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**An equation alternately of retarded and advanced type.**
*(English)*
Zbl 0628.34074

The present paper is related to the authors’ previous paper [J. Math. Anal. Appl. 99, 265-297 (1984; Zbl 0557.34059)] and the paper of S. M. Shah and the second author [Int. J. Math. Math. Sci. 6, 671-703 (1983; Zbl 0534.34067)]. The main results deal with the equation
\[
(1)\quad x'(t)=ax(t)+a_ 0x(2[(t+1)/2]),\quad x(0)=C_ 0,
\]
where [\(\cdot]\) is the greatest-integer function. The argument deviation \(\tau (t)=t-2[(t+1)/2]\) is a function of period 2 and equals t for \(-1\leq t<1\). It changes its sign in each interval \(2n-1\leq t<2n+1\). It was proved that (1) has a unique solution on [0,\(\infty)\) and a unique backward solution on (-\(\infty,0]\). Furthermore, some necessary and sufficient conditions for the asymptotical stability of the zero solutions are given. Moreover, the set of bounded solutions is characterized. Also the authors investigate the set of \((a,a_ 0)\) such that all nontrivial solutions have no zeros in (-\(\infty,\infty)\). Finally, the same equation with variable coefficients a(t), \(a_ 0(t)\) is examined, the condition for existence of a unique solution on [0,\(\infty)\) is determined, and conditions are found under which all solutions are oscillatory.

Reviewer: M.Kisielewicz

### MSC:

34K20 | Stability theory of functional-differential equations |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34K05 | General theory of functional-differential equations |

### Keywords:

first order differential equations; retarded arguments; asymptotical stability; bounded solutions
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\textit{K. L. Cooke} and \textit{J. Wiener}, Proc. Am. Math. Soc. 99, 726--732 (1987; Zbl 0628.34074)

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DOI

### References:

[1] | A. R. Aftabizadeh and Joseph Wiener, Oscillatory properties of first order linear functional-differential equations, Applicable Anal. 20 (1985), no. 3-4, 165 – 187. · Zbl 0553.34045 |

[2] | S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences , Academic Press, New York, 1982, pp. 179-187. · Zbl 0512.92018 |

[3] | Kenneth L. Cooke and Joseph Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984), no. 1, 265 – 297. · Zbl 0557.34059 |

[4] | S. M. Shah and Joseph Wiener, Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. Math. Sci. 6 (1983), no. 4, 671 – 703. · Zbl 0534.34067 |

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