A FURTHER GENERALIZATION OF NONEXPANSIVITY

作者：Dominguez Benavides T; Lorenzo Ramirez P

来源：Journal of Nonlinear and Convex Analysis, 2014, 15(2): 299-311.

• Fixed point
• generalized nonexpansive mappings
• multivalued nonexpansive mappings
• Banach spaces

Many different generalizations of the notion of nonexpansivity have appeared in the last years. Recently T. Suzuki defined: let M be a metric space. A mapping T: M -> M is said to satisfy condition (C) if 1/2d(x, Tx) <= d(x, y) double right arrow d(Tx, Ty) <= d(x, y). In this paper we extend this definition in the following way: a mapping T : M -> M is of Suzuki type if there exists a nondecreasing function psi : (0, infinity) -> (0, infinity) such that d(x, Tx) - psi(d(x, Tx)) <= d(x, y) double right arrow d(Tx, Ty) <= d(x, y), for all x, y is an element of M. We prove the existence of a fixed point for this class of mappings under similar assumptions as those used for mappings satisfying condition (C). We also extend this definition and the corresponding fixed point results to the case of multivalued mappings. In the last section we show an example of a mapping of Suzuki type which does not satisfy any previously considered condition extending nonexpansivity.

• 出版日期2014