Almost periodic solutions to one quasilinear systems with linear delay.

*(English. Russian original)*Zbl 0855.34086
Sib. Math. J. 35, No. 4, 683-688 (1994); translation from Sib. Mat. Zh. 35, No. 4, 768-773 (1994).

A quasilinear system (with a delay depending linearly on time) of the form (1) \(dx (t)/dt = Ax (t) + Bx (\mu t) + f(t) + \nu F(t,x (t), x (\mu t))\), \(\mu = \text{const}\), \(0 < \mu < 1\), \(t \geq t_0 > 0\), \(\nu > 0\), is considered, where \(A\) and \(B\) are constant \((m \times m)\)-matrices, \(x(t)\) is an \(m\)-dimensional vector-function of the time \(t\), \(f(t)\) is an almost periodic \(m\)-dimensional vector-function, \(F\) is a nonlinear vector-function. A theorem on existence of a unique almost periodic asymptotically stable solution of the equation (1) is proved.

Reviewer: S.G.Zhuravlev (Moskva)

##### MSC:

34K14 | Almost and pseudo-almost periodic solutions to functional-differential equations |

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34K25 | Asymptotic theory of functional-differential equations |

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\textit{B. G. Grebenshchikov} and \textit{V. I. Rozhkov}, Sib. Math. J. 35, No. 4, 683--688 (1994; Zbl 0855.34086); translation from Sib. Mat. Zh. 35, No. 4, 768--773 (1994)

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##### References:

[1] | V. I. Rozhkov and B. G. Grebenshchikov, ”Sufficient conditions for existence of almost periodic solutions to linear stationary systems with linear delay,” in: Abstracts: XXVII Scientific Conference of Fiz.-Math. Department of the Univ. ”Druzhby Narodov,” Moscow, 1991, p. 107. |

[2] | B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967). · Zbl 0155.41601 |

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[5] | B. G. Grebenshchikov, ”Stability with respect to the first approximation of systems with linear time-dependent delay,” Differentsial’nye Uravneniya,26, No. 2, 214–218 (1990). · Zbl 0698.34062 |

[6] | B. G. Grebenshchikov, ”On boundedness of solutions to a nonhomogeneous system with linear time-dependent delay,” in: Stability and Nonlinear Oscillations [in Russian], Ural’sk. Univ., Sverdlovsk, 1986, pp. 7–12. |

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