We show that if (J,,} is a sequence of uniformly LI-bounded functions on a measure space, and if.f, -fpointwise a.e., then lim,,_(I{lf,, 1 -IIf,, fII) If I,' for all 0 < p < oc. This result is also generalized in Theorem 2 to some functionals other than the L P norm, namely I. /( J,, -(f, f) f ) -1 0 for suitablej: C -C and a suitable sequence (fJ}. A brief discussion is given of the usefulness...

In this paper, the authors prove some existence results of solutions for a new class of generalized bi-quasi-variational inequalities (GBQVI) for quasi-pseudo-monotone type I operators in non-compact settings in locally convex Hausdorff topological vector spaces. In obtaining these results on GBQVI for quasi-pseudo-monotone type I operators in non-compact settings, we shall use the concept of e...

The Path Partition Conjecture for digraphs states that for every digraph D, and every choice of positive integers λ1, λ2 such that λ1 + λ2 equals the order of a longest directed path in D, there exists a partition of D in two subdigraphs D1,D2 such that the order of the longest path in Di is at most λi for i = 1, 2. We present sufficient conditions for a digraph to satisfy the Path Partition Co...

In this paper we study the Plateau problem for affine maximal hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. In particular we formulate the affine Plateau problem as a geometric variational problem for the affine area functional, and we prove the existence and regularity of maximizers. As a special case, we obtain corresponding exist...