In this work we will be concerned with the existence of almost homoclinic solutions for a Newtonian system $q̈+{\nabla }_{q}V(t,q)=f\left(t\right)$, where t ∈ ℝ, q ∈ ℝⁿ. It is assumed that a potential V: ℝ × ℝⁿ → ℝ is C¹-smooth and its gradient map ${\nabla }_{q}V:ℝ\times ℝⁿ\to ℝⁿ$ is bounded with respect to t. Moreover, a forcing term f: ℝ → ℝⁿ is continuous, bounded and square integrable. We will show that the approximative scheme due to J. Janczewska (see [J2]) for a time periodic potential extends to our case.

Let $p,q$ be analytic functions in the unit disk $𝒰$. For $\alpha \in [0,1)$ the authors consider the differential subordination and the differential equation of the Briot-Bouquet type: $$p^{1-\alpha }\left(z\right)+\frac{z{p}^{\text{'}}\left(z\right)}{\delta p^{\alpha }\left(z\right)+\lambda p\left(z\right)}\prec h\left(z\right),z\in 𝒰,$$