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Boundary-value problems for weakly nonlinear delay differential systems. (English) Zbl 1222.34075

Summary: Conditions are derived for the existence of solutions to nonlinear boundary-value problems for systems of $n$ ordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of the argument in the nonlinearity:

$\stackrel{˙}{z}\left(t\right)=Az\left(t-\tau \right)+g\left(t\right)+\epsilon Z\left(z\left({h}_{i}\left(t\right),t,\epsilon \right),\phantom{\rule{1.em}{0ex}}t\in \left[a,b\right],$

assuming that these solutions satisfy the initial and boundary conditions

$z\left(s\right):=\psi \left(s\right)\phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}s\notin \left[a,b\right],\phantom{\rule{1.em}{0ex}}\ell z\left(·\right)=\alpha \in {ℝ}^{m}·$

The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices leads to an explicit and analytical form of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functional $\ell$) does not coincide with the number of unknowns in the differential system with single delay.

##### MSC:
 34K10 Boundary value problems for functional-differential equations 34A45 Theoretical approximation of solutions of ODE