For a

nonempty set H a function [omicron] : H x H [right arrow] P*(H) is called a hyper operation on H.

Let C be a

nonempty, closed, convex subset of a real Hilbert space H and f : C x C [right arrow] R be a bifunction such that f (x, x) = 0 for all x [member of] C.

[1] Let X be a (

nonempty) set and s [greater than or equal to] 1 a given real number.

(1) A

nonempty subset A of S is called an AG-subgroupoid of S if [A.sup.2] [subset or equal to] A.

Let C be a

nonempty closed convex subset of a real Hilbert space H.

Let X be a

nonempty set (decision space) and denote by [??] a preorder (i.e., a reflexive and transitive binary relation) on X.

For two random mappings S, T: [OMEGA] x Y [right arrow] E with T([omega], Y) [subset or equal to] S([omega], Y) and C being a

nonempty closed convex subset of a separable Banach space E, there exists a real number [delta] [member of] [0, 1) and a monotone increasing function [phi]: [R.sup.+] [right arrow] [R.sup.+] with [phi](0) = 0, and for all x, y [member of] C, one has

Consider [mathematical expression not reproducible] the subfamily of [W.sup.n,1]([[0, T].sup.N]) consisting of relatively compact sets in the topology [[tau].sup.[omega]] and [mathematical expression not reproducible] the family of all

nonempty and bounded subsets (innorm) of [W.sup.n,1]([[0, T].sup.N]).

Let g : X [right arrow] X be a nonlinear operator on a metric space (X, d) and G : E [right arrow] BC(X) be a set-valued operator, where E [subset] X is a

nonempty subset and BC(X) is the family of

nonempty bounded closed subsets of X.

Let H be a

nonempty set and [less than or equal to] be an ordered relation on H.

We say that X has the CIPD if for any

nonempty closed subset A [approximately equal to] X, there exists a selfmap f : X [right arrow] X homotopic to the identity [1.sub.X] such that A = Fix(f) = {x [member of] X | f(x) = x}.