On Coupled System of Nonlinear Ψ-Hilfer Hybrid Fractional Differential Equations

Both the paper and the candidate solution aim to prove existence for the same Ψ–Hilfer hybrid coupled IVP under essentially identical hypotheses (H1)–(H2) and the same smallness condition 4σ{ |y0/u(0,y(0+))| + ((Ψ(T)−Ψ(0))^{μ+1−ξ}/Γ(μ+1)) ||g|| } + δ < 1. The paper’s route uses a three-operator fixed-point lemma on a Banach algebra and the integral equivalence (Lemma 3.1), while the model works in the neutral variables z and applies Schauder after a per-time contraction step. However, both arguments tacitly treat the factor 1/u(0, y(0+)) as if it were a uniform constant. Without an explicit a priori lower bound on |u(0,p)| over the relevant range (or an equivalent device), one cannot bound M = sup_{y∈S} ||F y|| or define a solution-independent ball for Schauder, since y(0+) is unknown and 1/u(0,y(0+)) may be arbitrarily large. This leaves a gap in both proofs. See the paper’s IVP statement and integral form (1.1)–(1.2) and Lemma 3.1, the main existence Theorem 3.2 with (3.1), and the proof steps where M and the set S are formed and bounded .