The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are K-(quasi)convex with respect to a convex cone K. In particular, we recover some known characterizations of K-(quasi)convex vector-valued functions, given by means of the polar cone of K.

Several interesting classes of k-uniformly close-to-convex functions and k-uniformly quasi-convex functions are defined here by using the DziokSrivastava operator. We provide necessary and sufficient coefficient conditions, extreme points, integral representations, and distortion bounds for functions belonging to each of these classes of k-uniformly close-to-convex functions and k-uniformly qua...

Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X, ‖ · ‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = R this construction yields a norm such that D(R) becomes a Banach space. Furthermore, we show that for a real Banach space...

A bstract . Graph transformations tha.t ma.y be reduced to the form x(l +l) = x(I)+vu(Ax(I)), with x(I),x(I+1 ) E R , u convex function and A symmet ric mat rix, are studied. In par t icular , a reduction to t he class is exhibited for some graph t ransformations recentl y introduced by Odlyzko and Randall. Fur ther result s on periods, quasi-periods, and pre-periods are presented. A class of m...

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is af...

In this paper, we introduce and study a class of generalized symmetric vector quasi-equilibrium problems in abstract convex spaces. By virtue of the properties of Γ-convex and KC-map, we give some sufficient conditions to guarantee the existence of solutions for the generalized symmetric vector quasi-equilibrium problems in abstract convex spaces. As application, we show an existence theorem of...