Some further stability and boundedness results of some differential equations of the fourth order

Abstract

In this paper sufficient conditions (Theorem 1 and Corollary 1) for the asymptotic stability (in the large) of the trivial solution x=0 of the differential equations $$D_1 (x) = x^{(4)} + f_1 (\ddot x)\dddot x + f_2 (\dot x,\ddot x) + g(\dot x) + h(x,\dot x) = 0$$ , and $$D_2 (x) = x^{(4)} + F_1 (\ddot x)\dddot x + F_2 (\dot x,\ddot x)\ddot x + G(\dot x)\dot x + H(x,\dot x)x = 0$$ are given. A result (Theorem 2) on the boundedness of the solutions of the differential equations D1(x)=p1(t) and D2(x)=p2(t) is also established. Further, the results which we obtain reduce to results which are more general than those obtained by Ezeilo [1] for the differential equation $$x^{(4)} + f_1 (\ddot x)\dddot x + a_2 \ddot x + g(\dot x) + a_4 x = p(t)$$ .