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Fixed point theorems of (a,b)-monotone mappings in Hilbert spaces

Research paper by Lai-Jiu Lin, Sung-Yu Wang

Indexed on: 07 Aug '12Published on: 07 Aug '12Published in: Fixed Point Theory and Applications



Abstract

We propose a new class of nonlinear mappings, called (a,b)-monotone mappings, and show that this class of nonlinear mappings contains nonspreading mappings, hybrid mappings, firmly nonexpansive mappings, and (a1,a2,a3,k1,k2)-generalized hybrid mappings with a1<1. We also give an example to show that a (a,b)-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 (a,b)-monotone mappings. As a special case of our result, we give an existence theorem of fixed points for (a1,a2,a3,k1,k2)-generalized hybrid mappings with a1<1. We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for (a,b)-monotone mappings. We give an example of (a,b)-monotone mappings which assures the Mann’s type weak convergence.