We study the weak bounded approximation property (weak BAP) of pairs and show that each of the spaces c0 and `1 has a subspace having the approximation property but failing the weak BAP.

Abstract In this paper we study a Dirichlet double phase problem with parametric superlinear right-hand side that has subcritical growth. Under very general assumptions on the data, prove existence of at least two nontrivial bounded weak solutions to such by using variational methods and critical point theory. contrast other works do not need suppose Ambrosetti–Rabinowitz condition.

We prove a pointwise gradient estimate for the bounded weak solution of the Cauchy problem associated to the quasilinear Fisher-KPP type equation ut = '(u)xx + (u) when ' satis…es that '(0)=0; and (u) is vanishing only for levels u = 0 and u = 1. As a …rst consequence we prove that the bounded weak solution becomes instantaneously a continuous function even if the initial datum is merely a disc...

We consider the simplest optimal control problem with one nonregular mixed constraint $G(x,u)\le0,$ i.e., such a that gradient $G_u(x, u)$ can vanish on surface $G = 0.$ Using Dubovitskii--Milyutin theorem approximate separation of convex cones, we prove first order necessary condition for weak minimum in form so-called local principle, which is formulated terms functions bounded variation, int...

This paper is concerned with the fast computation of a relation d on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of d is the relation δ, whose convex closure yields the product relation σ th...