A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of R . Under the assumption that the obstacle K is a closed convex and bounded subset of R with smooth boundary or it is a closed n-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In part...

Given x0, a point of a convex subset C of an Euclidean space, the two following statements are proven to be equivalent: (i) any convex function f : C → R is upper semi-continuous at x0, and (ii) C is polyhedral at x0. In the particular setting of closed convex mappings and Fσ domains, we prove that any closed convex function f : C → R is continuous at x0 if and only if C is polyhedral at x0. Th...

In this paper we introduce a notion of digital implicit surface in arbitrary dimensions. The digital implicit surface is the result of a morphology inspired digitization of an implicit surface {x ∈ R : f(x) = 0} which is the boundary of a given closed subset of R, {x ∈ R : f(x) ≤ 0}. Under some constraints, the digital implicit surface has some interesting properties, such as k-tunnel freeness,...

The Fermat-Weber location problem is to find a point in R that minimizes the sum of the weighted Euclidean distances from m given points in R. In this paper we consider the Fermat-Weber problem of one new facilitiy with respect to n unknown customers in order to minimizing the sum of transportation costs between this facility and the customers. We assumed that each customer is located in a none...

Throughout this paper, we denote byN and R the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space X. A mapping T : D → D is said to be nonexpansive, if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ D. Let N D and CB D denote the family of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on CB D is...

Sequences are denoted by x — although C functions are involved in the sequel, their derivative are not explicitly used so this notation for sequences will not interfere with the usual one for derivatives. A function f : R −→ P(R) is a subset of R. Notations and theorems related to manifolds are taken from Hirsch[1]. Points belonging to manifolds will be denoted with tildes. The closed n-dimensi...