On Equivalence of Some Iterations Convergence for Quasi-Contraction Maps in Convex Metric Spaces
© Zhiqun Xue et al. 2010
Received: 23 July 2010
Accepted: 9 September 2010
Published: 19 September 2010
We show the equivalence of the convergence of Picard and Krasnoselskij, Mann, and Ishikawa iterations for the quasi-contraction mappings in convex metric spaces.
Combining above three definitions, Zamfirescu  showed the following result.
Clearly, every quasi-contraction mapping is the most general of above mappings.
Later on, in 1992, Xu  proved that Ishikawa iteration can also be used to approximate the fixed points of quasi-contraction mappings in real Banach spaces.
Let be any nonempty closed convex subset of a Banach space and a quasi-contraction mapping. Suppose that for all and . Then the Ishikawa iteration sequence defined by (1)–(3) converges strongly to the unique fixed point of .
In this paper, we will show the equivalence of the convergence of Picard and Krasnoselskij, Mann, and Ishikawa iterations for the quasi-contraction mappings in convex metric spaces.
where , and as . Then as (see ).
2. Results for Quasi-Contraction Mappings
this is a contradiction.
it is a contradiction.
which is a contradiction.
The authors are extremely grateful to Professor B. E. Rhoades of Indiana University for providing useful information and many help. They also thank the referees for their valuable comments and suggestions.
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