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A note on a generalization of Diliberto's Theorem for certain differential equations of higher dimension

Ladislav Adamec (2005)

Applications of Mathematics

In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.

A note on an approximative scheme of finding almost homoclinic solutions for Newtonian systems

Robert Krawczyk (2014)

Banach Center Publications

In this work we will be concerned with the existence of almost homoclinic solutions for a Newtonian system q ̈ + ∇ q V ( t , q ) = f ( t ) , where t ∈ ℝ, q ∈ ℝⁿ. It is assumed that a potential V: ℝ × ℝⁿ → ℝ is C¹-smooth and its gradient map ∇ q V : ℝ × ℝ ⁿ → ℝ ⁿ 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.

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