Second Order Differential Inequalities via Aggregation

Abstract

Abstract. Let E be a Banach space ordered by a solid and normal cone. We introduce a polynorm with respect to a given selection of positive pairwise disjoint vectors Pi,---iPm, and derive monotonicity properties of solutions of second order differential inequalities under one-sided matrix Lipschitz conditions. IntroductionLet £ be a real Banach space, ordered by a cone K, that is K is a closed convex subset of E with \K C K (A > 0) and K D (-K) = {0}, and x < y :y -x € K. We will always assume that K is solid (that is Int K 0) and normal, and we write x y if y -x G Int K. Since K is solid, the set K* = {tp € E* : <p(x) >0(x> 0)} is a cone, the dual cone, in the space of all continuous linear functionals E*.Throughout the paper let m > 2 and let pi,... ,p m G K \ {0} have the following properties:(1) p : = pi + h Pro £ Int K;(2) ini{pi,pj} = 0 (i^j).

Nonlinear modes for the Gross–Pitaevskii equation—a demonstrative computation approach

“…In this situation theorem 2 of [26] (see also [27]) states that u(a) v(a) implies that u(t) v(t) for t ∈ [a; b]. This gives the assertion of comparison lemma.…”

“…(i) f (t, ξ), g(t, ξ ) are defined on [a; b] × [A; B] and locally Lipschitz continuous (see[27])with respect to ξ , ξ ∈ [A; B], (A,B, b may be finite or infinite); (ii) g(t, ξ ) f (t, ξ) for any t ∈ [a; b], ξ ∈ [A; B]; (iii) f (t, ξ)is monotone nondecreasing with respect to ξ , ξ ∈ [A; B]. Let us apply the comparison lemma by taking f (t, ξ) = − ξ + ξ 3 ; g(t, ξ ) = −(ω − V (t))ξ + ξ 3 ,…”