A generalized distance and enhanced Ekeland’s variational principle for vector functions

About Semicontinuity of Set-valued Maps and Stability of Quasivariational Inclusions

“…The following existence result is an immediate consequence of Theorem 2.1 of [21], and Lemma 3.8(iii), Theorem 4.1 of [22]. This proposition implies the following result for the lower semicontinuity of .…”

“…Now we investigate a particular scalar case of (QEP 1 λ ) and (SQEP 1 λ ), defined in Subsection 5.1, in connection with an application of versions of Ekeland's variational principle considered in [21] and [22]. Let (X, d) be a complete metric space, Λ a metric space and f : X × X × → R. For λ ∈ , we are concerned with the following scalar equilibrium problem…”

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

“…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:…”

“…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.…”