Enhanced Mathematical Framework for Non-Integer Order Functional Integro-Differential Equations through Advanced Contractivity Conditions and Dhage’s Sophisticated Fixed Point Theorem

This investigation develops advanced methodologies for analyzing solution existence and asymptotic convergence in non-integer order functional integro-differential equations within sophisticated algebraic frameworks. Our methodology employs enhanced contractivity conditions, advanced function characteristics, and asymptotic convergence principles. The primary theoretical advancement materializes through refined multipoint asymptotic convergence techniques established by Dhage’s innovative approach. Our mathematical structure provides a comprehensive foundation for studying sophisticated non-integer order equations exhibiting memory characteristics and functional interdependencies.