On the Global Stability Properties and Boundedness Results of Solutions of Third-Order Nonlinear Differential Equations

We studied the global stability and boundedness results of third-order nonlinear differential equations of the form ?⃛?+ψ(x, ?̇?, ?̈?)?̈?+ f(x, ?̇?, ?̈?) = P(t, x, ?̇?, ?̈?). Particular cases of this equation have been studied by many authors over years. However, this particular form is a generalization of the earlier ones. A Lyapunov function was used for the proofs of the two main theorems: one with P ≡ 0 and the other with P ̸ = 0. The results in this paper generalize those of other authors who have studied particular cases of the differential equations. Finally, a concrete example is given to check our results.

Global stabilities of some special cases of (1) have been studied by a number of authors.
In 1953, Šimanov [1] investigated the global stability of the zero solution of the equation where  and  are constants.Later, Ezeilo [2] and Ogurtsov [3] discussed the global stability of the zero solution of the equation of the form Then, Goldwyn and Narendra [4] studied on the same subject for the following differential equation: Recently, Qian [5] and Omeike [6] have discussed the global stability, and in a recent paper, Tunc ¸ [7] has investigated the boundedness of solutions of the following differential equations: Plus, Tunc ¸ [8] and Omeike [9,10] have studies on the global stability of solutions of the differential equation of the form Moreover, Tunc ¸and Omeike have studies on the asymptotic behavior of the following differential equations: respectively.Motivation of this study has been based on recent studies of Qian [5], Tunc ¸ [7,8,11], and Omeike [6,9,10].Equation ( 1) is a quite general third-order nonlinear differential equation.

Preliminaries
Before introducing our main results, we state some basic theorems and a Lyapunov function which will be required in future.Consider the autonomous system where  : Ω →   is continuous, with Ω being an open set in   containing the origin.Let (0) = 0 and () ̸ = 0 for  ̸ = 0.
Theorem 2. In addition to the assumptions of Theorem 1, if the origin is the only invariant subset of , then the zero solution of (4) is globally asymptotically stable.
Proof. is the set where V = 0, and assume that (0, 0) is to be the only invariant subset of ; then (0, 0) solution of ( 4) is globally asymptotically stable.
Theorem 3. In Theorem 1, if (ii) is replaced by the condition that V() is negative definite at all points  ∈   , then the zero solution of (4) is globally asymptotically stable.
It is well known that the stability is a very important problem in the theory and applications of differential equations.So far, the most effective method to study the stability of nonlinear differential equations is still Lyapunov's second method.The major advantage of this method is that stability in large can be obtained without any prior knowledge of solutions.Today, this method is widely recognized as an excellent tool not only in the study of differential equations but also in the theory of control systems, dynamical systems, systems with time lag, power system analysis, time-varying nonlinear feedback systems, and so on.Its chief characteristic is the construction of a scalar function, namely, the Lyapunov function.Unfortunately, it is sometimes very difficult to find a proper Lyapunov function for a given system.Therefore, in this work, we construct a suitable Lyapunov function which is an excellent tool in the proof of the main theorems.Here, this function,  = () = (, , ), is defined by Rewrite the function (, , ) as follows: where

Main Result
In the case  ≡ 0, we have the following.