Total Stability in Abstract Functional Differential Equations with Infinite Delay

Recently, authors [2] have discussed some equivalent relations for ρ-uniform stabilities of a given equation and those of its limiting equations by using the skew product flow constructed by quasi-processes on a general metric space. In 1992, Murakami and Yoshizawa [6] pointed out that for functional differential equations with infinite delay on a fading memory space B = B((−∞, 0];R) ρ-stability is a useful tool in the study of the existence of almost periodic solutions for almost periodic systems and they proved that ρ-total stability is equivalent to BC-total stability. The purpose of this paper is to show that equivalent relations established by Murakami and Yoshizawa [6] holds even for functional differential equations with infinite delay on a fading memory space B = B((−∞, 0];X) with a general Banach space X.