Abstract
In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.
Highlights
Differential equations of second order with and without delay(s) can find a wide range of applications in atomic energy, biology, chemistry, control theory, economy, engineering technique fields, information theory, medicine, physics, population dynamics, and so forth
We are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov
We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered
Summary
Differential equations of second order with and without delay(s) can find a wide range of applications in atomic energy, biology, chemistry, control theory, economy, engineering technique fields, information theory, medicine, physics, population dynamics, and so forth (see Burton [1], El’sgol’ts [2], Hale [3], Krasovskii [4], Smith [5], and Yoshizawa [6]). We would naturally be inclined to compute the solutions of differential equations of second order with and without delay(s) explicitly or numerically. It should be noted that finding analytical or explicit solutions of differential equations of second order with delay(s) is more difficult, even if, to the best of our knowledge, there is no general method in the literature to find the explicit solutions of those equations. The problem is to find convenient techniques that will be useful in obtaining some qualitative information such as stability, instability, convergence, global existence, integrability, boundedness of solutions, existence of periodic solutions, and so forth about the elusive solutions of ordinary or delay differential equations
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