Stability and Boundedness of Solutions of Certain Non-Autonomous Third Order Nonlinear Differential Equations

Abstract

In this paper, by defining an appropriate Lyapunov functional, we obtain sufficient conditions for which all solutions of certain real non-autonomous third order nonlinear differential equations are asymptotically stable and bounded. The results obtained improve and extend some known results in the literature.

Existence of Periodic Properties of Solutions of Certain Autonomous Third Order Nonlinear Differential Equations

“…Following the argument used in [5] it can be further verified thaṫ (7) where θ = r(t, x 1 , y 1 , z 1 + q) − r(t, x 2 , y 2 , z 2 + q) and δ 10 , δ 11 are finite constants.…”

“…x +φ(x,ẋ)ẍ + g(ẋ) + h(x) = p (t, x,ẋ,ẍ), (1) in which φ, g, h and p depend on the arguments displayed explicitly and dots denote differentiation with respect to t. Moreover, the existence and the uniqueness of solutions of (1) will be assumed. Equations of the form (1) do arise in some aspect of applied sciences such as after effect, nonlinear oscillations, biological systems and equations with deviating arguments (see [1], [2] and [3]) and an effective method for studying the qualitative properties of solutions of such nonlinear equations is still the Lyapunov's direct method (see [4], [5], [6], [7], [8], [9], [10], [11], [12]). Many of these results on stability, boundedness, convergence of solutions exist for more general or special cases of (1) and are summarized in [13].…”