Banach空间中渐近非扩张映射的不动点迭代

Fixed Point Iterations for Asymptotically Nonexpansive Mappings in Banach Spaces

设D是一致凸Banach空间X的非空闭凸子集 ,T∶D→D是渐近非扩张映射且kn ≥ 1 ,∑ ∞n =1(kn- 1 ) <∞ .设T的不动点集F(T) ≠ ,T是全连续的 (X满足Opial条件 ) ,{xn},{yn},{zn}由定义 2给出 ,如果 ∑∞n =1cn <∞ ,∑ ∞n =1c′n <∞ ,∑ ∞n =1c″n <∞ ,且下列条件之一满足 :(i)b″n ∈ [a ,b] ( 0 ,1 ) ;b′n ∈ [0 ,β];bn ∈[0 ,α],αβ+ β <1 ;(ii)b′n ∈ [a ,b] ( 0 ,1 ) ;b″n ∈ [a ,1 ];bn ∈ [0 ,b];(iii)bn ∈[a ,b] ( 0 ,1 ) ;b′n ∈ [a ,1 ],则 {xn},{yn},{zn}强收敛于T的不动点 .( {xn}弱收敛于T的不动点 ) .

Let X be a uniformly convex Banach space,and let D be a nonempty closed,bounded,and convex subset of X,let T be a completely continuous asymptotically nonexpansive self-map of D with {k n} satisfying k n≥1,∑∞ n=1(k n-1)<∞ and F(T)≠.Let {x n},{y n},{z n} be the sequence generated by definition 2 and ∑∞ n=1c n<∞,∑∞ n=1c′ n<∞,∑∞ n=1c″ n<∞ and (i) b″ n∈[a,b](0,1);b′ n∈[0,β];b n∈[0,α],αβ+β<1 or (ii) b′ n∈[a,b](0,1);b″ n∈[a,1];b n∈[0,b] or (iii) b n∈[a,b](0,1);b′ n∈[a,1],then {x n},{y n},{z n} converges strongly to some fixed point of T.