On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings

Abstract

Locally convex spaces, Weak τ-functions, Ekeland’s variational principle, K-lower semicontinity from above, Lower closedness,

On Ekeland’s Variational Principle for Pareto Minima of Set-Valued Mappings

“…A case where Corollary 4.2 is more suitable for applications than recent known results is given by the following: Example 4.5 Let X, Y, p, K, k 0 , and (x 0 , y 0 ) be as in Example 4.4, and Finally, we discuss some consequences for single-valued mappings. Using Theorems 4.2 and 4.4 and similar techniques of [12,13], we can obtain Corollaries 5.1-5.6 of [12] and Corollaries 7.6-7.8 of [13]. These corollaries bring some improvements to the corresponding results of [10,14,18,19,22,24,25], cf.…”

“…is lower semicontinuous, i.e., lim inf y→ȳ p(x, y) ≥ p(x,ȳ), a weak τ -function becomes a τ -function introduced in [10]. Observe further that (see [12,13]) the w-distance, Tataru's distance, τ -distance, and τ -function are all particular cases of a weak τ -function. Example 2.1 in [13] shows that being a weak τ -function may be strictly weaker than being a kind of the mentioned distances.…”

“…Definition 2.1 (See [12,13]) Let (X, d) be a metric space. A function p : X × X → R + is called a weak τ -function iff the following conditions hold:…”

“…A more relaxed notion of K-lower semicontinuity from above at a pointx ∈ X was proposed in [26] for single-valued scalar functions. In [12], we extended it to setvalued mappings, suitably for considering a Kuroiwa's minimum [27], which is different from a Pareto minimum, as follows: F : X ⇒ Y is called K-lower semicontinuous from above (K-lsca) atx ∈ X iff, for each x n →x with F (x n ) ⊆ F (x n+1 ) + K for all n ∈ N (the set of natural numbers), one has F (x n ) ⊆ F (x) + K for all n ∈ N. As any definition for a point is extended to a set, F is called K-lsca on A ⊆ X iff it is K-lsca at all x ∈ A. If A = dom F , we omit "on A" in the statement.…”

“…In [11], the concept of a Q-function, which is an extension of a τ -function but independent of τ -distance, was introduced. In [12,13], we proposed a definition of a weak τ -function, which generalizes both the τ -function and τ -distance. We also obtained improved formulations of the EVP for Pareto minimizers of a single-valued mapping and for Kuroiwa's minimizers of a set-valued mapping.…”