FIXED POINT THEORY OF MULTIVALUED WEIGHTED MAPS JACOBO PEJSACHOWICZ AND ROBERT SKIBA 1. Introduction Without doubt the golden age of fixed point theory for multival- ued map occurred in the post-war period. Motivated by the recently born disciplines of mathematical economics and game theory, using tools from the flourishing algebraic topology of that time, several well known fixed point theorems were proved. Most of the research was done in the area of fixed points for convex-valued maps, due to their importance in applications. However, the interest of topologists was immediately directed toward more general classes of maps. The purpose of this paper is to survey the fixed point theory of two very special classes of multivalued maps with weights which were found during the intense research activity of that time. The first, which we will call acyclic weighted carriers, was discovered by Gabriele Darbo in 1950. The second class was introduced by him in 1957 under the name of weighted maps. We will briefly explain how they were found, review the significant results and point out few applications of this theory. At first glance acyclic weighted carriers look similar to other cat- egories of acyclic maps for which a fixed point theory has been con- structed. However, this similarity is only apparent. For example, it is known that several classes of acyclic morphisms are homotopic to a continuous map by a homotopy in this category [68, 69]. This is far from being true for acyclic weighted carriers. In the presence of a non- trivial branching, an acyclic weighted carrier cannot be homotopic to any linear combination of continuous maps. On the other hand, weighted maps are closely related to maps into the n-th symmetric product of a space [85]. But the concept of weighted map is more flexible and as a consequence the category is considerably larger than that of maps into symmetric products. As we will see, weighted maps and weighted carriers arise quite naturally as solutions of parametrized families of nonlinear equations. This fact makes the corresponding fixed point and degree theories of considerable interest. Date : 29.03.2004. 1