Some fixed point results for generalized modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math>-contractions in

This paper introduces new generalizations of modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>F</mml:mi></mml:math>-contractions through <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>-admissible mappings. It expands the fixed-point results of modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>F</mml:mi></mml:math>-contractions within the framework of modular metric spaces. Several theorems are presented regarding the existence of solutions, along with a condition that ensures the uniqueness of a solution. These theorems are applied to specific mappings to verify the results. Additionally, the paper presents significant outcomes, demonstrating the existence of solutions to a general form of an integral equation. Finally, a particular example of an integral equation is discussed.