Document Type : Research Paper
Authors
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey.
Abstract
In this paper, we present some fixed point theorems for single valued mappings on $K$-complete, $M$-complete and Symth complete quasi metric spaces. Here, for contractive condition, we consider some altering distance functions together with functions belonging to $C$-class and $A$-class. At the same time, we will consider two different type $M$ functions in contractive conditions because the quasi metric does not provide the symmetry property. Finally, we show that our main results includes many fixed point theorems presented on both complete metric and complete quasi metric spaces in the literature. We also provide an illustrative example to show importance of our results.
Keywords
[2] I. Altun, M. Olgun and G. Minak, Classification of completeness of quasi metric space and some new fixed point results, Nonlinear Funct. Anal. Appl., 22 (2) (2017), pp. 371-384.
[3] A.H. Ansari, Note on $psi $-$varphi $ contractive type mappings and related fixed point, The Second Regional Conference on Mathematics and Applications, Payame Noor University, 2014, 377-380.
[4] A.H. Ansari, V.S. Cavic, T. Dosenovic, S. Radenovic, N. Saleem and J. Vujakovic, $C$-class functions on some fixed point results in ordered partial metric spaces via admissible mappings, Novi Sad J. Math., too appear in 2019.
[5] A.H. Ansari and A. Razani, Some fixed point theorems for $C$-class functions in $b$-metric spaces, Sahand Communications in Mathematical Analysis, 10 (1) (2018), pp. 85-96.
[6] A.H. Ansari, A. Razani and N. Hussain, Fixed and coincidence points for hybrid rational Geraghty contractive mappings in ordered $b$-metric spaces, Int. J. Nonlinear Anal. Appl., 8 (1) (2017), pp. 315-329.
[7] A.H. Ansari, A. Razani and N. Hussain, New best proximity point results through various auxiliary functions, J. Linear Topol. Algebra, 6 (1) (2017), pp. 73-89.
[8] A.H. Ansari, A. Razani and M. Abbas, Unification of coincidence point results in partially ordered $G$-metric spaces via $C$-class functions, J. Adv. Math. Stud., 10 (1) (2017), pp. 1-19.
[9] H. Aydi, W. Shatanawi, M. Postolache, Z. Mustafa, and N. Talat, Theorems for Boyd-Wong type contractions in ordered metric spaces, Abstr. Appl. Anal., (2012), ID:359054.
[10] M. Beygmohammadi and A. Razani, Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space, Int. J. Math. Math. Sci., (2010), ID:317107.
[11] L. Ciric, Periodic and fixed point theorems in a quasi-metric space, J. Australian Math. Soc. Ser. A, 54 (1993), pp. 80-85.
[12] S. Cobzas, Completeness in quasi-metric spaces and Ekeland variational principle, Topology Appl., 159 (2012), pp. 10-11.
[13] T. Dosenovic and S. Radenovic, Ansari's method in generalizations of some results in fixed point theory: Survey, Military Technical Courier, 66 (2) (2018), pp. 261-268.
[14] T. Dosenovic, S. Radenovic and S. Sedghi, Generalized metric spaces: Survey, TWMS J. Pure Appl. Math., 9 (1) (2018), pp. 3-17.
[15] P.N. Dutta and B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., (2008), ID:406368.
[16] W. Kirk, Fixed Point Theory in Distance Spaces, Springer International Publishing, Switzerland, 2014.
[17] M.S. Khan and M. Swaleh, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc., 30 (1) (1984), pp. 1-9.
[18] S.G. Matthews, Partial Metric Topology, Ann. New York Acad. Sci., 728 (1994), pp. 183-197.
[19] S. Moradi, Endpoints of multi-valued cyclic contraction mappings, Int. J. Nonlinear Anal. Appl., 9 (1) (2018), pp. 203-210.
[20] 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 $(psi ,varphi )$-weakly contractions in ordered $b$-metric spaces, Springer Plus, (2016) 5:802.
[21] I.L. Reilly, P.V. Subrahmanyam and M.K. Vamanamurthy, Cauchy sequences in quasi pseudo metric spaces, Monatsh. Math., 93 (1982), pp. 127-140.
[22] H. Simsek and M.T. Yalcin, Generalized $Z$-contraction on quasi metric spaces and a fixed point result, J. Nonlinear Sci. Appl., 10 (2017), pp. 3397-3403.
[23] W.A. Wilson, On quasi-metric spaces, Amer. J. Math., 53 (1931), pp. 675-684.
)