Stability and Boundedness Behaviour of Solutions of a Certain Second Order Non-Autonomous Differential Equations (original) (raw)

Ordinary differential equations On the stability and boundedness of solutions to certain second order differential equation

In this paper, we investigate by means of second method of Lyapunov, sufficient conditions that guarantee uniform-asymptotic stability of the trivial solution and ultimate boundedness of all solutions to a certain second order differential equation. We construct a complete Lyapunov function in order to discuss the qualitative properties mentioned earlier. The boundedness result in this paper is new and also complement some boundedness results in literature obtained by using an incomplete Lyapunov function together with a signum function. Finally, we demonstrate the correctness of our results with two numerical examples and graphical representation of the trajectories of solutions to the examples using Maple software.

On the boundedness of solution of the second order ordinary differential equation with damping term and involution

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS, 2021

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

On the boundedness of some nonlinear differential equation of second order

International Journal of Mathematics And its Applications, 2015

In this paper we study the boundedness of the solutions of some nonlinear differential equation using as a key tool the Second Lyapunov method, i.e. find sufficient conditions under which the solutions of this equation are bounded. Various particular cases and methodological remarks are included at the end of paper.

ASYMPTOTIC STABILITY AND BOUNDEDNESS CRITERIA FOR A CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, we examined some criteria for the stability and boundedness of solutions to certain second order nonlinear differential equation x ′′ + b(t) f (x, x ′) + c(t)g(x)h(x ′) = p(t, x, x ′), where b, c, f , g, h and p are real valued functions which depend on the argument displayed explicitly. By applying a suitable Lyapunov function to study the qualitative properties mentioned earlier, we are able to extablish the asymptotic stability and boundedness of solutions. Examples on the stability and boundednress of solutions are hereby included to corroborate our results.