... point theorems for new nonlinear mappings in a Hilbert spaces. J Nonlinear Convex Anal. 11,79–88 (2010)17. Takahashi, W, Yao, JC: Fixed point theorems and ergodic theorems for nonlinear mappings ... exactly one point. Remark 4.1. The conclusion of Lemma 4.1 is still true if T : C ® X is any one ofnonexpansive mapping, nonspreading mapping, TJ-1 mapping, TJ-2 mapping, andhybrid mapping. Theorem ... nonspreading mapping, TJ-1 mapping, TJ- 2 mapping, hybrid mapping, and nonexpansive mapping. Then, {Tnx } is bounded forsome x Î C if and only if F(T) ≠ ∅.4 Δ-convergent theorems In the sequel,...