for all . Such condition was introduced by [20] under the name of quasi-contraction.
Lemma 2.1 Let be a non-self-mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
.
Then, for all , there exists a sequence such that
Proof Let . Since , there exists such that . Again, since , there exists such that . Continuing this process, by induction, we can build a sequence satisfying (2.1). □
Definition 2.6 Under the assumptions of Lemma 2.1, any sequence satisfying (2.1) is called a proximal Picard sequence associated to .
For every , we denote by the set of all proximal Picard sequences associated to a.
Let and . For all , we define the following sets:
and
Definition 2.7 We say that is proximal T-orbitally complete iff every Cauchy sequence for some , converges to an element in .
If T is a self-mapping on A, then the requirement in the preceding definition reduces to the condition that A is T-orbitally complete (see [20]).
3 Some useful lemmas
The following preliminary results will be useful later.
Lemma 3.1 Let be a pair of closed subsets of a metric space . Suppose that the following conditions hold:
-
(i)
;
-
(ii)
B is approximatively compact with respect to A.
Then the set is closed.
Proof Let be a sequence in such that
for some . By the definition of , there exists a sequence in B such that
On the other hand, we have
Using (3.2), we get
Letting in the above inequalities and using (3.1), we obtain
Since B is approximatively compact with respect to A, the sequence admits a convergent subsequence . Let such that
From (3.1), (3.2), and (3.3), we get
which implies that . □
Lemma 3.2 Let be a pair of closed subsets of a complete metric space . Let be a non-self-mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
;
-
(iii)
T is a generalized proximal contraction of the second kind.
Then is proximal T-orbitally complete.
Proof Let and be a Cauchy sequence. Since is complete and A is closed, there exists such that
By definition of , for all , we have
Since T is a generalized proximal contraction of the second kind, for all , we have
Using the above inequality and the triangular inequality, we obtain
where (from )
Using a standard technique of iterations, one can show that is a Cauchy sequence. Since is complete and B is closed, there exists such that
Now, from (3.4) and (3.5), we have
which implies that . □
4 Main result
Our main result is giving by the following best proximity point theorem.
Theorem 4.1 Let be a pair of subsets of a metric space . Let be a giving mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
is proximal T-orbitally complete;
-
(iii)
;
-
(iv)
T is a proximal quasi-contraction.
Then T has a unique best proximity point . Moreover, for any , any sequence converges to .
Proof Let be an arbitrary point in . From Lemma 2.1, the set is nonempty. Let be a proximal Picard sequence associated to . So, we have
Clearly, if for some , from (4.1), will be a best proximity point. So, we can suppose that
The proof is divided into several steps.
Step 1. Giving , we claim that for every pair with , we have
Let with . Using the fact that T is a proximal quasi-contraction, from (4.1), we have
This proves our claim.
Step 2. We claim that
Let be fixed. By (4.2), there exists a pair with
such that
Suppose that . From (4.3), we have
that is a contradiction with . Then and
This proves our claim.
Step 3. We claim that
From (4.4), we have
for some . Now, using (4.3), we have
which proves our claim.
Step 4. We claim that is a Cauchy sequence.
Let with . Using (4.3), we have
On the other hand, from (4.4) we have
for some . Using (4.3), we obtain
Thus we have
From (4.6) and (4.7), we obtain
Continuing this process, by induction, we get
Thanks to (4.5), we obtain
which implies (since ) that the proximal Picard sequence is Cauchy.
Step 5. Existence of a best proximity point.
Since is proximal T-orbitally complete, the sequence converges to some element . Since , there exists such that
Since T is a proximal quasi-contraction, we have
Letting in the above inequality, we obtain
which holds only if , that is, . So, we have , which means that is a best proximity point of T.
Step 6. Uniqueness of the best proximity point.
Suppose that is another best proximity point, that is,
Using the fact that T is a proximal quasi-contraction, we obtain the following inequality:
which holds only if , that is, . □
Example 4.1 Consider the Euclidean space endowed with the standard metric:
Let us define
Clearly is a pair of closed subsets of with , and (see Figure 1). Moreover, since is a closed subset of the complete metric space , then is proximal T-orbitally complete for any mapping . Define the non-self-mapping by
where
We shall prove that T is a proximal quasi-contraction. Indeed, let
such that
It is easy to show that
and
We distinguish three cases.
Case 1. and . In this case, we have
Case 2. . In this case, we have
Case 3. . In this case, we have
Thus, we proved that T is a proximal quasi-contraction mapping with . Now, all the required hypotheses of Theorem 4.1 are satisfied, we deduce that T has a unique best proximity point. In this example, is the unique best proximity point of T.
5 Particular cases
In this section, we will show that many recent best proximity point theorems can be deduced from our main result.
The following result can easily be deduced from Theorem 4.1 (see Remark 2.1).
Corollary 5.1 Let be a pair of subsets of a metric space . Let be a giving mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
is proximal T-orbitally complete;
-
(iii)
;
-
(iv)
T is a generalized proximal contraction of the first kind.
Then T has a unique best proximity point . Moreover, for any , any sequence converges to .
Corollary 5.2 Let be a pair of closed subsets of a complete metric space . Let be a giving mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
B is approximatively compact with respect to A;
-
(iii)
;
-
(iv)
T is a proximal quasi-contraction.
Then T has a unique best proximity point . Moreover, for any , any sequence converges to .
Proof The result follows immediately from Theorem 4.1 and Lemma 3.1. Indeed, from Lemma 3.1, since B is approximatively compact with respect to A, then is a closed subset of the complete metric space , which implies that is proximal T-orbitally complete. So, we have only to apply Theorem 4.1 to get the desired result. □
The following result due to [9] is an immediate consequence of Corollary 5.2.
Corollary 5.3 Let be a pair of closed subsets of a complete metric space . Let be a giving mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
B is approximatively compact with respect to A;
-
(iii)
;
-
(iv)
T is a generalized proximal contraction of the first kind.
Then T has a unique best proximity point . Moreover, for any , any sequence converges to .
The following best proximity point result due also to [9] is a consequence of Corollary 5.1 and Lemma 3.2.
Corollary 5.4 Let be a pair of closed subsets of a complete metric space . Let be a giving mapping. Suppose that the following conditions hold:
-
(i)
;
-
(ii)
T is a generalized proximal contraction of the first and second kind.
Then T has a unique best proximity point . Moreover, for any , any sequence converges to .
Proof Since is complete and T is a generalized proximal contraction of the second kind, it follows from Lemma 3.2 that is proximal T-orbitally complete. Now, the desired result can be obtained from Corollary 5.1. □
Taking in Theorem 4.1, we obtain the famous Ćirić’s fixed point theorem for quasi-contractive mappings (see [20]).
Corollary 5.5 Let be a metric space and let be a quasi-contraction, that is,
for all , where is some constant. If X is T-orbitally complete, then T has a unique fixed point . Moreover, for any , the sequence converges to .
6 Conclusion
A new class of non-self-contractive mappings is introduced in this work. Under a proximal orbital completeness assumption, we established the existence and uniqueness of best proximity points for such mappings. We proved also that our proximal orbital completeness condition is weaker than the compactness condition and the proximal condition of second kind.
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Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no RGP-VPP-237.
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Jleli, M., Samet, B. An optimization problem involving proximal quasi-contraction mappings. Fixed Point Theory Appl 2014, 141 (2014). https://doi.org/10.1186/1687-1812-2014-141
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DOI: https://doi.org/10.1186/1687-1812-2014-141