Generalized Esclangon]Landau Conditions for Differential]Difference Equations

Let Jswa, `. for some a gR or JsR and denote by X a Banach space. We study the solutions on J of the equation U . V n. t.q nksy01 mjs1aj,k t.V k. tqtj. sb t., where t1-t2??? -tm, aj,k: JªL X. and b: JªX. We prove that if b and all of the coefficients are bounded and V is a bounded solution of U., then VX,VY, . . . ,V n. are bounded, provided one of the following holds: i. a j,pk. are bounded for all 1FpFkFny1; ii. Jswa, `.and tm F0; iii. Jswa, `. and 5 p. .5 r