Abstract

We propose a new class of nonlinear mappings, called -monotone mappings, and show that this class of nonlinear mappings contains nonspreading mappings, hybrid mappings, firmly nonexpansive mappings, and -generalized hybrid mappings with . We also give an example to show that a -monotone mapping is not necessary to be a quasi-nonexpansive mapping. We establish an existence theorem of fixed points and the demiclosed principle for the class of -monotone mappings. As a special case of our result, we give an existence theorem of fixed points for -generalized hybrid mappings with . We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for -monotone mappings. We give an example of -monotone mappings which assures the Mann’s type weak convergence.

Keywords:

fixed point; demiclosed principle; strong convergence; weak convergence; nonspreading mapping; hybrid mapping; nonexpansive mapping; - monotone mapping; Mann’s type iteration; CQ type iteration