Weak Convergence Theorems for Strict Pseudo-contractions in Banach spaces

作者：Zhou Hai Yun^*

来源：Acta Mathematica Sinica-English Series, 2014, 30(5): 755-766.

DOI：10.1007/s10114-014-3096-5

• Frechet differentiable Banach spaces
• Reich's inequality
• a countable family of lambda-strict pseudo-contractions
• normal Mann iteration algorithm
• weak convergence theorems

In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257-1265 (2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results (Lemma 3.4, Theorems 3.5-3.6, 3.8-3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages (2010)] do not hold in L-p for p > 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include L-p for p >= 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above.

• 出版日期2014-5