Conditional Lipschitz Shadowing for Ordinary Differential Equations

The candidate solution reproduces the paper’s two main sufficient conditions for conditional Lipschitz shadowing. For the semilinear case, it uses the exponential dichotomy Green operator and a contraction on a sup-norm ball, yielding κ = 2N/(λ − 2NL) and ε0 = δ(λ/(2N) − L), exactly as in Theorem 2 and its proof (see the theorem statement and the contraction estimates around (2.11)–(2.12) ; the dichotomy setup is as in Definition 3 ). For the nonlinear logarithmic-norm criterion, it introduces the averaged Jacobian E(t), invokes convexity of the logarithmic norm (Lemma 1), and obtains the same bound with κ = 1/m and ε0 = mδ, matching Theorem 5 and its proof via variation-of-constants and the log-norm bound . Minor stylistic differences (Dini derivative vs. variation-of-constants; choice of closed ball for the fixed-point map) are routine and mathematically equivalent. No missing hypotheses were detected relative to the paper’s statements.