In this paper, wยข first present constructing a Lyapunov function for (1.1) and then we show the asymptotic stability in :the large of the trivial solution x = 0 for case p = O, and the boundedness result of the solutions of (1.1)for ease p#O. These results improve several well-known results_ Key words nonlinear differential equations of the fourth order, Lyapunov function, stability, boundedness, intrinsic method
I. Int~'oduction
We shall be concerned here with fourth order differential equation of the form
- in which the functions q~, f, k, h and p depend only on the arguments displayed and the dots denote differentiation with respect to t. The functions q~, f, k, h and p are continuous for all values their respective arguments. Moreover the derivatives Of(X,ox ~) =f,(x, ~) ' --dxdh ~-ht(x) exist and are continuous. The boundedness and stability properties of solutions for various fourth order differential equations have been considered by many authors, namely, Afuwapetยข t-'l; Asmussen[41 BarbalattSl, Ezeilo[tl], Ezeilo and Tejumola 021, Hara [ul, D41, Harrowtm Lalli and Skrapek [17], Sinha and Holt p-ยฐ] to mention a few. They have obtained some results either for the equation xC'~+/~(~)~'+/2(~, ~.)+l~(~)+/,(x, ~)=p(t, x, ~, #, ~')
or for various special cases of this equation. Some of their papers have been summarized in [19]. Motivation for the study of (1.1) comes from the works of Ezeilo ~1], Harrow[tSl, Lalli and Skrapek [171 and Yu Yuanhong and Cheng Wendeng ~=j on the global asymptotic stability of the origin and the boundedness of solutions of the Cauchy problem. Our aim is to obtain similar results for the equation (1.1). For this, we know that the main tool is to construct a suitable Lyapunov function V(x, !1, z, w), Note that it is difficult to find a Lyapunov function V(x, 9, z, w) for the fourth order equations t6],t~"]. However, numerous methods have been purposed in the literature to derive Lyapunov function to study the boundednesg,and the