Generalized Esclangon–Landau Conditions for Differential–Difference Equations

AbstractLet J=[α,∞) for some α∈R or J=R and denote byXa Banach space. We study the solutions on J of the equation (*) Ω(n)(t)+∑n−1k=0∑mj=1aj,k(t)Ω(k)(t+tj)=b(t), wheret1<t2···<tm,aj,k: J→L(X) andb: J→X. We prove that ifband all of the coefficients are bounded and Ω is a bounded solution of (*), then Ω′,Ω″,…,Ω(n)are bounded, provided one of the following holds: (i)a(p)j,kare bounded for all 1≤p≤k≤n−1; (ii) J=[α,∞) andtm≤0; (iii) J=[α,∞) and ‖a(p)j,k(t)‖=O(eρ|t|) for some 0