Fixed Point Theory on Extension-type Spaces and Essential Maps on Topological Spaces

In Section 2, we begin by presenting most of the up-to-date results in the literature [3, 5, 6, 7, 8, 12] concerning fixed point theory in extension-type spaces. These results are then used to obtain a number of new fixed point theorems, one concerning approximate neighborhood extension spaces and another concerning inward-type maps in extensiontype spaces. Our first result was motivated by ideas in [12] whereas the second result is based on an argument of Ben-El-Mechaiekh and Kryszewski [9]. Also in Section 2 we present a new continuation theorem for maps defined between Hausdorff topological spaces, and our theorem improves results in [3]. For the remainder of this section we present some definitions and known results which will be needed throughout this paper. Suppose X and Y are topological spaces. Given a class of maps, (X ,Y) denotes the set of maps F : X → 2Y (nonempty subsets of Y) belonging to , and c the set of finite compositions of maps in . We let ( )= {Z : FixF = ∅ ∀F ∈ (Z,Z)}, (1.1)