Derkachova V. V., Vareh N. V., Atanova M. Y.

*Oles Honchar Dnipropetrovsk National University*

**RESEARCH OF THE SYSTEMS WITH THE EVEN AMOUNT OF DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT**

This work deals with the research of the systems of differential equations with deviations of arguments with the even amount of equations.

Let us consider the following system:

where α*i* – is relation of odd numbers,

The results of research of solutions of the system (1) (τi (t) ≡ t, *i*=1, n-1), when αi = 1, *i = *1,2,3; 0 < α <1 have been presented in the paper [1]. Unlike of that work the delays of arguments are included in every equation in this one. We'll adduce one of the got results.

__Theorem 1.__ Let the conditions be executed: then every solution of the system (1) either oscillates strongly, or each its component tends to zero or to infinity at *t*→.

The condition is very severe, that’s why we improved it with additional conditions .

__Theorem2__. Let the conditions be executed:

1)

3) (2)

4) (3)

5)(4)

6)(5)

Then every solution of the system (1) either oscillates strongly, or each its component tends to zero or to infinity as *t*→.

The formulated result relates to researches on finite interval.

**The list of references:**

1. Varekh N.V., Gorshkova P.G., Kukoyashna T.G., Marchenko A.V. “Research of differential equations with aftereffect”, “International Scientific Conference Mathematicalproblems of Technical Mechanics – 2011”, Dnepropetrovsk-Dneprodzerzhinsk, p.61.