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DEMONSTRATIO MATHEMATICAVol. XXXVIINo 42004Gerd HerzogONE-SIDED ESTIMATES FOR QUASIMONOTONESYSTEMS OF BOUNDARY VALUE PROBLEMSAbstract. We prove existence and uniqueness theorems for Dirichlet boundary valueproblems of the form u" + f{t,u) — 0, u(0) = uo, u(l) = ui in ordered finite dimensionalBanach spaces, involving one-sided estimates and quasimonotonicity.1. IntroductionLet £ be a finite dimensional real vector space, ordered by a cone K.A cone K is a closed convex subset of E with A K Ç K (A > 0), andK D (—K) — {0}. As usual x < y : <i=>- y — x € K. We will always assumethat if is a solid, that is K° ^ 0, and we write x <C y for y — x E K°. SinceK is the solid, the setK* = {<P e E* : <p{x) >0(x>0)}is a cone, the dual cone, in the space of all linear functionals E*.A function / : E —> E is quasimonotone increasing (qmi), in the senseof Volkmann [18], ifx,yeE,x < y, <peK*, <p(x) = tp(y) => <p(f(z)) < <p(f(y)),and a function / : [0,1] x E —* E is called qmi if x i-+ f(t, x) is qmi for eacht e [0,1].For a function / : [0,1]
Demonstratio Mathematica – de Gruyter
Published: Oct 1, 2004
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