FIXED POINT THEOREMS AND CONVERGENCE THEOREMS FOR GENERALIZED HYBRID NON-SELF MAPPINGS IN HILBERT SPACES

In this paper, we first prove a fixed point theorem for normal generalized hybrid non-self mappings in a Hilbert space. In the proof, we show that widely more generalized hybrid mappings are deduced from normal generalized hybrid non-self mappings. Next, we prove a weak convergence theorem of Mann's type [20] for widely more generalized hybrid non-self mappings in a Hilbert space. For the proof, we use the demi-closedness property for widely more generalized hybrid non-self mappings. Finally, using an idea of mean convergence by Shimizu and Takahashi [21] and [22], we prove a mean strong convergence theorem for widely more generalized hybrid mappings in a Hilbert space. This theorem generalizes Hojo and Takahashi's mean strong convergence theorem [9] for generalized hybrid mappings.