| Summary: | We study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mi>f</mi><mfenced separators="" open="(" close=")"><mi>x</mi><mo>,</mo><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mfenced><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mi>x</mi><mo>+</mo><mi>g</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><msup><mi>ε</mi><mn>2</mn></msup><msub><mi>p</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>+</mo><msup><mi>ε</mi><mn>3</mn></msup><msub><mi>p</mi><mn>2</mn></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <i>n</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∈</mo><msup><mi mathvariant="double-struck">N</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></semantics></math></inline-formula> are of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">C</mi><mn>3</mn></msup><mo>,</mo><mspace width="4pt"></mspace><msup><mi mathvariant="script">C</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula> in a neighborhood of the origin, respectively, the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula> are of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">C</mi><mn>0</mn></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>π</mi><mo>−</mo></mrow></semantics></math></inline-formula>periodic in the variable <i>t</i>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> is a small parameter as usual. The mathematical tool that we have used is the averaging theory of dynamical systems of second order.
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