We consider the Darboux problem for a functional differential equation: $\left(\partial ²u\right)/\left(\partial x\partial y\right)(x,y)=f(x,y,u_{(x,y)},u(x,y),\partial u/\partial x(x,y),\partial u/\partial y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function $u_{(x,y)}:[-a₀,0]\times [-b₀,0]\to {ℝ}^k$ 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 for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

The paper deals with a weakly coupled system of functional-differential equations ${\partial }_t{u}_i(t,x)=f_i(t,x,u(t,x),u,{\partial }_x{u}_i(t,x),{\partial }_{xx}{u}_i(t,x))$, i ∈ S, where (t,x) = (t,x₁,...,xₙ) ∈ (0,a) × G, $u={u_i}_{i\in S}$ and S is an arbitrary set of indices. Initial boundary conditions are considered and the following questions are discussed: estimates of solutions, criteria of uniqueness, continuous dependence of solutions on given functions. The right hand sides of the equations satisfy nonlinear estimates of the Perron type with respect to the unknown functions. The results are...