EXISTENCE OF BOUNDED ASYMPTOTIC SOLUTIONS OF AUTONOMOUS DIFFERENTIAL EQUATIONS

The paper proves the main results rigorously using sums of commuting operators and an analytic-semigroup hypothesis to invert D̃−Ã on the appropriate quotient space, then interprets this as existence/uniqueness of asymptotic mild solutions with sp(w) ⊂ sp(f) (Theorem 3.7), and handles the resonant case via a Riesz projection onto σi(A)\sp(f) when that set (or σi(A)) is bounded (Theorem 3.13) . By contrast, the candidate constructs an inverse R(D̃,A) through a Dunford–Riesz integral over a contour enclosing i·sp(f). This requires a bounded contour surrounding i·sp(f), which is not available if sp(f) is unbounded, and the argument does not address how to define the integral (or pass to limits) in that case. The candidate also asserts the analyticity assumption on T(t) is unnecessary without proof. Hence the candidate solution is incomplete/incorrect in general, while the paper’s argument is consistent and complete under its stated assumptions.