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Existence and uniqueness of special solutions of linear differential equations with deviated arguments. (English) Zbl 0611.34061

Consider the linear equation with deviating argument \[ \dot x(t)=A(t)x(t)+\sum^{n}_{1}B_ i(t)x(t+\tau_ i(t))+f(t) \] where X is a Banach space, \(\tau_ i: R\to R\), \(f: R\to X\), \(A,B_ i: R\to L(X)\) are given. Here L(X) is the space of linear bounded operators from X to X. The so-called special solutions - defined on the whole real axis and passing through a given point of \(R\times X\)- are considered. Some representation formulas for the solutions are given. For the case of small coefficients and small deviations existence and uniqueness of special solutions are considered under the assumption that for certain numbers \(a\geq 0\), \(b_ i>0\), \(\eta_ i>0\) the following holds \(\| A\| <a\), \(\| B_ i\| <b_ i\), \(| \tau_ i| <\eta_ i\) together with some additional conditions on a, \(b_ i\), \(\eta_ i\). Among them, the condition \(a+\sum^{n}_{1}b_ ie^{\eta_ i\gamma}<\gamma\) is discussed, in comparison with a condition belonging to Driver, namely \((a+\sum^{n}_{1}b_ i)e(\max \eta_ i)<1.\)
Reviewer: V.Rǎsvan

MSC:

34K05 General theory of functional-differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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