tailieunhanh - Continuous regularization method for ill posed operator equations of hammerstein type
The aim of this paper is to study a method of approximating a solution of the operator equation of Hammerstein type x + F2F1(x) = f on the base of constructing a system of differential equations of the first order, where Fi , i = 1, 2, are the continuous monotone operators in real Hilbert space H. Then this method is considered in connection with finite-dimentional approximations for H. | ’ Tap ch´ Tin hoc v` Diˆu khiˆn hoc, , (2007), 99–108 ı a ` e e . . . CONTINUOUS REGULARIZATION METHOD FOR ILL-POSED OPERATOR EQUATIONS OF HAMMERSTEIN TYPE NGUYEN BUONG1 , DANG THI HAI HA2 1 Viˆn Cˆng nghˆ thˆng tin, Viˆn Khoa hoc v` Cˆng nghˆ Viˆt Nam e o e o e e e . . . . a o . . 2 Vietnamese Forestry University, Xuan Mai, Ha Tay Abstract. The aim of this paper is to study a method of approximating a solution of the operator equation of Hammerstein type x + F2 F1 (x) = f on the base of constructing a system of differential equations of the first order, where Fi , i = 1, 2, are the continuous monotone operators in real Hilbert space H . Then this method is considered in connection with finite-dimentional approximations for H. ´ ´ ’ o a a ’ e ınh T´m t˘t. Muc d´ cua b`i b´o l` nghiˆn c´.u mˆt ph´p xˆp xı nghiˆm cua tr` o a a a a e u . . . ıch ’ ´ to´n tu. loai Hammerstein x + F2 F1 (x) = f trˆn viˆc xˆy hˆ tr` vi phˆn cˆp a ’ . e e a e ınh a a . . . . ’. dˆy c´c to´n tu. Fi , i = 1, 2, l` diˆu v` liˆn tuc trong khˆng gian Hilbert H . Sau d´, ’ a a e a e . o o mˆt, o a a o . . ´ ´ ` ’ e e e o a a e a ’ u e ph´p n`y x´t liˆn kˆt v´.i viˆc xˆ p xı h˜.u han chiˆu cua H. . . . 1. INTRODUCTION Let H be a real Hilbert space with norm and scalar product denoted by . and x∗ , x , respectively. Let Fi , i = 1, 2, be monotone, in general nonlinear, bounded (. image of any bounded subset is bounded) and continuous operators. Our main aim of this paper is to study a stable method of finding an approximative solution for the equation of Hammerstein type x + F2 F1 (x) = f, f ∈ R(I + F2 F1 ), () where I and R(A) denote the identity operator in H and the range of the operator A, respectively. Note that the solution set of (), denoted by S0 , is closed convex (see [1]). Fih Usually instead of Fi , i = 1, 2, and f we know their monotone continuous approximations and fδ such that h F1 (x) − F1 (x) h F2 (x)
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