Implicit Fractional Differential Equations in Banach Spaces via Picard and Weakly Picard Operator Theory

Both the paper and the model correctly set up the Caputo IVP as a fixed-point problem in z(t) and derive a contraction under the smallness condition M2 T^α/Γ(α+1) + M3 < 1. However, both arguments incorrectly promote the Hölder-α regularity of I^α z to Lipschitz continuity to keep T invariant on CL(J, B_R). In particular, the paper’s Step 3 uses a step that effectively treats |t2−t1|^α as O(|t2−t1|) for 0<α<1, which is false, and the model makes the same mistake via the inequality |t1 − t2|^α ≤ T^{α−1}|t1 − t2|. The existence–uniqueness part can be repaired by working directly on C(J,X) with the sup norm (or with the Bielecki norm on C(J,X)) without requiring CL invariance, and the identity I^{1−α}I^α = I^1 suffices to conclude c_0D_t^α x = z when x = x0 + I^α z. But as written, each proof has a gap.