Linking

Abstract

It is interesting that the concept of linking is of importance in critical point theory. To the average person two objects are said to be linked if they cannot be pulled apart. This is basically the idea we shall use in finding critical points. Let E be a Banach space. We introduce the set Ф of mappings Γ(t) ∈ C(E x [0, 1], E) with the following properties:

• a) for each t ∈ [0, 1), Γ(t) is a homeomorphism of E onto itself and Γ(t)^-1 is continuous on E x [0, 1)

• b) Γ(0) = I

• c) for each Γ(t) ∈ Ф there is a u₀ ∈ E such that Γ(1)u = u₀ for all u ∈ E and Γ(t)u → u₀ as t → 1 uniformly on bounded subsets of E.