On partial differential inequalities of the first order with a retarded argument

Abstract

This paper deals with first order partial differential inequalities of the form

where $x=(x_1,\dots ,x_n),z_x(t,x)=(z_{x_1}(t,x),\dots ,z_{x_n}(t,x))$ and

We assume that (i) is of the Volterra type. Let

and

Assume that $u,v\in C(E_0\bigcup E,\mathbf{R})$ satisfy on $E$ the Lipschitz condition with respect to $(t,x)$. Suppose that $u$ and $v$ satisfy almost everywhere on $E$ the differential inequalities

and the initial inequality $u(t,x)\le v(t,x)$ on $E_0$. In the paper we prove that under certain assumptions concerning the functions $f,\alpha ,\beta$, the inequality $u(t,x)\le v(t,x)$ is satisfied on $E$.