Iterative solution of nonlinear equations of the monotone and dissipative types

Abstract

Suppose X = L_p,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 {x_n}^∞ _n=0 by x_o εK, x_n+1=x_n + λr_n, for n ≥,where λ= ⊂[(p-1)L² _*]^-1] L* = (1+L) , and r_n = f-x_n -Tx_nThen,{x_n}^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

Keywords:

• Iteration methods
• monotone nonlinear equations

AMS(MOS):

• 47H15

_*Research supported by a grant from The Third world Academy of Sciences (TWASRG MP.88-14),I.C.T.P., Trieste, ITALY

_*Research supported by a grant from The Third world Academy of Sciences (TWASRG MP.88-14),I.C.T.P., Trieste, ITALY

Notes

_*Research supported by a grant from The Third world Academy of Sciences (TWASRG MP.88-14),I.C.T.P., Trieste, ITALY