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 . 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 and the other with . 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.
We will consider here the equations of the form Now, (1) has an equivalent system where , , and . We also assume that the real functions , , and depend only on the arguments displayed explicitly. The dots denote differentiation with respect to .
Global stabilities of some special cases of (1) have been studied by a number of authors.
In 1953, Šimanov  investigated the global stability of the zero solution of the equation where and are constants.
Later, Ezeilo  and Ogurtsov  discussed the global stability of the zero solution of the equation of the form Then, Goldwyn and Narendra  studied on the same subject for the following differential equation: Recently, Qian  and Omeike  have discussed the global stability, and in a recent paper, Tunç  has investigated the boundedness of solutions of the following differential equations: Plus, Tunç  and Omeike [9, 10] have studies on the global stability of solutions of the differential equation of the form Moreover, Tunç 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 , Tunç [7, 8, 11], and Omeike [6, 9, 10]. Equation (1) is a quite general third-order nonlinear differential equation. Equations (3a), (3b), (3c), (3d), (3e), (3f), and (3g) are some special cases of (1), and our study is reducible to the studies in [1–11], but the inversions are not possible. Thus, the studies which have been done in [1–11] are some special cases of our study. Hence, our results extend and include those results obtained in [1–11].
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 is continuous, with being an open set in containing the origin. Let and for .
Proof. See .
Proof. is the set where , and assume that is to be the only invariant subset of ; then solution of (4) is globally asymptotically stable.
Proof. See .
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
3. Main Result
In the case , we have the following.
Theorem 4. Let , , , and be positive constants such that is sufficiently small, and , and assume that the following conditions are satisfied:(i),(ii),(iii) for all , , and ,(iv) for all , , and ,(v), for all , and ,(vi).Then, the zero solution of (1) is globally asymptotically stable.
Proof. From conditions (i)–(iv) of Theorem 4, we obtain
where . It follows that there exists a constant small enough that
Hence, is a positive definite function (see Global Asymptotic Stability on page 223, Theorem 5.2.12 that of Rao ).
Now, we show that the derivative of with respect to along the solution path of system (2) is negative semidefinite.
Let where First, from condition (v) of Theorem 4, we obtain Next, from conditions (iii), (v), and (vi) of Theorem 4 we obtain that .
Hence In addition, we can conclude that as (see Global Asymptotic Stability on page 223, Theorem 5.2.12 of Rao ).
The whole discussions (conditions of Theorems 1, 2, 3, and 4) show that the zero solution of system (2) is globally asymptotically stable (also see Theorem 1.5 that of Reissing et al. ). Then, the rest of the proof may now follow as in . Thus, the proof of Theorem 4 is completed.
In the case , we have the following.
Theorem 5. Suppose that the following conditions are satisfied:(i) all the conditions of Theorem 4 hold;(ii), where , is a space of integrable Lebesgue functions.Then, there is a finite positive constant such that every solution of system (2) satisfies
Proof. Consider the function defined as above. Since , then the total derivative of can be revised as Let . Then, we have Using the inequality we obtain From (9), we have where Integrating (19) from to , we obtain Setting then, we have Using Gronwall-Bellman inequality (see Rao ) yields The proof of Theorem 5 is complete.
Case 1 (). With , , , from condition (vi) of Theorem 4, we have
Hence, condition (vi) of Theorem 4 is satisfied.
Then, it is easy to check that all the other conditions [(i)–(vi)] of Theorem 4 are satisfied. Hence, the trivial solution of (25) is globally asymptotically stable.
A. I. Ogurtsov, “On the stability of the solutions of some non-linear differential equations of third and fourth order,” Izvestiya Vysshikh Uchebnykh Zavedenii, vol. 10, pp. 2000–2009, 1959 (Russian).View at: Google Scholar
M. Goldwyn and S. Narendra, Stability of Certain Nonlinear Differential Equations Using the Second Method of Lyapunov, Craft Laboratory, Harvard University, Cambridge, Mass, USA, 1963.
M. R. M. Rao, Ordinary Differential Equations, Affiliated East-West Press, New Delhi, India, 1980.View at: MathSciNet
R. Reissing, G. Sansone, and R. Conti, Nonlinear Differential Equations of Higher Order, Noordhoff International Publishing, Leyden, The Netherlands, 1974.