A well known theorem proved in 1960 by Browder states that the set of fixed points of a family of continuous functions \(f_{p}:C\longrightarrow C\) , C being a compact and convex subset of \(\mathbb {R}^{d}\) , depending continuously on a parameter \(p\in [0,1]\) , i.e. the set \(C_{f}:=\{(p,x)\in [0,1]\times C: f(p,x)=x\}\) , has a connected component whose projection on the first coordinate is [0, 1]. The same result remains true if C is a closed and convex subset of a Banach space and f is compact. In the present paper, by using the so-called degree of nondensifiability (DND), we introduce a new class of mappings called parametric DND-condensing (which is a generalization of the class of compact mappings) and prove a generalization of Browder’s theorem. Moreover, in the proposed generalization, we can replace the parameter space [0, 1] by an arbitrary Peano Continuum of a Banach space.
