On the Mittag-Leffler stability of mixed-order fractional homogeneous cooperative delay systems

The paper proves local Mittag–Leffler stability for the mixed-order Caputo delay system under (H1),(H2),(S) with a precise barrier/first-contact argument (Theorem 3.5), carefully handling heterogeneous fractional orders via β := min_i α_i/p and a time-ratio estimate that controls the lag terms even when some components of g^{(j)}(v) are negative; see the statement and proof skeleton in Theorem 3.5 and inequalities (11)–(15) with Lemma 2.9 and the Caputo–Mittag–Leffler identity, as well as the boundedness/positivity results that underpin the first-contact setup . By contrast, the candidate solution’s key comparison inequality (scaling all lag terms by σ^p irrespective of the sign of g^{(j)}(v)) is false for components where g^{(j)}_i(v) ≤ 0, so the proposed upper-solution inequality can fail. Moreover, it incorrectly requires σ ≥ 2^{-1/p}, which contradicts the local-smallness regime. The paper’s proof avoids these pitfalls by using ratio bounds with E_β(−c(t−τ)^β)/E_β(−cr^β) and restricting the first touch to t* > 1, making the supremum in (11) finite and the contradiction argument rigorous .