By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by B. Djafari Rouhani [J. Differential Equations, 229 (2006) 412-425] 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 $du/dt +Au\ni f$, where $A$ is an accretive (possibly multivalued) operator in $X\times X$, and for some $f_{\infty}\in X$ and $1\leq p<\infty$ we have $g\in L^p((1,+\infty);X)$, so that $g(\theta)=\frac 1{\theta}{(f(\theta)-f_{\infty})}$, $(\forall \theta> 1)$. Our results extend and improve many previously known results. Moreover, we answer an open question raised by B. Djafari Rouhani.
