In this paper, we first introduce a family $mathcal{S}={S_n: C o C}$ of non-Lipschitzian mappings, called {it total asymptotically nonexpansive},(briefly, TAN) on a nonempty closed convex subset $C$ of a real Banach space $X$, and next give necessary and sufficient conditions for strong convergence of the sequence ${x_n}$ defined recursively by the algorithm $x_{n+1}=S_n x_n, ;ngeq 1$, starting from an initial guess $x_1in C$, to a common fixed point for such a continuous TAN family $mathcal{S}$ in Banach spaces. Finally, some applications to a finite family of TAN self mappings are also added.

Keywords: Common fixed points, non-Lipschitzian mappings, total asymptotically nonexpansive mappings, strong convergence