Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation

The paper correctly reduces the chemostat with CFDS to a 1D periodic FDE, establishes the integral representation, proves uniqueness of the washout under D(t) > μmax with KY > 1 (and L ≥ T), and outlines uniqueness of a non-trivial periodic solution under pointwise and averaged bounds around s*. However, its existence claim for a non-trivial periodic solution (Theorem 2.2) relies on Schauder’s fixed point plus Theorem 2.1 and does not actually rule out the trivial solution; the step that converts “some fixed point in X” into “a non-trivial solution” is not justified. For uniqueness under the averaged condition (ii)(b), the argument that solutions remain in [0, s*] is not rigorously established. The model’s solution adds a clean period-average identity for CFDS and supplies a strong L2-type uniqueness argument; it gives a coherent existence/uniqueness framework under the pointwise bound D(t) ≤ ν(s*). But its (ii)(b) existence/uniqueness argument also lacks a fully rigorous a priori bound ensuring solutions stay in [0, s*]. Hence both are incomplete: the paper’s existence proof for non-trivial solutions is under-justified, and the model’s averaged-case (ii)(b) argument leaves a gap.