Infinite Systems of First-Order Partial-Differential Functional Inequalities

Abstract

This paper deals with an infinite system of differential-functional inequalities of the form

$$ u_x^i(x,y) \leqslant {f^i}(x,y,u(x,y),u,u_y^i(x,y))\quad (i = 1,2,...)$$

in

$$ P = \left\{ {\left( {x,y} \right);{x_0} \leqslant x \leqslant {x_0} + b,\;\left| {{y_k}} \right| \leqslant {b_k} - c(x - {x_0})} \right\}$$

. Here

$$\begin{array}{*{20}{c}} {y = ({{y}_{1}},...,{{y}_{n}}),\quad u = ({{u}^{1}},{{u}^{2}}...):} \\ {P \ni (x,y) \to u(x,y) = ({{u}^{1}}(x,y),{{u}^{2}}(x,y),...) \in {{\ell }^{\infty }}} \\ \end{array}$$

is the unknown function,

$$u_y^i(x,y) = gra{d_y}{u^i}(x,y)$$

, and fⁱ is a functional of the map u.