Quasimonotonicity as a Tool for Differential and Functional Inequalities

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 R. 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 R , 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 presented in the next paragraph; for simplicity we only consider the semilinear case. Similarly as in [7], a functional dependence with retarded argument will be admitted, which for N = 0 in the case of absence of all derivatives leads to a theorem on functional inequalities; cf. also the joint paper with Baron [1]. Finally the survey article by Herzog [2] on quasimonotonicity has to be mentioned.