  The aim of this paper in to introduce a large class of mappings, called {\it
enriched Kannan mappings}, that includes all Kannan mappings and some
nonexpansive mappings. We study the set of fixed points and prove a convergence
theorem for Kransnoselskij iteration used to approximate fixed points of
enriched Kannan mappings in Banach spaces. We then extend further these
mappings to the class of enriched Bianchini mappings. Examples to illustrate
the effectiveness of our results are also given. As applications of our main
fixed point theorems, we present two Kransnoselskij projection type algorithms
for solving split feasibility problems and variational inequality problems in
the class of enriched Kannan mappings and enriched Bianchini mappings,
respectively.
