JournalszaaVol. 2, No. 2pp. 135–144

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

• ### Zdzisław Kamont

University of Gdansk, Poland
• ### Stanisław Zacharek

Technical University of Gdansk, Poland

## Abstract

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

$(i) \: z_t(z, x) \leq f (t,x, z (t, x), z(\alpha (t, x), \beta (t, x)), z(t, x))$

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

$z(\alpha (t, x), \beta (t, x)) = (z(\alpha_1 (t, x), \beta_1 (t, x)), \dots, z(\alpha_m (t, x), \beta_m (t, x))).$

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

$E = \{(t, x): 0 \leq t \leq a, \quad |x_i - \dot{x}_i | \leq b_i = M_i t \quad (i = 1, \dots, n)\}$

and

$E_0 = \{(t, x): -\tau \leq t \leq 0, \quad |x_i - \dot{x}_i | \leq b_i \quad (i = 1, \dots, n)\}.$

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

$u_t (t, x) \leq f(t, x, u (t, x), u(\alpha (t, x), \beta (t, x)), u_x (t, x))$
$v_t (t, x) \geq f(t, x, v (t, x), v(\alpha (t, x), \beta (t, x)), v_x (t, x))$

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