Fixed point and weak convergence theorems for point-dependent λ-hybrid mappings in Banach spaces

The purpose of this article is to study the fixed point and weak convergence problem for the new defined class of point-dependent lambda-hybrid mappings relative to a Bregman distance D (f) in a Banach space. We at first extend the Aoyama-Iemoto-Kohsaka-Takahashi fixed point theorem for lambda-hybrid mappings in Hilbert spaces in 2010 to this much wider class of nonlinear mappings in Banach spaces. Secondly, we derive an Opial-like inequality for the Bregman distance and apply it to establish a weak convergence theorem for this new class of nonlinear mappings. Some concrete examples in a Hilbert space showing that our extension is proper are also given. 2010 MSC: 47H09; 47H10.