ASYMPTOTIC BEHAVIOR FOR ALMOST-ORBITS OF ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN A METRIC SPACE

Abstract

Let $(M, \rho)$ be a metric space and $\tau$ a Hausdorff topology on $M$ such that $\{ M, \tau \}$ is sequentially compact. Let $T$ be a $\rho$-asymptotically nonexpansive type self-mapping of $M$ and $u = \{ x_n \}$ a $\rho$-bounded almost-orbit of $T$. We study the $\tau$-convergence of $u$ in $M$ when the triplet $\{ M, \rho, \tau \}$ satisfies various types of $\tau$-Opial conditions. Our results, which also hold for the continuous case of one-parameter semigroups, extend and unify many previously known results [1, 5, 9, 10, 12-19, 21, 22], and answers affirmatively an open question of S. Reich [20, p.550] in the very general context of a metric space.