Abstract
LetA be a (nonlinear) operator in an ordered linear spaceX with resolvantJ λ=(I+λA)-1 well-defined onX and non-decreasing for any smallλ>0, andν ∈X. We define sub-potential ofν with respect toA, as anyu ∈X satisfyingu≧J λ(u+λv) for smallλ>0, and show that this coincides with the notion of sub-solution of the equationAu∋ν in some abstract cases where such notion is defined in a natural way. At last, we give some general properties of sub-potentials, in particular an extension of the Kato inequality whenX is a lattice, and, for good set of constraintsU, existence of a largest solution for the control problem:u ∈U andu is a sub-potential ofν with respect toA.
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