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The iterative solution of the equation fx+Tx for a monotone operator T in L p spaces. (English) Zbl 0606.47067
Suppose T is a multivalued monotone operator (not necessarily continuous) with open domain D(T) in L p (2p<), fR(I+T) and the equation fx+Tx has a solution qD(T). Then there exists a neighbourhood BD(T) of q and a real number r 1 >0 such that for any rr 1 , for any initial guess x 1 B, and any single-valued section T 0 of T, the sequence {x n } n=1 generated from x 1 by x n+1 =(1-C n )x n +C n (f-T 0 x n ) remains in D(T) and converges strongly to q with x n -q=O(n - ). Furthermore, for X=L p (E), μ(E)<, μ=Lebesgue measure and 1<p<2, suppose T is a single-valued locally Lipschitzian monotone operator with open domain D(T) in X. For fR(I+T), a solution of the equation x+Tx=f is obtained as the limit of an iteratively constructed sequence with an explicit error estimate.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
65J15Equations with nonlinear operators (numerical methods)