Dynamics of a Model of Polluted Lakes via Fractal-Fractional Operators with Two Different Numerical Algorithms

Both the paper and the candidate solution reformulate the FF–ML initial value problem into an equivalent Volterra-type integral equation with the same Atangana–Baleanu (Mittag–Leffler) integral operator, and both use a Schauder-type fixed point framework. The paper uses the Leray–Schauder alternative and excludes the homotopy-on-boundary case via (P2), while the candidate applies the (classical) Schauder fixed point theorem on an invariant ball. However, the paper’s compactness (equicontinuity) argument relies on a bound involving v^{σ−1}−s^{σ−1} and v^{θ+σ−1}−s^{θ+σ−1}, which does not yield uniform equicontinuity on [0,S] when 0<σ<1; moreover, the local term s^{σ−1}Q(s,K(s)) in the FFMLI operator is singular at s=0 unless additional regularity is imposed. These issues are not addressed in the paper, while the candidate solution explicitly adds a mild regularity assumption at the origin (e.g., σ=1 or Q(s,K(s))=o(s^{1−σ}) as s→0^+) to ensure that FFMLI maps C(J) into C(J) and to obtain equicontinuity needed for Arzelà–Ascoli. With that added hypothesis, the candidate’s existence proof is sound. See the paper’s definitions of FFMLD/FFMLI and the integral reformulation (Def. 1–2 and Eq. (14)) , the existence framework and conditions (P1), (P2) (Theorem 4) , and the equicontinuity estimate used in the paper’s compactness argument (Eq. (19)) .