The purpose of this paper is to derive some criteria for vectorial prequasi-invex type functions via Jensen type inequalities. It is shown that a Jensen type inequality is sufficient and necessary for a vector function to be prequasi-invex under the condition of lower levelclosedness, cone lower semicontinuity, cone upper semicontinuity and semistrict prequasi-invexity, respectively. Analogous ...

The aim of this work is to prove a chain rule and an L1-lower semicontinuity theorems for integral functional defined on BV (Ω). Moreover we apply this result in order to obtain new relaxation and Γ-convergence result without any coerciveness and any continuity assumption of the integrand f(x, s, p) with respect to the variable s. Sunto. L’obiettivo di questo lavoro é quello di dimostrare una n...

The concept of equi-outer semicontinuity allows us to relate the pointwise and the graphical convergence of set-valued-mappings. One of the main results is a compactness criterion that extends the classical Arzelà-Ascol̀ı Theorem for continuous functions to this new setting; it also leads to the exploration of the notion of continuous convergence. Equi-lower semicontinuity of functions is relate...

We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. We also give several equivalent characterizations of lower semicontinuity. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set....

In this paper, dedicated to Neil Trudinger in the occasion of his 70 birthday, we propose a relatively elementary proof of the weak lower semicontinuity in W 1,p of a general integral of the Calculus of Variations of the type (1.1) below with a quasiconvex density function satisfying p-growth conditions. Several comments and references on the related literature, and a subsection devoted to some...

Let Ω be an open set of R and let f : Ω × R × R be a nonnegative continuous function, convex with respect to ξ ∈ R. Following the well known theory originated by Serrin [14] in 1961, we deal with the lower semicontinuity of the integral F (u,Ω) = ∫ Ω f (x, u(x), Du(x)) dx with respect to the Lloc (Ω) strong convergence. Only recently it has been discovered that dependence of f (x, s, ξ) on the ...

A concept of weak $f$-property for a set-valued mapping is introduced‎, ‎and then under some suitable assumptions‎, ‎which do not involve any information‎ ‎about the solution set‎, ‎the lower semicontinuity of the solution mapping to‎ ‎the parametric‎ ‎set-valued vector equilibrium-like problems are derived by using a density result and scalarization method‎, ‎where the‎ ‎constraint set $K$...