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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 2, Issue 2, 1983, pp. 135–144
DOI: 10.4171/ZAA/55

Published online: 1983-04-30

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

Zdzisław Kamont and Stanisław Zacharek

(1) University of Gdansk, Poland
(2) Technical University of Gdansk, Poland

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 $E$.