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An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces. (English) Zbl 0801.47040

Summary: Suppose X is an s-uniformly smooth Banach space (s>1). Let T:XX be a Lipschitzian and strongly accretive map with constant k(0,1) and Lipschitz constant L. Define S: XX by Sx=f-Tx+x. For arbitrary x 0 X, the sequence {x n } n=1 is defined by

x n+1 =(1-α n )x n +α n Sy n ,y n =(1-β n )x n +β n Sx n ,n0,

where {α n } n=0 , {β n } n=0 are two real sequences satisfying:

(i) 0α n p-1 2 -1 s(k+kβ n -L 2 β n ) (w+h) -1 for each n,

(ii) 0β n p-1 min{k/L 2 ,sk/(w+h)} for each n,

(iii) n α n =,

where w=b(1+L) s and b is the constant appearing in a characteristic inequality of X, h=max{1,s(s-1)/2}, p=min{2,s}. Then {x n } n=1 converges strongly to the unique solution of Tx=f. Moreover, if p=2, α n =2 -1 s(k+kβ-L 2 β) (w+h) -1 , and β n =β for each n and some 0βmin{k/L 2 ,sk/(w+h)}, then

x n+1 -qρ n/s x 1 -q,

where q denotes the solution of Tx=f and

ρ=1 - 4 -1 s 2 (k+kβ-L 2 β) 2 (w+h) -1 (0,1)·

A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X. Suppose X is an m- uniformly convex Banach space (m>1) and c is the constant appearing in a characteristic inequality of X, two similar results are showed in the cases of L satisfying (1-c 2 )(1+L) m <1+c-cm(1-k) or (1-c 2 )L m <1+c-cm(1-s).

47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
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