Convergence of iterative methods for unbounded operator equations

Abstract

w h e r e ll'll i s t h e norm o f X, II ' i [= t h e norm o f Y. We examine the existence of a strong solution of Eq. (i) and the convergence to it of iterative processes. Definition. Element xe~X is called a strong solution of Eq. (i) if there is a sequence {z,} • ~ (A), convergent to x* such that I[A(xn)Ii2 ~ 0 as n ~ ~. It is well known that in essence iterative methods find the solution of a complex problem. as the limit of the solutions of simpler problems. Since, for linear equations, we use the principle of superposition, and the solution can often be found explicitly, we basically choose as the simpler problems linearizations of the initial equation. Linearization of an unbounded operator leads naturally to unbounded linear operators. We introduce the concept of linearizer of opera~or A at point ~(A), which generalizes the Frechet derivative [i, p. 637] to unbounded unsmooth operators. Definition. Linear operator C: X + Y is called a linearizer of operator A at point ~ ~(A) if ~ (A) ~ (C) and there exist numbers e ~ 0, 6 > 0 such that I[A(x)--A(~)--C(x--~)I[,<eI[X~--~IIt for all x ~(A) satisfying [Ix x[[1 < 6. We call e and 6 the linearizer constants; they indicate, respectively, the accuracy and radius of the neighborhood of approximation of ~he initial operator by the linear operator C at point x. If A has a Frechet derivative at point ~ ~ (A), then it is the linearizer of A at this point. But as distinct from the Frechet derivative, it is not required in the definition of linearizer that: a) operator C be bounded, and b) that numbers e and 6 be dependent. For Frechet derivatives C ~Z (X, Y) (the space of linear bounded operators from X into Y), we have E(6) ~ 0 as 6 ~ 0. In the future, we shall assume that ~ (C) and R(C) (the domain of definition and range of values of linearizer Cx) are independent of variation of x in ~ (A) and that ~ (A) = (C) n S( x~ r), R(C) = Y. THEOREM i. Let operator A have at every point x ~U ~ a linearizer Cx with constants gx, 6x which satisfy the conditions: i) Cx is invertible and [[Cilt[~(y.x)<~x<?, 2) Ex'Tx ~ O < I, 3) 7x. llA(x)ll2/6x < K, K is constant, 4) .llA(xO)ll=/<l o ) < r . Institute of Mathematics and Mechanics, Academy of Sciences of the Kazakh SSR. Translated from Matematicheskie Zametki, Vol. 41, No. 5, pp. 637-645, May, 1987. Original article submitted May 28, 1986.