On an open question of Takahashi for nonspreading mappings in Banach spaces

In this paper, we first introduce a new class of mappings called asymptotically nonspreading mappings and establish weak and strong convergence theorems of the iterative sequences generated by these mappings in a real Banach space. We modify Halpern's iterations for finding a fixed point of an asymptotically nonspreading mapping and provide an affirmative answer to an open problem posed by Kurokawa and Takahashi in their final remark of (Kurokawa and Takahashi in Nonlinear Anal. 73:1562-1568, 2010) for nonspreading mappings. Furthermore, we investigate the approximation of common fixed points of asymptotically nonspreading mappings and nonexpansive mappings and derive a strong convergence theorem by a new hybrid method for these mappings. Our results improve and generalize many known results in the current literature. MSC: 47H10, 37C25.