Abstract
This paper provides a new, symmetric, nonexpansiveness condition to extend the classical Suzuki mappings. The newly introduced property is proved to be equivalent to condition (E) on Banach spaces, while it leads to an entirely new class of mappings when going to modular vector spaces; anyhow, it still provides an extension for the modular version of condition (C). In connection with the newly defined nonexpansiveness, some necessary and sufficient conditions for the existence of fixed points are stated and proved. They are based on Mann and Ishikawa iteration procedures, convenient uniform convexities and properly selected minimizing sequences.
Highlights
The idea of looking for new contractive conditions to lead to wider and wider classes of mappings, as well as the effort of extending the metric setting, are two of the main directions in fixed point theory
This paper provides a new contribution related to these directions by defining a new nonexpansiveness property, on Banach spaces initially, and extending it afterwards to modular vector spaces
This paper provides an answer by defining a new property, called condition (CDE), which is proved to be equivalent to condition (E) on Banach spaces and to include the classes of Suzuki-type generalized nonexpansive mappings, as well as the class of mappings satisfying condition (Da)
Summary
The idea of looking for new contractive conditions to lead to wider and wider classes of mappings, as well as the effort of extending the metric setting, are two of the main directions in fixed point theory. An important step toward analyzing a more general nonexpansiveness condition on Banach spaces was performed by Suzuki in [14] He defined the so called condition (C), which provided a wider class of mappings than the nonexpansive mappings and stronger than the class of quasinonexpansive mappings. In [13], the class of Suzuki-type generalized nonexpansive mappings on Banach spaces was changed in connection with an admissible pair of parameters This new property extends Suzuki’s condition, but remains subordinated to quasinonexpansiveness. This paper provides an answer by defining a new property, called condition (CDE), which is proved to be equivalent to condition (E) on Banach spaces and to include the classes of Suzuki-type generalized nonexpansive mappings, as well as the class of mappings satisfying condition (Da).
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