Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces

De La Sen, M.; Ibeas, Asier

doi:https://doi.org/10.1155/2014/914915

Abstract and Applied Analysis

On this page

Abstract Introduction Acknowledgments References Copyright Related Articles

Special Issue

Variational Analysis, Optimization, and Fixed Point Theory 2014

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 914915 | https://doi.org/10.1155/2014/914915

Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces

M. De La Sen¹and Asier Ibeas²

Academic Editor: Adrian Petrusel

Received09 May 2014

Revised22 Jul 2014

Accepted05 Aug 2014

Published15 Oct 2014

Abstract

This paper presents some results concerning the properties of distances and existence and uniqueness of best proximity points of p-cyclic proximal, weak proximal contractions, and some of their generalizations for the non-self-mapping , where and , , are nonempty subsets of which satisfy , such that is a metric space. The boundedness and the convergence of the sequences of distances in the domains and in their respective image sets of the cyclic proximal and weak cyclic proximal non-self-mapping, and of some of their generalizations are investigated. The existence and uniqueness of the best proximity points and the properties of convergence of the iterates to such points are also addressed.

1. Introduction

The characterization and study of existence and uniqueness of best proximity points is an important tool in fixed point theory concerning cyclic nonexpansive mappings including the problems of (strict) contractions, asymptotic contractions, contractive, weak-contractive mappings, and cyclic mappings and also in related problems of proximal contractions, weak proximal contractions, and approximation results and methods [1–15]. The application of the theory of fixed points in stability issues of dynamic systems, [16–21] has been proved to be a very useful tool. See, for instance, [22–26] and references therein. Some best approximation problems in semiconvex and locally convex structures and Hyers-Ulam type stability in multivalued functions and in additive-quadratic functional equations are investigated in [27–30] and some of the references therein. Recent trends concerning best proximity points and related problems are dealt with in [31–35] and some references therein. In particular, the problem of best proximity points of two mappings in a cyclic disposal is investigated in [31] under a nonlinear contractive condition. In [32], several results are obtained for proximal and weak proximal contractions of several types as well as for generalized proximal nonexpansive mappings. A modified Suzuki proximal contraction is proposed and discussed in [33] and “ad hoc” best proximity and fixed point results are obtained. Generalizations of proximal contractions of first and second kinds are given in [34, 35] for non-self-mappings and related optimal approximate solution theorems are obtained.

This paper is devoted to formulating and proving some results being concerned with the boundedness and convergence properties of distances and the convergence of the built iterated sequences to unique existing best proximity points of -cyclic proximal and weak proximal contractions of the form () where and , for all , are nonempty subsets of which satisfy , for all , with being a metric space. In the most general case, all the and pairs of subsets, for all , are assumed to be pairwise disjoint. The results are also extended to a class of generalized -cyclic proximal and weak proximal contractions in the sense that the contractiveness constraints are referred to finite sets of consecutive iterations rather than to each iteration. The boundedness and convergence of the sequences of distances in the domains and image sets of the cyclic proximal and weak cyclic proximal non-self-mappings are investigated. The existence and uniqueness of the best proximity points and their allocation as limit points, or limit cycles of best proximity points, are also addressed. These last properties are achieved if the metric space is complete under approximative compactness’ assumptions of the image subsets of the cyclic mapping with respect to the domain subsets.

2. -Cyclic Proximal Contractions, Extensions, Boundedness, and Convergence of Distances

Consider the metric space and subsets and of for , where with . Consider also a non-self-mapping , satisfying , for all . Assume that , , and , for all by assuming also that and , for all , for all . If the pair satisfies for any , then and are best proximity points in and with respect to .

In the following, the fact that the best proximity points are best proximity points with respect to the mapping is not mentioned explicitly.

is the set of best proximity points of and is the set of best proximity points of . Through the paper, it is assumed that and , for all . An important remark is that the above statement can be considered for the particular case that which is well known in the context of -cyclic self-mappings with for all . However, the proposed statement is more general in the sense of the following illustrative example.

Example 1. Consider a metric space and such that are nonempty with , , , and for . Assume also that , , , and . Then, we can formulate the following simple 2-cyclic proximal-type problem. Fix as a best proximity point of and then compute and , best proximity points of and , such that (2-cyclic proximal constraint, first step); (2-cyclic proximal constraint, second step); (2-cyclic best proximity constraints); (2-cyclic associate best proximity constraints for the images).
Note that there are four potentially distinct constraints related to , , , and which can be distinct so that the problem is more general than the simple use of for for the 2-cyclic self-mapping . A variant proximal-type problem arises if and and the best proximity points are taken as follows:
, , , , , and then .

The following definitions will be then used through the paper.

Definition 2. is said to be a -cyclic proximal contraction with respect to its domain () if there are real constants , for all , such that any two sequences and , for all , satisfy the constraints provided that , for any given with if and that , for all .

Definition 3. is said to be a weak -cyclic proximal contraction with respect to its domain () if there are real constants , for all , subject to , such that any two sequences and , for all , satisfy the constraints (1) and (2) provided that for any given and that , for all .

Definition 4. is said to be a generalized -cyclic proximal contraction with respect to its domain () if there are bounded real functions , for all , such that any sequences and , for all , satisfy the constraints (1) and (2) with the replacements , for all , provided that for any given and that , for all .

Definition 5. is said to be a generalized weak -cyclic proximal contraction with respect to its domain () if there are bounded real functions , for all , and a strictly increasing sequence of integers , subject to , , and , where such that any two sequences and , for all , provided that for any given , satisfy the constraints and the constraints (1) and (2) provided that for any given and that , for all .

The following assertions are obvious without proof from Definitions 2–5.

Assertions 1. If is , then it is .
If is , then it is .
If is , then it is .

Note that the converse implications of those in Assertions 1 are not true in general. The relevant distances satisfy the following convergence and boundedness result.

Lemma 6. Consider a metric space with subsets and a -cyclic mapping which is , subject to , , and , with , for all such that and are nonempty, for all . Consider also any sequences which satisfy , for all . Then, the following properties hold.
(i) The sequences of distances , and they are bounded for any given initial points , for any given .
If, furthermore, is continuous in , for all , then and and both sequences of distances are bounded for any given initial points , for any given .
If the sets of best proximity points and , for all , are bounded, then the sequences , , , and are uniformly bounded for any initial points for some .
(ii) The sequences , for all (note that and for ) are Cauchy sequences for any initial points any given initial point for any arbitrary given . The corresponding image sequences , for all , are also Cauchy sequences if , for all , and is continuous in , for all .

Proof. Note that, for any , implies and implies . Take sequences with initial points such that and for some . The problem of boundedness and convergence of distances can be described equivalently from initial points (i.e., both initial conditions at the same set), with and denoting for some (in particular, if for the set of both initial points) since . One has from (1)-(2) and (3)–(6) that, for any sequences and fulfilling and , for all , such that if and for all since , and with .
Thus, , for all from (9a) and the distance subsequence , for all is bounded from (9b) for any given initial points . Also, one gets from (6), subject to (3), that for all . Those results also imply that the sequences of distances . It is now proved by contradiction that and . Assume that, for each given for some and any , there are some , some , some , sequences of integers , and sequences of best proximity points and for , such that does not converge to zero so that it has some subsequence which does not converge either: since , for all . This implies that while does not converge to zero.
Since , , and is continuous in , for all and any given , then . Also, , for all and any given ; then , . Thus, and if , for any given .
On the other hand, and are bounded, since is bounded from (6) because is and since one has for some positive real constant that If the sets of best proximity points for all are bounded, then the sequences of distances , , , and are uniformly bounded for any initial best proximity points for some which follows by taking . Property (i) has been fully proved.
To prove Property (ii), take any sequences and , for all , for given initial points for some . Note, from (6) for , for all , that with the given upper bound being independent of the integers and . Thus, one has for any that where , with and , since Definition 5 holds for . Thus, one gets from (15) that, for any given real , , for all for any and any given integers if . Thus, the sequences , for all , are Cauchy sequences for any given initial point and any . This implies also that the sequences of images of the above points are also Cauchy sequences since is contractive and then continuous.

From Assertions 1, we also have the subsequent parallel result to Lemma 6.

Lemma 7. Assume that is either or or with the assumptions of Lemma 6, and consider any sequences which satisfy , for all . Then, the following properties hold.
(i) The sequences of distances , and they are bounded for any given initial points for any given .
If, furthermore, , for all , and is continuous in , for all , then and and it is bounded for any initial points any given initial points , for some .
If the sets of best proximity points and , for all , are bounded, then the sequences , , , and are uniformly bounded for any initial best proximity points for some .
(ii) The sequences , for all , are Cauchy sequences for any given initial point for any given . The corresponding image sequences , for all , are also convergent, then Cauchy sequences if , for all , and is continuous in , for all .

Definition 8. is said to be a -cyclic proximal contraction with respect to its image () if there are real constants , for all , such that any two sequences and , for all , being point-to-point images of sequences and for any given which satisfy , for all , where if such that the initial points are the images of points , for any given , satisfy the constraints

Definition 9. is said to be a weak -cyclic proximal contraction with respect to its image () if there are real constants , for all , subject to , such that any two sequences and , for all , being point-to-point images of sequences and for any given , where for , for all , such that the initial points are the images of points , for any given , satisfy constraints (16) and (17).

Definition 10. is said to be a generalized -cyclic proximal contraction with respect to its image () if there are bounded real functions , for all , such that any sequences and , for all , being point-to-point images of sequences and for any given , where for , for all , such that the initial points are the images of points , satisfy the constraints (16) and (17) with the replacements , for all .

Definition 11. is said to be a generalized weak -cyclic proximal contraction with respect to its image () if there are bounded real functions , for all , and a strictly increasing sequence of integers , subject to , , and , where such that any two sequences and , for all , being point-to-point images of sequences and for any given , where for , for all , such that the initial points are the images of points , for any given , satisfy the following constraints: and constraints (16) and (17) with the replacements , for all .

The following assertions are obvious without proof from Definitions 8–11 and are a parallel result to Assertions 1.

Assertions 2. If is , then it is .
If is , then it is .
If is , then it is .

Note that the converse implications of those in Assertions 1 are not true in general.

The relevant distances satisfy the following convergence and boundedness result which is a counterpart of Lemmas 6 and 7. Its proof is close to that of Lemma 6 and Assertions 2 by using (16) and (17) for Definition 11 and their variants for Definitions 8–10.

Lemma 12. Assume that is either or or or under the assumptions of Lemma 6, and consider any sequences which satisfy , for all . Then, the following properties hold.
(i) The sequences of distances , and they are bounded for any given initial points , for some .
If, furthermore, , for all , and is continuous in , for all , then and and it is bounded any given initial points for some .
If the sets of best proximity points and , for all , are bounded, then the sequences , , , and are uniformly bounded for any initial best proximity points for some .
(ii) The sequences , for all , are Cauchy sequences for any given initial point for any given . The corresponding image sequences , for all , are also convergent; then Cauchy sequences if , for all , and are continuous in , for all .

Remark 13. The result of Lemma 12, as well as Lemma 12 (ii), obtained under the assumption that is continuous in and also holds without such a continuity assumption if the contractive conditions (16) and (17) become modified to the right limits as follows: provided that any discontinuity points in , if any, are of first-class finite-jump type under right best proximity constraints In the same way, the result of Lemmas 6 and 7, as well as their properties (ii) obtained under the assumption that , is continuous in and also holds under finite-jump discontinuities in for sequences , , and satisfying the contractive proximal conditions (1) and (2) if Definition 2, or their counterparts of Definitions 3–5 for right values and under right best proximity constraints (24).

We first recall the subsequent useful definition [2–4, 7] as follows.

Definition 14. Let and be two nonempty subsets of a metric space and let for . is said to be approximately compact with respect to if each sequence satisfying for some has a convergent subsequence.

Note that if the sets of best proximity points and are nonempty if Definition 14 holds, then is approximately compact with respect to if every sequence such that for some has a convergent subsequence since . Note that every set is approximately compact with respect to itself and that every compact set is approximately compact with respect to any nonempty subset of a metric space. Also, if is compact and is approximately compact with respect to , each sequence has a convergent sequence. If and are nonempty and closed and is approximately compact with respect to , then is closed. See, for instance, [2–4, 7]. A result on existence and uniqueness of best proximity points follows for -cyclic proximal contraction fulfilling Definitions 2–5 under Lemmas 6 and 7 follows.

Theorem 15. Consider a complete metric space with nonempty closed subsets and a -cyclic mapping being either or or or , subject to set distances , , and , for all such that is nonempty and is approximately compact with respect to and , for all . The following properties hold.(i) has a unique best proximity point at such that , for all , and all the sequences converge to a unique limit cycle .(ii)Furthermore, if either is continuous, respectively, has eventual finite-jump discontinuity points, then , for all , respectively, , for all , are unique best proximity points such that , for all , respectively, , for all , and all the sequences converge to a unique limit cycle .

Proof. Since is nonempty and , for all , then and are nonempty, for all . Also, is closed since is approximately compact with respect to . Consider any sequences which satisfy One gets, from Lemma 6(i), if the mapping is and, from Lemma 7(i), if the mapping is either or or that, since , for all , for some and since and , for all , and some subsequences of the sequences and , for all , respectively, for any given initial points for any given . The following results hold.
(1) From (26) and by taking and , for all for for any given , one gets , since is closed, for all and, from (27), , for all .
(2) Again, from (26) , for all for for any given , , for all .
(3) Combining results (1) and (2) with (26), it follows that , for all .
(4) Results (1)–(3) hold irrespective of the subset for where the initial conditions of the sequences belong to, so for any (see the beginning of the proof of Lemma 6). Thus, from result , there are unique limit points at each subset of all the sequences such that any such sequence converges to a unique limit cycle consisting of best proximity points of adjacent subsets , for all .
(5) Since is closed and approximately compact with respect to , for all , one gets from (28) that a subsequence of is convergent for each ; say , for all . Since , all its subsequences converge to the same limit so that and then is unique, since each is unique, within each , for all and, again, from (28), , for all if is continuous at and if has a finite-jump discontinuity at , then (see Remark 13). The result has been proved.

A further result on the existence and uniqueness of best proximity points follows for -cyclic proximal contractions subject to Definitions 8–11 under Lemma 12 and whose proof is very close to that of Theorem 15.

Theorem 16. Consider a complete metric space with nonempty subsets and a -cyclic mapping being either or or or , subject to set distances , , and , for all , such that is nonempty and closed and is nonempty and is approximately compact with respect to and , for all . The following properties hold.(i) has a unique best proximity point at such that , for all , and all the sequences converge to a unique limit cycle .(ii)Furthermore, if either is continuous, respectively, has eventual finite-jump discontinuity points, then , for all , respectively, , for all , are unique best proximity points such that , for all , respectively, , for all , and all the sequences converge to a unique limit cycle .

Proof. Since , for all , then and are nonempty, since is nonempty, for all . is, furthermore, closed since is approximately compact with respect to . Thus, , , and are nonempty and closed, . Consider any sequences which satisfy One gets from Lemma 12 that, since , , for some , since and , , and some subsequences of the sequences and , , respectively, for any given initial points for any given . The following results hold.
(6) From (31) and by taking , , for for any given , one gets , since is closed, and, from (32), , .
(7) Again from (31), , for for any given , so that , .
(8) Combining results (6) and (7) with (31), it follows that , .
(9) Results (6)–(9) hold irrespective of the subset for where the initial conditions of the sequences belong to then for any . Thus, considering result , there are unique limit points at each subset of all the sequences such that any such sequence converges to a unique limit cycle consisting of best proximity points of adjacent subsets , .
Since is closed and approximately compact with respect to , , one gets from (33) that a subsequence of is convergent for each ; say , . Since , all its subsequences converge to the same limit so that and fulfilling which is unique. Assume not so that there are for some such that . Assume a sequence with and a sequence with initial point and some for some and some nonnegative integer . But then does not converge to zero so that is unique, . The distance convergence properties are independent of the fact that for the initial condition is as equal or distinct as , as discussed in Lemma 6. If and another sequence. Since each is unique, within each , and, again, from (33), , if is continuous at and if has a finite-jump discontinuity at , then (see Remark 13). The result has been proved.

Example 17. Consider a -cyclic proximal contraction with respect to its domain: , where and for are nonempty closed subsets of . Take any sequences , being subsequences of defined either by , or by , , , and subject to the constraints below under the Euclidean metric , for some contractive real constant such that and is a complete metric space and also a Banach space. Assume that and with and and and with and , , , , and , so that , with and is a (Definition 2 with ) if the subsequent constraints hold for all : In particular, (37)–(40a) and (40b) are satisfied if, for all , Parallel results for the case when is , , or (Definitions 3–5 with ) can be discussed in the same way with the appropriate extensions for the contractive constant or function. It follows that , , , , , , and, according to (40a), (40b), and (35)-(36), , since , , and .

Example 18. Consider Example 17 in the case that is , , , or (Definitions 8–11 with ); (40a) using (37) can be reformulated accordingly. In particular, if it is , then one gets for some real constant Then, , , , , , , , and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the Spanish Government for its support of this research trough Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. Finally, they thank the reviewers for their useful suggestions towards the improvement of the first version of this paper.

References

S. Sadiq Basha, “Best proximity points: optimal solutions,” Journal of Optimization Theory and Applications, vol. 151, no. 1, pp. 210–216, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | 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 | 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
M. Gabeleh, “Best proximity points for weak proximal contractions,” Bulletin of the Malaysian Mathematical Society. In press.
View at: Google Scholar
B. Samet, “Some results on best proximity points,” Journal of Optimization Theory and Applications, vol. 159, no. 1, pp. 281–291, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. Jleli and B. Samet, “Best proximity points for $α$ - $ψ$ -proximal contractive type mappings and applications,” Bulletin des Sciences Mathematiques, vol. 137, no. 8, pp. 977–995, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. de la Sen, “Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 572057, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
C.-M. Chen and C. H. Lin, “Best periodic proximity points for cyclic weaker Meir-Keeler contractions,” Journal of Applied Mathematics, vol. 2012, Article ID 782389, 7 pages, 2012.
View at: Publisher Site | Google Scholar
M. de la Sen, “Some combined relations between contractive mappings, Kannan mappings, reasonable expansive mappings, and T-stability,” Fixed Point Theory and Applications, vol. 2009, Article ID 815637, 2009.
View at: Google Scholar | MathSciNet
M. de la Sen and R. P. Agarwal, “Some fixed point-type results for a class of extended cyclic self-mappings with a more general contractive condition,” Fixed Point Theory and Applications, vol. 2011, article 59, 2011.
View at: Google Scholar | MathSciNet
M. de la Sen, R. P. Agarwal, and R. Nistal, “Non-expansive and potentially expansive properties of two modified p-cyclic self-maps in metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 14, no. 4, pp. 661–686, 2013.
View at: Google Scholar | MathSciNet
M. De la Sen and R. P. Agarwal, “Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type,” Fixed Point Theory and Applications, vol. 2011, article 102, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
N. Hussain, “Common fixed points in best approximation for Banach operator pairs with Ćirić type $I$ -contractions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1351–1363, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
C. Mongkolkeha, Y. J. Cho, and P. Kumam, “Best proximity points for generalized proximal $C$ -contraction mappings in metric spaces with partial orders,” Journal of Inequalities and Applications, vol. 2013, article 94, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
S. S. Basha and N. Shahzad, “Best proximity point theorems for generalized proximal contractions,” Fixed Point Theory and Applications, vol. 2012, article 42, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
V. M. Marchenko and J. J. Luazo, “On the stability of hybrid differential-difference systems,” Differential Equations, vol. 45, no. 5, pp. 743–756, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
V. M. Marchenko, “Hybrid discrete-continuous systems: II. Controllability and reachability,” Differential Equations, vol. 49, no. 1, pp. 112–125, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
V. M. Marchenko, “Observability of hybrid discrete-continuous systems,” Differential Equations, vol. 49, no. 11, pp. 1389–1404, 2013.
View at: Google Scholar
A. Bilbao-Guillerna, M. de la Sen, A. Ibeas, and S. Alonso-Quesada, “Robustly stable multiestimation scheme for adaptive control and identification with model reduction issues,” Discrete Dynamics in Nature and Society, vol. 2005, no. 1, pp. 31–67, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
S. Alonso-Quesada and M. De la Sen, “Robust adaptive control of discrete nominally stabilizable plants,” Applied Mathematics and Computation, vol. 150, no. 2, pp. 555–583, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
M. de la Sen, “Adaptive stabilization of first-order systems using estimates modification based on a Sylvester determinant test,” Computers and Mathematics with Applications, vol. 37, no. 10, pp. 51–62, 1999.
View at: Publisher Site | Google Scholar
M. De la Sen, “About robust stability of dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2008, Article ID 480187, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
S. N. Mishra, S. L. Singh, and R. Pant, “Some new results on stability of fixed points,” Chaos, Solitons & Fractals, vol. 45, no. 7, pp. 1012–1016, 2012.
View at: Publisher Site | Google Scholar
M. De la Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2011, Article ID 867932, 19 pages, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
M. de la Sen, “Preserving positive realness through discretization,” Positivity, vol. 6, no. 1, pp. 31–45, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
M. de la Sen and N. Luo, “Design of linear observers for a class of linear hybrid systems,” International Journal of Systems Science, vol. 31, no. 9, pp. 1077–1090, 2000.
View at: Publisher Site | Google Scholar
H. K. Pathak and M. S. Khan, “Common fixed point results and its applications to best approximation in ordered semi-convex structure,” Bulletin of Mathematical Analysis and Applications, vol. 1, no. 1, pp. 30–48, 2009.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. K. Nashine and M. S. Khan, “An application of fixed point theorem to best approximation in locally convex space,” Applied Mathematics Letters, vol. 23, no. 2, pp. 121–127, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
H. A. Kenary, H. Rezaei, Y. Gheisari, and C. Park, “On the stability of set-valued functional equations with the fixed point alternative,” Fixed Point Theory and Applications, vol. 2012, article 81, 2012.
View at: Publisher Site | Google Scholar
T. Z. Xu and J. M. Rassias, “A fixed point approach to the stability of an AQ-functional equation on β-Banach modules,” Fixed Point Theory and Applications, 2012:32, 14 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
W. Shatanawi, “Best proximity point on nonlinear contractive condition,” Journal of Physics: Conference Series, vol. 435, no. 1, Article ID 012006, 2013.
View at: Publisher Site | Google Scholar
M. Gabeleh, “Best proximity points theorems via proximal non-self mappings,” Journal of Optimization Theory and Applications, 2014.
View at: Publisher Site | Google Scholar
N. Hussain, A. Latif, and P. Salimi, “Best proximity point results for modified Suzuki α-ψ-proximal contractions,” Fixed Point Theory and Applications, vol. 2014, article 10, 2014.
View at: Google Scholar
W. Sanhan, C. Mongkolkeha, and P. Kumam, “Generalized proximal ψ-contraction mappings and best proximity points,” Abstract and Applied Analysis, vol. 2012, Article ID 896912, 19 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
W. Sintunavarat and P. Kumam, “The existence theorems of an optimal approximate solution for generalized proximal contraction mappings,” Abstract and Applied Analysis, vol. 2013, Article ID 375604, 8 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2014 M. De La Sen and Asier Ibeas. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

860

Downloads

1144

Citations

Abstract and Applied Analysis

Variational Analysis, Optimization, and Fixed Point Theory 2014

Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces

Abstract

1. Introduction

2. -Cyclic Proximal Contractions, Extensions, Boundedness, and Convergence of Distances

3. Best Proximity Points and Related Convergence Results

Conflict of Interests

Acknowledgments

References

Copyright