On a sharpened form of the Schauder fixed-point theorem

IfK is a compact convex subset of a locally convex topological vector spaceX , we consider a continuous mappingf ofK intoX . A fixed-point theorem is proved for such a mapf under the assumption that for a given continuous realvalued functionp onK ×X withp (x,y ) convex iny and for each pointx inK not fixed byf , there exists a pointy in the inward setIK (x ) generated byK atx withp(x,y - f(x)) less thanp(x,x - f(x)) . ForX a Banach space, in particular, this yields a sharp extension and a drastic simplification of the fixed point theory of weakly inward (and weakly outward) mappings. The result comes close in the domain of mappings of compact convex sets to the thrust of fixed point conditions of the Leray-Schauder type for compact maps of sets with interior inX .