Iterative solution of nonlinear equations of the monotone and dissipative types

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Abstract
Suppose X = Lp,P≥2, and K is a non-empty closed convex subset of X. Suppose T:K → K is a monotonic Lipschitzian mapping with Lipschitz constant L≥1. Define the sequence {xn}∞ n=0 by xo εK, xn+1=xn + λrn, for n ≥,where λ= ⊂[(p-1)L2 *]-1] L* = (1+L) , and rn = f-xn -TxnThen,{xn}α n=0 converges strongly to a solution of x+Tx = f in K. Convergence is at least as fast as a geometric progression with ratio (1-λ)1/2. A related result deals with the iterative solution of the equation x-λAx = f where A:X → X is a Lipschitzian dissipative operator and λ>0 is a real constant

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