This paper is concerned with the best proximity pair problem in Hilbert spaces. Given two subsets $A$ and $B$ of a Hilbert space $H$ and the set-valued maps $F:A o 2^ B$ and $G:A_0 o 2^{A_0}$, where $A_0={xin A: |x-y|=d(A,B)~~~mbox{for some}~~~ yin B}$, best proximity pair theorems provide sufficient conditions that ensure the existence of an $x_0in A$ such that $$d(G(x_0),F(x_0))=d(A,B).$$

Passivity and dissipativity, in Jan Willems’s sense, are known to be powerful tools for stability analysis feedback control design. For instance, passive systems negative interconnection with slope-restricted, static, single-valued, smooth nonlinearities, as Lur’e systems, have been thoroughly studied automatic control, yielding the so-called absolute problem (with famous Popov circle criteria)...

If A is compact, then A and B have a coincidence, that is, there exists (z0, Vo) € X x Y with y0 G A(x0) D B(x0). [Recall that a nonempty compact space is acyclic if all its reduced Cech homology groups over Q vanish.] The proof of Theorem 0 relies on a Lefschetz-type fixed point theorem of Gorniewicz and Granas [14] which is itself based on sophisticated homological machinery. One of our conce...

In this paper we deene the Extended Linear Complementarity Problem (ELCP), an extension of the well-known Linear Complementarity Problem (LCP). We study the general solution set of an ELCP and we present an algorithm to nd all its solutions. Finally we show that the ELCP can be used to solve some important problems in the max algebra.

It is well known that variational inequalities are systematically used in the theory of many practical problems related to ”equilibrium”. The equilibrium is an important state considered in Physics, Engineering, Sciences and Economics and even in Biology, [5], [37], [38], [24], [56]. When a variational inequality is defined on a closed convex set, which is in particular a closed convex cone, in...