The oscillatory solutions of multi-order fractional differential equations

Both the paper and the candidate reduce the Caputo equation to an equivalent Volterra-type integral identity using the standard inversion I^α CD^α x = x − T_{n−1} and I^α CD^β x = I^{α−β}(x − T_{m−1}), separate early-/late-time contributions, multiply by the weight t^{β−α−n+1}, and invoke (A) to fix signs and (B) to force a contradiction, hence oscillation. The paper derives a bound of the form t^{β−α−n+1}x(t) ≤ c + t^{β−α−n+1}I^α g(t) and reaches a contradiction with (B) (see Theorem 3.1 and its proof, especially the identity preceding (9) and inequality (9) ; assumptions (A)–(B) are given with (3)–(4) ). The candidate’s proof is essentially the same framework, differing only in whether one rearranges for x or I^α g; both rely on the same identities and estimates.