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 .
