The aim of this paper is to extend the theory of lower and upper solutions to the periodic problem associated with planar systems of differential equations. We generalize previously given definitions and we are able to treat both the well-ordered case and the non-well-ordered case. The proofs involve topological degree arguments, together with a detailed analysis of the solutions in the phase plane.

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

In a measure space (X , A, μ), we consider two measurable functions f , g : E → R, for some E ∈ A. We prove that the property of having equal p-norms when p varies in some infinite set P ⊆ [1, +∞) is equivalent to the following condition: μ({x ∈ E : | f (x)| > α}) = μ({x ∈ E : |g(x)| > α}) for all α ≥ 0.

No abstract