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Abstract
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In this work we study the existence of homoclinic orbits of the planer system
of Li´enard type
𝑥˙ = 𝐾(𝑦 − 𝐹(𝑥)), 𝑦˙ = −𝑔(𝑥),
where 𝐾 is strictly increasing and 𝐾(±∞) = ±∞. We present sufficient and necessary
conditions for this system to have a positive and a negative semiorbit which starts at a
point on the curve 𝑦 = 𝐹(𝑥) and approaches the origin without intersecting the 𝑥-axes.
The conditions obtained are very sharp.
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