Some research perspectives in nonlinear functional analysis

Some research perspectives in nonlinear functional analysis BIAGIO RICCERI The object of this lecture is to propose a series of conjectures and problems in different fields of analysis. They have been formulated with the aim of introducing some innovative methods in the study of classical topics, as open mappings, fixed points, critical points, global minima, control theory. We start recalling the following definition. Let (E, ·) be a real normed space. A non-empty set A ⊂ E is said to be antiproximinal with respect to · if, for every x ∈ E \ A and every y ∈ A, one has x − y > inf z∈A x − z. CONJECTURE 1.-There exists a non-complete real normed space E with the following property: for every non-empty convex set A ⊂ E which is antiproximinal with respect to each norm on E, the interior of the closure of A is non-empty. The main reason for the study of Conjecture 1 is to give a contribution to open mapping theory in the setting of non-complete normed spaces. Actually, making use of Theorem 4 of [8], one can prove the following result. THEOREM 1.-Let X, E be two real vector spaces, C a non-empty convex subset of X, F a multifunction from C onto E, with non-empty values and convex graph. Then, for every non-empty convex set A ⊆ C which is open with respect to the rela-tivization to C of the strongest vector topology on X, the set F (A) is antiproximinal with respect to each norm on E. We now present CONJECTURE 2.-Let E be a real Banach space, and let J : E → R be a continuously Gâteaux differentiable functional. Assume that there are r > 0 and x 0 , x 1 ∈ E, with x 0 − x 1 > r, such that inf x−x 0 =r J(x) ≥ max{J(x 0), J(x 1)}. Put c = inf u∈A sup t∈[0,1] J(u(t)) where A denotes the set of all continuous functions u : [0, 1] → E such that u(0) = x 0 , u(1) = x 1 .