Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

Abstract

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov's second (or direct method). Our results include and improve some well known results existing in the literature.

On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument

“…Such equations are frequently encountered as mathematical models of most dynamical processes in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, etc. Especially, since 1960s, many good books and papers have been published and are still being published on the delay differential equations (see, for example, Burton [1,2], Burton and Zhang [3], Èl'sgol'ts [4], Èl'sgol'ts and Norkin [5], Gopalsamy [6], Hale [7], Hale and Verduyn Lunel [8], Kolmanovskii and Myshkis [9], Kolmanovskii and Nosov [10], Krasovskii [11], Sadek [15], Sinha [16], Tejumola and Tchegnani [17], Tunç [19,20,[22][23][24][25], Yoshizawa [26], Zhu [27] and the references thereof). It is also worth mentioning that the use of the Lyapunov direct method [12] for equations with delays encountered some principal difficulties.…”

“…in which b 1 , b 2 and b 3 are not necessarily constants (see the book of Reissig et al [14] as a survey and the papers of Tunç [18], Tunç and Ateç [21]). Meanwhile, it should be noted that, in 1969, Palusinski et al [13] applied an energy metric algorithm for the generation of the Lyapunov function for third order ordinary nonlinear differential equation without delay:…”

Uniform Ultimate Boundedness of Solutions of Third-Order Nonlinear Delay Differential Equations

“…The most effective method to study the uniform boundedness and uniform ultimate boundedness of (1.1) is the Lyapunov's direct (or second) method. See [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein. Thus, by using a more general Lyapunov function, our result improves some known results in [12] and [15], and revise a result in [14].…”

Boundedness Results for Solutions of Certain Nonlinear Differential Equations of Second Order

“…In particular, there have been extensive results on the boundedness of solutions of various nonlinear differential equations of second order in the literature. For example, for a comprehensive treatment of the subject on the boundedness of solutions, we refer the readers to books of Ahmad and Rama Mohana Rao [1], Krasovskii [11], Yoshizawa [22] and the papers of Graef [3,4], Graef and Spikes [5,6,7], Huang et al [9], Jin [10], Murakami [12], Saker [13], Sun [14], Tunç [15,16,17], Tunç and Şevli [18], Tunç and Tunç [19], Villari [20], Ye et al [21], Zhao [24], Waltman [25], Wong [26], Wong and Burton [27] and the references contained in these sources. Thus, it is worthwhile to continue to the investigation of the boundedness of solutions of nonlinear differential equations of second order in this case.…”