Almost contraction mappings in cone $b$-metric spaces over Banach algebras

Year 2020, Volume: 49 Issue: 6, 1965 - 1973, 08.12.2020

Faruk Develi Muttalip Özavşar

https://doi.org/10.15672/hujms.486939

Cited By: 2

Abstract

We introduce the condition of being Cauchy for a sequence in cone $b$-metric spaces (cbms) over Banach algebras. Based on this result, we extend almost mappings in cone metric spaces over Banach algebras to cbms over Banach algebras and prove the related fixed point theorem. In addition, we apply our results to some applications to illustrate their usability.

Keywords

Cone b-metric spaces over Banach algebras, spectral radius, common fixed point

References

• [1] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (1), 416–420, 2008.
• [2] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (1), 43–53, 2004.
• [3] W.S. Du, A note on cone metric fixed theory and its equivalence, Nonlinear Anal. 72 (5), 2259–2261, 2010.
• [4] L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2) 1468–1476, 2007.
• [5] H. Huang and S. Radenovic, Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications, J. Non. Sci. Appl. 8 (5), 787–799, 2015.
• [6] H. Huang, S. Hu, B.Z. Popovic and S. Radenovic, Common fixed point theorems for four mappings on cone b-metric spaces over Banach algebras, J. Non. Sci. Appl. 9 (6), 3655–3671, 2016.
• [7] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83 (4), 261–263, 1976.
• [8] G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci. 4 (2), 199–215, 1996.
• [9] Z. Kadelburg and S. Radenovic, A note on various types of cones and fixed point results in cone metric spaces, Asian J. Math. Appl. 2013, Article ID:ama0104, 2013.
• [10] H. Liu and S. Xu, Cone metric spaces with Banach algebras and Fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. 2013 (320), 2013.
• [11] M. Ozavsar, Fixed point theorems for (k, l)-almost contractions in cone metric spaces over Banach algebras, Mathematical Advances in Pure and Applied Sciences, 1 (1), 46–51, 2018.
• [12] Y. Piao, Unique common fixed points for two mappings with Kannan-Chatterjea type conditions on cone metric spaces over Banach algebras without normality, Adv. Inequal. Appl. 2016, Article ID: 1, 2016.
• [13] S. Radenovic and B.E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl. 57 (10), 1701–1707, 2009.
• [14] W. Rudin, Functional Analysis and its Applications, McGraw-Hill, New York, 1991.
• [15] T. Suzuki, Basic inequality on a b-metric space and its applications, J. Inequal. Appl. 2017, 256, 2017.
• [16] S. Xu and S. Radenovic, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl. 2014, 102, 2014.