Remarks on a recent paper titled: “On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces”

Abstract

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 < κ < 1 2 $0<\kappa < \frac{1}{\sqrt{2}}$ , 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 $E=l_{p}$ , 2 ≤ p < ∞ $2\leq p<\infty $ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0 < κ ≤ 1 2 $0<\kappa \leq \frac{1}{\sqrt{2}}$ , then E is necessarily l 2 $l_{2}$ , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

Keywords

• Fixed point
• Accretive
• Uniformly smooth