The purpose of this paper is to give a theoretical analysis of Tikhonov for solving a system of illposed equations involving linear and bounded mappings in real Hilbert spaces under perturbative operators and right hand side. Then, an example of finding a common solution of N systems of linear algebraic equations with singular matrixes is given. | Nguyễn Đình Dũng và đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 83(07): 73 - 79 REGULARIZATION FOR A COMMON SOLUTION OF A SYSTEM OF ILLPOSED EQUATIONS INVOLVING LINEAR BOUNDED MAPPINGS WITH PERTURBATIVE DATA Nguyen Dinh Dung1*, Nguyen Buong2 1 Thai Nguyen University, Thai Nguyen, Viet Nam Vietnamese Academy of Science and Technology Institute of Information Technology, Ha Noi, Viet Nam 2 ABSTRACT The purpose of this paper is to give a theoretical analysis of Tikhonov for solving a system of illposed equations involving linear and bounded mappings in real Hilbert spaces under perturbative operators and right hand side. Then, an example of finding a common solution of N systems of linear algebraic equations with singular matrixes is given. Key words: Tikhonov regularization, ill-posed problem INTRODUCTION* h where, A j Let X and Y j be Hilbert spaces with scalar C 0 . With the above condition on Aj , each j product and norm of X denoted by the symbols .,. and . X respectively. Let A j , j 1,., N , equation (1) is ill-posed. By this we mean that the solution set S j does not depend continuously on be N linear bounded mappings from X to Y j . Consider the following problem: Find an ~ x X such that Aj ~ x f j , j 1,., N , (1) Sj . X . There fore S is also closed convex in X . We are especially interested in the situation where the data f j and A j are not exactly in the know. j We have only j approximation f j of the data f j and A hj of Aj satisfying j Yj h Yj Yj x x* 2 X , (4) small parameter of regularization. For the case h where Aj Aj , it is proved in [1] that each j minimization problem in (4) hase unique solution j2 , 0 then x j j converges to a x j and if ~ solution x j satisfying ~ x j x* min x x* X , j 1,., N . X , j x S j Our problem: Finding h , x ( hj, j ) such that x ( hj, j ) ~ x as , h, j 0 , a relation h, j h , such that x j ~ x as h, 0 and .
