Some remarks on linear functional differential inequalities of hyperbolic type

Abstract

It is proved that for the validity of a theorem on differential inequalities for the hyperbolic equation $$ \frac{\partial^2u(t, x)}{\partial t \partial x} = l(u)(t,x)+q(t,x)$$ with a nonincreasing linear operator $l: {C}([a, b]\times [c,d];\mathbb{R})\rightarrow{L}([a, b]\times [c,d];\mathbb{R})$, it is necessary that the operator indicated be an $(a, c)$-Volterra operator.