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

Abstract

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

(i)zt(z,x)f(t,x,z(t,x),z(α(t,x),β(t,x)),z(t,x))(i) \: z_t(z, x) \leq f (t,x, z (t, x), z(\alpha (t, x), \beta (t, x)), z(t, x))

where x=(x1,,xn),zx(t,x)=(zx1(t,x),,zxn(t,x))x = (x_1, \dots, x_n), z_x (t, x) = (z_{x_1} (t, x), \dots, z_{x_n} (t, x)) and

z(α(t,x),β(t,x))=(z(α1(t,x),β1(t,x)),,z(αm(t,x),βm(t,x))).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):0ta,xix˙ibi=Mit(i=1,,n)}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

E0={(t,x):τt0,xix˙ibi(i=1,,n)}.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,vC(E0E,R)u,v \in C(E_0 \bigcup E, \mathbf R) satisfy on EE the Lipschitz condition with respect to (t,x)(t, x). Suppose that uu and vv satisfy almost everywhere on EE the differential inequalities

ut(t,x)f(t,x,u(t,x),u(α(t,x),β(t,x)),ux(t,x))u_t (t, x) \leq f(t, x, u (t, x), u(\alpha (t, x), \beta (t, x)), u_x (t, x))
vt(t,x)f(t,x,v(t,x),v(α(t,x),β(t,x)),vx(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)v(t,x)u(t, x) \leq v(t, x) on E0E_0. In the paper we prove that under certain assumptions concerning the functions f,α,βf, \alpha, \beta, the inequality u(t,x)v(t,x)u(t, x) \leq v(t, x) is satisfied on EE.

Cite this article

Zdzisław Kamont, Stanisław Zacharek, On partial differential inequalities of the first order with a retarded argument. Z. Anal. Anwend. 2 (1983), no. 2, pp. 135–144

DOI 10.4171/ZAA/55