| Summary: | In this manuscript, we are interested in studying the homoclinic solutions of fractional Hamiltonian system of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mmultiscripts><mi mathvariant="fraktur">D</mi><mo>∞</mo><mi>α</mi><mi>ς</mi><mrow></mrow></mmultiscripts><mrow><mo>(</mo><mmultiscripts><mi mathvariant="fraktur">D</mi><mi>ς</mi><mi>α</mi><mrow><mo>−</mo><mo>∞</mo></mrow><mrow></mrow></mmultiscripts><mi>Z</mi><mrow><mo>(</mo><mi>ς</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>−</mo><mi mathvariant="script">A</mi><mrow><mo>(</mo><mi>ς</mi><mo>)</mo></mrow><mi>Z</mi><mrow><mo>(</mo><mi>ς</mi><mo>)</mo></mrow><mo>+</mo><mo>∇</mo><mi>ω</mi><mrow><mo>(</mo><mi>ς</mi><mo>,</mo><mi>Z</mi><mrow><mo>(</mo><mi>ς</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>∈</mo><msup><mi>H</mi><mi>α</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>C</mi><mn>1</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are not periodic in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ς</mi></semantics></math></inline-formula>. The characteristics of the critical point theory are used to illustrate the primary findings. Our results substantially improve and generalize the most recent results of the proposed system. We conclude our study by providing an example to highlight the significance of the theoretical results.
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