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A mixed neutral system. (English) Zbl 0553.34042
Consider a mixed type neutral system (1) $x'(t)=f(t,x)+\sum\sp{m}\sb{j=1}F\sb j(t,x)x'(t+P\sb j)$, (2) $x(0)=x\sb 0$, where f is a continuous n-vector-valued functional, each $F\sb j$ is a continuous $n\times n$ matrix-valued function defined on $R\times C(R,R\sp n)$, each $P\sb j$ is a constant real number and $x\sb 0\in R\sp n$. It is also assumed that $\vert \cdot \vert$ is a norm in $R\sp n$, $\Vert \cdot \Vert$ the induced matrix norm and P, $M\sb f,M\sb F,K\sb f$ and $K\sb F$ are positive constants such that $\vert f\vert \le M\sb f$, each $\Vert F\sb j\Vert \le M\sb F$ on $R\times C(R,R\sp n)$, $P=\max\sb{j} \vert P\sb j\vert$ and for all $t\in R$, with x, $\tilde x\in C(R,R\sp n)$, $\vert f(t,x)-f(t,\tilde x)\vert \le K\sb f\max\sb{t- p\le s\le t+p}\vert x(s)-\tilde x(s)\vert$ and $\Vert F(t,x)-F(t,\tilde x)\Vert \le K\sb F\max\sb{t-p\le s\le t+p}\vert x(s)-\tilde x(s)\vert.$ The author proves that if $P,M\sb f,M\sb F,K\sb f$ and $K\sb F$ are sufficiently small and for any constant $a>0$ $$ e\sp{ap}[(1/a)(K\sb f+(mK\sb FM\sb f)/(1-mM\sb F))+mM\sb F]<1 $$ then (1) and (2) have a unique solution such that $\int\sp{t+1}\sb{t}\vert x'(s)\vert ds$ is bounded for all t. An example is given, to illustrate the theory.
Reviewer: O.Akinyele

MSC:
34K05General theory of functional-differential equations
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References:
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