The dichotomy spectrum approach for a global nonuniform asymptotic stability problem: Triangular case via uniformization

Abstract We introduce a new conjecture of global asymptotic stability for nonautonomous systems, which is fashioned along the nonuniform exponential dichotomy spectrum and whose restriction to the autonomous case is related to the classical Markus–Yamabe Conjecture: we prove that the conjecture is fulfilled for a family of triangular systems of nonautonomous differential equations satisfying boundedness assumptions. An essential tool to carry out the proof is a necessary and sufficient condition ensuring the property of nonuniform exponential dichotomy for upper block triangular linear differential systems. We also obtain some byproducts having interest on itself, such as the diagonal significance property in terms of the above-mentioned spectrum.