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We consider the following Darboux problem for the functional differential equation $\partial ²u/\partial x\partial y(x,y)=f(x,y,u_{(x,y)},\partial u/\partial x(x,y),\partial u/\partial y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]$0,a\left]\times \right(0,b],$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 prove a theorem on existence of the Carathéodory solutions of the above problem.

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.

We deal with monotone iterative method for the Darboux problem for the system of hyperbolic partial functional-differential equations $$\left\{\begin{array}{c}\frac{{\partial }^{2}u}{\partial x\partial y}(x,y)=f(x,y,u_{(x,y)},u_{(x,y)},\text{a.e.}\text{in}[0,1]\times [0,b]\hfill \\ u(x,y)=\psi (x,y),\text{on}[-a_0,a]\times [-b_0,b]\setminus (0,a]\times (0,b],\hfill \end{array}\right.$$ where the function $u_{(x,y)}:[-a_0,0]\times [-b_0,0]\to {ℝ}^k$ is defined by $u_{(x,y)}(s,t)=u(s+x,t+y)$ for $(s,t)\in [-a_0,0]\times [-b_0,0]$.

We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in $W_{loc}^{1,\infty }$ topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.

We consider a second order semilinear functional evolution equation with infinite delay in a Banach space. We prove the existence of mild solutions for this equation using the measure of noncompactness technique and the Schauder fixed point theorem.