Convergence theorems for image-strict pseudo-contractions in 2-uniformly smooth Banach spaces Original Research Article

In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real 2-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 324 (2007) 336–349]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in CC. This result extends a recent result of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] both from nonexpansive mappings to λλ-strict pseudo-contractions and from Hilbert spaces to 2-uniformly smooth Banach spaces.