Hojat Ansari, A., Dosenovic, T., Radenovic, S., Ume, J. (2019). $C$-class Functions and Common Fixed Point Theorems Satisfying $\varphi $-weakly Contractive Conditions. Sahand Communications in Mathematical Analysis, 13(1), 17-30. doi: 10.22130/scma.2018.59792.211

Arslan Hojat Ansari; Tatjana Dosenovic; Stojan Radenovic; Jeong Sheok Ume. "$C$-class Functions and Common Fixed Point Theorems Satisfying $\varphi $-weakly Contractive Conditions". Sahand Communications in Mathematical Analysis, 13, 1, 2019, 17-30. doi: 10.22130/scma.2018.59792.211

Hojat Ansari, A., Dosenovic, T., Radenovic, S., Ume, J. (2019). '$C$-class Functions and Common Fixed Point Theorems Satisfying $\varphi $-weakly Contractive Conditions', Sahand Communications in Mathematical Analysis, 13(1), pp. 17-30. doi: 10.22130/scma.2018.59792.211

Hojat Ansari, A., Dosenovic, T., Radenovic, S., Ume, J. $C$-class Functions and Common Fixed Point Theorems Satisfying $\varphi $-weakly Contractive Conditions. Sahand Communications in Mathematical Analysis, 2019; 13(1): 17-30. doi: 10.22130/scma.2018.59792.211

$C$-class Functions and Common Fixed Point Theorems Satisfying $\varphi $-weakly Contractive Conditions

^{1}Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

^{2}Faculty of Technology, Bulevar cara Lazara 1, University of Novi Sad, Serbia.

^{3}Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia.

^{4}Department of Mathematics, Changwon National University, Changwon, 641-773, Korea.

Abstract

In this paper, we discuss and extend some recent common fixed point results established by using $\varphi-$weakly contractive mappings. A very important step in the development of the fixed point theory was given by A.H. Ansari by the introduction of a $C-$class function. Using $C-$class functions, we generalize some known fixed point results. This type of functions is a very important class of functions which contains almost all known type contraction starting from 1922. year, respectively from famous Banach contraction principle. Three common fixed point theorems for four mappings are presented. The obtained results generalizes several existing ones in literature.We finally propose three open problems.

[1] M. Abbas and D. Doric, Common fixed point theorem for four mappings satisfying generalized weak contractive condition, Filomat 24 (2010), pp. 1-10.

[2] M. Abbas and M.A. Khan, Common fixed point theorem of two mappings satisfying a generalized weak contractive condition, Int. J. Math. Math. Sci., Article ID 131068 (2009), pp. 1-9.

[3] R.P. Agarwal, R.K. Bisht, and N. Shahzad, A comparison of various noncommuting conditions in metric fixed point theory and their applications, Fixed Point Theory Appl., 1 (2014), pp. 1-33 .

[4] Lj. Ciric, Generalized contractions and fixed point theorems, Pub. Inst. Math., 12 26 (1971), pp. 19-26.

[5] D. Doric, Common fixed point for generalized $varphi ,psi $-weak contractions, Appl. Math. Lett., 22 (2009), pp. 1896-1900.

[6] T. Dosenovic, M. Postolache, and S. Radenovic, On multiplicative metric spaces, Survey, Fixed Point Theory Appl., 92 (2016), pp. 1-17.

[7] T. Dosenovic, and S. Radenovic, Some critical remarks on the paper: ''An essential remark on fixed point results on multiplicative metric spaces'', J. Adv. Math. Stud., 10 (2017), pp. 20-24.

[8] P.N. Dutta and B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., Article ID 406368 (2008), pp. 1-8.

[9] D. Gopal, M. Imdad, and M. Abbas, Metrical common fixed point theorems without completeness and closedness, Fixed Point Theory Appl., 1 (2012), pp. 1-9.

[10] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal., 74 (2011), pp. 768-774.

[11] M. Jleli, V.'C. Rajic, B. Samet, and C. Vetro, Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations, J. Fixed Point Theory Appl., 12 (2012), pp. 175–-192.

[12] G. Jungck and B.E. Rhoades, Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math., 29 (1998), pp. 227-238.

[13] A. Latif, H. Isik, and A.H. Ansari, Fixed points and functional equation problems via cyclic admissible generalized contractive type mappings, J. Nonlinear Sci. Appl., 9 (2016), pp. 1129-1142.

[14] Z. Liu, Compatible mappings and fixed points, Acta Sci. Math., 65 (1999), pp. 371-383.

[15] Z. Liu, X. Zhang, J.S. Ume, and S.M. Kang, Common fixed point theorems for four mappings satisfying $psi $-weakly contractive conditions, Fixed Point Theory Appl., 20 (2015), pp.1-22.

[16] S. Moradi, Z. Fathi, and E. Analouee, The common fixed point of single-valued generalized $f-$weakly contractive mappings, Appl. Math. Lett., 24 (2011), pp. 771-776.

[17] Z. Mustafa, H. Huang, and S. Radenovic, Some remarks on the paper ''Some fixed point generalizations are not real generalizations'', J. Adv. Math. Stud., 9 (2016), pp. 110-116.

[18] Z. Mustafa, M.M.M. Jaradat, A.H. Ansari, B.Z. Popovic, and H.M. Jaradat, $C-$class functions with new approach on coincidence point results for generalized $(psi, phi)-$weakly contractions in ordered $b-$metric spaces, SpringerPlus, 5 (2016), pp. 1-18.

[19] L. Pasicki, Fixed point theorems for contracting mappings in partial metric spaces, Fixed Point Theory Appl., 185 (2014), pp. 1-16.

[20] 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 Appl., 1 (2013), pp. 1-17.

[21] O. Popescu, Fixed points for $varphi, $$psi $-weak contractions, Appl. Math. Lett., 24 (2011), pp. 1-4.

[22] S. Radenovic, and Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comp.Math. Appl., 60 (2010), pp. 1776-1783.

[23] S. Radenovic, Z. Kadelburg, D. Jandrlic, and A. Jandrlic, Some results on weakly contractive maps, Bulletin of the Iranian Mathematical Society, 3 (2012), pp. 625-645.

[24] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), pp. 257-290.

[25] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), pp. 2683-2693.