Fixed point theorems for the class Q(X, Y)

Kuratowski [6] showed that a continuous compact map f : X → X defined on a closed convex subset X of a Banach space has a fixed point. This theorem has enormous influence on fixed point theory, variational inequalities, and equilibrium problems. In 1968, Goebel [5] established the well-known coincidence theorem, and then there had been a lot of generalization and application (see, [1, 2, 5]). Let X be a subset of a Hausdorff topological vector space E and Y a Hausdorff topological vector space, we define a new class Q(X ,Y) of set-valued maps from X into Y as follows. T ∈Q(X ,Y) implies that for any compact convex subset K of X and any continuous function f : T(K) → K , the composition f (T|K ) : K → 2K has a fixed point. Subclasses of Q(X ,Y) are the class of continuous functions C(X ,Y), the class of the Kakutani maps K(X ,Y) (with convex values and codomains being convex spaces), the class of the acyclic maps V(X ,Y) (with acyclic values), and the class of the approachable maps 0(X ,Y) (whose domains and codomains are subsets of topological vector spaces), and so forth. A nonempty subset X of a Hausdorff topological vector space E is said to be nearly convex (see Wu [7]) if for every compact subset A of X and every neighborhood V of the origin 0 of E, there is a continuous mapping h : A→ X such that x− h(x) ∈ V for all x ∈ A and h(A) is contained in some convex subset of X .