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A mixed neutral system. (English) Zbl 0553.34042

Consider a mixed type neutral system (1) ${x}^{\text{'}}\left(t\right)=f\left(t,x\right)+{\sum }_{j=1}^{m}{F}_{j}\left(t,x\right){x}^{\text{'}}\left(t+{P}_{j}\right)$, (2) $x\left(0\right)={x}_{0}$, where f is a continuous n-vector-valued functional, each ${F}_{j}$ is a continuous $n×n$ matrix-valued function defined on $R×C\left(R,{R}^{n}\right)$, each ${P}_{j}$ is a constant real number and ${x}_{0}\in {R}^{n}$. It is also assumed that $|·|$ is a norm in ${R}^{n}$, $\parallel ·\parallel$ the induced matrix norm and P, ${M}_{f},{M}_{F},{K}_{f}$ and ${K}_{F}$ are positive constants such that $|f|\le {M}_{f}$, each $\parallel {F}_{j}\parallel \le {M}_{F}$ on $R×C\left(R,{R}^{n}\right)$, $P={max}_{j}|{P}_{j}|$ and for all $t\in R$, with x, $\stackrel{˜}{x}\in C\left(R,{R}^{n}\right)$, $|f\left(t,x\right)-f\left(t,\stackrel{˜}{x}\right)|\le {K}_{f}{max}_{t-p\le s\le t+p}|x\left(s\right)-\stackrel{˜}{x}\left(s\right)|$ and $\parallel F\left(t,x\right)-F\left(t,\stackrel{˜}{x}\right)\parallel \le {K}_{F}{max}_{t-p\le s\le t+p}|x\left(s\right)-\stackrel{˜}{x}\left(s\right)|·$ The author proves that if $P,{M}_{f},{M}_{F},{K}_{f}$ and ${K}_{F}$ are sufficiently small and for any constant $a>0$

${e}^{ap}\left[\left(1/a\right)\left({K}_{f}+\left(m{K}_{F}{M}_{f}\right)/\left(1-m{M}_{F}\right)\right)+m{M}_{F}\right]<1$

then (1) and (2) have a unique solution such that ${\int }_{t}^{t+1}|{x}^{\text{'}}\left(s\right)|ds$ is bounded for all t. An example is given, to illustrate the theory.

Reviewer: O.Akinyele
##### MSC:
 34K05 General theory of functional-differential equations