Strong convergence of a modified Halpern-type iteration for asymptotically quasi-∅-nonexpansive mappings

In this paper, the problem of modifying Halpern iteration for approximating a common fixed point of a family of asymptotically quasi- ∅-nonexpansive mappings is considered. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper mainly improve the corresponding results announced in [Y.J. Cho, X. Qin, S.M. Kang, Strong convergence of the modified Halpern- type iterative algorithms in Banach spaces, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009) 51-68].