We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator Π g C and the relative projection operator P g C in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X . We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a...

A lower semicontinuity result is proved in the space of special functions of bounded deformation for a fracture energetic model ∫ Ju Ψ([u], νu)dH , [u] · νu ≥ 0 H − a. e. on Ju. A characterization of the energy density Ψ, which ensures lower semicontinuity, is also given.

In this paper, we first present some sufficient conditions for the upper semicontinuity and/or the continuity of the Bregman farthest-point map QgC and the relative farthest-point map S g C for a nonempty D-maximally approximately compact subset C of a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for a Klee set to be singleton in a Banach space ...

Using the homotopy invariance property of the degree and a newly introduced concept of the interior-point-ε-exceptional family for continuous functions, we prove an alternative theorem concerning the existence of a certain interior-point of a continuous complementarity problem. Based on this result, we develop several sufficient conditions to assure some desirable properties (nonemptyness, boun...

and of the corresponding extension (1.3) to BV (Ω), with the aim of understanding which are the minimal assumptions on f that guarantee lower semicontinuity. Here, the starting point is the result, proved in [20], stating that if the integrand f(x, u, ξ) is a nonnegative, continuous function from Ω× IR× IR , convex in ξ, and such that the (classical) derivatives ∇xf,∇ξf,∇x∇ξf exist and are cont...

C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785– 795.] defined three concepts of semicontinuity for self-adjoint elements of A∗∗, the enveloping von Neumann algebra of a C∗-algebra A. We give the basic properties of the analogous concepts for elements of pA∗∗p, where p is a closed projection in A∗∗. In other words, in place of affine functionals on Q, the quasi–state space of A, we c...