Convergence Results for Jungck-type Iterative Processes in Convex Metric Spaces

Olatinwo, Memudu Olaposi

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 1, pp. 79-87.

Résumé

In this paper, the convergence results of [V. Berinde; A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45 (1) (2007), 33–41], [V. Berinde; On the convergence of Mann iteration for a class of quasi-contractive operators, Preprint, North University of Baia Mare (2003)] and [V. Berinde; On the Convergence of the Ishikawa Iteration in the Class of Quasi-contractive Operators, Acta Math. Univ. Comenianae 73 (1) (2004), 119–126] are extended from arbitrary Banach space setting to the convex metric space by weakening further the conditions on the parameter sequence $\lbrace \alpha _n\rbrace \subset [0,1]$. We establish the convergence of Jungck–Mann and Jungck–Ishikawa iterative processes for two nonselfmappings in a convex metric space setting by employing a general contractive condition. Similar results are also deduced for the Mann and Ishikawa iterations. Our results generalize, extend and improve a multitude of results in the literature including those of Berinde mentioned above.

Détail

Classification : 47H10, 54H25
Keywords: arbitrary Banach space setting; Jungck–Mann and Jungck–Ishikawa iterative processes; convex metric space