This note extends our earlier existence and uniqueness results in AIMS Mathematics, 10 (2025), 19280–19299. Using sharper estimates for the squared Green’s functions and Rus’s fixed point theorem with the unweighted L2 metric, we obtain conditions for the classical problem with 1 < α ≤ 2 and the sequential problem with 1/4 < θ ≤ 1. A weighted L2 metric covers the full sequential range 0 < θ ≤ 1 and gives a less restrictive condition than the Banach condition based on the maximum L1 integral. We also compare the Rus and Banach conditions and illustrate the improvements with examples.