THREE-STEP ITERATIVE SEQUENCES WITH ERRORS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS IN CONVEX METRIC SPACES (Nonlinear Analysis and Convex Analysis)
J.K. Kim, K.H. Kim, K.S. Kim · Kyoto University Research Information Repository (Kyoto University) · 2004
In this paper, we will give some necessary and sufficient conditions for three- step iterative sequences with errors to converge to a fixed point for asymptotically quasi- nonexpansive mappings in convex metric spaces.The results of this paper are general- izations and improvements of the corresponding results of Chang, Kim et $\mathrm{a}/.$, Liu and Xu-Noor. 1 $x$ , $y\in$ D(T).(2) The mapping $T$ is said to be quasi-nonexpansive if(3) The mapping $T$ is said to be asymptotically nonexpansive if there exists a se-(4) The mapping $T$ is said to be asymptotically quasi-nonexpansive if there exists a sequence $k_{n}'\in[0, \infty)$ with $\lim_{narrow\infty}k_{n}=0$ such that(2) The mapping $T$ is said to be quasi-nonexpansive if) The mapping $T$ is said to be asymptotically nonexpansive if there exists a sequence $k_{n}^{\wedge}\in[0, \infty)$ with $\lim_{narrow\infty}k_{n}^{\wedge}=0$ such that $d(T^{n}x, T^{n}y)\leq(1+k_{n})d(x, y)$ , $\forall x$ , $y\in D(T)$ , $\forall n\in$ N.(4) The mapping $T$ is said to be asymptotically quasi-nonexpansive if there exists a sequence $k_{n}'\in[0, \infty)$ with $\lim_{narrow\infty}k_{n}=0$ such that $d(T^{n}x, p)\leq(1+k_{n}^{\sim})d(x, p)$ , $\forall x\in D(T)$ , $\forall p\in F(T)$ , $\forall n\in$ N.