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On unbounded nonoscillatory solutions of systems of neutral differential equations. (English) Zbl 0758.34055
The author considers the system $(d^ n/dt^ n)[x_ i(t)+(-1)^ ra_ i(t)x_ i(h_ i(t))]=\sum^ N_{j=1}p_{ij}(t)f_{ij}(x_ j(g_{ ij}(t))),\tag{1}$ $$i=1,\dots,N$$, where $$a_ i:[t_ 0,\infty)\to[0,\beta_ i]$$, $$t_ 0\geq 0$$, $$0<\beta_ i<1$$, $$r\in\{0,1\}$$; $$h_ i,p_{ij},g_{ij}:[t_ 0,\infty)\to R$$, $$f_{ij}:R\to R$$, $$1\leq i\leq N$$ are continuous functions, $$h_ i(t)<t$$ for $$t\geq t_ 0$$, $$\lim h_ i(t)=\infty$$, $$\lim g_{ij}(t)=\infty$$ as $$t\to\infty$$, $$f_{ij}(u)u>0$$ for $$u\neq 0$$, $$f_{ij}$$ nondecreasing, $$1\leq i,$$ $$j\leq N$$; $$\lim_{t\to\infty}a_ i(t)(h_ i(t)/t)^ k=\overline a_{ik}a_{ik}\in[0,\beta_ i]$$, $$1\leq i\leq N$$ and every $$k\in\{1,2,\dots,n-1\}$$. In two theorems, sufficient conditions are given for the system (1) to possess nonoscillatory solutions $$X=(x_ 1,\dots,x_ N)$$ with the asymptotic behavior $$\lim_{t\to\infty}(x_ i(t)/t^{k_ i})=c_ i\neq 0$$, $$\text{sgn} c_ i=\text{sgn} c_ 1$$ or $$\lim_{t\to\infty}(x_ i(t)/t^{k_ i})=0$$, $$\lim_{t\to\infty}(x_ i(t)/t^{k_ i-1})=\infty$$ for $$k_ i\in\{1,2,\dots,n-1\}$$, $$1\leq i\leq N$$.

##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34K40 Neutral functional-differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34K25 Asymptotic theory of functional-differential equations
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##### References:
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