Remarks on a recent paper titled: \u201cOn the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces\u201d

C E Chidume

doi:10.1186/s13660-021-02576-0

Abstract

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition 0<kappa < frac{1}{sqrt{2}}, as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is E=l_{p}, 2leq p<infty . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0<kappa leq frac{1}{sqrt{2}}, then E is necessarily l_{2}, a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

Highlights

Let H1 and H2 be two real Hilbert spaces, C and Q be nonempty closed and convex subsets of H1 and H2, respectively
Studies of split feasibility problem (SFP) and split common fixed point problem (SCFPP) in real Banach spaces more general than Hilbert spaces are very scanty in the literature
In 2015, Tang et al [12] studied the SFP and SCFPP in real Banach spaces more general than Hilbert spaces and by using hybrid methods and Halpern-type methods, they proved strong and weak convergence of the sequence generated by their algorithm to solutions of these problems

Summary

Introduction

Let H1 and H2 be two real Hilbert spaces, C and Q be nonempty closed and convex subsets of H1 and H2, respectively. These problems and their generalizations have been studied in real Hilbert spaces and iterative methods for approximating their solutions, assuming existence, in this setting abound in the literature (see, for example, Tang et al [12], and the references therein).

Results

Conclusion