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We consider the linear and nonlinear problem for partial functional differential equations ∂ 2u ∂x∂y (x,y) = C(x,y)u(x,y)+P(x,y)u(x,y) a.e. in [0,a]× [0,b] ∂ 2u ∂x∂y (x,y) = f (x,y,u(x,y) ,u(x,y)) a.e. in [0,a]× [0,b] with Darboux condition u(x,y) = ψ(x,y) on [−a0,a]× [−b0,b]\ (0,a]× (0,b] where the Hale operator u(x,y) : [−a0,0]× [−b0,0] → Rn is defined by u(x,y)(s,t) = u(s+ x,t + y) for (s,t) ∈ [−a0,0]× [−b0,0] . We give a few theorems about weak and strong inequalities and the existence theorem for the nonlinear problem. Mathematics subject classification (2010): 35L70, 35R10, 35R45.

The Darboux problem for hyperbolic functional differential equations and inequalities in the sense of Carathéodory