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On boundedness and stability results for the solution of certain fourth-order differential equations via the intrinsic method. (English) Zbl 0880.34051

Summary: We first construct a Lyapunov function for

x (4) +ϕ(x ¨)x +f(x,x ˙)x ¨+k(x)x ˙+h(x)=p(t,x,x ˙,x ¨,x )(1)

in which the functions ϕ, f, k, h and p depend only on the arguments displayed and the dots denote differentiation with respect to t. The functions ϕ, f, k, h and p are continuous for all values of their respective arguments. Moreover, the derivatives f(x,x ˙) x=f x (x,x ˙), dh dx=h ' (x) exist and are continuous.

We show the asymptotic stability in the large of the trivial solution x=0 for the case p0, and give a boundedness result for the solutions of (1) for the case p0. These results improve several well-known results.

MSC:
34D20Stability of ODE
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