Asymptotic properties of some non-autonomous systems in Banach spaces

Let X be a reflexive Banach space. We introduce the notion of weakly almost nonexpansive sequences (xn)n⩾0 in X, and study their asymptotic behavior by showing that the nonempty weak ω-limit set of the sequence (xn/n)n⩾1 always lies on a convex subset of a sphere centered at the origin of radius d=limn→∞‖xn/n‖. Subsequently we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system , where A is an accretive (possibly multivalued) operator in X×X, and f−f∞∈Lp((0,+∞);X) for some f∞∈X and 1⩽p<∞. These results extend recent results of J.S. Jung and J.S. Park [J.S. Jung, J.S. Park, Asymptotic behavior of nonexpansive sequences and mean points, Proc. Amer. Math. Soc. 124 (1996) 475–480], and J.S. Jung, J.S. Park, and E.H. Park [J.S. Jung, J.S. Park, E.H. Park, Asymptotic behaviour of generalized almost nonexpansive sequences and applications, Proc. Nonlinear Funct. Anal. 1 (1996) 65–79], as well as our results cited below containing previous results by several authors.