Vector Lyapunov functions andp-th-order conditional stability and boundedness

Summary In this paper we introduce the concept of p-th-order conditional stability and boundedness for ordinary differential equations in Banach spaces. Using vector Lyapunov functions and the comparison technique we give sufficient conditions for p-th-order conditional stability and boundedness properties of the abstract differential equation. Our results generalize in many ways various known results of stability and boundedness. Entrata in Redazione il 23 luglio 1977.