# Common fixed points of generalized rational contractions on a closed ball in partial metric spaces

Volume 10, Issue 10, pp 5261--5270
Publication Date: October 19, 2017 Submission Date: March 25, 2017
• 1059 Views

### Authors

Muhammad Nazam - Department of Mathematics, International Islamic University, Islamabad, Pakistan. Muhammad Arshad - Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan. Choonkil Park - Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea. Sungsik Yun - Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Republic of Korea.

### Abstract

The notion of generalized contractions of rational type on a closed ball is introduced and used to establish some common fixed point theorems for two, three and four mappings in complete ordered partial metric spaces. These results improve several well-known, primary and conventional results. We give an example to illustrate the main idea of our results that there are mappings which have only fixed points inside or on the closed ball instead of whole space.

### Keywords

• Common fixed point
• closed ball
• generalized contraction
• partial metric space

•  47H09
•  47H10
•  54H25

### References

• [1] T. Abdeljawad, Meir-Keeler $\alpha$-contractive fixed and common fixed point theorems , Fixed Point Theory Appl., 2013 (2013), 10 pages.

• [2] T. Abdeljawad, E. Karapınar, K. Taş , Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett., 24 (2011), 1900–1904.

• [3] A . Almeida, A. F. Roldán-López-de-Hierro, K. Sadarangani , On a fixed point theorem and its application in dynamic programming, Appl. Anal. Discrete Math., 9 (2015), 221–244.

• [4] I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl., 2011 (2011), 10 pages.

• [5] I. Altun, S. Romaguera, Characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point, Appl. Anal. Discrete Math., 6 (2012), 247–256.

• [6] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl., 157 (2010), 2778–2785.

• [7] M. Arshad, A. Azam, P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl., 2009 (2009), 11 pages.

• [8] M. Bukatin, R. Kopperman, S. Matthews, H. Pajoohesh, Partial metric spaces , Amer. Math. Monthly, 116 (2009), 708–718.

• [9] M. A. Bukatin, S. Y. Shorina, Partial metrics and co-continuous valuations, Foundations of software science and computation structures, Lisbon, (1998), Lecture Notes in Comput. Sci., Springer, Berlin, 1378 (1998), 125–139.

• [10] B. K. Dass, S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math., 6 (1975), 1455–1458.

• [11] ´I. M. Erhan, E. Karapinar, D. Türkoğlu, Different types Meir-Keeler contractions on partial metric spaces, J. Comput. Anal. Appl., 14 (2012), 1000–1005.

• [12] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393.

• [13] R. H. Haghi, S. Rezapour, N. Shahzad , Some fixed point generalizations are not real generalizations, Nonlinear Anal., 74 (2011), 1799–1803.

• [14] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71–76.

• [15] K. N. Leibovic, The principle of contraction mapping in nonlinear and adaptive control systems, IEEE Trans. Automatic Control, 9 (1964), 393–398.

• [16] S. G. Matthews, Partial metric topology, Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., New York, 728 (1994), 183–197.

• [17] H. K. Nashine, E. Karapinar , Fixed point results in orbitally complete partial metric spaces, Bull. Malays. Math. Sci. Soc., 36 (2013), 1185–1193.