Document Type : Research Paper


Department of Pure Mathematics, Payame Noor University (PNU), P. O. Box: 19395-3697, Tehran, Iran.


In this article, we will study the existence and uniqueness of optimal common fixed points for self-mappings in metric spaces with w-distance. We obtain generalizations of the Kocev and Rako\v{c}evi'{c} fixed point theorems. The obtained results do not require the continuity or the condition $(C;k)$ of maps,  but require the weaker condition $(W)$. We also improve some of our results when the metric space is equipped with a w$_0$-distance. In this way, we get new existence results for non-cyclic quasi-contraction mappings of the Fisher type.


[1] A. Bagheri Vakilabad, A common fixed point theorem using an iterative method, Sahand Commun. Math. Anal., 17 (1) (2020), pp. 91-98.
[2] Lj. B. Ciric, On mappings with contractive iteration, Publ. de l'Institut Math, 26 (40) (1979), pp. 79-82.
[3] Lj. B. Ciric, H. Lakzian and V. Rakocevic, Fixed point theorems for w-cone distance contraction mappings in TVS-cone metric spaces, Fixed Point Theory Appl., 2012 (1) (2012), pp. 1-9.
[4] B. Fisher, Results on common fixed points on complete metric spaces, Glasg. Math. J., 21 (1980), pp. 165-167.
[5] Ch. Garodia and I. Uddin, On Approximating Fixed Point in CAT(0) Spaces, Sahand Commun. Math. Anal., 18 (4) (2021), pp. 113-130.
[6] E. Graily and S.M. Vaezpour, Generalized distance and fixed Point theorems for weakly contractive Mappings, Int. j. basic appl. sci., 4 (1) (2013), pp. 161-164.
[7] D. Ilic and V. Rakocevic, Common fixed points for maps on metric space with w-distance, Appl. Math. Comput., 199 (2) (2008), pp. 599-610.
[8] O. Kada, T. Suzuki and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Jpn. J. Math., 44 (1996), pp. 381-391.
[9] D. Kocev, E. Karapinar and V. Rakocevic, Quasi-contraction mappings of Ciric and Fisher type via w-distance, Quaest. Math., 42 (1) (2019), pp. 1-14.
[10] D. Kocev and V. Rakocevic, On a theorem of Brain Fisher in the framework of w-distance, Carpathian J. Math., 33 (2) (2017), pp. 199-205.
[11] A. Kostic, E. Karapinar and V. Rakocevic, Best proximity points and fixed points with R-functions in the framework of w-distance, Bull. Aust. Math. Soc., 99 (2019), pp. 497-507.
[12] V. Rakocevic, Fixed point results in w-distance spaces, Chapman and Hall/CRC, New York, 2021.
[13] A. Safari-Hafshejani, A. Amini-Harandi and M. Fakhar, Best proximity points and fixed points results for non-cyclic and cyclic Fisher quasi-contractions, Numer. Funct. Anal. Optim., 40 (5) (2019), pp. 603-619.
[14] T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. J., 44 (1997), pp. 61-72.