International Journal of AnalysisVolume 2013 (2013), Article ID 654659, 7 pageshttp://dx.doi.org/10.1155/2013/654659

Research Article

Common Fixed Point Theorems for --Contractive Type Mappings

Jalal Hassanzadeasl

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 8 November 2012; Revised 5 March 2013; Accepted 11 March 2013

Academic Editor: Leo G. Rebholz

Copyright © 2013 Jalal Hassanzadeasl. 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.

Abstract.

Recently, Samet et al. (2012) introduced the notion of --contractive type mappings. They established some fixed point theorems for these mappings in complete metric spaces. In this paper, we introduce the notion of a coupled --contractive mapping and give a common fixed point result about the mapping. Also, we give a result of common fixed points of some coupled self-maps on complete metric spaces satisfying a contractive condition.

1. Introduction

We know fixed point theory has many applications and was extended by several authors from different views (see, e.g., [1–33]). Recently, Samet et al. introduced the notion of --contractive type mappings [3]. Denote with the family of nondecreasing functions such that for all , where is the th iterate of . It is known that for all and [3]. Let be a metric space, a self map on , and . Then, is called a --contractive mapping whenever for all . Also, we say that is -admissible whenever implies for all [3]. Also, we say that has the property () if is a sequence in such that for all and , then for all . Let be a complete metric space and let a -admissible --contractive mapping on . Suppose that there exists such that . If is continuous or has the property (), then has a fixed point (see [3]; Theorems 2.1 and 2.2). Finally, we say that has the property () whenever for each there exists such that and . If has the property () in the Theorems  2.1 and 2.2, then has a unique fixed point ([3]; Theorem 2.3). It is considerable that the results of Samet et al. generalize similar ordered results in the literature (see the results of the third section [3]). The aim of this paper is introducing the notion of generalized coupled --contractive mappings and give a common fixed point result about the mappings.

Definition 1. Let    the family of functions satisfy:(i) and for all ;(ii) is continuous;(iii) is nondecreasing on ;(iv) for all .

Definition 2. Let the family of functions satisfy is nondecreasing;, for all .

These functions are known in the literature as ()-comparison functions. It is easily proved that if is a ()-comparison function, then for all .

Definition 3. Let be a metric space, and let with given coupled mappings. Let , , , and let for all coupled mappings and . One says that ,   are generalized coupled -contractive mappings whenever for all .

Definition 4. Let , and let . One says that ,   are coupled -admissible if for all .

Definition 5. Let be a compete metric space. For two subsets ,   of , one marks , if for all , there exists such that .

Definition 6. A partial metric on a nonempty set is a function such that for all :;;;.

A partial metric space is a pair such that is a nonempty set, and is a partial metric on . It is clear that if , then from and , . But if , may not be . A basic example of a partial metric is the pair , where for all . If is a partial metric on , then the function given by is a metric on .

Example 7. Let endowed with the standard metric for all . Define the coupled mappings by We define the mapping by If Similarly, . This shows that ,   are coupled -admissible.

Lemma 8. Let be a metric space. Suppose that are generalized coupled --contractive mappings. Then, .

Proof. We first show that any fixed point of is also a fixed point of and conversely. Define for all . Since , we may assume there exists such that , but . Since , we have This contradiction establishes that . A similar argument establishes the reverse containment, and therefore .

2. Main Results

Now, we are ready to state and prove our main results.

Theorem 9. Let be a complete metric space. Suppose that are generalized coupled --contractive mappings and satisfy the following conditions: ,   are coupled -admissible; there exists such that ; or there exists such that ; or is continuous.Then ,   have common fixed point . Further, for each , the iterated sequence with and converges to the common fixed point of .

Proof. By Lemma 8, we have . Let such that . Define the sequence in by and for all . If for some , then are a common fixed point for ,  . So, we can assume that and for all . Since ,   are coupled -admissible, we have Inductively, we have for all . We obtain Now, if So, in general, which is a contradiction since . Thus, Similarly, if we have for all . By induction, we get for all . Fix , and let such that Let with . Using the triangle inequality, we obtain Thus we proved that is a Cauchy sequence in the metric space .Since is a complete metric space, there exists such that as . From the continuity of , it follows that as , then . Similarly if is continuous, we have .

Corollary 10. Let be a complete metric space. Suppose that is a generalized --contractive mapping and satisfies the following conditions: is -admissible; there exists such that ; or there exists such that ; is continuous.Then, has a fixed point . Further, for each , the iterated sequence with converges to the fixed point of .

Example 11. Let endowed with the standard metric for all . Define the coupled mappings by We define the mapping by If and for all , we have for all . Thus, ,   are generalized coupled --contractive mappings. Moreover, there exists such that . In fact, for , we have . Obviously, is continuous, and so it remains to show that ,   are coupled -admissible. To do so, let such that . This implies that by the definition of . We then have , and . Then, ,   are coupled -admissible. Now, all the hypotheses of Theorem 9 are satisfied.Consequently, ,   have common fixed points. In this example, is at least one common fixed point of and .

Now, we omit the continuity hypothesis of and .

Theorem 12. Let be a complete metric space. Suppose that are generalized coupled --contractive mappings and satisfy the following conditions: ,   are coupled -admissible; there exists such that ; or there exists such that ; if is a sequence in such that for all and as , then there exists a subsequence of such that for all ; or if is a sequence in such that for all and as , then there exists a subsequence of such that for all .Then, ,   have common fixed point . Further, for each , the iterated sequence with and converges to the common fixed point of .

Proof. Following the proof of Theorem 9, we know that is a Cauchy sequence in the complete metric space . Then, there exists such that as . From Theorem 9 and condition (iii), there exists a subsequence of such that for all .Applying Theorem 9, for all , we get that On the other hand, we have Letting , in the above equality, we get that Suppose that . From (25), for large enough, we have , which implies that Thus, from (23), we have . Letting in the above inequality, and using (25), we obtain that which is a contradiction. Thus, we have ; that is, . Similarly, it can be shown that .

Corollary 13. Let be a complete metric space. Suppose that is generalized --contractive mapping and satisfies thefollowing conditions: is -admissible; there exists such that ; or there exists such that ; if is a sequence in such that for all and as , then there exists a subsequence of such that for all ; or if is a sequence in such that for all and as , then there exists a subsequence of such that for all .Then, has a fixed point . Further, for each , the iterated sequence with converges to the fixed point of .

Example 14. Let endowed with the standard metric for all . Define the coupled mappings by We define the mapping by If and for all , we have for all . Thus, ,   are generalized coupled --contractive mappings. Moreover, there exists such that . In fact, for , we have . Let ,   and for all . By the definition of we have, for all , so and . It remains to show that ,   are coupled -admissible. In doing so, let such that . This implies that by the definition of . We have ,  , and . Then ,   are coupled -admissible. Now, all the hypotheses of Theorem 12 are satisfied. Consequently, ,   have common fixed points. In this example, is at least one common fixed point of and .

3. Fixed Point Theorems on Ordered Metric Space

Theorem 15. Let be a complete ordered metric space, , , and ,   be coupled mappings on such that and for all with . Suppose that there exists such that or , and if is a sequence in such that or for all and , then or for all . If implies   or   or   or  , then have common fixed points.

Proof. Define by whenever , and define whenever ⋠ . It is easy to check that ,   are a coupled -admissible and generalized coupled --contractive mappings on . Now, by using Theorem 9, ,   have common fixed points.

Corollary 16. Let be a complete ordered metric space, , , and a mapping on such that for all with . Suppose that there exists such that or and if is a sequence in such that or for all and , then or for all . If implies or , then has a fixed point.

4. Fixed Point Theorems on Metric Spaces Endowed with Partial Metric

If we substitute a partial metric instead the metric in Theorem 9, it is easy to check that a similar result holds for the partial metric space case as follows. We define for all coupled mappings and .

Theorem 17. Let be a complete partial metric space,   a function, , , and ,   self maps on such that for all . Suppose that ,   are coupled -admissible and there exists such that or . Assume that if is a sequence in such that or for all and as , then there exists a subsequence of such that or for all . Then have common fixed points.

Corollary 18. Let be a complete partial metric space,   a function, , , and a self map on such that for all . Suppose that is -admissible and there exists such that or . Assume that if is a sequence in such that or for all and as , then there exists a subsequence of such that or for all . Then has a fixed point.

Example 19. Let endowed with the partial metric , for all . Define the mapping , by We define the mapping by since is -admissible. If , we have Suppose that is a sequence in such that for all and as ; by definition of , we have ,   and , on the other hand ; then there exists a subsequence of such that for all . There exists such that . This show that all conditions of Corollary 18 are satisfied, and so has a fixed point in .

Acknowledgment

The author would like to thank Tehran Science and Research Branch, Islamic Azad University, for the financial support of this research, which is based on a research project contract.

References

M. Abbas, T. Nazir, and S. Radenović, “Common fixed points of four maps in partially ordered metric spaces,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1520–1526, 2011.

L. Zhilong, “Fixed point theorems in partially ordered complete metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 69–72, 2011.

B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for –-contractive type mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.

S. M. A. Aleomraninejad, S. Rezapour, and N. Shahzad, “On fixed point generalizations of Suzuki's method,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1037–1040, 2011.

S. M. A. Aleomraninejad, S. Rezapour, and N. Shahzad, “Fixed points of hemi-convex multifunctions,” Topological Methods in Nonlinear Analysis, vol. 37, no. 2, pp. 383–389, 2011.

S. M. A. Aleomraninejad, S. Rezapour, and N. Shahzad, “Some fixed point results on a metric space with a graph,” Topology and its Applications, vol. 159, no. 3, pp. 659–663, 2012.

M. A. Alghamdi, S. H. Alnafei, S. Radenović, and N. Shahzad, “Fixed point theorems for convex contraction mappings on cone metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2020–2026, 2011.

I. Altun, B. Damjanović, and D. Djorić, “Fixed point and common fixed point theorems on ordered cone metric spaces,” Applied Mathematics Letters, vol. 23, no. 3, pp. 310–316, 2010.

H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered -metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2443–2450, 2011.

H. Aydi, H. K. Nashine, B. Samet, and H. Yazidi, “Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6814–6825, 2011.

V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 18, pp. 7347–7355, 2011.

V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 15, pp. 4889–4897, 2011.

M. Borcut and V. Berinde, “Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5929–5936, 2012.

M. Derafshpour, S. Rezapour, and N. Shahzad, “Best proximity points of cyclic -contractions in ordered metric spaces,” Topological Methods in Nonlinear Analysis, vol. 37, no. 1, pp. 193–202, 2011.

H.-S. Ding and L. Lu, “Coupled fixed point theorems in partially ordered cone metric spaces,” Filomat, vol. 25, no. 2, pp. 137–149, 2011.

J. H. Asl, S. Rezapour, and N. Shahzad, “On fixed points of -contractive multifunctions,” Fixed Point Theory and Applications, vol. 2012, article 212, 2012.

J. Jachymski, “Equivalent conditions for generalized contractions on (ordered) metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 768–774, 2011.

E. Karapınar and B. Samet, “Generalized - contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, Article ID 793486, 17 pages, 2012.

N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 983–992, 2011.

H. K. Nashine and B. Samet, “Fixed point results for mappings satisfying -weakly contractive condition in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 6, pp. 2201–2209, 2011.

H. K. Nashine, B. Samet, and C. Vetro, “Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 712–720, 2011.

H. K. Nashine and W. Shatanawi, “Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1984–1993, 2011.

S. Rezapour and P. Amiri, “Some fixed point results for multivalued operators in generalized metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2661–2666, 2011.

S. Rezapour, R. H. Haghi, and B. E. Rhoades, “Some results about -stability and almost -stability,” Fixed Point Theory, vol. 12, no. 1, pp. 179–186, 2011.

S. Rezapour, R. H. Haghi, and N. Shahzad, “Some notes on fixed points of quasi-contraction maps,” Applied Mathematics Letters, vol. 23, no. 4, pp. 498–502, 2010.

S. Rezapour and J. H. Asl, “A simple method for obtaining coupled fixed points of --contractive type mappings,” International Journal of Analysis, vol. 2013, Article ID 438029, 7 pages, 2013.

M.-D. Rus, “Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 5, pp. 1804–1813, 2011.

B. Samet and C. Vetro, “Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 12, pp. 4260–4268, 2011.

M. Shen and S. Hong, “Common fixed points for generalized contractive multivalued operators in complete metric spaces,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1864–1869, 2009.

W.-S. Du, “Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi's condition in quasiordered metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 876372, 9 pages, 2010.

W. Shatanawi and B. Samet, “On -weakly contractive condition in partially ordered metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 8, pp. 3204–3214, 2011.

X.-Q. Hu and X.-Y. Ma, “Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6451–6458, 2011.

X. Zhang, “Fixed point theorems of multivalued monotone mappings in ordered metric spaces,” Applied Mathematics Letters, vol. 23, no. 3, pp. 235–240, 2010.