# The uniform lower bound of solutions of nonlinear system of differential equations

### Abstract

We study non-linear system of differential equations in finite-dimensional vector space with the method of the first approximation. A system of the first approximation were considered, i.e. a linear system of differential equations with continuous and tending to zero coefficients on infinite interval. Singular exponents of a linear system of differential equations, in this case taking the critical ie, zero values, so are unsuitable for use. The paper defined the generalized special lower rate of a linear system of differential equations with continuous and tending to zero coefficients. We present an equivalent definition of a generalized special low index of a linear system of differential equations with continuous and tending to zero coefficients. Applying a generalized lower specific indicator of a linear system of differential equations with continuous and tending to zero coefficients obtained uniform lower estimate of solutions of differential equations solving the nonlinear system in a certain class of nonlinear differential systems. Powered sufficient condition for the instability of the zero solution of nonlinear differential equations.### References

[1] Persidskii K.P. On the stability of motion in the first approximation // Matematical collection. - 1933. - V. 40, No3. – p. 284-292.(in Russian)

[2] Bohl P. Uber Differentialgleichungen // J. Reine und angew. Math. – 1913. –B. 144. – p. 284-318.

[3] Bylov B.F., Vinograd R.E., Grobman D.M., Nemyskii V.V. The theory of Lyapunov exponents and its application to stability issues. - M., 1966.- 576 p. (in Russian)

[4] Daletskii Y.L., Krein M.G. Stablitity of solutions of differential equations in a Banach space. - M., Nauka, 1970. – 536 p. (in Russian)

[5] Izobov N.A. Linear systems of ordinary differential equations // Matematical analysis. The results of science and technology. - 1974. – Vol. 12. – p. 71-146. (in Russian)

[6] Aldibekov T.M. Generalized central and generalized singular exponents of the system of differential equations // Mathematical journal IM MES RK. 2003.- Vol.3, No.1(7). - p.15-18.

[7] Aldibekov T.M. ,Mirzakulova A.E., Aldazharova M.M. Generalized singular exponent’s linear system of differential equations // KazNU BULLETIN. Mathematics, Mechanics and Computer Science Series. 1(88) 2016. P.47-54.

[2] Bohl P. Uber Differentialgleichungen // J. Reine und angew. Math. – 1913. –B. 144. – p. 284-318.

[3] Bylov B.F., Vinograd R.E., Grobman D.M., Nemyskii V.V. The theory of Lyapunov exponents and its application to stability issues. - M., 1966.- 576 p. (in Russian)

[4] Daletskii Y.L., Krein M.G. Stablitity of solutions of differential equations in a Banach space. - M., Nauka, 1970. – 536 p. (in Russian)

[5] Izobov N.A. Linear systems of ordinary differential equations // Matematical analysis. The results of science and technology. - 1974. – Vol. 12. – p. 71-146. (in Russian)

[6] Aldibekov T.M. Generalized central and generalized singular exponents of the system of differential equations // Mathematical journal IM MES RK. 2003.- Vol.3, No.1(7). - p.15-18.

[7] Aldibekov T.M. ,Mirzakulova A.E., Aldazharova M.M. Generalized singular exponent’s linear system of differential equations // KazNU BULLETIN. Mathematics, Mechanics and Computer Science Series. 1(88) 2016. P.47-54.

Published

2018-07-18

How to Cite

MOLDABEK, Zh. Т.; ALDIBEKOV, T. М..
The uniform lower bound of solutions of nonlinear system of differential equations.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 92, n. 4, p. 46-54, july 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/453>. Date accessed: 23 may 2019.
Section

Mathematics

Keywords
linear differential systems, singular exponents, nonlinear differential systems, bound of the solutions