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

✍ Scribed by C.E. Chidume

• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 253 KB
• Volume: 116
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1016/s0022-247x(86)80017-8

✦ Synopsis

Suppose T is a multivalued monotone operator (not necessarily continuous) with open domain D(T) in L e (2 ~< p < co), f~ R(I + T) and the equation fe x + Tx has a solution qcD(T). Then there exists a neighbourhood BcD(T) of q and a real number rt > 0 such that for any r >~ rl, for any initial guess xl ~ B, and any singlevalued section To of T, the sequence {xn},~_l generated from x 1 by xn+~=

(1--C,,)x,+C,(f--Tox,) remains in D(T) and converges strongly to q with Irx, -q]] =O(n-~/2). Furthermore, for X= Lp(E), /,t(E) < o~, # = Lebesgue measure and l <p < 2, suppose T is a singe-valued locally Lipschitzian monotone operator with open domain D(T) in X. Forf~ 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.