This paper studies the existence of nonuniform polynomial dichotomy for discrete evolution families in Banach spaces. Some necessary and sufficient conditions for the nonuniform polynomial dichotomy concept of discrete evolution families with respect to invariant projections sequence and strongly invariant projections sequence are obtained, respectively. Additionally, we consider two notions of Lyapunov functions and use our results to obtain the relations between (generalized) Lyapunov functions and nonuniform polynomial dichotomy.

The aim of this paper is to give some discrete and continuous versions of Barbashin type theorems for the uniform polynomial stability and instability of evolution families in Banach spaces, by using Banach sequence spaces in ℋ ⁢ ( ℕ ≥ 1 ) and Banach function spaces in ℋ ⁢ ( ℝ ≥ 1 ) , respectively. Variants for uniform polynomial stability and instability of some well-known results in the exponential stability theory and polynomial stability theory are obtained.

The aim of this paper is to give some Datko and Barbashin type characterizations for the uniform \(h\)-instability of evolution families in Banach spaces, by using some important sets of growth rates. We prove four characterization theorems of Datko type and two characterization theorems of Barbashin type for uniform \(h\)-instability. Variants for uniform \(h\)-instability of some well-known results in stability theory (Barbashin (1967), Datko (1972)) are obtained.

In this paper, we investigate several two-parameter weak Orlicz–Hardy martingale spaces generated by the $p$-convex and $q$-concave functions, and establish their atomic decomposition theorems. Using the atomic decomposition, we obtain a sufficient condition for the boundedness of a sublinear operator defined on the two-parameter weak Orlicz–Hardy martingale spaces. Furthermore, the dual spaces of the two-parameter weak Orlicz–Hardy martingale spaces are considered.

In this paper, we consider the problem of exponential instability behavior of random dynamical systems described by cocycles in Banach spaces. We prove some continuous and discrete versions of Datko type theorem for the exponential instability in average of cocycles. In addition, two characterizations of the exponential instability in average in terms of Lyapunov functions are given.