A Best Proximity Point Theorem for Generalized Non-Self-Kannan-Type and Chatterjea-Type Mappings and Lipschitzian Mappings in Complete Metric Spaces

Ungchittrakool, Kasamsuk

doi:https://doi.org/10.1155/2016/9321082

Journal of Function Spaces

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Research Article | Open Access

Volume 2016 | Article ID 9321082 | https://doi.org/10.1155/2016/9321082

A Best Proximity Point Theorem for Generalized Non-Self-Kannan-Type and Chatterjea-Type Mappings and Lipschitzian Mappings in Complete Metric Spaces

Kasamsuk Ungchittrakool^1,2

Academic Editor: Tomonari Suzuki

Received01 Feb 2016

Revised05 May 2016

Accepted07 Jun 2016

Published14 Jul 2016

Abstract

The purpose of this paper is to provide and study a best proximity point theorem for generalized non-self-Kannan-type and Chatterjea-type mappings and Lipschitzian mappings in complete metric spaces. The significant mapping in a unified form which related to contractive mappings, Kannan-type mappings, and Chatterjea-type mappings is established. We also provide some examples to illustrate the situation corresponding to the main theorem. The main result of this paper can be viewed as a general and unified form of several previously existing results.

1. Introduction

Fixed point theory can be looked upon as an important model that can be used in several real world problems and it is in close relationship with other branches of mathematics. It furnishes unified treatment and is a vital tool for solving equations of form , where is a self-mapping defined on a subset of some suitable spaces such as a metric space, a normed linear space, or a topological vector space. However, in case the mapping is not a self-mapping, the fixed point theorems are not specified to provide the existence of a solution for the equation . On the other hand, the best approximation theorems and the best proximity point theorems play an important rule to solve an approximate solution to the equation when is a non-self-mapping, in which case a solution does not necessarily exist. For some interesting best approximation theorems, let us refer to [1–6]. For a non-self-mapping , a best proximity point theorem investigates the situations which lead to the existence of an element nearest to . It can be said on the other hand that a best proximity point theorem explores an element for which the value is minimum in the setting of metric spaces. This means that it is to study the global minimization of the real valued function . A best proximity point theorem succeeds in finding the global minimum of by constraining an approximate solution of the equation to satisfy the condition that . The solutions of the equation are called best proximity points of the mapping . Furthermore, if is a self-mapping, then all best proximity points turn into the fixed points of . With all these reasons, the study on best proximity point theorems is interesting and will cover the fixed point theorems implicitly.

For the previous research involving best proximity point theorems for several types of some mappings and contractions, one can refer to [7–17]. Best proximity point theorems for set valued mappings have been found in [18–28]. Moreover, in case of common best proximity point theorems, there are some interesting results in [29–32].

In 2013, Sadiq Basha et al. [33] established some best proximity point theorems for non-self-nonexpansive mappings, non-self-Kannan-type mappings, and non-self-Chatterjea-type mappings. Their results are quite interesting and there are examples to illustrate the main results.

Motivated and inspired by the research mentioned above, we are interested in studying a best proximity point theorem for generalized non-self-Kannan-type and Chatterjea-type mappings and Lipschitzian mappings in complete metric spaces with the way of an optimal approximate solution for and , where is a generalized non-self-Kannan-type and Chatterjea-type mapping and is a Lipschitzian mapping, respectively. The results obtained in this paper can be viewed as some extensions of the corresponding ones announced by Sadiq Basha et al. [33], Kannan [34], Chatterjea [35], and many others.

2. Non-Self-Lipschitzian Mappings and Generalized Non-Self-Kannan-Type and Chatterjea-Type Mappings

In this section, some definitions related to Lipschitzian mappings and their special forms are given and will be used in the sequel. The new mappings called generalized non-self-Kannan and Chatterjea mappings are established which are wider than non-self-Kannan mappings and non-self-Chatterjea mappings. Some other notions are provided and will be used in the next section.

Definition 1. Let and be nonempty subsets of a metric space . An element in is said to be a best proximity point of a mapping if .

It is noticed that best proximity becomes a fixed point if the underlying mapping is a self-mapping. Moreover, in light of the fact that for all in , the function attains its global minimum at a best proximity point.

Definition 2. Let and be two metric spaces. A mapping is said to be a non-self-Lipschitzian mapping if there exists a constant such thatfor all .

The smallest number for which (1) holds is called the Lipschitz constant of .

Definition 3. Let and be two metric spaces. A Lipschitzian mapping with the Lipschitz constant is said to be a non-self-nonexpansive mapping.

Definition 4. Let and be two metric spaces. A Lipschitzian mapping with the Lipschitz constant is said to be a non-self-contractive mapping.

Definition 5. Let and be nonempty subsets of a metric space . A mapping is said to be (1)a non-self-Kannan mapping (see [34] for the self-mapping case) if there exists a constant such that for all ;(2)a non-self-Chatterjea mapping (see [35] for the self-mapping case) if there exists a constant such that for all .

Definition 6. Let and be nonempty subsets of a metric space . A mapping is said to be a generalized non-self-Kannan and Chatterjea mapping if there exist nonnegative constants such that andfor all .

It is obvious that (4) is in a generalized form of (2) and (3).

Definition 7. Let and be nonempty subsets of a metric space and let be a mapping. A mapping is said to be (1)a non-self-Kannan mapping with respect to the mapping if there exists a constant such that for all ;(2)a non-self-Chatterjea mapping with respect to the mapping if there exists a constant such that for all .

Definition 8. Let and be nonempty subsets of a metric space and let be a mapping. A mapping is said to be a generalized non-self-Kannan and Chatterjea mapping with respect to the mapping if there exist nonnegative constants such that andfor all .

It is clear that (7) is in a generalized form of (5) and (6).

Definition 9 (see [33]). Let and be nonempty subsets of a metric space . Given mappings and , the pair is said to form a weak -cyclic contraction if there exists a nonnegative real number such that for all and .

Definition 10 (see [33]). Let and be nonempty subsets of a metric space. Given mappings and , the pair is said to form a -cyclic contraction if there exists a nonnegative real number such that for all and .

It is easy to observe that every -cyclic contraction is a weak -cyclic contraction.

3. Main Results

In this section, we establish and prove a best proximity point theorem for generalized non-self-Kannan-type and Chatterjea-type mappings and Lipschitzian mappings in complete metric spaces. Before going into the main theorem, it is useful to know the following observation.

Remark 11. Let , and be nonnegative real numbers with and satisfyThen the following hold: (i).(ii).(iii).(iv) and .

Proof. Notice that (i) is directly obtained from (10). For (ii), we observe that (iii) and (iv) are not hard to verify from (ii).

Theorem 12. Let and be nonempty closed subsets of a complete metric space. Let and satisfy the following conditions: (a) is a Lipschitzian mapping with Lipschitz constant .(b)There exist nonnegative constants such that (c) for all (d)The pair forms a weak -cyclic contraction.Then, there exist elements and such thatIf is any fixed element in , , and , then the sequences and converge to some best proximity points of and , respectively. Further, if is another best proximity point of , then

Proof. Let be any fixed element in . Then, it can generate the sequences and by for all and for all , respectively. It is observed thatIt follows from (14) and Remark 11(i) that By mathematical induction, we obtain By using (b), it is not hard to verify that is a Cauchy sequence in and hence converges to some element in . Similarly, we observe thatIt follows from (17) and Remark 11(i) that Therefore, mathematical induction yields It follows from (b) that is a Cauchy sequence in and hence converges to some element in . Furthermore, it can be observed thatLetting in (20), it yields Notice that ; it is not hard to verify that and then .
On the other hand, we also found thatLetting in (22), it yields This implies that and hence . Since the pair forms a weak -cyclic contraction, it follows that there exists such that Hence This shows that is a best proximity point of and is a best proximity point of . Similarly, if we suppose that is another best proximity point of , it can be proved that Consequently, it can be observed thatIt follows from (29) and Remark 11(iii) that This completes the proof of the theorem.

To achieve a better understanding of the situation of the main theorem even more, let us consider the following example to illustrate.

Example 13. Let with supremum norm. Define the following two sets:It is not hard to verify that and . Define the mappings and by for all as shown in Figure 1.
Then, (1)All the conditions are consistent with all of the assumptions in Theorem 12.(2)The mapping is a non-self-Lipschitzian mapping with constant which is not a non-self-nonexpansive mapping.(3)There exist such that . That is, the mapping is not a non-self-Kannan-type mapping with respect to the mapping .(4)There exist such that . That is, the mapping is not a non-self-Chatterjea-type mapping with respect to the mapping .

Solution 1. () It is found thatOn the other hand, let us considerFor convenience for writing, we will let and . From (29), (34), and (35), it is sufficient to show thatfor all . Now, we let (1).(2) + for all . The two surfaces and can be illustrated as in Figure 2.
The idea to prove (36) is to divide the unit square of -plane into six parts as shown in Figure 3.
() We will show that (36) is true on the area . It is clear that , , and for all . Since , then . It implies that , , and and then , , and . Thus, we haveThus it is sufficient to show that or equivalently to show thatDefine for all . It is not hard to compute that for all . Thus is an increasing function on . This implies thatfor all . On the other hand,for all . It is found that is the critical point of providing the absolute maximum valueon . It is not hard to see that is an increasing function on and is a decreasing function on . Notice that the starting point provides the value and then it implies that the end point provides the absolute minimum value on . Therefore, for all . Thus, we have for all .
() For the area , one can study from the proof of the area to verify that (36) holds for all .
() For the area , one can study from the proof of the area to conclude that (36) is true for all .
Notice that the graph of the two surfaces and has symmetry. Therefore, (36) is also true for the areasHence we can conclude that (36) holds for all . This shows that is a generalized non-self-Kannan and Chatterjea mapping with respect to the mapping with constants , .
Further, it can be verified thatWe observe that the pair is a weak -cyclic contraction. For this fact, let us consider and then Therefore, for any it can be seen that It can be observed that the mapping is a non-self-Lipschitzian mapping with constant but it is not a non-self-nonexpansive mapping as follows: Therefore, is a non-self-Lipschitzian mapping with constant . We note that the mapping is not a non-self-nonexpansive mapping. Indeed, if we let and , then() The mapping is not a non-self-Kannan-type mapping with respect to the mapping .
To guarantee this truth, we let and , and it follows from (33) thatand by using (34), we haveComparing (53) and (54), we obtain for all . Therefore, is not a non-self-Kannan-type mapping with respect to the mapping .
() The mapping is not a non-self-Chatterjea-type mapping with respect to the mapping .
In order to guarantee this fact, we let and ; it follows from (35) thatComparing (53) and (56), we obtainfor all . Therefore, is not a non-self-Chatterjea-type mapping with respect to the mapping .

If the mapping in Example 13 is a nonexpansive mapping, then the mapping in the following example still cannot be both a non-self-Kannan-type mapping with respect to the mapping and a non-self-Chatterjea-type mapping with respect to the mapping . However, the mapping is still to maintain its general property; that is, is a generalized non-self-Kannan and Chatterjea mapping with respect to the mapping .

Example 14. Let , , and be as in Example 13. Define by for all . Then, (i) All the conditions are consistent with all of the assumptions in Theorem 12.(ii) The mapping is a non-self-nonexpansive mapping.(ii) There exist such that . That is, the mapping is not a non-self-Kannan-type mapping with respect to the mapping .(iv) There exist such that . That is, the mapping is not a non-self-Chatterjea-type mapping with respect to the mapping .

Solution 2. By Example 13, it can be verified analogously that (i) and (ii) hold. For the proof of (iii) and (iv), we considerLet and in (59). Then, we obtain respectively. Comparing (53) and (60), we obtainfor all . Therefore, is not a non-self-Kannan-type mapping with respect to the mapping . Similarly, comparing (53) and (61), we obtain for all . Therefore, is not a non-self-Chatterjea-type mapping with respect to the mapping .

Corollary 15 (see [33, Theorem ]). Let and be nonempty closed subsets of a complete metric space. Let and satisfy the following conditions for some nonnegative number : (i_K) is nonexpansive.(ii_K) for all .(iii_K)The pair forms a weak -cyclic contraction. Then, there exist elements and such that If is any fixed element in , , and , then the sequences and converge to some best proximity points of and , respectively. Further, if is another best proximity point of , then

Proof. Letting , , and in Theorem 12, then we have the desired result.

Corollary 16 (see [33, Theorem ]). Let and be nonempty closed subsets of a complete metric space. Let and satisfy the following conditions for some nonnegative number : (i_C) is nonexpansive.(ii_C) for all .(iii_C)The pair forms a weak -cyclic contraction. Then, there exist elements and such that If is any fixed element in , , and , then the sequences and converge to some best proximity points of and , respectively. Further, if is another best proximity point of , then

Proof. Letting , , and in Theorem 12, then we have the desired result.

Remark 17. From the discovery of Examples 13 and 14, they are something beyond some assumptions and conditions of Corollaries 15 and 16. For this reason, the situations appearing above are important in causing Theorem 12.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

The author would like to acknowledge Naresuan University for financial support.

References

K. Fan, “Extensions of two fixed point theorems of F. E. Browder,” Mathematische Zeitschrift, vol. 112, pp. 234–240, 1969.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. B. Prolla, “Fixed-point theorems for set-valued mappings and existence of best approximants,” Numerical Functional Analysis and Optimization, vol. 5, no. 4, pp. 449–455, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
S. Reich, “Approximate selections, best approximations, fixed points, and invariant sets,” Journal of Mathematical Analysis and Applications, vol. 62, no. 1, pp. 104–113, 1978.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. M. Sehgal and S. P. Singh, “A generalization to multifunctions of Fan's best approximation theorem,” Proceedings of the American Mathematical Society, vol. 102, no. 3, pp. 534–537, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. M. Sehgal and S. P. Singh, “A theorem on best approximations,” Numerical Functional Analysis and Optimization, vol. 10, no. 1-2, pp. 181–184, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
V. Vetrivel, P. Veeramani, and P. Bhattacharyya, “Some extensions of Fan's best approximation theorem,” Numerical Functional Analysis and Optimization, vol. 13, no. 3-4, pp. 397–402, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3665–3671, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3790–3794, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. P. Farajzadeh, S. Plubtieng, and K. Ungchittrakool, “On best proximity point theorems without ordering,” Abstract and Applied Analysis, vol. 2014, Article ID 130439, 5 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
S. Karpagam and S. Agrawal, “Best proximity point theorems for p-cyclic Meir-Keeler contractions,” Fixed Point Theory and Applications, vol. 2009, Article ID 197308, 9 pages, 2009.
View at: Publisher Site | Google Scholar
B. Piatek, “On cyclic Meir-Keeler contractions in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 1, pp. 35–40, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, “Best proximity point theorems,” Journal of Approximation Theory, vol. 163, no. 11, pp. 1772–1781, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, “Best proximity point theorems generalizing the contraction principle,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 5844–5850, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, “Best proximity points: global optimal approximate solutions,” Journal of Global Optimization, vol. 49, no. 1, pp. 15–21, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, “Global optimal approximate solutions,” Optimization Letters, vol. 5, no. 4, pp. 639–645, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
C. Vetro, “Best proximity points: convergence and existence theorems for $p$ -cyclic mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 7, pp. 2283–2291, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Al-Thagafi and N. Shahzad, “Best proximity sets and equilibrium pairs for a finite family of multimaps,” Fixed Point Theory and Applications, vol. 2008, Article ID 457069, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Al-Thagafi and N. Shahzad, “Best proximity pairs and equilibrium pairs for Kakutani mul-timaps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 3, pp. 1209–1216, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
W. K. Kim, S. Kum, and K. H. Lee, “On general best proximity pairs and equilibrium pairs in free abstract economies,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2216–2227, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
W. A. Kirk, S. Reich, and P. Veeramani, “Proximinal retracts and best proximity pair theorems,” Numerical Functional Analysis and Optimization, vol. 24, no. 7-8, pp. 851–862, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha and P. Veeramani, “Best approximations and best proximity pairs,” Acta Scientiarum Mathematicarum (Szeged), vol. 63, no. 1-2, pp. 289–300, 1997.
View at: Google Scholar
S. Sadiq Basha and P. Veeramani, “Best proximity pair theorems for multifunctions with open fibres,” Journal of Approximation Theory, vol. 103, no. 1, pp. 119–129, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Sadiq Basha, P. Veeramani, and D. V. Pai, “Best proximity pair theorems,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 8, pp. 1237–1246, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. S. Srinivasan, “Best proximity pair theorems,” Acta Scientiarum Mathematicarum, vol. 67, no. 1-2, pp. 421–429, 2001.
View at: Google Scholar | MathSciNet
K. W{\l}odarczyk, R. Plebaniak, and A. Banach, “Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3332–3341, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
K. Wlodarczyk, R. Plebaniak, and A. Banach, “Erratum to: ‘Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces’ [Nonlinear Anal. 70 (2009) 3332–3341. doi:10.1016/j.na.2008.04.037],” Nonlinear Analysis, vol. 71, pp. 3583–3586, 2009.
View at: Google Scholar
K. Włodarczyk, R. Plebaniak, and C. Obczyński, “Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 794–805, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, “Common best proximity points: global minimization of multi-objective functions,” Journal of Global Optimization, vol. 54, no. 2, pp. 367–373, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, “Common best proximity points: global minimal solutions,” TOP, vol. 21, no. 1, pp. 182–188, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
S. Sadiq Basha, N. Shahzad, and R. Jeyaraj, “Common best proximity points: global optimization of multi-objective functions,” Applied Mathematics Letters, vol. 24, no. 6, pp. 883–886, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
N. Shahzad, S. Sadiq Basha, and R. Jeyaraj, “Common best proximity points: global optimal solutions,” Journal of Optimization Theory and Applications, vol. 148, no. 1, pp. 69–78, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Sadiq Basha, N. Shahzad, and R. Jeyaraj, “Best proximity points: approximation and optimization,” Optimization Letters, vol. 7, no. 1, pp. 145–155, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. K. Chatterjea, “Fixed point theorems,” Comptes Rendus de l'Academie Bulgare des Sciences, vol. 25, pp. 727–730, 1972.
View at: Google Scholar

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Copyright © 2016 Kasamsuk Ungchittrakool. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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