Existence and stability analysis of solution for fractional delay differential equations

The paper proves existence–uniqueness for the Caputo delay IVP cD^α υ(t)=f(t,υ(t),υ(g(t))) with history on [−h,0] under a global Lipschitz condition in the present and delayed variables, using a Bielecki (exponentially weighted) norm to make the associated Volterra operator a contraction; the contraction constant is 2L/τ^α and can be made <1 by choosing τ large, independently of T (see the statement of the problem and Theorem 1 with its Bielecki-norm proof and inequality (3) in the manuscript ). The candidate model solution reaches the same existence–uniqueness conclusion via an alternative weighted norm built from the Mittag–Leffler function and the identity I^α[E_α(ω t^α)]=(E_α(ω t^α)−1)/ω, yielding a contraction constant 2L/ω. Thus both are correct; the proofs are essentially the same fixed-point strategy but with different weights (exponential vs. Mittag–Leffler).