Dichotomies for Linear Evolutionary Equations in Banach Spaces

On the uniform exponential stability of linear skew‐product semiflows

“…In fact, in the last decade many contributions were done in the study of linear skew-product flows with Banach fibers over a compact metric space [1,2,3]. Also, many applications were done in [2,9,10,11,25].…”

“…In the finite dimensional context the Sacker-Sell spectrum offered an interesting way to describe all of these properties (see [9,23,24,25]). This kind of approach was extended in the last decade to the case of normcontinuous cocycles on infinite dimensional Banach spaces by Latushkin and Stepin.…”

“…Many of these equations are obtained when linearize the nonlinear differential equations (see for instance [25]). …”

ON ${\bf (a,b)}$ -DICHOTOMY FOR EVOLUTIONARY PROCESSES ON A HALF-LINE

“…In last few years, several results about exponential stability and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh [13,14], F. Räbiger [13], Y. Latushkin [2,6,7,8], M. Pinto [1,18], T. Randolph [7,8], R.Rau [22], R. Sacker [24], G. Sell [24] and R. Schnaubelt [8,13,25], W. Zhang [26]. All these results are given for the uniform case.…”

A biographical sketch of Robert J. Sacker

“…(First and foremost, the linear operator Fðy; tÞ is now defined only for t $ 0, and pðtÞ is a semiflow and not a flow [45].) Eventually these challenges were met, as seen in [40], and the alternative theorem is valid in this setting.…”