On the asymptotic properties of non-autonomous systems

By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system ${du/dt +Auni f}By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system du/dt +Au ' f{du/dt +Auni f}, where A is an accretive (possibly multivalued) operator in X × X, and for some f_￥ ? X{f_{infty}in X} and 1 ≤ p < ∞ we have g ? L^p((1,+￥);X){gin L^p((1,+infty);X)}, so that g(θ) = (f(θ) − f _∞)/θ, ${(forall theta > 1)}${(forall theta > 1)}. Our results extend and improve many previously known results. Moreover, we answer an open question raised by B. Djafari Rouhani.