Discrete integral conditions for weak trajectories and asymptotic stability for operators in Banach spaces

Let t↦φ(t):R+→R+ be a nondecreasing function which is positive on (0,∞) and let A be a bounded linear operator acting on a complex Banach space X. Let X′ denote the strong dual of X. We prove that if for x∈X and x′∈X′ one has(0.1)sup‖x‖≤1,‖x′‖≤1⁡∑n=0∞φ(|〈Anx,x′〉|)<∞, then there exists a positive number ν=ν(A,φ)∈(0,1) such that the spectral radius of A, denoted by r(A), satisfies the inequality r(A)≤ν. Some theorems on existence for discrete evolution equations in Banach spaces are presented and extensions of these results to discrete evolution families are also made.