Approximation Theorems and Fixed Point Theorems for Various Classes of 1-set-contractive Mappings in Banach Spaces

In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with intK0. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors.