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