Abstract
The purpose of this paper is to introduce and study the general viscosity approximation methods for quasi-nonexpansive mappings in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converge strongly to a fixed point of quasi-nonexpansive mappings in Hilbert spaces, which is also the unique solution of some variational inequality. Then this result is used to study the split equality fixed point problems, the split equality common fixed point problems, the split equality null point problems, etc. Our results improve and generalize many results in the literature and they should have many applications in nonlinear science.
Highlights
Let C be a nonempty subset of a real Hilbert space H
The viscosity approximation method for nonlinear mappings was first introduced by Moudafi [1]
We prove that the sequences generated by the proposed new algorithm converge strongly to a fixed point of quasi-nonexpansive mappings in Hilbert spaces, which is the unique solution of some variational inequality
Summary
Let C be a nonempty subset of a real Hilbert space H. It is proved that if the sequences {αn}n∈N, {βn}n∈N, and {sn}n∈N satisfy appropriate conditions, the sequence {xn}n∈N generated by (1.6) converges strongly to the unique solution of the variational inequality (1.2), where Fix(T) is the fixed point set of a quasi-nonexpansive mapping T. We prove that the sequences generated by the proposed new algorithm converge strongly to a fixed point of quasi-nonexpansive mappings in Hilbert spaces, which is the unique solution of some variational inequality. This result is used to study the split equality fixed point problems, the split equality common fixed point problems, the split equality null point problems, etc.
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