Mann type iterative methods for finding a common solution of split feasibility and fixed point problems

The purpose of this paper is to study and analyze three different kinds of Mann type iterative methods for finding a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann’s iterative method and the extragradient method, we first propose Mann type extragradient-like algorithm for finding an element of the set $${{{\rm Fix}}(S) \cap \Gamma}$$ ; moreover, we derive the weak convergence of the proposed algorithm under appropriate conditions. Second, we combine Mann’s iterative method and the viscosity approximation method to introduce Mann type viscosity algorithm for finding an element of the $${{{\rm Fix}}(S)\cap \Gamma}$$ ; moreover, we derive the strong convergence of the sequences generated by the proposed algorithm to an element of set $${{{\rm Fix}}(S) \cap \Gamma}$$ under mild conditions. Finally, by combining Mann’s iterative method and the relaxed CQ method, we introduce Mann type relaxed CQ algorithm for finding an element of the set $${{{\rm Fix}}(S)\cap \Gamma}$$. We also establish a weak convergence result for the sequences generated by the proposed Mann type relaxed CQ algorithm under appropriate assumptions.