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- W2029405004 abstract "Let A be a closed (linear) operator with domain D ( A ) and range R ( A ) in a Banach space X . Let { A x } and { B α } be two nets of bounded operators on X such that ∥ A α ∥ ⩽ M , R ( A α ) ∪ R ( B α ) ⊂ D ( A ) and B α A ⊂ AB α = A α − I , w − lim α AA α x = 0 for all x ϵ X , and s-lim α A α Ax = 0 for all x ϵ D ( A ). We first formulate an abstract mean ergodic theorem which contains as special cases many known or new mean ergodic theorems. The theorem is then used to study the approximate solutions of the functional equation Ax = y . One of the results says that if y ϵ A(D(A) ∩ R(A) ) , then {− B α y } converges to the unique solution x in R(A) . The general results provide a unified approach to many specific methods for solving Ax = y . The cases treated here include: (1) A = I − T with T ϵ B ( X ); (2) A is the generator of a (Y) semigroup; (3) A is the generator of a strongly continuous cosine operator function; (4) the operator equation A 1 E + EA 2 = B ." @default.
- W2029405004 created "2016-06-24" @default.
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- W2029405004 date "1989-12-01" @default.
- W2029405004 modified "2023-10-17" @default.
- W2029405004 title "Mean ergodic theorems and linear functional equations" @default.
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- W2029405004 doi "https://doi.org/10.1016/0022-1236(89)90018-9" @default.
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