Almost limited sets in Banach lattices

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak^?${\mathrm{weak}}^{?}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L  -weakly compact sets coincide. In particular, in terms of almost Dunford–Pettis operators into _c0$c_0$, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E   is almost limited if and only if every continuous linear operator T:E→_c0$T:E\to c_0$ is an almost Dunford–Pettis operator.     

Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities

For Ω$\Omega$, an open bounded subset of R^N${\mathbb{R}}^N$ with smooth boundary and 1<p<∞$1<p<\infty$, we establish W^1,p(Ω)$W^{1,p}\left(\Omega \right)$a priori bounds and prove the compactness of solution sets to differential inequalities of the form

|divA(x,∇u)|≤F(x,u,∇u),$\left|\text{div}A(x,\nabla u)\right|\le F(x,u,\nabla u)\text{,}$

which are bounded in L^∞(Ω)$L^{\infty }\left(\Omega \right)$. The main point in this work is that the nonlinear term F$F$ may depend on ∇u$\nabla u$ and may grow as fast as a power of order p$p$ in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein–Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.