The iterative solution of the equation f ∈ x + Tx for a monotone operator T in L p spaces

Abstract

Suppose T is a multivalued monotone operator (not necessarily continuous) with open domain D ( T ) in L p (2⩽ p -∞), f ∈ R ( I + T ) and the equation f ∈ x + Tx has a solution q ∈ D ( T ). Then there exists a neighbourhood B ∋ D ( T ) of q and a real number r 1 > 0 such that for any r ⩾ r 1 , for any initial guess x 1 ∈ B , and any singlevalued 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 ‖ = 0( n −1/2 ). Furthermore, for X = L p ( E ), μ( E ) < ∞, μ = Lebesgue measure and 1 < p < 2, suppose T is a singe-valued locally Lipschitzian monotone operator with open domain D ( T ) in X . For f ∈ R ( 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.