A Quasi-Residual Principle in Regularization for a Common Solution of a System of Nonlinear Monotone Ill-Posed Equations

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A Quasi-Residual Principle in Regularization for a Common Solution

of a System of Nonlinear Monotone Ill-Posed Equations

Nguyen B u o n g 1'*, Tran Thi H uong2**”, and Nguyen Thi Thu T huy3****

1 Vielnam Academy of Science and Technology

18 Hoang Quoc Vie I, Hanoi, Vielnam

Thainguyen N a tio n a l U niversity, Thainguyen Ci.li/, Vielnurn

: Thainguyen College of Sciences, Tliainguyen University, Vielnam

Received March 23, 2013

Abstract—In this paper we study the Browder—Tikhonov regularization method for finding a

common solution for a system of nonlinear ill-posed equations with potential, hemicontinuous and

monotone mappings in Banach spaces. We give a principle, named quasi-residual, to choose a value

oi the regularization parameter and an estimate of convergence rates for the regularized solutions.

DOI: 1 0 . 3 1 0 3 /S 1 0 6 6 3 6 9 X 1 6 0 3 0 0 6 3

Keywords: monotone operators, hemicontinuous, strictly convex Banach space, Fréchet dif￾ferentiable. the Browcler—Tikhonov regularization method.

INTRO DUCTIO N

Let E he a real rellexive Banach space and E* be its dual space, which both are assumed to be strictly

convex. For Ihe sake of simplicity, norms of E and E* are denoted by the symbol || ■ ¡|. We use the

symbol (x*,x) to denote the value of the linear and continuous functional x* G E* at the point x & E.

In addition, we assume that E possesses the property: Weak convergence and convergence of norms lor

any sequence in E imply its strong convergence.

Consider the problem of finding a common solution for a system of the following equations

Ai(x) = fi, f i e E * , i = 0 A , . . . , N , (I)

where N is a fixed positive integer and A.-L is a potential, hemicontinuous and monotone mapping with

domain 'D(A) = E for ¿ = 0 .1 ,..., N . Recall that a mapping A of domain 'D(A) Ç E into E* is called

A-inverse strongly-monotone if and only if for any x, y e V{A)

(A{x) - A (y),x - y) > A||.4(x) - -4(j/)||2,

where A is some positive constant. II is called monotone if and only il it satisfies the following condition

{A(x) - A(y),x — y ) > 0:

strictly monotone at a point y G D(A) if and only if the symbol “ = ” in the latter inequality implies x := y,

and potential, if and only if A(x) = <p'(x), the Gâteaux derivative of a convex functional if(x). Denote by

N

S-i the set of solutions fo ri-th equation in ( 1 ). Throughout this paper, we assume that S := ri «S'» ^ 0-

" The text was submitted by the authors in English.

E-lliail: nbuongaioit. ac . vn.

E-mail: [email protected].

E-lliail: [email protected].

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