Quasimonotonicity as a Tool for Differential and Functional Inequalities
In the context of differential inequalities, the name “quasimonotonicity” had been introduced by Wolfgang Walter . In this monograph also the basic comparison theorems involving ordinary and parabolic differential inequalities, respectively, are treated, the latter being a generalization by Mlak  of a theorem of Nagumo  to functions having values in ℝn.
For both comparison theorems versions are known, where the functions have values in ordered topological vector spaces; cf.  for ordinary differential inequalities and the joint paper with Simon  for parabolic inequalities. When restricting  to the semilinear case, then functions f(x,t,ξ) are involved, whereas in  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.
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