Properties of fixed point spaces

Fixed point theorems and convergence theorems for some generalized nonexpansive mappings

“…1 The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Theorem 1 is meaningful because contractions do not characterize the metric completeness while Caristi and Kannan mappings do; see [4,14,17]. Since lim r→1−0 θ(r) = 1/2, it is very natural to consider the following condition.…”

A Suzuki-type common fixed point theorem

“…It has been generalized by many authors in many different directions; see ( [2], [3], [4], [13], [15], [16] and references therein). Connell [5] gave an example of a metric space such that X is not complete even every contraction on X has a fixed point. Hence, Banach theorem cannot characterize the metric completeness of X.…”

Kannan type mapping in TVS-valued cone metric spaces and their application to Urysohn integral equations

“…The Banach contraction theorem [4] is an extremely dynamic tool in mathematical analysis. However, the Kannan fixed point theorem [13] is imperative because it characterizes completeness of metric spaces [18], while Banach theorem cannot characterize the metric completeness of X [7]. the Banach type contractive condition (i.e.…”