Ein Existenzsatz für Gewöhnliche Differentialgleichungen in Geordneten Banachräumen

Lemmert, Roland; Redheffer, Ray; Volkmann, Peter

doi:10.1007/978-3-0348-7192-1_30

Cited by 8 publications

(14 citation statements)

References 3 publications

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“…(Further discussion of flow-invariance can be found in §6 of [22].) Some other results on quasimonotonicity and more references can be found in [11], [15], [24].…”

Section: Discussion Of the Hypothesesmentioning

confidence: 99%

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

Biles

Schechter

2000

Proc. Amer. Math. Soc.

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Abstract. This paper proves the existence of solutions to the initial value problemwhere f : [0, 1] × R M → R M may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set M can be arbitrarily large (finite or infinite); our theorem is new even for card(M ) = 2. The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.

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“…(Further discussion of flow-invariance can be found in §6 of [22].) Some other results on quasimonotonicity and more references can be found in [11], [15], [24].…”

Section: Discussion Of the Hypothesesmentioning

confidence: 99%

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

Biles

Schechter

2000

Proc. Amer. Math. Soc.

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“…Since Theorem 4.1 yields not just a fixed point but a smallest or greatest one, we can combine it with the fixed point theorem of Lemmert [8]. However, K must be strict now.…”

Section: Quasimonotone Increasing Mappings With Upward Jumpsmentioning

confidence: 99%

“…Here we combine our main result with the fixed point theorem of Lemmert [8] for monotone increasing mappings of ordered sets.…”

Section: Introductionmentioning

confidence: 99%

Smallest and greatest fixed points of quasimonotone increasing mappings

Uhl

2003

Mathematische Nachrichten

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“…by the restricted hypothesis (2). Thus the value of ϕ at the left side of (8) is nonnegative, hence 0 ≤ ϕ(u(t 2 ) + d) − ϕ(u(t 3 )), and therefore ϕ(u(t 3 )) ≤ 0.…”

mentioning

confidence: 95%

“…It should be pointed out that in some ordered Banach spaces the quasimonotonicity of f implies the existence of solutions to (1), in fact extremal solutions, without any Lipschitz or compactness condition. Some of these results are based on the fixed point theorem of Lemmert [2]. See the survey [10].…”

Section: Introductionmentioning

confidence: 99%

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

Uhl¹

1998

Proc. Amer. Math. Soc.

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Abstract. A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side f (t, x) was carried over by Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of f is a necessary condition here if f is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in ∞ by taking coordinate functionals).

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Ein Existenzsatz für Gewöhnliche Differentialgleichungen in Geordneten Banachräumen

Cited by 8 publications

References 3 publications

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

Smallest and greatest fixed points of quasimonotone increasing mappings

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

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