# Generalized $\mathit{Z}$-contraction on quasi metric spaces and a fixed point result

Volume 10, Issue 7, pp 3397--3403 Publication Date: July 20, 2017       Article History
• 1103 Views

### Authors

Hakan Şimşek - Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey. Menşur Tuğba Yalçin - Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey.

### Abstract

The simulation function is defined by Khojasteh et al. [F. Khojasteh, S. Shukla, S. Radenović, Filomat, 29 (2015), 1189–1194]. Khojasteh introduced the notion of Z-contraction which is a new type of nonlinear contractions defined by using a specific simulation function. Then, they proved existence and uniqueness of fixed points for Z-contraction mappings. After this work, studies involving simulation functions were performed by various authors [H. H. Alsulami, E. Karapınar, F. Khojasteh, A. F. Roldán-López-de-Hierro, Discrete Dyn. Nat. Soc., 2014 (2014), 10 pages], [M. Olgun, Ö. Biçer, T. Alyildiz, Turkish J. Math., 40 (2016), 832–837]. In this paper, we introduce generalized simulation function on a quasi metric space and we present a fixed point theorem.

### Keywords

• Quasi metric space
• left K-Cauchy sequence
• simulation functions
• fixed point.

•  47H10
•  54H25

### References

• [1] H. H. Alsulami, E. Karapınar, F. Khojasteh, A. F. Roldán-López-de-Hierro, A proposal to the study of contractions in quasi-metric spaces, Discrete Dyn. Nat. Soc., 2014 (2014), 10 pages.

• [2] I. Altun, G. Mınak, M. Olgun, Classification of completeness of quasi metric space and some new fixed point results, J. Nonlinear Funct. Anal., (Submitted),

• [3] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.

• [4] F. E. Browder, W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72 (1966), 571–575.

• [5] V. W. Bryant, A remark on a fixed-point theorem for iterated mappings, Amer. Math. Monthly, 75 (1968), 399–400.

• [6] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215 (1976), 241–251.

• [7] L. B. Ćirić, On a common fixed point theorem of a Greguštype, Publ. Inst. Math. (Beograd) (N.S.), 49 (1991), 174–178.

• [8] S. Cobzaş, Completeness in quasi-metric spaces and Ekeland Variational Principle, Topology Appl., 158 (2011), 1073– 1084.

• [9] K. Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (1992)

• [10] M. Edelstein, A theorem on fixed points under isometries, Amer. Math. Monthly, 70 (1963), 298–300.

• [11] T. L. Hicks, Fixed point theorems for quasi-metric spaces, Math. Japon., 33 (1988), 231–236.

• [12] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), 381–391.

• [13] R. Kannan, Some results on fixed points, II, Amer. Math. Monthly, 76 (1969), 405–408.

• [14] F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189–1194.

• [15] M. Olgun, Ö. Biçer, T. Alyildiz, A new aspect to Picard operators with simulation functions, Turkish J. Math., 40 (2016), 832–837.

• [16] I. L. Reilly, P. V. Subrahmanyam, M. K. Vamanamurthy, Cauchy sequences in quasipseudometric spaces, Monatsh. Math., 93 (1982), 127–140.

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

• [18] A. F. Roldan Lopez de Hierro, B. Samet, $\varphi$-admissibility results via extended simulation functions, J. Fixed Point Theory Appl., 2016 (2016), 19 pages.

• [19] J. L. Sieber, W. J. Pervin, Completeness in quasi-uniform spaces, Math. Ann., 158 (1965), 79–81.

• [20] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math., 80 (1975), 325–330.

• [21] W. A. Wilson, On quasi-metric spaces, Amer. J. Math., 53 (1931), 675–684.