Linear hyperbolic functional-differential equations with essentially bounded right-hand side

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem u 1,1 (t, x) = l 0 (u) (t, x) + l 1 (u 1,0) (t, x) + l 2 (u 0,1)(t, x) + q (t, x), u (t, c) = (t) for t [ a, b ], u (a, x) = (x) for x [ c, d ] given on the rectangle [ a, b ] [ c, d ] are established, where the linear operators l 0, l 1, l 2 map suitable function spaces into the space of essentially bounded functions. General results are applied to the hyperbolic equations with essentially bounded coefficients and argument deviations.