Stability criteria of nonlinear generalized proportional fractional delayed systems

The paper’s Theorem 2 establishes finite-time stability (FTS) for the nonlinear GPFS with delay under the generalized Lipschitz condition (3), supplying the explicit constants ψ and ϕ and the sufficient condition (13). Its proof proceeds from the mild form (2), applies Hölder with k=1+α and r=1+1/α, estimates the kernel via Γ(α^2)/k^{α^2} to define ω, then uses Jensen and a Grönwall step to derive inequality (19) and hence (13) for FTS . The candidate solution follows the same blueprint: mild form; Lipschitz splitting into current and delayed terms; raising to power r with (x+y+z)^r ≤ 3^{1/α}(…); Hölder with the same conjugate exponents; the same ω, μ^k and Γ_r(α) constants; splitting the delay into history/live parts to get the exp(−rτ) and (1−e^{−rτ}) coefficients; and a Grönwall-type closure that yields precisely the bound stated as (19), implying condition (13) and FTS. The only discrepancy is a notational slip in the candidate’s Step 1 (using μ^{1−α} instead of μ^α in the mild integral prefactor), but from Step 4 onward the constants match those in the paper and the final result coincides exactly with (19) and (13). Thus both are correct, and the arguments are substantially the same, differing only in the presentation of the final Grönwall step (the candidate uses a supremum-and-convolution trick while the paper applies Jensen and then Grönwall directly). Definitions and the FTS notion are as in the paper’s Section 2 and Definition 3 , with the system specified in (1) .