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An iterative process for nonlinear Lipschitzian strongly accretive mappings in L p spaces. (English) Zbl 0724.65058

A very accurate description of the content of the paper under review is given in the author’s abstract:

“Suppose X=L p (or l p ), p2. 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. Let {C n } n=1 be a real sequence satisfying:

(i) 0<C n k[(p-1)L 2 +2k-1] -1 for each n,

(ii) n=0 C n =·

Then, for arbitrary x 0 X, the sequence

x n+1 =(1-C n )x n +C n Sx n ,n0,

converges strongly to the unique solution of Tx=f. Moreover, if C n =k[(p-1)L 2 +2k-1] -1 for each n, then

x n+1 -qρ n/2 x 1 -q, where q denotes the solution of Tx=f and

ρ=(1-k[(p-1)L 2 +2k-1] -1 )(0,1)·

A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X.”

Reviewer: I.Marek (Praha)

65J15Equations with nonlinear operators (numerical methods)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)