Infinite Systems of Strong Parabolic Differential–functional Inequalities

We investigate a weakly coupled infinite system of nonlinear strong parabolic differential–functional inequalities of the following form (1) ∂tz (t, x) < f (t, x, z(t, x), ∂xz (t, x), ∂ xxz (t, x), z), i ∈ S, in an arbitrary domain D. The right-hand sides f i of these inequalites are functionals of an unknown function z and Volterra functionals only will be regarded in this paper. We give a fundamental theorem on strong parabolic differential–functional inequalities, generalizing the well-known Nagumo–Westphal lemma to encompass the case of an infinite system. This paper continues and, in a way, concludes Szarski’s research on various generalizations of the theorem on strong differential inequalities.