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Publishing house "Academy of Natural History"

**Editorial Board:**

Asgarov S. (Azerbaijan), Alakbarov M. (Azerbaijan), Aliev Z. (Azerbaijan), Babayev N. (Uzbekistan), Chiladze G. (Georgia), Datskovsky I. (Israel), Garbuz I. (Moldova), Gleizer S. (Germany), Ershina A. (Kazakhstan), Kobzev D. (Switzerland), Kohl O. (Germany), Ktshanyan M. (Armenia), Lande D. (Ukraine), Ledvanov M. (Russia), Makats V. (Ukraine), Miletic L. (Serbia), Moskovkin V. (Ukraine), Murzagaliyeva A. (Kazakhstan), Novikov A. (Ukraine), Rahimov R. (Uzbekistan), Romanchuk A. (Ukraine), Shamshiev B. (Kyrgyzstan), Usheva M. (Bulgaria), Vasileva M. (Bulgar).

**Materials of the conference "EDUCATION AND SCIENCE WITHOUT BORDERS"**

**1. ****Introduction. Basic definitions and
limiting equations.**

Suppose _{}_{ } is a real axis, _{} is a real linear space of n-vectors x with a norm |x|, _{} is a real number, _{} is the Banach space of continuous functions _{} with a norm _{}, _{} is a space _{}. For a continuous function _{} and every _{}, the function _{} is defined by the equality _{} A right-hand derivative is denoted by _{}.

The functional differential equation with a finite delay

_{} (1)

is considered,
where _{} is a continuous function which satisfies the assumptions 1-3 [1, 2].

**2. Basic results. Stability theorems.**

We will investigate the problem of the stability on the base of Lyapunov constant-sign functionals. We shall use the following definitions.

**Definition 1. **The solution
_{} of Eq.(1) is stable with respect to set _{}, if, for any _{} one can get _{}_{ } , so that for _{}_{ } it is true that _{}_{ }for each solution _{} of Eq.(1) for any _{}.

**Definition 2. **The solution
_{} of Eq.(1) is uniformly asymptotically stable with respect to set _{}_{, }if it is stable with respect to _{}_{ }and a _{} exists, so that for any _{} one can get _{} so that for every _{}_{ }it is true that _{} for any _{}_{.}

**Definition 3. **The solution** **_{} is a point of uniform attraction for the whole family of limiting equations
_{} with respect to set _{}, if a _{} exists, so that for any _{} there is _{} so that for any solution _{} _{} of any equation _{} for any _{} the inequality _{} holds._{ }

Suppose _{} is a certain continuous functional, _{} is a certain solution of Eq.(1). Along this solution the functional
V is a continuous time-dependent function _{}. For this function it is possible to define an upper right-hand
derivative _{}.

Let us denote
as _{} continuous strictly monotonically increasing functions _{}.

**Definition
4. **Let
us define a set for the functional _{}:

_{}

_{}

The definitions which have been introduced enable us to derive the sufficient conditions of stability and asymptotic stability when a non-negative functional with a non-positive derivative exists.

**Theorem 1.** Suppose that:

**1)
**a continuous functional _{} exists,
so that _{};

**2)
**the solution _{} is a point of uniform attraction for
solutions _{} with respect to the set _{}.

**Then
the solution **_{} is stable by Lyapunov.

**Theorem 2. **We will assume that:

**1)
**the continuous functional exists _{}such that:

_{}

**2)
**the solution * x=*0
is asymptotically stable uniformly with respect to the set _{}

**Then
the solution x=0 of equation (1)
is uniformly stable by Lyapunov.**

**3. Conclusion**. There is the
development of the method of Lyapunov constant -sign functionals with using of
the limit equations in the work. The obtained theorems 1,2 develop and expand some results from [2].

2. PAVLIKOV, S.V., Lyapunov constant- sign functional in the problem of stability of a functional-differential equation. PMM., 2007, 3, 377 – 388.

Pavlikov S. V., Savin I. A About the investigation of the stability of functional differential equations of retarded type. International Journal Of Applied And Fundamental Research. – 2014. – № 2 –

URL: www.science-sd.com/457-24571 (20.08.2019).