About convergence rates in regularization for ill posed operator equations of hammerstein type

The aim of this paper is to study convergence rates of the regularized solutions in connection with the finite-dimensional approximations for the operator equation of Hammerstein type x+ F2F1(x)=f in reflexive Banach spaces under the perturbations for not only the operators Fi,i=1,2, but also f. | ’ Tap ch´ Tin hoc v` Diˆu khiˆ n hoc, , (2007), 50—58 ı e e . . a ` . ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED OPERATOR EQUATIONS OF HAMMERSTEIN TYPE NGUYEN BUONG1 , DANG THI HAI HA2 1 Vietnamse Academy of Science and Technology, Institute of Information Technology 2 Vietnamese Forestry University, Xuan Mai, Ha Tay Abstract. The aim of this paper is to study convergence rates of the regularized solutions in connection with the finite-dimensional approximations for the operator equation of Hammerstein type x + F2 F1 (x) = f in reflexive Banach spaces under the perturbations for not only the operators Fi , i = 1, 2, but also f . The conditions of convergence and convergence rates given in this paper for a class of inverse-strongly monotone operators Fi , i = 1, 2, are much simpler than those in the past papers. ´ ´ . . ’ T´m t˘t. Muc d´ cua b`i b´o n`y l` nghiˆn c´.u tˆc dˆ hˆi tu cua nghiˆm hiˆu chınh d˜ o a ıch ’ a a a a e u o o o . ’ e e a . . . . ` ´p xı h˜.u han chiˆu cho tr` to´n tu. loai Hammerstein x + F2 F1 (x) = f trong khˆng ’ . ’ u e ınh a o xˆ a . ˜ ` ’ o ’ a ’ a ’ ’ e e o . e o a ’ gian Banach phan xa v´.i nhiˆu khˆng chı c´ o. c´c to´n tu. Fi , i = 1, 2 m` ca o. f . Diˆu kiˆn hˆi tu . . . o ´ . . ` ´ v` tˆc dˆ hˆi tu trong b`i b´o n`y cho to´n tu. diˆu manh Fi , i = 1, 2 l` yˆu nhiˆu a o o o . e a a a a ’ e a e . . . .i c´c kˆt qua tru.´.c. ´ ’ so v´ a e o o 1. INTRODUCTION Let X be a reflexive real Banach space, and X ∗ be its dual which both are strictly convex. For the sake of simplicity the norms of X and X ∗ are denoted by the symbol . . We write x∗ , x or x, x∗ instead of x∗ (x) for x∗ ∈ X ∗ and x ∈ X . Concerning the space X , in addition assume that it possesses the property: the weak convergence and convergence of norms for any sequence follows its strong convergence. Let F1 : X → X ∗ and F2 : X ∗ → X be monotone, in general nonlinear, bounded (. image of any bounded subset is .