In the pointfree topological context of locales and frames, real functions on a locale L are represented as localic morphisms S(L)→ L(R) (i.e. frame homomorphisms L(R) → S(L)) where S(L) stands for the frame of all sublocales of L and L(R) denotes the frame of reals. This is reminiscent of dealing with (not necessarily continuous) real functions X → R as with continuous functions D(X)→ R where ...

In this paper, we introduce and investigate a new regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. The normalization argument analysis enables us to study existence as well stability of solutions these problems. We prove Frank–Wolfe type theorem regular an Eaves non-regular pseudoconvex Moreover, under condition, show results on su...

A strategic-form game is a potential game if the incentive of all players to change their strategy can be expressed in one global function, called the game’s potential. Potential games have many applications in Economics and other disciplines (cf. Rosenthal 1973, Monderer and Shapley 1996, Ostrovsky and Schwarz 2005, Armstrong and Vickers 2001, Myatt and Wallace 2009, inter alia). Potential gam...

This paper is motivated by a problem suggested in Müller [11] that concerns the weak lower semicontinuity of a smooth integral functional I(u) on a Sobolev space along all its weakly convergent minimizing sequences. Here we study a restricted weak lower semicontinuity of I(u) along all weakly convergent Palais-Smale sequences (that is, sequences {uk} satisfying I′(uk)→ 0). In view of Ekeland’s ...

We study a restricted weak lower semicontinuity property, which we call the (PS)-weak lower semicontinuity, for a smooth integral functional on the Sobolev space along all weakly convergent Palais-Smale sequences of the functional. By the Ekeland variational principle, the (PS)-weak lower semicontinuity is sufficient for the existence of minimizers under the usual coercivity assumption. In gene...

This paper sets out a theoretical framework for approximating the attractor A of a semigroup S(t) deened on a Banach space X by a q{step semi{discretization in time with constant step{length k. Using the one{step theory of Hale, Lin and Raugel, suucient conditions are established for the existence of a family of attractors fA k g X q , for the discrete semigroups S n k deened by the numerical m...