On some open problems in cone metric space over Banach algebra

Subject Areas : Fixed point theory

A. Ahmed ¹ , Z. D. Mitrovic ² , J. N. Salunke ³

1 - Department of Humanities and Basics Sciences, School of Engineering, Matoshri Pratishthan Group of Institutions, Nanded, India
2 - University of Banja Luka, Faculty of Electrical Engineering, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina
3 - School of Mathematical Sciences, Swami Ramanandh Teerth Marathwada University, Nanded, India

Received: 2017-10-24 Accepted : 2017-12-10 Published : 2017-12-01

Keywords: Cone metric space over Banach algebra, fixed points, Lipschitz mapping, c-sequence,

Abstract :

In this paper we prove an analogue of Banach and Kannan fixed point theorems by generalizing the Lipschitz constat $k$, in generalized Lipschitz mapping on cone metric space over Banach algebra, which are answers for the open problems proposed by Sastry et al, [K. P. R. Sastry, G. A. Naidu, T. Bakeshie, Fixed point theorems in cone metric spaces with Banach algebra cones, Int. J. of Math. Sci. and Engg. Appl. (6) (2012), 129-136].

References:

[1] S. Banach, Sure les operations dans les ensembles abstraits et leur applicaiton aux equations integrales, Fund. Math. 3 (1922), 133-181.

[2] H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. 320 (2013), 1-10.

[3] Z. D. Mitrovic, On an open problem in rectangular b-metric space, J. Anal. 25 (1) (2017), 135-137.

[4] K. P. R. Sastry, G. A. Naidu, T. Bakeshie, Fixed point theorems in cone metric spaces with Banach algebra cones, Int. J. Math. Sci. and Engg. Appl. 6 (2012), 129-136.

[5] S. Xu, S. Radenovi´c, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebra without assumption of normality, Fixed Point Theory Appl. 102 (2014), 1-12.

[6] S. Radenovic, B. E. Rhoades, Fixed point theorems for two non-self mappings in cone metric spaces, Comput. Math. Appl. 57 (2009), 1701-1707.

[7] H. Huang, S. Radenovi´c, Some fixed point results of generalized Lipschitz mappings on cone b-metric spaces over banach algebra, J. Comput. Anal. Appl. 20 (2016), 566-583.