Unified common fixed point theorems under weak reciprocal continuity or without continuity
Keywords:metric space, compatible mappings, $R$-weakly commuting mappings, weak reciprocal continuity, coincidentally commuting, implicit relation
AbstractThe purpose of this paper is two fold. Firstly, using the notion of weak reciprocal continuity due to Pant et al. Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57(1), 181-190 (2011)], we prove unified common fixed point theorems for various variants of compatible and $R$-weakly commuting mappings in complete metric spaces employing an implicit relation which covers a multitude of contraction conditions yielding thereby known as well as unknown results as corollaries. Secondly, we point out that more natural results can be proved under relatively tighter conditions if we replace the completeness of the space by completeness of suitable subspaces. The realized improvements in our results are also substantiated using appropriate examples.
J. Ali and M. Imdad, An implicit function implies several contraction conditions, Sarajevo J. Math. 4, no. 2 (2008), 269-285.
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
S. Chauhan, M. Imdad and C. Vetro, Unified metrical common fixed point theorems in 2-metric spaces via an implicit relation, J. Operat. Vol. 2013, Article ID 186910, 11 pages. https://doi.org/10.1155/2013/186910
Lj. B. Ciric, Generalized contractions and fixed point theorems, Publ. Inst. Math. (Beograd) (N.S.) 12(26) (1971), 19-26.
B. C. Dhage, On common fixed points of coincidentally commuting mappings in $D$-metric spaces, Indian J. Pure Appl. Math. 30, no. 4 (1999), 395-406.
S. A. Husain and V. M. Sehgal, On common fixed points for a family of mappings, Bull. Austral. Math. Soc. 13, no. 2 (1975), 261-267. https://doi.org/10.1017/S000497270002445X
M. Imdad and J. Ali, Reciprocal continuity and common fixed points of nonself mappings, Taiwanese J. Math. 13, no. 5 (2009), 1457-1473. https://doi.org/10.11650/twjm/1500405553
M. Imdad, J. Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces, Chaos Solitons & Fractals 42, no. 5 (2009), 3121-3129. https://doi.org/10.1016/j.chaos.2009.04.017
M. Imdad and S. Chauhan, Employing common limit range property to prove unified metrical common fixed point theorems, Internat. J. Anal. vol. 2013, Article ID 763261, 10 pages. https://doi.org/10.1155/2013/763261
M. Imdad and Q. H. Khan, Six mappings satisfying a rational inequality, Rad. Mat. 9, no. 2 (1999), 251-260.
M. Imdad, M. S. Khan and S. Sessa, On some weak conditions of commutativity in common fixed point theorems, Internat. J. Math. Math. Sci. 11, no. 2 (1988), 289-296. https://doi.org/10.1155/S0161171288000353
M. Imdad, S. Kumar and M. S. Khan, Remarks on some fixed point theorems satisfying implicit relations. Rad. Mat. 11, no. 1 (2002), 135-143.
G. Jungck, Commuting mappings and fixed point, Amer. Math. Monthly 83, no. 4 (1976), 261-263. https://doi.org/10.1080/00029890.1976.11994093
G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9, no. 4 (1986), 771-779. https://doi.org/10.1155/S0161171286000935
G. Jungck and B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29, no. 3 (1998), 227-238.
Z. Kadelburg, S. Radenovic and N. Shahzad, A note on various classes of compatible-type pairs of mappings and common fixed point theorems, Abstr. Appl. Anal. vol. 2013 Article ID 697151, 6 pages. https://doi.org/10.1155/2013/697151
M. S. Khan and M. Imdad, A common fixed point theorem for a class of mappings, Indian J. Pure Appl. Math. 14 (1983), 1220-1227.
S. Kumar and R. Chugh, Common fixed points theorem using minimal commutativity and reciprocal continuity conditions in metric space, Sci. Math. Japon. 56, no. 2 (2002), 269-275.
P. P. Murthy, Important tools and possible applications of metric fixed point theory, Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000), Nonlinear Anal. 47, no. 5 (2001), 3479-3490. https://doi.org/10.1016/S0362-546X(01)00465-5
R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), 436-440. https://doi.org/10.1006/jmaa.1994.1437
R. P. Pant, Common fixed points of four mappings, Bull. Cal. Math. Soc. 90 (1998), 281-286.
R. P. Pant, Noncompatible mappings and common fixed points, Soochow J. Math. 26 (2000), 29-35.
R. P. Pant, R. K. Bisht and D. Arora, Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57, no. 1 (2011), 181-190. https://doi.org/10.1007/s11565-011-0119-3
H. K. Pathak, Y. J. Cho and S. M. Kang, Remarks on $R$-weakly commuting mappings and common fixed point theorems, Bull. Korean Math. Soc. 34, no. 2 (1997), 247-257.
H. K. Pathak and M. S. Khan, A comparison of various types of compatible maps and common fixed points, Indian J. Pure Appl. Math. 28, no. 4 (1997), 477-485.
V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math. 32, no. 1 (1999), 157-163. https://doi.org/10.1515/dema-1999-0117
V. Popa, M. Imdad and J. Ali, Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces, Bull. Malays. Math. Sci. Soc. (2)33, no. 1 (2010), 105-120.
V. Popa, M. Imdad and J. Ali, Fixed point theorems for a class of mappings governed by strictly contractive implicit function, Southeast Asian Bulletin of Math. 34, no. 5 (2010), 941-952.
S. Sessa, On a weak commutativity condition in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153.
S. Sessa, M. S. Khan and M. Imdad, A common fixed point theorem with a weak commutativity condition, Glas. Mat. Ser. III 21(41)(1) (1986), 225-235.
S. L. Singh and A. Tomar, Weaker forms of commuting maps and existence of fixed points, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10, no. 3 (2003), 145-161.
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.