Almost Contraction Mappings and $(S,T)$-Stability of Jungck Iteration in Cone Metric Spaces over Banach Algebras

In this work, we first introduce almost contraction mappings for a pair of two mappings in cone metric spaces over Banach algebras (CMSBA). Then, we observe that the class of such mappings in this setting contains those of many well known mappings. Finally, we prove some fixed point theorems, and obtain $(S,T)$-stability results of Jungck iterations for some mappings in CMSBA.

Keywords:

Banach algebra, cone metric space jungck iteration, stability,

PDF

___

Abbas, M., Jungck, G., Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341(2008), 416–420.
Berinde, V. ,Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9(2004), 43–53.
Develi, F., Ozavsar, M., Almost contraction mappings in cone b-metric spaces over Banach algebras, Hacettepe J. Math. Stat., 49(2020), 1965–1973.
Hardy, G.E., Rogers, T.D., A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16(1973), 201–206.
Harder, A.M., Hicks, T.L., Stability results for fixed point iteration procedure, Math. Japonica, 33(1988), 693–706.
Huang, L.G., Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2007), 1468–1476.
Huang, H., Radenovic S., Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras andapplications, J. Non. Sci. Appl., 8(2015), 787–799.
Huang, H., Xu, S., Liu, H., Radenovic, S., Fixed point theorems and T -stability of Picard iteration for generalized Lipschitz mappings in conemetric spaces over Banach algebras, J. Comput. Anal. Appl., 20(2016), 869–888.
Huang, H., Deng, G., Radenovic S., Some topological properties and fixed point results in cone metric spaces over Banach algebras, Positivity, 23(2019), 21–34.
Jungck, G., Commuting mappings and fixed points, Amer. Math. Monthly, 83(1976), 261–263.
Jungck, G., Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci., 4 (1996), 199–215.
Jankovic, S., Kadelburg, Z., Radenovic, S., On cone metric spaces: a survey, Nonlinear Anal., 74 (2011), 2591–2601.
Kadelburg, Z., Radenovic, S.,A note on various types of cones and fixed point results in cone metric spaces, Asian J. Math. Appl., 2013, ArticleID ama0104, 7 pages.
Liu, H., Xu, S., Cone metric spaces with Banach algebras and Fixed point theorems of generalized Lipschitz mappings, Fixed Point TheoryAppl., 2013(2013), 1–10.
Ozavsar, M., Fixed point theorems for (k,l)-almost contractions in cone metric spaces over Banach algebras, Mathematical Advances in Pure and Applied Sciences, 1(2018), 46–51.
Osilike, M.N.,Stability results for fixed point iteration procedures, J. Nigerian Math. Soc., 14(1995), 17–29.
Rudin, W., Functional Analysis. 2ndedn., McGraw-Hill, New York, 1991.
Singh, S.L., Bhatnagar, C., Mıshra, S.N., Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci., 2005(2005), 3035–3043.
Xu, S., Radenovic, S., Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014(2014), 1–12.