### Infinite systems of hyperbolic differential-functional inequalities.

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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...

In the paper we deal with the Darboux problem for hyperbolic functional differential equations. We give the sufficient conditions for the existence of the sequence $\left\{{z}^{\left(m\right)}\right\}$ such that if $\tilde{z}$ is a classical solution of the original problem then $\left\{{z}^{\left(m\right)}\right\}$ is uniformly convergent to z$\tilde{z}$. The convergence that we get is of the Newton type.

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