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On the oscillation of nonlinear differential systems with retarded arguments. (English) Zbl 0599.34092
The author studies the oscillatory properties of the systems of nonlinear differential inequalities with retarded arguments of the form \[ y_ i'(t)-f_ i(t,y_{i+1}(t),y_{i+1}(h_{i+1}(t)))=0,\quad i=1,2,...,n- 1, \]
\[ \{y_ n'(t)+f_ n(t,y_ 1(t),y_ 1(h_ 1(t)))\}sgn y_ 1(h_ 1(t))\leq 0, \] where the following conditions are always assumed: (a) \(h_ i: [a,\infty)\to R\) \((i=1,2,...,n)\) are continuous and \(h_ i(t)\leq t\) for \(t\geq a\), \(\lim_{t\to \infty}h_ i(t)=\infty\), \((i=1,2,...,n)\); (b) \(f_ i: [a,\infty)\times R^ 2\to R\) \((i=1,2,...,n)\) are continuous, \(vf_ i(t,u,v)\geq 0\) \((i=1,2,...,n)\) for \(uv>0\) and not identically zero on any subinterval of [a,\(\infty)\); \(f_ i(t,u,v)\) \((i=1,2,...,n-1)\) are nondecreasing in u and v for each fixed \(t\in [a,\infty)\).

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34A34 Nonlinear ordinary differential equations and systems
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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