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Best proximity points for Geraghty’s proximal contraction mappings
Fixed Point Theory and Applications volume 2013, Article number: 180 (2013)
Abstract
In this paper, we generalized the notion of proximal contractions of the first and second kinds by using Geraghty’s theorem and establish best proximity point theorems for proximal contractions. Our results improve and extend the recent results of Sadiq Basha and some others.
MSC:47H09, 47H10.
1 Introduction
Several problems can be modeled as 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 nonself-mapping from A to B, then the aforementioned equation does not necessarily admit a solution. In this case, it is contemplated to find 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 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 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.
A classical best approximation theorem was introduced by Fan [1], that is, if A is a non-empty compact convex subset of a Hausdorff locally convex topological vector space B and is a continuous mapping, then there exists an element such that . Afterward, several authors, including Prolla [2], Reich [3], Sehgal and Singh [4, 5], derived the extensions of Fan’s theorem in many directions. Other works on the existence of a best proximity point for contractions can be seen in [6–14].
In 1922, Banach proved that every contractive mapping in a complete metric spaces has a unique fixed point, which is called Banach’s fixed point theorem or Banach’s contraction principle. Since Banach’s fixed point theorem, many authors have extended, improved and generalized this theorem in several ways. Some applications of Banach’s fixed point theorem can be found in [15–18]. One of such generalizations is due to Geraghty [19] as follows.
Theorem 1.1 [19]
Let be a complete metric space and let f be a self-mapping on X such that for each satisfying
where , is the family of functions from into which satisfies the condition
Then the sequence converges to the unique fixed point of f in X.
In 2005, Eldred et al. [20] obtained best proximity point theorems for relatively nonexpansive mappings. Best proximity point theorems for several types of contractions were established in [21–25].
Recently, Sadiq Basha in [26] gave necessary and sufficient conditions to claim the existence of a best proximity point for proximal contractions of the first kind and the second kind, which are non-self mapping analogues of contraction self-mappings, and also established some best proximity and convergence theorems.
The aim of this paper is to introduce the new classes of proximal contractions, which are more general than a class of proximal contractions of the first and second kinds, by giving the necessary condition to have best proximity points, and we also give some illustrative example of our main results. The results of this paper are extension and generalizations of the main result of Sadiq Basha in [26] and some results in the literature.
2 Preliminaries
Given nonempty subsets A and B of a metric space , we recall the following notations and notions that will be used in what follows.
If , then and are nonempty. Further, 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 [27]).
Definition 2.1 [26]
A mapping is called a proximal contraction of the first kind if there exists such that
for all .
It is easy to see that a self-mapping that is a proximal contraction of the first kind is precisely a contraction. However, a nonself-proximal contraction is not necessarily a contraction.
Definition 2.2 [26]
A mapping is called a proximal contraction of the second kind if there exists such that
for all .
Definition 2.3 Let and be mappings. The pair is called a proximal cyclic contraction pair if there exists such that
for all and .
Definition 2.4 Let and be an isometry. The mapping S is said to preserve the isometric distance with respect to g if
for all .
Definition 2.5 A point is called a best proximity point of the mapping if it satisfies the condition that
It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.
3 Main results
In this section, we introduce a new class of proximal contractions, the so-called Geraghty’s proximal contraction mappings, and prove best proximity theorems for this class.
Definition 3.1 A mapping is called Geraghty’s proximal contraction of the first kind if, there exists such that
for all .
Definition 3.2 A mapping is called Geraghty’s proximal contraction of the second kind if, there exists such that
for all .
It is easy to see that if we take , where , then Geraghty’s proximal contraction of the first kind and Geraghty’s proximal contraction of the second kind reduce to a proximal contraction of the first kind (Definition 2.1) and a proximal contraction of the second kind (Definition 2.2), respectively.
Next, we extend the result of Sadiq Basha [26] and Banach’s fixed point theorem to the case of nonself-mappings satisfying Geraghty’s proximal contraction condition.
Theorem 3.3 Let be a complete metric space and let A, B be nonempty closed subsets of X such that and are nonempty. Let , and satisfy the following conditions:
-
(a)
S and T are Geraghty’s proximal contractions of the first kind;
-
(b)
g is an isometry;
-
(c)
the pair is a proximal cyclic contraction;
-
(d)
, ;
-
(e)
and .
Then there exists a unique point and there exists a unique point such that
Moreover, for any fixed , the sequence defined by
converges to the element x. For any fixed , the sequence defined by
converges to the element y.
On the other hand, a sequence in A converges to x if there exists a sequence of positive numbers such that
where satisfies the condition that .
Proof Let be a fixed element in . In view of the fact that and , it follows that there exists an element such that
Again, since and , there exists an element such that
By the same method, we can find in such that
So, inductively, one can determine an element such that
Since and , S is Geraghty’s proximal contraction of the first kind, g is an isometry and the property of β, it follows that for each
which implies that the sequence is non-increasing and bounded below. Hence there exists such that . Suppose that . Observe that
which implies that . Since , we have which is a contradiction and hence
Now, we claim that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then there exists and subsequences , of such that for any
for any . For each , let . Then we have
and so it follows from (3.2) and (3.3) that
Notice also that
and so
Taking in the above inequality, by (3.2), (3.4) and , we get , which is a contradiction. So we know that the sequence is a Cauchy sequence. Hence converges to some element .
Similarly, in view of the fact that and , we can conclude that there exists a sequence such that it converges to some element . Since the pair is a proximal cyclic contraction and g is an isometry, we have
Taking in (3.5), it follows that
and so and . Since and , there exist and such that
From (3.1) and (3.7), since S is Geraghty’s proximal contraction of the first kind of S, we get
Letting in the above inequality, we get and so . Therefore, we have
Similarly, we can show that and so
From (3.6), (3.9) and (3.10), we get
Next, to prove the uniqueness, suppose that there exist and with and such that
Since g is an isometry and S is Geraghty’s proximal contraction of the first kind, it follows that
and hence
which is a contradiction. Thus we have . Similarly, we can prove that .
On the other hand, let be a sequence in A and be a sequence of positive real numbers such that
where satisfies the condition that
By (3.1) and (3.12), since S is Geraghty’s proximal contraction of the first kind and g is an isometry, we have
For any , choose a positive integer N such that for all . Observe that
Since is arbitrary, we can conclude that for all the sequence is non-increasing and bounded below and hence converges to some nonnegative real number . Since the sequence converges to x, we get
Suppose that . Since
it follows from inequalities (3.11), (3.13) and (3.14) that
which implies that and so , that is,
which is a contradiction. Thus and hence is convergent to the point x. This completes the proof. □
If g is the identity mapping in Theorem 3.3, then we obtain the following.
Corollary 3.4 Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let , and be the mappings satisfying the following conditions:
-
(a)
S and T are Geraghty’s proximal contractions of the first kind;
-
(b)
, ;
-
(c)
the pair is a proximal cyclic contraction.
Then there exists a unique point and there exists a unique point such that
If we take , where , we obtain the following corollary.
Corollary 3.5 [26]
Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let , and be the mappings satisfying the following conditions:
-
(a)
S and T are proximal contractions of the first kind;
-
(b)
g is an isometry;
-
(c)
the pair is a proximal cyclic contraction;
-
(d)
, ;
-
(e)
and .
Then there exists a unique point and there exists a unique point such that
Moreover, for any fixed , the sequence defined by
converges to the element x. For any fixed , the sequence defined by
converges to the element y.
If g is the identity mapping in Corollary 3.5, we obtain the following corollary.
Corollary 3.6 Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let , and be the mappings satisfying the following conditions:
-
(a)
S and T are proximal contractions of the first kind;
-
(b)
, ;
-
(c)
the pair is a proximal cyclic contraction.
Then there exists a unique point and there exists a unique point such that
Next, we establish a best proximity point theorem for nonself-mappings which are Geraghty’s proximal contractions of the first kind and the second kind.
Theorem 3.7 Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let and be the mappings satisfying the following conditions:
-
(a)
S is Geraghty’s proximal contraction of the first and second kinds;
-
(b)
g is an isometry;
-
(c)
S preserves isometric distance with respect to g;
-
(d)
;
-
(e)
.
Then there exists a unique point such that
Moreover, for any fixed , the sequence defined by
converges to the element x.
On the other hand, a sequence in A converges to x if there exists a sequence of positive numbers such that
where satisfies the condition that .
Proof Since and , as in the proof of Theorem 3.3, we can construct the sequence in such that
for each . Since g is an isometry and S is Geraghty’s proximal contraction of the first kind, we see that
for all . Again, similarly, we can show that the sequence is a Cauchy sequence and so it converges to some . Since S is Geraghty’s proximal contraction of the second kind and preserves the isometric distance with respect to g, we have
which means that the sequence is non-increasing and bounded below. Hence there exists such that
Suppose that . Observe that
Taking in the above inequality, we get . Since , we have which is a contradiction and thus
Now, we claim that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then there exists and subsequences , of such that, for any ,
for any . For each , let . Then we have
and so it follows from (3.17) and (3.18) that
Notice also that
So, it follows that
and so . Since , we have , that is, , which is a contradiction. So, we obtain the claim and then it converges to some . Therefore, we can conclude that
which implies that . Since , we have for some and then . By the fact that g is an isometry, we have . Hence and so . Since , there exists such that
Since S is Geraghty’s proximal contraction of the first kind, it follows from (3.16) and (3.19) that
for all . Taking in (3.20), it follows that the sequence converges to a point u. Since g is continuous and , we have as . By the uniqueness of the limit, we conclude that . Therefore, it follows that .
The uniqueness and the remaining part of the proof follow from the proof of Theorem 3.3. This completes the proof. □
If g is the identity mapping in Theorem 3.7, then we obtain the following.
Corollary 3.8 Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let be the mappings satisfying the following conditions:
-
(a)
S is Geraghty’s proximal contraction of the first and second kinds;
-
(b)
.
Then there exists a unique point such that
Moreover, for any fixed , the sequence defined by
converges to the best proximity point x of S.
If we take in Theorem 3.7, where , we obtain the following.
Corollary 3.9 [26]
Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let and be the mappings satisfying the following conditions:
-
(a)
S is a proximal contraction of the first and second kinds;
-
(b)
g is an isometry;
-
(c)
S preserves isometric distance with respect to g;
-
(d)
;
-
(e)
.
Then there exists a unique point such that
Moreover, for any fixed , the sequence defined by
converges to the element x.
If g is the identity mapping in Corollary 3.9, then we obtain the following.
Corollary 3.10 Let be a complete metric space and let A, B be nonempty closed subsets of X. Further, suppose that and are nonempty. Let be a mapping satisfying the following conditions:
-
(a)
S is a proximal contraction of the first and second kinds;
-
(b)
.
Then there exists a unique point such that
Moreover, for any fixed , the sequence defined by
converges to the best proximity point x of S.
4 Examples
Next, we give an example to show that Definition 3.1 is different from Definition 2.1; moreover, we give an example which supports Theorem 3.3. First, we give some proposition for our example as follows.
Proposition 4.1 Let be a function defined by . Then we have the following inequality:
for all .
Proof If , we have done. Suppose that . Then since we have
it follows that . In the case , by a similar argument, we can prove that inequality (4.1) holds. □
Proposition 4.2 For each , we have that the following inequality holds:
Proof Since
so that
□
Example 4.3 Consider the complete metric space with Euclidean metric. Let
Then . Define the mappings as follows:
First, we show that S is Geraghty’s proximal contractions of the first kind with defined by
Let , , and be elements in A satisfying
Then we have for . If , we have done. Assume that . Then, by Proposition 4.1 and the fact that the function is increasing, we have
Thus S is Geraghty’s proximal contraction of the first kind.
Next, we prove that S is not a proximal contraction of the first kind. Suppose S is a proximal contraction of the first kind, then for each satisfying
there exists such that
From (4.2), we get and and so
Letting , we get
which is a contradiction. Thus S is not a proximal contraction of the first kind.
Example 4.4 Consider the complete metric space with metric defined by
for all . Let
Define two mappings , and as follows:
Then , , and the mapping g is an isometry.
Next, we show that S and T are Geraghty’s proximal contractions of the first kind with defined by
Let , , and be elements in A satisfying
Then we have
If , we have done. Assume that , then, by Proposition 4.2, we have
Thus S is Geraghty’s proximal contraction of the first kind. Similarly, we can see that T is Geraghty’s proximal contraction of the first kind. Next, we show that the pair is a proximal cyclic contraction. Let and be such that
Then we get
In the case , clear. Suppose that , then we have
where . Hence the pair is a proximal cyclic contraction. Therefore, all the hypotheses of Theorem 3.3 are satisfied. Further, it is easy to see that and are the unique elements such that
5 Conclusions
This article has investigated the existence of an optimal approximate solution, the so-called best proximity point, for the generalized notion of proximal contractions of the first and second kinds, which were defined by Sadiq Basha in [26]. Furthermore, an algorithm for computing such an optimal approximate solution and example which supports our main results have been presented.
References
Fan K: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 1969, 112: 234–240. 10.1007/BF01110225
Prolla JB: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 1982–1983, 5: 449–455.
Reich S: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 1978, 62: 104–113. 10.1016/0022-247X(78)90222-6
Sehgal VM, Singh SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 1988, 102: 534–537.
Sehgal VM, Singh SP: A theorem on best approximations. Numer. Funct. Anal. Optim. 1989, 10: 181–184. 10.1080/01630568908816298
Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70(10):3665–3671. 10.1016/j.na.2008.07.022
Eldred AA, Veeramani P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081
Caballero J, Harjani J, Sadarangani K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 231 10.1186/1687-1812-2012-231
Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 2008, 69(11):3790–3794. 10.1016/j.na.2007.10.014
Karpagam S, Agrawal S: Best proximity point theorems for p -cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009., 2009: Article ID 197308
Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 2009, 71: 2918–2926. 10.1016/j.na.2009.01.173
Vetro C: Best proximity points: convergence and existence theorems for p -cyclic mappings. Nonlinear Anal. 2010, 73(7):2283–2291. 10.1016/j.na.2010.06.008
Wlodarczyk K, Plebaniak R, Banach A: Erratum to: ’Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces’. Nonlinear Anal. 2009, 71: 3583–3586.
Wlodarczyk K, Plebaniak R, Obczynski C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 2010, 72: 794–805. 10.1016/j.na.2009.07.024
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87(1):109–116. 10.1080/00036810701556151
Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9
Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 93 10.1186/1687-1812-2011-93
Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5
Eldred AA, Kirk WA, Veeramani P: Proximinal normal structure and relatively nonexpanisve mappings. Stud. Math. 2005, 171(3):283–293. 10.4064/sm171-3-5
Amini-Harandi A: Best proximity points for proximal generalized contractions in metric spaces. Optim. Lett. 2012. 10.1007/s11590-012-0470-z
Al-Thagafi MA, Shahzad N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008., 2008: Article ID 457069
Kim WK, Kum S, Lee KH: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 2008, 68(8):2216–2227. 10.1016/j.na.2007.01.057
Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012. 10.1007/s10957-012-9991-y
Wlodarczyk K, Plebaniak R, Banach A: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 2009, 70(9):3332–3341. 10.1016/j.na.2008.04.037
Sadiq Basha S: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 2011, 74: 5844–5850. 10.1016/j.na.2011.04.017
Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415
Acknowledgements
Mr. Chirasak Mongkolkeha was supported from the Thailand Research Fund through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for the Ph.D. Program at KMUTT, Thailand. This research was partially finished at Department of Mathematics Education, Gyeongsang National University, Republic of Korea. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant No. 2012-0008170). The third author was supported by the Commission on Higher Education, the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant No. MRG5580213).
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Mongkolkeha, C., Cho, Y.J. & Kumam, P. Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point Theory Appl 2013, 180 (2013). https://doi.org/10.1186/1687-1812-2013-180
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DOI: https://doi.org/10.1186/1687-1812-2013-180