Fixed point and common fixed point theorems on ordered cone $b$-metric space over Banach algebra

Volume 13, Issue 1, pp 22--33
Publication Date: August 28, 2019 Submission Date: April 07, 2019 Revision Date: July 01, 2019 Accteptance Date: July 23, 2019
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Authors

Sharafat Hussain - Department Mathematics, Quaid i Azam University, Islamabad, Pakistan. - Department of Mathematics, Women University of Azad Jammu and Kashmir Bagh, AJK, Pakistan.

Abstract

In this work, we obtain some fixed point and common fixed point theorems of comparable maps satisfying certain contractive conditions on partially ordered cone b-metric space over Banach algebras. Some examples are also provided to illustrate the main results presented in this paper, which extend and generalize several known results in cone b-metric spaces.

Keywords

• Fixed point
• partially ordered set
• cone b-metric space
• Banach algebra

•  47H10
•  54H25

References

• [1] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008), 416--420

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

• [3] I. Altun, G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo (2), 58 (2009), 319--325

• [4] I. Altun, H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl., 2010 (2010), 17 pages

• [5] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), 26--37

• [6] F. Bojor, Fixed point of $\phi$-contraction in metric spaces endowed with a graph, An. Univ. Craiova Ser. Mat. Inform., 37 (2010), 85--92

• [7] N. Bourbaki, Topologie générale, Herman, Paris (1971)

• [8] H. Çakallı, A. Sönmez, Ç. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett., 25 (2012), 429--433

• [9] S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5--11

• [10] W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 72 (2010), 2259--2261

• [11] J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72 (2010), 1188--1197

• [12] H. P. Huang, S. Radenović, Common fixed point theorems of generalized Lipschitz mappings in cone $b$-metric spaces over Banach algebras and applications, J. Nonlinear Sci. Appl., 8 (2015), 787--799

• [13] L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468--1476

• [14] N. Hussain, A. M. Al-Solami, M. A. Kutbi, Fixed points of $\alpha$-admissible mappings in cone $b$-metric spaces over Banach algebra, J. Math. Anal., 8 (2017), 89--97

• [15] N. Hussian, M. H. Shah, KKM mappings in cone $b$-metric spaces, Comput. Math. Appl., 62 (2011), 1677--1684

• [16] Z. Kadelburg, M. Pavlović, S. Radenović, Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl., 59 (2010), 3148--3159

• [17] Z. Kadelburg, S. Radenovic, A note on various types of cones and fixed point results in cone metric spaces, Asian J. Math. Appl., 2013 (2013), 7 pages

• [18] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73 (2010), 3123--3129

• [19] M. A. Kutbi, J. Ahmed, A. E. Al-Mazrooei, N. Hussain, Multivalued fixed point theorem in cone $b$-metric space over Banach algebra with Application, J. Math. Anal., 9 (2018), 52--64

• [20] H. Liu, S. Y. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013 (2013), 10 pages

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

• [22] T. Rasham, A. Shoaib, M. Arshad, Fixed Point Results for Locally Hardy Rogers-Type Contractive Mappings for Dislocated Cone Metric Space, TWMS J. Pure Appl. Math., 10 (2019), 76--82

• [23] S. Rezapour, R. Hamlbarani, Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings", J. Math. Anal. Appl., 345 (2008), 719--724

• [24] W. Rudin, Functional Analysis, McGraw-Hill, New York (1991)

• [25] A. Shoaib, P. Kumam, A. Shahzad, S. Phiangsungnoen, Q. Mahmood, Fixed point results for fuzzy mappings in a $b$-metric space, Fixed Point Theory Appl., 2014 (2014), 12 pages

• [26] Q. Yan, J. D. Yin, T. Wang, Fixed point and common fixed point theorems on ordered cone metric spaces over Banach algebras, J. Nonlinear Sci. Appl., 9 (2016), 1581--1589