Contractions, inwardness, tool theorems

Abstract: The paper is devoted to the fixed point theory in four aspects: of contractions, nonexpansive mappings, generalized inward mappings, and of the tool theorems. The manuscript was written about ten years ago. At first Nadler's concept of contraction for multivalued mappings is replaced here by a more general, and yet elegant condition: for some $\alpha + \epsilon <1$, and each $x \in X$ there exists a $y \in F(x)$ such that $d(F(y),y) \leq \alpha d(y,x) \leq (\alpha + \epsilon) d(F(x),x)$}. For nonexpansive'' mappings we apply bead spaces that are more general than uniformly convex spaces, and our requirements on mappings are weaker than nonexpansivity in the sense of the Hausdorff distance. In the last, third section the Caristi theorem is replaced by more specializedtools'', and we apply them to obtain stronger fixed point theorems on generalized inward mappings. In particular, if for each $x \in X$ a nearest point of $F(x)$ belongs to the generalized inward set, then the values of $F$ need not to be closed.