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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.

##### 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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##### 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 [3] A. Halanau and D. Wexler, The Qualitative Theory of Systems with Impulse [Russian translation], Mir, Moscow (1971). [4] B. G. Grebenshchikov, ”Stability of systems with linear time-dependent delay,” in: Stability and Nonlinear Oscillations [in Russian], Ural’sk. Univ., Sverdlovsk, 1983, pp. 25–34. [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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