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, ${I_K}(x)$, of x relative to K is defined as follows: ${I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}$. A mapping $T:K \to X$ is said to be inward if $Tx \in {I_K}(x)$ for each $x \in K$, and weakly inward if Tx belongs to the closure of ${I_K}(x)$ for each $x \in K$. 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.