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Letf : IR IR N ! IR N be a continuous function and leth : IR ! IR be a continuous and strictly positive function. A sucien t condition such that the equation _ x = f (t;x) admits solutions x : IR ! IR N satisfying the inequality jx (t)j k h (t); t 2 IR; k > 0, where jj is the euclidean norm in IR N ; is given. The proof of this result is based on the use of a special function of Lyapunov type, which is often called guiding function. In the particular case h 1, one obtains known results regarding the existence of bounded solutions.

Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions