Fixed Point Theorems for Mappings Satisfying Inwardness Conditions

Let X be a normed linear space and let K be a convex subset of X. The inward set, , of x relative to K is defined as follows: <!-- MATH ${I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}$ --> . A mapping is said to be inward if <!-- MATH $Tx \in {I_K}(x)$ --> for each , and weakly inward if Tx belongs to the closure of for each . In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.