Abstract and Applied Analysis

Volume 2013 (2013), Article ID 697151, 6 pages

http://dx.doi.org/10.1155/2013/697151

## A Note on Various Classes of Compatible-Type Pairs of Mappings and Common Fixed Point Theorems

^{1}Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia^{2}Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia^{3}Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 8 September 2013; Accepted 27 September 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Zoran Kadelburg et al. 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

Several authors have introduced various conditions which can be used in order to prove common fixed point results. However, it became clear recently that some of these conditions, though formally distinct from each other, actually coincide in the case when the given mappings have a unique point of coincidence. Hence, in fact, new common fixed point results cannot be obtained in this way. We make a review of such connections and results in this paper.

#### 1. Introduction

The simplest common fixed point results for mappings can be obtained if and commute (Jungck, [1]). Obviously, this condition is too strong, and so it is natural to seek for weaker assumptions. Hence, several authors have introduced various other conditions (we will call them compatible-type conditions) which can be used in order to prove common fixed point results. Some of these conditions were given in [2–19]. These (and other) conditions were used in other papers cited in the references. A review of the relationship between various compatible-type conditions introduced until 2001 was given in [20].

However, it became clear recently that some of these conditions, though formally distinct from each other, actually coincide in the case when the given mappings have a unique point of coincidence. Hence, in fact, new common fixed point results cannot be obtained in this way. We make a review of such connections and results in this paper.

#### 2. Definitions and Relations between Various Types of Pairs

Most of the notions and results that follow can be formulated and proved in various types of spaces—metric, symmetric, cone metric, -metric, probabilistic metric, and so forth. For the sake of simplicity, we will stay within the framework of metric spaces.

Let be a metric space, and let . We will denote by the set of coincidence points (CP) of and , that is, by the set of points of coincidence (POC) of and , that is, and by the set of sequences in satisfying , that is,

The following are definitions of some of the multitude of compatible-type conditions, introduced and used for establishing common fixed point results in recent decades.

*Definition 1. *It is said that the pair is(1)weakly commuting [2] if, for all , ;(2)said to satisfy the property (E.A) [7] if (i.e., if there exists a sequence in such that );(3)compatible [3] if, for all , implies that ;(4)noncompatible [4] if, for some , or does not exist;(5)subcompatible [15] if, for some , ;(6)conditionally compatible [6] if implies that for some , ;(7)weakly compatible [5] if, for all , implies that (i.e., if, for all , implies that );(8)occasionally weakly compatible [8] (see also [21–23]) if, for some , and (i.e., if, for some , );(9)conditionally commuting [9] if implies that there exists such that, for all , ;(10)faintly compatible [16] if it is conditionally compatible and conditionally commuting (i.e., (6) and (9) hold);(11)reciprocally continuous [13] if, for all , implies that and ;(12)subsequentially continuous [15] if, for some , and , ;(13)conditionally reciprocal continuous [19] if implies that, for some , , , and ;(14)a -operator pair [10] if, for some , ;(15)a -operator pair [11] if, for some , ;(16)a generalized -operator pair of order [12] if, for some , ;(17)a -operator pair [14] if, for some , ;(18)-biased [17] if, for all , , where or ;(19)weakly -biased [17] if, for all , implies that ;(20)occasionally weakly -biased [11, 18] if, for some , ;

Note that the conditions (7), (8), and (9) are purely set-theoretical and do not depend on the metric structure of . All other conditions are metrical and could change if the metric of the space is changed (or some other structure of the space is applied).

In Table 1, we state which of these properties trivially hold (or can never hold) if one (or both) of the sets and is empty or nonempty. Also, some of implications between these conditions obviously hold in some of these cases. Note that (and hence, ).

Counter examples for some of the reverse implications are given in [8, Example in Section 2], [14, Example 3.1], [21, Example], and [24, Example 2.12].

The following are some other implications (mostly clear from definitions) that hold between the introduced notions. When it is not obvious that the reverse implication does not hold, a reference for a counterexample is given. . For the reverse implication see [3, Examples 2.1 and 2.2]. . For the reverse implication see [25, Example 1]. and . For the reverse implications see [26, Example 2.3]. . For the reverse implication see [16, Example 1.2]. (for the reverse implication take arbitrary mappings satisfying ) and . (2) and (7) are independent of each other (see [27, Examples 2.1-2.2]). . For the reverse implication see, for example, [28, Example 2.3] (note that a part of the definition of the mapping in this example is missing; it should be, for example, for ). (6) does not imply (7) (see [16, Example 1.3]). (3) and (8) are independent (in one direction take any pair with , and for the reverse see [9, Example 1.1]). (4) and (8) are independent of each other (see [9, Examples 1.1–1.3]). . For the reverse implication see [15, Example 1.2]. and . For the reverse implications take arbitrary mappings satisfying (and hence, ). (9) does not imply (7) (see [9]). . For the reverse implication see [16, Example 1.4]. (4) and (10) are independent of each other. An example of a noncompatible pair which is not faintly compatible is [16, Example 1.5] (see also [9, Example 1.1]). Any pair of commuting that maps with is faintly compatible and not noncompatible. . For the reverse implication see [16, Example 1.7] or [29, Example 2.1]. (11) and (12) are independent conditions, (in one direction, take arbitrary pair satisfying , and in the other see [15, Example 1.4]). and . For the reverse implications see [19, Example 6]. and . Indeed, it follows from (8) that there exists such that ; that is, . But then . (14) and (15) (together) do not imply either (7), or (8) (see [10, Example A]), or (17) (see [14, Example 3.3]). (for the reverse implication and that (16) does not imply (8) see [12, Example 3.2]). (17) does not imply (14), and (17) does not imply (15) (see [14, Example 3.4]). . For the reverse implication see [17]. . For the reverse implication see [17]. and . For the reverse implications see [30, Example 2.3] and [18, Example 3.2]. (19) does not imply (8) (see [30, Example 2.4]) and (20) does not imply (19) (see [18, Example 3.2]). Finally, if is a singleton, then [31, Proposition 2.2], [32, Proposition 2.22], and [24, Proposition 2.11].

#### 3. Reducing Common Fixed Point Results to the Case of Weak Compatibility

The following simple result can be used to show that several common fixed point theorems obtained recently are actually not generalizations of previously known results.

Proposition 2. *Let be a metric space, and let . Let the pair have exactly one point of coincidence; that is,
**
Then conditions (7), (8), (9), and (17) are equivalent, and equivalent with the condition that the pair has a unique common fixed point. *

*Proof. *Note first that implies that and (just take , where , for all ) Consider the following. holds because . follows by [31, Proposition 2.2]. holds because . was proved in [14], and follows because .

In the case that has a unique POC, it was proved in [22] that condition (8) implies that has a unique common fixed point. The converse is obvious.

An easy example of mappings and on , when has two elements, shows that the condition that is a singleton cannot be removed from the previous proposition, neither does this proposition hold when , as [19, Example 6] shows.

When considering two pairs of mappings, the following is a direct consequence of Proposition 2.

Corollary 3. *Let and be two pairs of self-maps on a metric space , satisfying
**
Then the following conditions are equivalent. *(i)* and both satisfy condition (7).*(ii)* and both satisfy condition (8).*(iii)* and both satisfy condition (17).*(iv)*, , , and have a unique common fixed point. *

Applying Proposition 2 or Corollary 3, it is easy to show that a lot of the results of papers cited in the references are actually not generalizations of previously known ones.

As a sample, consider Theorems 2.1 and 2.2 of [16]. in these assertions is a singleton. Further, mappings and have a unique common fixed point. By Proposition 2, it follows that the pair is weakly compatible (condition (4)). Hence, using formally weaker assumption (10) does not produce a more general assertion.

Similarly, in Theorems 2.1 and 2.2 and Corollaries 2.1–2.6 of [33], applying Corollary 3, we get that the pairs and are weakly compatible.

In the same way, it can be concluded that the following results are actually not generalizations of previously known ones: [6, Theorems 1.4 and 1.5]; [8, Theorem 2.1]; [19, Theorems 1–3 of Section 2]; [11, Theorems 2.8–2.12]; [14, Theorems 4.1, 4.4, 4.6, 4.8, 4.10, 4.12 and 5.1]; [18, Theorems 4.1, 4.6, 5.6, 6.2; Corollaries 4.3, 4.4]; [22, Lemma 1; Theorems 1–5; Corollaries 1–5]; [30, Theorem 2.5; Corollary 2.7]; [34, Theorems 1–4 and Corollaries 1–3]; [35, Theorems 3.1, 3.4, and 3.6; Corollaries 3.9, 3.10, and 3.11]; [36, Theorems 2.2 and 2.6]; [37, Theorems 2.1–2.3 and 3.1–3.4; Corollary 2.1]; [38, Theorem 1.1]; [39, Theorems 2.1 and 2.3; Corollaries 2.2, 2.4 and 2.5]; [40, Theorems 3.1–3.3]; [41, Theorems 2.1 and 2.2; Corollaries 2.1–2.3]; [42, Theorems 3.1–3.3 and 4.1–4.3]; [43, Theorems 2.1 and 2.2]; [44, Theorem 3.2; Corollaries 3.1 and 3.2]; [45, Theorems 2.3]; [46, Theorem 11; Corollary 13]; [47, Theorems 2.2 and 2.3]; [48, Theorems 2.1–2.5]; [49, Theorems 2.1, 2.4, and 2.10]; [50, Theorems 4.1, 4.2, and 5.1–5.5].

A different kind of conclusions can be made for the results from [28, 51, 52].

Again, as a sample, consider [52, Theorem 2.1], which (abbreviated) reads as follows.

Let and be two pseudoreciprocal continuous self-mappings of a complete metric space such that and satisfying certain contractive condition. If the pair is conditionally sequential absorbing, then and have a unique common fixed point.

It can be reformulated as follows.

Under the previous conditions, the pair is weakly compatible (i.e., satisfies condition (7)).

Indeed, the proof of [52, Theorem 2.1] shows that and have a unique common fixed point. The contractive condition easily implies that they also have a unique POC (i.e., is a singleton). Then, Proposition 2 implies that is weakly compatible (and occasionally weakly compatible, of the type PD, conditionally commuting). Hence, weak compatibility is again a natural (and the weakest possible) assumption for this kind of results.

Similar conclusions can be made for the following results: [28, Corollary 2.1]; [51,Theorems 1–3]; [52, Theorems 2.2 and 2.3].

Several very interesting results were also obtained for multivalued mappings. We just note [53–55].

It is interesting that in the case of hybrid pairs of mappings (one single-valued and one multivalued) conclusions similar to those from this paper cannot be made. Namely, it was shown in [31, Example 2.5] that in this case Proposition 2 no longer holds. Hence, for example, results from the papers [56–59] cannot be directly obtained from previously known ones.

#### Conflict of Interests

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

#### Acknowledgment

The first and the second authors are grateful to the Ministry of Education, Science and Technological Development of Serbia. The third author acknowledges with thanks the Deanship of Scientific Research (DSR), King Abdulaziz University Jeddah, for financial support.

#### References

- G. Jungck, “Commuting mappings and fixed points,”
*The American Mathematical Monthly*, vol. 83, no. 4, pp. 261–263, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,”
*Publications de l'Institute Mathematique*, vol. 32, no. 46, pp. 149–153, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Compatible mappings and common fixed points,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 9, no. 4, pp. 771–779, 1986. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Pant, “Common fixed point theorems for contractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 226, no. 1, pp. 251–258, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,”
*Far East Journal of Mathematical Sciences*, vol. 4, no. 2, pp. 199–215, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Pant and R. K. Bisht, “Occasionally weakly compatible mappings and fixed points,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 19, no. 4, pp. 655–661, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 181–188, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Al-Thagafi and N. Shahzad, “Generalized $I$-nonexpansive selfmaps and invariant approximations,”
*Acta Mathematica Sinica*, vol. 24, no. 5, pp. 867–876, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - V. Pant and R. P. Pant, “Common fixed points of conditionally commuting maps,”
*Fixed Point Theory*, vol. 11, no. 1, pp. 113–118, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Pathak and N. Hussain, “Common fixed points for $P$-operator pair with applications,”
*Applied Mathematics and Computation*, vol. 217, no. 7, pp. 3137–3143, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - N. Hussain, M. A. Khamsi, and A. Latif, “Common fixed points for $JH$-operators and occasionally weakly biased pairs under relaxed conditions,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 6, pp. 2133–2140, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - W. Sintunavarat and P. Kumam, “Common fixed point theorems for generalized $JH$-operator classes and invariant approximations,”
*Journal of Inequalities and Applications*, vol. 2011, article 67, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Pant, “Common fixed points of Lipschitz type mapping pairs,”
*Journal of Mathematical Analysis and Applications*, vol. 240, no. 1, pp. 280–283, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Pathak and Deepmala, “Common fixed point theorems for $PD$-operator pairs under relaxed conditions with applications,”
*Journal of Computational and Applied Mathematics*, vol. 239, pp. 103–113, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Bouhadjera and C. Godet-Thobie, “Common fixed theorems for pairs of subcompatible maps,” http://arxiv.org/abs/0906.3159.
- R. K. Bisht and N. Shahzad, “Faintly compatible mappings and common fixed points,”
*Fixed Point Theory and Applications*, vol. 2013, article 156, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - G. Jungck and H. K. Pathak, “Fixed points via ‘biased maps’,”
*Proceedings of the American Mathematical Society*, vol. 123, no. 7, pp. 2049–2060, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Bouhadjera and A. Djoudi, “Fixed point for occasionally weakly biased maps,”
*Southeast Asian Bulletin of Mathematics*, vol. 36, no. 4, pp. 489–500, 2012. View at Google Scholar · View at MathSciNet - R. P. Pant and R. K. Bisht, “Common fixed point theorems under a new continuity condition,”
*Annali dell’ Universita di Ferrara*, vol. 58, no. 1, pp. 127–141, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - P. P. Murthy, “Important tools and possible applications of metric fixed point theory,”
*Nonlinear Analysis*, vol. 47, pp. 3479–3490, 2001. View at Google Scholar - M. A. Al-Thagafi and N. Shahzad, “A note on occasionally weakly compatible maps,”
*International Journal of Mathematical Analysis*, vol. 3, no. 1–4, pp. 55–58, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck and B. E. Rhoades, “Fixed point theorems for occasionally weakly compatible mappings,”
*Fixed Point Theory*, vol. 7, no. 2, pp. 287–296, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck and B. E. Rhoades, “Erratum: Fixed point theorems for occasionally weakly compatible mappings,”
*Fixed Point Theory*, vol. 9, pp. 383–384, 2008, Fixed Point Theory, vol. 7, no. 2, pp. 287–296, 2006. View at Google Scholar - M. A. Alghamdi, S. Radenović, and N. Shahzad, “On some generalizations of commuting mappings,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 952052, 6 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-X. Fang and Y. Gao, “Common fixed point theorems under strict contractive conditions in Menger spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 1, pp. 184–193, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Rouzkard, M. Imdad, and H. K. Nashine, “New common fixed point theorems and invariant approximation in convex metric spaces,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 19, no. 2, pp. 311–328, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Pathak, R. Rodríguez-López, and R. K. Verma, “A common fixed point theorem using implicit relation and property (E.A) in metric spaces,”
*Filomat*, vol. 21, no. 2, pp. 211–234, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Gopal, M. Imdad, and M. Abbas, “Metrical common fixed point theorems without completeness and closedness,”
*Fixed Point Theory and Applications*, vol. 2012, article 18, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Imdad, J. Ali, and M. Tanveer, “Remarks on some recent metrical common fixed point theorems,”
*Applied Mathematics Letters*, vol. 24, no. 7, pp. 1165–1169, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Hussain and H. K. Pathak, “Subweakly biased pairs and Jungck contractions with applications,”
*Numerical Functional Analysis and Optimization*, vol. 32, no. 10, pp. 1067–1082, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Đorić, Z. Kadelburg, and S. Radenović, “A note on occasionally weakly compatible mappings and common fixed points,”
*Fixed Point Theory*, vol. 13, no. 2, pp. 475–479, 2012. View at Google Scholar · View at MathSciNet - M. Abbas, D. Gopal, and S. Radenović, “A note on recent introduced commutative conditions,”
*Indian Journal of Mathematics (IJM) Bulletin of the Allahabad Mathematical Society*. In press. - M. K. Jain, B. E. Rhoades, and A. S. Saluja, “Fixed point theorems for occasionally weakly compatible expansive mappings,”
*Journal of Advanced Mathematical Studies*, vol. 5, no. 2, pp. 54–58, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Abbas and B. E. Rhoades, “Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition,”
*Mathematical Communications*, vol. 13, no. 2, pp. 295–301, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Aliouche and V. Popa, “Common fixed point theorems for occasionally weakly compatible mappings via implicit relations,”
*Filomat*, vol. 22, no. 2, pp. 99–107, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. V. R. Babu and G. N. Alemayehu, “Common fixed point theorems for occasionally weakly compatible maps satisfying property (E.A) using an inequality involving quadratic terms,”
*Applied Mathematics Letters*, vol. 24, no. 6, pp. 975–981, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Bhatt, H. Chandra, and D. R. Sahu, “Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 73, no. 1, pp. 176–182, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. K. Bisht and R. P. Pant, “A critical remark on ‘Fixed point theorems for ocasionally weakly compatible mappings’,”
*Journal of the Egyptian Mathematical Society*, 2013. View at Publisher · View at Google Scholar - H. Bouhadjera, “On unique common fixed point theorems for three and four self mappings,”
*Filomat*, vol. 23, no. 3, pp. 115–123, 2009. View at Google Scholar - H. Chandra and A. Bhatt, “Fixed point theorems for occasionally weakly compatible maps in probabilistic semi-metric space,”
*International Journal of Mathematical Analysis*, vol. 3, no. 9–12, pp. 563–570, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Chauhan and B. D. Pant, “Common fixed point theorems for occasionally weakly compatible mappings using implicit relation,”
*The Journal of the Indian Mathematical Society*, vol. 77, no. 1–4, pp. 13–21, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. B. Ćirić, B. Samet, and C. Vetro, “Common fixed point theorems for families of occasionally weakly compatible mappings,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 631–636, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Hussain, M. H. Shah, and S. Radenović, “Fixed points of weakly contractions through occasionally weak compatibility,”
*Journal of Computational Analysis & Applications*, vol. 13, no. 3, pp. 532–543, 2012. View at Google Scholar - M. Imdad, J. Ali, and V. Popa, “Impact of occasionally weakly compatible property on common fixed point theorems for expansive mappings,”
*Filomat*, vol. 25, no. 2, pp. 79–89, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - M. Imdad and A. H. Soliman, “Some common fixed point theorems for a pair of tangential mappings in symmetric spaces,”
*Applied Mathematics Letters*, vol. 23, no. 4, pp. 351–355, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. D. Pant and S. Chauhan, “Common fixed point theorem for occasionally weakly compatible mappings in Menger space,”
*Surveys in Mathematics and its Applications*, vol. 6, pp. 1–7, 2011. View at Google Scholar · View at MathSciNet - R. P. Pant and R. K. Bisht, “Common fixed point of pseudo compatible maps,”
*Revista de la Real Academia de Ciencias, Fisicas y Naturales A*, 2013. View at Publisher · View at Google Scholar - K. P. R. Sastry, G. A. Naidu, P. V. S. Prasad, V. Madhavi Latha, and S. S. A. Sastri, “A critical look at fixed point theorems for occasionally weakly compatible maps in probabilistic semi-metric spaces,”
*International Journal of Mathematical Analysis*, vol. 4, no. 25–28, pp. 1341–1348, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. H. Shah, S. Simić, N. Hussain, A. Sretenović, and S. Radenović, “Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces,”
*Journal of Computational Analysis and Applications*, vol. 14, no. 2, pp. 290–297, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Deepmala and H. K. Pathak, “Some common fixed point theorems for D-operator pair with applications to nonlinear integral equations,”
*Nonlinear Functional Analysis & Applications*, vol. 18, pp. 205–218, 2013. View at Google Scholar - R. P. Pant, R. K. Bisht, and D. Arora, “Weak reciprocal continuity and fixed point theorems,”
*Annali dell'Universitá di Ferrara. Sezione VII. Scienze Matematiche*, vol. 57, no. 1, pp. 181–190, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. K. Patel, P. Kumam, and D. Gopal, “Some discussion on the existence of common fixed points for a pair of maps,”
*Fixed Point Theory and Applications*, vol. 2013, article 187, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - W.-Sh. Du, E. Karapinar, and N. Shahzad, “The study of fixed point theory for various multivalued non-self maps,”
*Abstract & Applied Analysis*, vol. 2013, Article ID 938724, 9 pages, 2013. View at Publisher · View at Google Scholar - F. Khojasteh and V. Rakočević, “Some new common fixed point results for generalized contractive multi-valued non-self-mappings,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 287–293, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. D. Rouhani and S. Moradi, “Fixed point of multi-valued generalized Φ-weak contractive mapping,”
*Fixed Point Theory & Applications*, vol. 2010, Article ID 708984, 2010. View at Publisher · View at Google Scholar - M. Abbas and B. E. Rhoades, “Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type,”
*Fixed Point Theory and Applications*, vol. 2007, Article ID 54101, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Abbas and B. E. Rhoades, “Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 274793, 2 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - A. Aliouche and V. Popa, “General common fixed point theorems for occasionally weakly compatible hybrid mappings and applications,”
*Novi Sad Journal of Mathematics*, vol. 39, no. 1, pp. 89–109, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Chauhan and P. Kumam, “Common fixed point theorem for occasionally weakly compatible mappings in probabilistic metric spaces,”
*Thai Journal of Mathematics*, vol. 11, no. 2, pp. 285–292, 2013. View at Google Scholar · View at MathSciNet