Some remarks on linear functional differential inequalities of hyperbolic type - Šremr, J. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
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 $l: C([a, b]× [c,d];\mathbb{R})→L([a, b]× [c,d];\mathbb{R})$, it is necessary that the operator indicated be an $(a, c)$-Volterra operator.