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Nonexistence of homoclinic orbits for a class of Hamiltonian systems. (English) Zbl 1296.34120

Summary: We give several sufficient conditions under which the first-order nonlinear Hamiltonian system \[ x'(t)= \alpha(t) x(t)+ f(t, y(t)),\quad y'(t)=-g(t, x(t))- \alpha(t) y(t) \] has no solution \((x(t),y(t))\) satisfying condition \[ 0< \int^{+\infty}_{-\infty} [|x(t)|^\nu+ (1+ \beta(t))|y(t)|^\mu]\,dt< +\infty, \] where \(\mu,\nu>1\) and \((1/\mu)+ (1/\nu)= 1\), \(0\leq xf((t,x)\leq \beta(t)|x|^\mu\), \(xg(t,x)\leq \gamma_0(t)|x|^\nu\), \(\beta(t)\), \(\gamma_0(t)\geq 0\), and \(\alpha(t)\) are locally Lebesgue integrable real-valued functions defined on \(\mathbb{R}\).

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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