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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:

z ˙(t)=Az(t-τ)+g(t)+εZ(z(h i (t),t,ε),t[a,b],

assuming that these solutions satisfy the initial and boundary conditions

z(s):=ψ(s)ifs[a,b],z(·)=α 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 ) does not coincide with the number of unknowns in the differential system with single delay.

34K10Boundary value problems for functional-differential equations
34A45Theoretical approximation of solutions of ODE