Fixed Points of Contractive Type Maps in Cone Metric Space over Banach Algebra

Main Article Content

Ashfaque Ur Rahman
Geeta Modi
K. Qureshi
Manoj Ughade

Abstract

Our goal in this paper is to prove some fixed point and common fixed theorems for contractive type maps in a CMS over Banach algebra, which unify, extend and generalize most of the existing relevant fixed point theorems from Shaoyuan Xu and Stojan Radenovic [1]. We provide illustrative example to verify our results.

Keywords:
CMS over Banach algebra, c-sequence, contractive mapping, fixed point

Article Details

How to Cite
Rahman, A. U., Modi, G., Qureshi, K., & Ughade, M. (2019). Fixed Points of Contractive Type Maps in Cone Metric Space over Banach Algebra. Journal of Advances in Mathematics and Computer Science, 33(1), 1-11. https://doi.org/10.9734/jamcs/2019/v33i130166
Section
Original Research Article

References

Shaoyuan Xu, Stojan Radenovic. Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl. 2014;12.

Huang LG, Zhang X. Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007;332:1468-1476.

Rezapour S, Hamlbarani R. Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2008;345:719-724.

Jiang S, Li Z. Extensions of Banach contraction principle to partial cone metric spaces over a non-normal solid cone. Fixed Point Theory Appl. 2013;250.

Abbas M, Rajic VC, Nazir T, Radenovic S. Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces. Afr. Math; 2013.
DOI: 10.1007/s13370-013-0185-z

Al-Khaleel M, Al-Sharifa S, Khandaqji M. Fixed points for contraction mappings in generalized cone metric spaces. Jordan J. Math. Stat. 2012;5(4):291-307.

Gajic L, Rakocevic V. Quasi-contractions on a no normal cone metric space. Funct. Anal. Appl. 2012;46(1):75-79.

Ilic D, Rakocevic V. Quasi-contraction on a cone metric space. Appl. Math. Lett. 2009;22(5):728-731.

Kadelburg Z, Radenovic S, Rakocevic V. Remarks on ‘Quasi-contraction on a cone metric space’. Appl. Math. Lett. 2009;22(11):1674-1679.

Radenovic S, Rhoades BE. Fixed point theorem for two non-self-mappings in cone metric spaces. Comput. Math. Appl. 2009;57:1701-1707.

Jankovic S, Kadelburg Z, Radenovic S. On the cone metric space: A survey. Nonlinear Anal. 2011; 74:2591-2601.

Cakallı H, Sonmez A, Genc C. On an equivalence of topological vector space valued cone metric spaces and metric spaces. Appl. Math. Lett. 2012;25:429-433.

Du WS. A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 2010; 72(5):2259-2261.

Kadelburg Z, Radenovic S, Rakocevic V. A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 2011;24:370-374.

Feng Y, Mao W. The equivalence of cone metric spaces and metric spaces. Fixed Point Theory. 2010;11(2):259-264.

Liu H, Xu S. Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory Appl. 2013;320.

Zoto K, Vardhami I. Common fixed point results for generalized contractive mappings and applications. Journal of Function Spaces; 2018.

Rudin W. Functional analysis. McGraw-Hill, New York; 1991.

Liu H, Xu S. Fixed point theorem of quasi-contractions on cone metric spaces with Banach algebras. Abstr. Appl. Anal. 2013, Article ID 187348.

Kadelburg, Z, Pavlovic, M, Radenovic, S: Common fixed point theorems for ordered contractions and quasi-contractions in ordered cone metric spaces. Comput. Math. Appl. 2010;59:3148-3159.

Abbas M, Jungck G. Common fixed point results for non-commuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 2008;341:416-420.