Existence, Uniqueness and Continuous Dependence for Deformable Derivative Initial Value Problems

This paper establishes a comprehensive theoretical framework for initial-value problems involving deformable derivatives. Local existence and uniqueness of solutions are proved via the Banach fixed-point theorem under Lipschitz continuity assumptions. These results are extended to global existence under linear growth conditions, continuous dependence on initial conditions and parameters is rigorously demonstrated. Furthermore, the theory is generalized to systems of equations. Collectively, these findings position the deformable derivative as a continuous bridge between integer-order and fractional-order calculus, offering a mathematically tractable and flexible tool for modeling complex physical systems with intermediate dynamics and memory effects.