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Zbl 0713.34055
Battelli, Flaviano; Lazzari, Claudio
Exponential dichotomies, heteroclinic orbits, and Melnikov functions. (English)
[J] J. Differ. Equations 86, No.2, 342-366 (1990). ISSN 0022-0396

The authors consider an n-dimensional perturbed system (*) $dz/dt=g(z)+h(t,z,\epsilon),$ where the perturbation term h(t,z,$\epsilon$) is bounded, $\epsilon$ being a multidimensional parameter, and they give, using the method of Lyapunov-Schmidt, a sufficient condition for the existence of a bounded solution of (*) as the solvability condition of a system of bifurcation equations whose coefficients depend only on the solutions of the unperturbed system of (*), $dz/dt=g(z)$, and the perturbation term h(t,z,$\epsilon$) (Theorem 4). This gives generalizations of the results obtained by {\it J. Gruendler} [SIAM J. Math. Anal. 16, 907-931 (1985; Zbl 0601.70017)] and {\it K. J. Palmer} [J. Differ. Equations 55, 225-256 (1984; Zbl 0508.58035)].
[S.Kono]

MSC 2000:: *34D10 Stability perturbations of ODE
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: method of Lyapunov-Schmidt; bounded solution; bifurcation equations

Citations: Zbl 0601.70017; Zbl 0508.58035

Cited in: Zbl 0967.34042 Zbl 0812.34046 Zbl 0812.34045 Zbl 0742.34031

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