Mild and classical solutions to fractional Cauchy problem on time scales

Area of Science:

• Mathematics
• Differential Equations
• Time Scale Theory

Background:

• Fractional calculus extends traditional calculus to non-integer orders.
• Cauchy problems are fundamental in modeling various dynamic systems.
• Time scales provide a unified framework for studying differential and difference equations.

Purpose of the Study:

• To investigate the existence and uniqueness of solutions for fractional Cauchy problems with impulsive non-local conditions on time scales.
• To analyze both classical and mild solutions within this framework.

Main Methods:

• Application of the Gronwall inequality tailored for time scales.
• Utilization of the Banach fixed-point theorem in the context of time scale analysis.

Main Results:

• Establishment of the conditions for the existence and uniqueness of solutions.
• Demonstration of the applicability of the theoretical results through a concrete example.

Conclusions:

• The employed mathematical tools effectively address the complexities of fractional Cauchy problems on time scales.
• The findings contribute to the theoretical understanding and practical application of impulsive fractional dynamics.