Boundedness of solutions of the first-order linear multidimensional difference equations

The paper’s Theorem 3.6 gives necessary and sufficient conditions for bounded solutions of x(n+1)=J_λ x(n)+y(n) (|λ|=1) under the hypothesis y_m(n)=\tilde y_m(n)/n^{m-1} with bounded partial sums of \tilde y_m(n)λ^{-n}, yielding the initial-value relations x_m(1)=−∑_{r=0}^{M−m}(−1)^r∑_{k=1}^∞((k)[r]/r!)λ^{−k−r}y_{m+r}(k), m=2,…,M. The candidate solution derives the same formula via the standard variation-of-constants representation, the decomposition J_λ=λI+N, the binomial/Jordan-power expansion, and an equivalent combinatorial identity (paper’s Lemma 3.4) plus Dirichlet/Abel tail bounds. The structure—rewriting x_m(n) as a finite sum of binomials in n with n-independent coefficients and showing boundedness iff all coefficients at C(n−1,t) for t≥1 vanish—is the same as in the paper’s proof (Proposition 3.5 and Theorem 3.6). Minor differences are present in presentation (e.g., a generating-function proof of the combinatorial identity and an explicit remainder estimate O(n^{1−m})), but the logical path and conditions coincide. Hence both are correct and substantially the same proof. Key steps and formulas correspond directly to Proposition 3.5, Lemma 3.4, and the proof of Theorem 3.6 in the paper .