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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:
[1] J. Jaroš and T. Kusano: Asymptotic Behavior of Nonoscillatory Solutions of Nonlinear Functional Differential Equations of Neutral Type. Func. Ekvac. 32 (1989), 251-263. · Zbl 0705.34078
[2] J. Jaroš and T. Kusano: Oscillation theory of higher order linear functional differential equations of neutral type. Hiroshima Math. J. 18 (1988), 509-531. · Zbl 0693.34038
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[5] J. Ruan: Types and criteria of nonoscillatory solutions for second order linear neutral differential equations. Chinese Ann. Math. Ser. A 8 (1987), 114-124. · Zbl 0632.34071
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