TAILIEUCHUNG - Discrepancy principle and ill-posed equations with M-accretive perturbations

In this paper, on the base of the discrepancy principle for regularization parameter choice, the convergence rates of the regularized solution as well as its Galerkin approximations for nonlinear ill-posed problems with m-accretive perturbations are established without demanding the weak continuity of the duality mapping of the Banach spaces. | Journal of Science of Hanoi National University of Education Natural sciences Volume 52 Number 4 2007 pp. 47- 54 DISCREPANCY PRINCIPLE AND ILL-POSED EQUATIONS WITH M- ACCRETIVE PERTURBATIONS Dr. Nguyen Buong Vietnamse Academy of Science and Technology Institute of Information Technology 18 Hoang Quoc Viet Cau Giay Ha Noi E-mail nbuong@ Quang Hung Ministry of Industry 54 Hai Ba Trung Street Hoan Kiem Ha Noi hungvq@ Abstract. In this paper on the base of the discrepancy principle for regulariza- tion parameter choice the convergence rates of the regularized solution as well as its Galerkin approximations for nonlinear ill-posed problems with m-accretive per- turbations are established without demanding the weak continuity of the duality mapping of the Banach spaces. 1 2 1 Introduction Let X be a real uniformly convex Banach space having the property of approximations see 11 andX the dual space of X be strickly convex. For the sake of simplicity the norms of X and X will be denoted by the symbol . We write hx x i instead of x x for x X and x X . Let A be an m-accretive operator in X . see 11 i hA x h A x J h i 0 x h X where J is the normalized dual mapping of X the mapping from X onto X satisfies the condition hx J x i kJ x kkxk kJ x k kxk x X and ii R A λI X for each λ gt 0 where R A denotes the range of A and I is the identity operator inX. 1 Key words Accretive operators strictly convex Banach space Fr chet differentiable and Tikhonov regularization. 2 2000 Mathematics Subject Classification 47H17 CR . 47 NGUYEN BUONG AND VU QUANG HUNG We are interested in solving the operator equation A x f f X where A is an m-accretive operator in X. Note that if A is accretive satisfies condition i and demi-continuous or weak continuous then it is m-accretive see 3 12 . The existence of solution of is shown in 7 11 . Without additional conditions on the structure of A such as strongly or uniformly accretive property .

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