It has been applied to many classes of nonlinear mappings to find a fixed point.
From the above definitions, we note that the class of demicontractive mappings is fundamental; it includes many kinds of nonlinear mappings such as the directed mappings, the quasi-nonexpansive mappings, and the strictly pseudo-contractive mappings with fixed points as special cases.
The class of nonexpansive mappings as a class of important nonlinear mappings finds many applications in signal processing, image reconstruction and so on.
Recently, hybrid projection algorithms have been studied for the fixed point problems of nonlinear mappings by many authors; see, for example, [8 24].
Because of a simple construction, Halpern's iterations are widely used to approximate a solution of fixed points for nonexpansive mappings and other classes of nonlinear mappings by many authors in different styles.
Recently, Mann-type iterative algorithm and Ishikawa-type iterative algorithm (implicit and explicit) have been considered for the approximation of common fixed points of nonlinear mappings by many authors, see, for example, [6 24].
Iterative approximation of fixed points and zeros of nonlinear mappings has been studied extensively by many authors to solve nonlinear mapping equations as well as variational inequality problems and their generalizations (see, e.g., [11 19]).
Subsequently, many authors investigated fixed point problems of nonlinear mappings based on the methods in the framework of Hilbert spaces.
The fixed point property (fpp) of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology.
During the past five decades, many famous existence theorems of fixed points of nonlinear mappings were established.
