Quasimonotonicity as a Tool for Differential and Functional Inequalities

  • Peter Volkmann
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 161)


In the context of differential inequalities, the name “quasimonotonicity” had been introduced by Wolfgang Walter [8]. In this monograph also the basic comparison theorems involving ordinary and parabolic differential inequalities, respectively, are treated, the latter being a generalization by Mlak [3] of a theorem of Nagumo [4] to functions having values in ℝn.

For both comparison theorems versions are known, where the functions have values in ordered topological vector spaces; cf. [6] for ordinary differential inequalities and the joint paper with Simon [5] for parabolic inequalities. When restricting [5] to the semilinear case, then functions f(x,t,ξ) are involved, whereas in [6] functions f(t,ξ) occur. Here x is a variable in ℝN, t is a real variable, and ξ is a variable in an ordered topological vector space E; the values of f are in E.

Now it turns out that the comparison theorem from [6] can be considered as a special case from [5], when allowing N=0. This will be...


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Peter Volkmann
    • 1
    • 2
  1. 1.Institut für AnalysisKITKarlsruheGermany
  2. 2.Instytut MatematykiUniwersytet Śla̧skiKatowicePoland

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