# Stability, boundedness and periodic solutions to certain second order delay differential equations.

## DOI:

https://doi.org/10.4067/S0716-09172017000200257## Keywords:

Second order, Nonlinear differential equation, Uniform stability, Uniform ultimate boundedness, Existence of a unique periodic solutions## Abstract

Stability, boundedness and existence of a unique periodic solution to certain second order nonlinear delay differential equations is discussed. By employing Lyapunov's direct (or second) method, a complete Lyapunov functional is constructed and used to establish sufficient conditions, on the nonlinear terms, that guarantee uniform asymptotic stability, uniform ultimate boundedness and existence of a unique periodic solution. Obtained results complement many outstanding recent results in the literature. Finally, examples are given to show the effectiveness of our method and correctness of our results.## References

ADEMOLA, A. T. (2015) Boundedness and stability of solutions to certain second order differential equations. EN: Differential Equations and Control Processes N 3. [s.l.: s.n.].

ADEMOLA, A. T. (2016) Stability and boundedness of solutions to a certain second order non autonomous stochastic differential equation. EN: International Journal of Analysis. [s.l.: s.n.]. Article ID 2843709, 10 pages, doi.org/10.1155/2016/2843709.

ADEMOLA, A. T. (2016) Periodicity, stability, and boundedness of solutions to certain second order delay differential equations. EN: International Journal of Differential Equations. [s.l.: s.n.]. Article ID 2843709, 10 pages, doi.org/10.1155/2016/2843709.

ALABA, J. G. (2015) On stability and boundedness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equation. EN: Kragujevac Journal of Mathematics Volume 39 (2). [s.l.: s.n.], 255 - 266.

BURTON, T. A. (1985) Stability and periodic solutions of ordinary and functional differential equations. EN: Mathematics in Science and Engineering, 178. Orlando, FL: Academic Press. Inc.

BURTON, T. A. (1983) Volterra integral and differential equations. New York: Academic Press.

CAHLON, B. (2004) Stability criteria for certain second order delay differential equations with mixed coefficients, J. Comput. Appl. Math. 170. [s.l.: s.n.], 79 - 102.

DRIVER, R. D. (1977) Ordinary and delay differential equations. New York: Springer Verlag.

DOMOSHNITSKY, A. (2014) Non oscillation, maximum principles, and exponential stability of second order delay differential equations without damping term. EN: Domoshnitsky Journal of Inequalities and Applications 361. [s.l.: s.n.].

GRIGORYAN, G. A. (2013) Boundedness and stability criteria for linear ordinary differential equations of the second order. EN: Russian Mathematics (Iz.VUZ), 57(12). [s.l.: s.n.], 8 – 15.

HALE, J. K. (1977) Theory of functional differential equations. New York: Springer-Verlag.

JIN, Z. (2001) On the global asymptotic behaviour of solutions to a non autonomous generalized Lienard system. EN: J. Math. Res. Expo. 21 No.3. [s.l.: s.n.], 410 – 414.

KOLMANOVSKII, V. (1992) Applied theory of functional differential equations. [s.l.]: Springer.

KROOPNICK A. J. (2010) Bounded solutions to x00 + q(t)b(x) = f(t). EN: Int. J. Math. Edu. Sci. Tec. 41, 6. [s.l.: s.n.], 829 - 836.

KUAG, Y. (1985) Delay differential equations with applications in populations dynamics. EN: Mathematics in Science and Engineering, 191. Orlando, FL: Academic Press. Inc.

LAKSHMIKANTHAM, V., (1994) Theory of differential equations with unbounded delay. [s.l.]: Springer.

OGUNDARE, B. S. (2016) On the qualitative behaviour of solutions to certain second order nonlinear differential equation with delay. EN: Ann Univ Ferrara. [s.l.: s.n.], 1 -21.

OGUNDARE, B. S. (2014) Boundedness and stability properties of solutions of generalized Lienard equation. EN: Kochi J. Math., 9. [s.l.: s.n.], 97 - 108.

OGUNDARE, B. S. (2007) Boundedness, periodicity and stability of solution x + a(t)g(x ) + b(t)h(x) = p(t; x, x ). EN: Math. Sci. Res. J., 11(5). [s.l.: s.n.], 432 – 443.

TUNÇ, C. (2014) A note on the bounded solutions to x00+c(t, x, x0)+q(t)b(x) =f(t). EN: Appl. Math. Inf. Sci., 8(1). [s.l.: s.n.], 393 - 399.

TUNÇ, C. (2010) Boundedness analysis for certain two-dimensional differential systems via a Lyapunov approach. EN: Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 53(101). [s.l.: s.n.], 61 – 68.

TUNÇ, C. (2013) New Results on the existence of periodic solutions for Rayleigh equation with state-dependent delay. EN: J. Math. Fund. Sci, 45(2). [s.l.: s.n.], 154 - 162.

TUNÇ, C. (2013) Stability and boundedness in multi delay vector Liénard equation. EN: Filomat 27(3). [s.l.: s.n.], 435 – 445. DOI 10.2298/FIL1303435T.

TUNÇ, C. (2011) Stability and boundedness of solutions of non-autonomous differential equations of second order. EN: J Comput. Anal. Appl. 13(6). [s.l.: s.n.], 1067 - 1074.

TUNÇ, C. (2011) Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations. EN: Appl. Comput. Math. 10(3). [s.l.: s.n.], 449 – 462.

WANG, F. (2015) Existence, uniqueness and stability of periodic solutions of a duffing equation under periodic and anti-periodic eigenvalues conditions. EN: Taiwanese Journal of Mathematics 19(5). [s.l.: s.n.], 1457 – 1468. DOI:10.11650/tjm.19.2015.3992

XU, AN SHI. (1988) Boundedness and stability of solutions to second-order delay and non delay differential equations. EN: Chinese Ann. Math. Ser. A 9, No. 5. [s.l.: s.n.], 615—622.

YENICERIOGLU, A. F. (2007) The behavior of solutions of second order delay differential equations. EN: J. Math. Anal. Appl. 332. [s.l.: s.n.], 1278 - 1290.

YENICERIOGLU, A. F. (2008) Stability properties of second order delay integrodifferential equations. EN: Computers and Mathematics with Applications 56. [s.l.: s.n.], 3109-3117.

YOSHIZAWA, T. (1959) Liapunov’s function and boundedness of solutions. EN: Funkcial. Ekvac. 2. [s.l.: s.n.], 95-142.

YOSHIZAWA, T. (1975) Stability theory and existence of periodic solutions and almost periodic solutions. New York: Spriger-Verlag.

YOSHIZAWA, T. (1966) Stability theory by Liapunov’s second method. EN: The Mathematical Society of Japan. [s.l.: s.n.].

ZHU, Y. F. (1992) On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. EN: Annals of Diff. Equations 8(2). [s.l.: s.n.], 249-259.

## Published

## How to Cite

*Proyecciones (Antofagasta, On line)*, vol. 36, no. 2, pp. 257-282, Jun. 2017.