We show the existence of homoclinic type solutions of second order Hamiltonian systems of the type \(\ddot{q}(t)+\nabla _{q}V(t,q(t))=f(t)\), where \(t\in \mathbb {R}\), the \(C^1\)-smooth potential \(V:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}\) satisfies a relaxed superquadratic growth condition, its gradient is bounded in the time variable, and the forcing term \(f:\mathbb {R}\rightarrow \mathbb {R}^n\) is sufficiently small in the space of square integrable functions. The idea of our proof is to approximate the original system by time-periodic ones, with larger and larger time-periods. We prove that the latter systems admit periodic solutions of mountain-pass type, and obtain homoclinic type solutions of the original system from them by passing to the limit (in the topology of almost uniform convergence) when the periods go to infinity.

中文翻译：

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一类非齐次二阶哈密顿系统的同宿型解的存在性

我们显示了\（\ ddot {q}（t）+ \ nabla _ {q} V（t，q（t））= f（t）\）类型的二阶哈密顿系统的同宿型解的存在，其中\（t \ in \ mathbb {R} \），\（C ^ 1 \）-平滑势\（V：\ mathbb {R} \ times \ mathbb {R} ^ n \ rightarrow \ mathbb {R} \）满足一个松弛的超二次生长条件，其梯度以时间变量为界，并且强迫项\（f：\ mathbb {R} \ rightarrow \ mathbb {R} ^ n \）在平方可积函数的空间内足够小。我们证明的思想是按时间周期近似原​​始系统，时间周期越来越大。我们证明了后者的系统接纳山口型的周期解，并通过当周期达到无穷大时达到极限（在几乎均匀收敛的拓扑结构中），从而从中获得原始系统的同斜型解。