Abstract
In this paper, we first introduce a cyclic generalized contraction map in metric spaces and give an existence result for a best proximity point of such mappings in the setting of a uniformly convex Banach space. Then we give an existence and uniqueness best proximity point theorem for non-self proximal generalized contractions. Moreover, an algorithm is exhibited to determine such a unique best proximity point. Some examples are also given to support our main results. Our results extend and improve certain recent results in the literature.
MSC: 46N40, 47H10, 54H25, 46T99.
Similar content being viewed by others
Dedication
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
1 Introduction and preliminaries
Fixed point theory is indispensable for solving various equations of the form for self-mappings T defined on subsets of metric spaces. Given nonempty subsets A and B of a metric space and a non-self mapping , the equation does not necessarily have a solution, which is known as a fixed point of the mapping T. However, in such circumstances, it may be speculated to determine an element x for which the error is minimum, in which case x and Tx are in close proximity to each other. Best approximation theorems and best proximity point theorems are relevant in this perspective. One of the most interesting results in this direction is due to Fan [1] and can be stated as follows.
Theorem F Let K be a nonempty compact convex subset of a normed space E and let be a continuous non-self-mapping. Then there exists an x such that .
Many generalizations and extensions of this theorem appeared in the literature (see [2–6] and references therein).
On the other hand, though best approximation theorems ensure the existence of approximation solutions, such results need not yield optimal solutions. But, best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error , and hence the existence of a consummate approximate solution to the equation . Indeed, in view of the fact that for all x, a best proximity point theorem offers sufficient conditions for the existence of an element x, called a best proximity point of the mapping T, satisfying the condition that . Further, it is interesting to observe that best proximity point theorems also emerge as a natural generalization of fixed point theorems for a best proximity point reduces to a fixed point if the mapping under consideration is a self-mapping. Best proximity point theory of cyclic contraction maps has been studied by many authors; see [7–15] and references therein. Investigation of several variants of contractions for the existence of a best proximity point can be found in [16–19]. Best proximity point theorems for multivalued mappings are available in [20, 21].
2 Best proximity points for cyclic generalized contractions
Let A and B be nonempty subsets of a metric space , , and . We say that
-
(a)
T is cyclic contraction [10] if
for some , where
-
(b)
is a best proximity point for T if .
We first introduce the following new class of cyclic generalized contraction maps.
Definition 2.1 Let A and B be nonempty subsets of a metric space . A map is a cyclic generalized contraction map if , and
for each and , where satisfies for each .
If for each , where is constant, then T is a cyclic contraction.
A Banach space X is said to be uniformly convex if there exists a strictly increasing function such that the following implication holds for all , and :
Theorem 2.1 (Geraghty [22])
Let be a complete metric space and let be a map satisfying
where satisfies for each . Then T has a fixed point.
Now, we are ready to state our main result in this section.
Theorem 2.2 Let A and B be nonempty closed and convex subsets of a uniformly convex Banach space X and let be a cyclic generalized contraction map. Then T has a best proximity point.
Proof Suppose that , then the theorem follows from the above mentioned Geraghty fixed point theorem. Therefore, we may assume that . Let and let for each . Then from (2.1) we have
for each . Since and , so we have
From (2.2) and (2.3), we get
for each . Then from (2.4) we get for each , and so is a nonnegative nonincreasing sequence in ℝ. Hence converges to some real number . On the contrary, assume that . Since and , there exist and such that for all . We can take such that for all . Then
Let and let . Then from (2.2) and the above inequality, we get (note that )
for each and . So, we get
Letting , (2.5) implies , a contradiction. Then
Now, we show that
and
To show that , on the contrary, assume that there exists such that for each there exists such that
Choose such that and choose ϵ such that
By (2.6) there exists such that
Also, there exists such that
Let . It follows from (2.9)-(2.11) and the uniform convexity of X that
for all . As , the choice of ϵ and the fact that δ is strictly increasing imply that
a contradiction. A similar argument shows as . Hence, (2.7) and (2.8) hold.
Now we show that for each , there exists such that for all ,
On the contrary, assume that there exists such that for each there is satisfying
and
It follows from (2.13), (2.14) and the triangle inequality that
Letting , (2.7) implies
Let . Since and , there exist and such that for all . Thanks to (2.15), we can take such that for all . Then
and so from (2.2) we get
From (2.4) and (2.16), we get
for each . Letting and using (2.7), (2.8), (2.15) and (2.17), we get
a contradiction. Thus (2.12) holds.
Now we show that is a Cauchy sequence in A. To show the claim, we assume the contrary. Then there exists such that for each , there exist such that
Choose such that and choose such that
By (2.6) there exists such that
By (2.12) there exists such that
Let . It follows from (2.18)-(2.20) and the uniform convexity of X that
By the choice of ϵ and the fact that δ is strictly increasing, we have
a contradiction. Thus is a Cauchy sequence in A. Now the completeness of X and the closedness of A imply that
Since (note that and )
it follows from (2.6) and (2.21) that
Since
for each , then from (2.21)-(2.23) we get . Therefore, T has a best proximity point. □
Now we illustrate our main result by the following example.
Example 2.1 Consider the uniformly convex Banach space with Euclidean metric. Let and . Then A and B are nonempty closed and convex subsets of X and .
Let be defined as
We show that T is a generalized cyclic contraction map with for . To show the claim, notice first that the function , is convex, and so for . For each , we have
Thus all of the hypotheses of Theorem 2.2 are satisfied and then T has a best proximity point ( is a best proximity point of T in A).
Now we provide the following example to show that Theorem 2.2 is an essential extension of Theorem 3.10 of Eldred and Veeramani [10].
Example 2.2 Consider the uniformly convex Banach space with Euclidean metric. Let and . Then A and B are nonempty closed and convex subsets of X and .
Let be defined as
We first show that T is not a cyclic contraction map. To show the claim, on the contrary, assume that there exists such that
for each . Then
for each . Letting , we get
for each . Then
a contradiction. Now, we show that T is a cyclic generalized contraction, where for . Notice first that the function is increasing and concave and so is subadditive, that is, for each . For each with , we have (note that for we have and so (2.1) trivially holds for )
Thus all of the hypotheses of Theorem 2.2 are satisfied and then T has a best proximity point ( is a best proximity point of T in A). But since T is not a cyclic contraction, we cannot invoke the main result of [10] to show the existence of the best proximity point for T.
3 Best proximity points for generalized contraction
Given nonempty subsets A and B of a metric space, we recall the following notations and notions, which will be used in the sequel.
The set B is said to be approximatively compact with respect to A if every sequence in B, satisfying the condition that for some x in A, has a convergent subsequence. It is trivial to note that every set is approximatively compact with respect to itself, and that every compact set B is approximatively compact with respect to A.
A mapping is said to be a proximal contraction if there exists a non-negative number such that for all , , , in A,
To establish our results, we introduce the following new class of proximal contractions.
Definition 3.1 Let , be two maps. Let satisfy
Then T is said to be a -proximal contraction if
for all , , , in A.
Now, we are ready to state our first main result in this section.
Theorem 3.1 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.
Proof Let be a fixed element in . Since and , then there exists an element such that . Proceeding in this manner, having chosen , we can find satisfying
Since T is a -proximal contraction, then from (3.1) we have
We shall show that is a Cauchy sequence. Let . From (3.2) we get that the sequence is non-increasing (note that for all ). Therefore, there is some such that . We show that . Suppose, to the contrary, that . Then from (3.2) we get
a contradiction. Thus , that is,
Suppose, to the contrary, that is not a Cauchy sequence. Then there exists an and two subsequences of integers and , with
We may also assume
by choosing to be the smallest number exceeding for which (3.4) holds. From (3.4), (3.5) and by the triangle inequality,
Taking the limit as , we get (note that )
By the triangle inequality
From (3.2), we have
Then from (3.7) and (3.8), we have
Letting and using (3.2) and (3.6), we get
a contradiction. Therefore is a Cauchy sequence. Since is uniformly continuous and is a Cauchy sequence, then we get that is also a Cauchy sequence. Since X is complete and is closed, there exists such that . Further, it can be noted
Since g is continuous and , then . Therefore from the above, as . Since B is approximatively compact with respect to A, it follows that the sequence has a subsequence converging to some element . Thus and hence . Since , for some . Therefore . Since , then
From (3.1), (3.2) and (3.9), we have
Therefore
Hence
Suppose that there is another such that
Then from (3.10) and (3.11) we get
which implies that . □
The following theorem, which is the main result of Sadiq Basha [18], is immediate.
Theorem 3.2 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 an isometry,
-
(d)
.
Then there exists a unique element such that . Further, for any fixed element , the sequence defined by converges to x.
Now we illustrate our best proximity point theorem by the following example.
Example 3.1 Consider the complete metric space with Euclidean metric. Let and . Then , and . Let be defined as . Then g is a one-to-one continuous map, is uniformly continuous and .
Let be defined as . Let for each . Then it is easy to see that T is -proximal contraction. So, all the hypotheses of Theorem 3.1 are satisfied. Further, it is easy to see that is the unique element satisfying the conclusion of Theorem 3.1. However, we cannot invoke the above mentioned Theorem 3.2 of Sadiq Basha to show the existence of a best proximity point because g is not an isometry.
The following are immediate consequences of Theorem 3.1.
Theorem 3.3 Let A and B be nonempty closed subsets of a complete metric space such that B is compact. Moreover, assume that is 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.
Theorem 3.4 Let be a complete metric space and let satisfy the following conditions.
-
(a)
T is a -contraction,
-
(b)
is a one-to-one, onto continuous map such that is uniformly continuous.
Then there exists a unique element such that , that is, has a coincidence point x. Further, for any fixed element , the sequence defined by converges to x.
Theorem 3.5 Let be a complete metric space and let be a φ-contraction. Then T has a unique fixed point . Further, for any fixed element , the sequence defined by converges to x.
References
Fan K: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 1969, 112: 234–240. 10.1007/BF01110225
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
Prolla JB: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 1982/1983, 5: 449–455.
Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 2010, 11(3):475–489.
Hussain N, Khan AR: Applications of the best approximation operator to ∗-nonexpansive maps in Hilbert spaces. Numer. Funct. Anal. Optim. 2003, 24(3–4):327–338. 10.1081/NFA-120022926
Takahashi W: Fan’s existence theorem for inequalities concerning convex functions and its applications. In Minimax Theory and Applications. Edited by: Ricceri B, Simons S. Kluwer Academic, Dordrecht; 1998:597–602.
Amini-Harandi A: Best proximity points theorems for cyclic strongly quasi-contraction mappings. J. Glob. Optim. 2012. 10.1007/s10898-012-9953-9
Karpagam S, Agrawal S: Best proximity point theorems for p -cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009., 2009: Article ID 197308
Sadiq Basha S: Best proximity points: global optimal approximate solution. J. Glob. Optim. 2011, 49: 15–21. 10.1007/s10898-009-9521-0
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
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
Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 2008, 69: 3790–3794. 10.1016/j.na.2007.10.014
Abkar A, Gabeleh M: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 2011, 151: 418–424. 10.1007/s10957-011-9818-2
Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022
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
Amini-Harandi A: Best proximity points for proximal generalized contractions in metric spaces. Optim. Lett. 2013. 10.1007/s11590-012-0470-z
Amini-Harandi A: Common best proximity points theorems in metric spaces. Optim. Lett. 2012. 10.1007/s11590-012-0600-7
Sadiq Basha S: Best proximity points: optimal solutions. J. Optim. Theory Appl. 2011, 151: 210–216. 10.1007/s10957-011-9869-4
Abkar A, Gabeleh M: Best proximity points of non-self mappings. Top 2013. 10.1007/s11750-012-0255-7
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
Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380
Geraghty G: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5
Acknowledgements
This work was supported by the University of Shahrekord. The first author would like to express thanks for this support. The first author was also partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR, KAU for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Amini-Harandi, A., Hussain, N. & Akbar, F. Best proximity point results for generalized contractions in metric spaces. Fixed Point Theory Appl 2013, 164 (2013). https://doi.org/10.1186/1687-1812-2013-164
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-164