Common fixed point theorems for $(\psi, \varphi)$-weak contractive conditions in metric spaces

Year 2019, Volume: 48 Issue: 5, 1398 - 1408, 08.10.2019

Y. Mahendra Singh G. A. Hirankumar Sharma M. R. Singh

Abstract

The intent of this paper is to prove common fixed point theorems under $(\psi, \varphi)$ -weak contractive conditions satisfying the common limit range property  and the common property-(E.A) of four self mappings in the setting of metric spaces. Illustrative examples are also furnished to justify the validity of our results.

Keywords

altering distance function, $(\psi;\varphi)$-weak contractive conditions, weakly compatible, property-(E.A), common limit range property, common fixed point

References

• [1] M. Aamri and D. EL Moutawakil, Some new common fixed points under strict con- tractive conditions, J. Math. Anal. Appl. 270, 181-188, 2002.
• [2] M. Abbas and D. Doric, Common fixed point theorem for four mappings satisfying generalized weak contractive conditions, Filomat 24 (2), 1-10, 2010.
• [3] M. Abbas and M.A. Khan, Common fixed point theorems of two mappings satisfying a generalized weak contractive condition, Int. J. Math. Math. 2009, 9 pages, 2009.
• [4] A.P. Agarwal, M.A. Alghamdi and N. Shahzad, Fixed point theory for cyclic genralized contractions in partial metric spaces, Fixed Point Theory Appl. 2012, 11 pages, 2012.
• [5] I.Y. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert space, in: New Results in Operator Theory and its Appl. 98, 7-22, Birkhäuser, Basel, Switzerland, 1997.
• [6] I. Beg and M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory Appl. 2006, Article ID 74503, 7 pages, 2006.
• [7] B.S. Choudhury, P. Konar, B.E. Rhoades and N. Metiya, Fixed point theorems for generalized weakly contractive mapppings, Nonlinear Anal.: Theory Appl. 74 (6), 2116-2126, 2011.
• [8] H.S. Ding, Z. Kadelburg, E. Karapinar and S. Radenovic, Common fixed points of weak contractions in cone metric spaces, Abstract Appl. Anal. 2012, 18 pages, 2012.
• [9] D. Doric, Common fixed point for generalized $(\psi, \varphi)$-weak contractions, Appl. Math. Lett. 22, 1896-1900, 2009.
• [10] P.N. Dutta and B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. 2008, 8 pages, 2008.
• [11] M. Imdad, S. Chouhan and Z. Kadelburg, Fixed point theorems for mappings with common limit range property satisfying generalized $(\psi, \varphi)$-weak contractive conditions, Math. Sci., 7 (16), 1-8, 2013.
• [12] M. Imdad, B.D. Pant and S. Chouhan, Fixed point theorems in Menger spaces using the $(CRL_ST)$ property and applications, J. Nonlinear Anal. Optm. 3 (2), 225-237, 2012.
• [13] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83, 261-263, 1976.
• [14] G. Jungck, Compatible mappigs and common fixed points, Int. J. Math. Math. Sci. 9, 771-779, 1986.
• [15] G. Jungck and B.E. Rhoades, Fixed points for set-valued functions without continuity, Indian J. Pure Appl. Math. 29 (3), 227-238, 1998.
• [16] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30 (1), 1-9, 1981.
• [17] Y. Liu, J. Wu and Z. Li, Common fixed points of single-valued and multi-valued maps, Int. J. Math. Math. Sci. 19, 3045-3055, 2005.
• [18] P.P. Murthy, K. Tas and U.D. Patel, Common fixed point theorems for generalized $(\psi, \varphi)$-weak contraction condition in complete metric spaces, J. Ineq. Appl., 2015, Article 139, 2015.
• [19] R.P. Pant, Common fixed points of contractive maps, J. Math. Anal. Appl., 226, 251-258, 1998.
• [20] R.P. Pant, R-weakly commutativity and common fixed points, Soochow J. Math. 25, 37-42, 1999.
• [21] S. Radenovic and Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Computers Math. Appl. 60, 1176-1783, 2010.
• [22] S. Radenovic, Z. Kadelburg, D. Jandrlic and A Jandrlic, Some results on weakly contractive maps, Bull. Iranian Math. Soc. 38 (3), 625-645, 2012.
• [23] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal.: Theory, Method and Appl. 47 (4), 2683-2693, 2001.
• [24] B.D. Rouhani and S. Moradi, Common fixed point of multivalued generalized $\varphi$-weak contractive mappings, Fixed Point Theory Appl. 2010, 13 pages, 2010.
• [25] K.P.R. Sastry and I.R.S. Krishna Murthy, Common fixed points of two partially com- muting tangential selfmaps on a metric space, J. Math. Anal. Appl. 250, 731-734, 2000.
• [26] W. Sintunavarat and P Kumam, Common fixed point theorems for a pair of weakly compatible mappings in Fuzzy metric spaces, J. Appl. Math. 2011, 14 pages, 2011.
• [27] Y. Song, Coincidence Points for Non commuting $f$-Weakly Contractive Mappings, Int. J. Computational Appl. Math. 2 (1), 51-57, 2007.
• [28] Q. Zhang, Y. Song, Fixed point theory for generalized $\varphi$-Weak Contractions, Appl. Math. Lett. 22, 75-78, 2009.