A new extension of parameter continuation method for solving operator equations of the second kind

No 21 June 2021 |p 157 165 157 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO ISSN 2354 1431 http //tckh daihoctantrao edu vn/ A NEW EXTENSION OF PARAMETER CONTINUATION METHOD FOR SOLVING OPERATOR EQUATIONS OF THE[.]

No.21_June 2021 |p.157-165 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO ISSN: 2354 - 1431 http://tckh.daihoctantrao.edu.vn/ A NEW EXTENSION OF PARAMETER CONTINUATION METHOD FOR SOLVING OPERATOR EQUATIONS OF THE SECOND KIND Ngo Thanh Binh1,* * Nam Dinh University of Technology Education, Vietnam Email address: ntbinhspktnd@gmail.com https://doi.org/ 10.51453/2354-1431/2021/521 Article info Recieved: 27/3/2021 Accepted: 3/5/2021 Abstract: In this paper, we propose an extension of the parameter continuation method for solving operator equations of the second kind By splitting of the operator into a sum of two operators: one monotone, Lipschitz-continuous and one contractive, the applicability of the method is broader The suitability of the proposed approach is presented through an example Keywords: Parameter continuation method, Monotone operator, Contractive operator, Operator equations of the second kind, Approximate solution 157 No.21_June 2021 |p.157-165 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO ISSN: 2354 - 1431 http://tckh.daihoctantrao.edu.vn/ MỘT SỰ MỞ RỘNG MỚI CỦA PHƯƠNG PHÁP THÁC TRIỂN THEO THAM SỐ GIẢI PHƯƠNG TRÌNH TỐN TỬ LOẠI HAI Ngơ Thanh Bình1,* * Trường Đại học Sư phạm Kỹ thuật Nam Định, Việt Nam Địa email: ntbinhspktnd@gmail.com https://doi.org/ 10.51453/2354-1431/2021/521 Thông tin viết Ngày nhận bài: 27/3/2021 Ngày duyệt đăng: 3/5/2021 Tóm tắt Trong báo này, đề xuất mở rộng phương pháp thác triển theo tham số giải phương trình tốn tử loại hai Bằng cách tách tốn tử thành tổng hai toán tử: toán tử đơn điệu, liên tục Lipschitz toán tử co, khả áp dụng phương pháp mở rộng Sự phù hợp cách tiếp cận đề xuất trình bày thơng qua ví dụ Từ khóa: Phương pháp thác triển theo tham số, Toán tử đơn điệu, Toán tử co, Phương trình tốn tử loại hai, Giải xấp xỉ Introductions Parameter continuation method (PCM) was suggested and developed by Bernstein [1] and Schauder [3] which is the inclusion of the equation P( x)  into the one-parametric family of space X The monotone operator in Banach space is defined as follows connecting the Definition 1.1 [2, Definition 2] The mapping A , which operates in the Banach space X is called given equation (  1) with a solvable equation monotone if for any elements x1 , x2  X and any (  0) and study the dependence of the solution   the following inequality holds equations G( x,  )  0,   [0,1] from parameter The PCM is a powerful technique for solving operator equations, see for example [5– 7] Gaponenko [2] introduced the PCM for solving operator equations of the second kind x  A( x)  f , (1) where A is a Lipschitz-continuous and monotone operator, which operates in an arbitrary Banach 158 x1  x2    A( x1 )  A( x2 )  x1  x2 (2) Remark 1.1 [2, Remark 1] If X is Hilbert space then the condition of monotonicity (2) is equivalent to the classical condition A( x1 )  A( x2 ), x1  x2  0, x1 , x2  X , N.T.Binh/ No.21_Jun 2021|p.157-165 where ,  is an inner product in the Hilbert space sequence of approximate {x(n, N )}, n  1, 2, X We obtain the following result from the definition above Lemma 1.1 [2, Lemma] Assume that A is a monotone mapping which operates in the Banach constructed solutions by iteration process (3) converges to the exact solution x of the equation (1) Moreover, the following estimates hold q n 1 eqN  1  q eq  x(n, N )  x  space X Then for any elements x1 , x2  X and f (5) any positive numbers 1 ,  ,  1    , the where L is Lipschitz coefficient of the operator following inequality holds A, N is the smallest natural number such that x1  x2  1  A( x1 )  A( x2 )  x1  x2    A( x1 )  A( x2 ) The results obtained by Gaponenko are summarized in Theorem and Theorem q L  1, n  1, 2, N MAIN RESULTS Consider the general operator equation (1) Theorem 1.1 [2, Theorem 1] Suppose that the x  A( x)  f , mapping A, which operates in the Banach space X is Lipschitz - continuous and monotone Then the equation (1) has a unique solution for any space X into X , f is a given function in X Without loss of generality, one can express the element f  X operator A as a composition of two operators A1 The following iteration process is constructed to find approximate solutions of the equation (1) xi 1   where A is a nonlinear operator from a Banach 1 A( xi )  A( x j )  N N  and A2 Then the equation (1) can be rewritten as follow A( x p )  f , i, j, , p  0, 1,  3 N x  A1 ( x)  A2 ( x)  f (6) N terms Theorem 2.1 Assume that A1 is a Lipschitz- The symbolic notation (3) should be understood as continuous the following iteration processes, which consist of contractive operator Then the equation (1) has a N iteration processes unique solution xi 1   A( xi )  x(1) j ,i x(1) j 1  0,1, 2, , (2)   AG11 ( x(1) j )  xl , j  0,1, 2, , , x(pN11)   AG11 GN11 ( x(pN 1) )  f , p  0,1, 2,  4a   4b   4c   4d  and monotone operator, A2 is Proof We take a minimal natural number that q1   L  1,   a N such , where L is the Lipschitz N coefficient of the operator A1 The equation (6) can be written as the following form For simplicity, assume that A(0)  and the x  N A1 ( x)  A2 ( x)  f (7) number of steps in each iteration scheme of the iteration process (3) is the same and equals n Denoting x(n, N )  xn as the approximate solutions of the equation (1), which is constructed by the iteration process (3) In this case, Gaponenko received the error estimations of approximate solutions of the equation (1), which are presented in the following theorem Theorem 1.2 [2, Theorem 2] Assume that the conditions of Theorem 1.1 are satisfied Then the Consider the following subsidiary problems Problem ( N  ) Consider the operator equation x   A1 ( x)  A2 ( x)  f (8) We shall carry out a change of variable x(1)  x   A1 ( x)  G1 ( x) (9) We have 159 N.T.Binh/ No.21_Jun 2021|p.157-165  A1 ( x)   A1 ( x )   L x  x Hence  A1G11 is a contractive operator with  q1 x  x , x, x  X equation (12b) has a unique solution for any Hence  A1 is a contractive operator with contraction coefficient equal to q1   L  Then the x equation (9) has a unique solution for any (1) X , i.e., the operator contraction coefficient equal to q1  Then the G11 ( x(1) ) x(2)  X , i.e., the operator G21 is determined in the whole space X By Lemma 1.1, for any x(2) , x (2)  X , we have is determined in the whole space X By virtue of the G21 ( x(2) )  G21 ( x (2) )  x(1)  x (1) monotonicity of the operator A1 , the operator G11  x  x   [ A1 ( x)  A1 ( x )]  x  x  2 [ A1 ( x)  A1 ( x )] is Lipschitz - continuous with Lipschitz coefficient equal to Indeed, for any x(1) , x (1)  X , we have G11 ( x(1) )  G11 ( x (1) )  x  x  x(1)  x (1)   [ A1G11 ( x(1) )  A1G11 ( x (1) )]  x(2)  x (2)  x  x    A1 ( x)  A1 ( x ) Thus the operator G21 is Lipschitz - continuous  x(1)  x (1) with Lipschitz coefficient equal to After changing the variables (12a) and (12b), the equation (11) will take the following form After changing the variable (9), the equation (8) will take the following form P2 ( x(2) )  x(2)  A2G11G21 ( x(2) )  f   P1 x(1)  x(1)  A2G11 ( x (1) )  f (10) For any x(1) , x (1)  X , we have (13) For any x(2) , x (2)  X , we have A2G11G21 ( x(2) )  A2G11G21 ( x (2) )  q2 x(2)  x (2) A2G11 ( x(1) )  A2G11 ( x (1) )  q2 G11 ( x (1) )  G11 ( x (1) )  q2 x(1)  x (1) , where q2 is contraction coefficient of the operator A2 Thus A2G11 is a contractive operator with equation (10) has a unique solution for any f  X Consequently, the equation (8) has a unique ( N  ) Consider the operator equation (11) We shall carry out two changes of variables 12a  x(1)  x   A1 ( x)  G1 ( x), x x (1) equation (13) has a unique solution for any f  X x(2 ) for any f  X Problem ( N  ) Consider the operator x  N A1 ( x)  A2 ( x)  x  A1 ( x)  A2 ( x)  f (14) We shall carry out N   A1G11 ( x(1) )  G2 ( x(1) ) For any x , x 12b  (1) x(2)  x(1)   A1G11 ( x(1) )  G2 ( x(1) ), , x( N )  x( N 1)   A1G11 GN11 ( x( N 1) )  GN ( x( N 1) ) we G31 , G41 , , GN1  X , we have  A1G11 ( x(1) )   A1G11 ( x (1) )   L x(1)  x (1)  q1 x(1)  x (1) show that the 15a  15b  15c  15d  operators are determined in the whole space X and are Lipschitz - continuous with Lipschitz coefficients equal to Hence after the change of variables (15a)-(15d) the equation (14) will take the following form 160 changes of variables x  x   A1 ( x)  G1 ( x), Similarily, (1) N (1) x  2 A1 ( x)  A2 ( x)  f (2) contraction coefficient equal to q2  Then the equation solution x( ) for any f  X with Therefore the equation (11) has a unique solution contraction coefficient equal to q2  Then the Problem Thus A2G11G21 is a contractive operator N.T.Binh/ No.21_Jun 2021|p.157-165 PN ( x( N ) )  x( N )  A2G11 GN1 ( x( N ) )  f solutions of the integral equation (11) can be found by (16) iteration processes For any x( N ) , x ( N )  X , we have A2G11 GN1 ( x( N ) )  A2G11 xi 1   A1 ( xi )  x (1) j , i  0,1, 2, , GN1 ( x ( N ) ) x(1) j 1  q2 x( N )  x ( N ) Thus A2G11  (2)  A1G11 ( x(1) j )  xl , j  0,1, 2, , (2) 1 1 (2) xl(2) 1   A2 G1 G2 ( xl )  f , l  0,1, 2, , x0  GN1 is a contractive operator with contraction coefficient equal to q2  Then the integral equation (16) has a unique solution for any f  X Consequently, the equation (14) has a unique solution x( N )  x  X for any f  X , i.e the equation (1) has a unique solution x  X for any f  X This completes the proof 18a  18b  f 18c  Finally, we construct the iterative algorithm to find approximate solution of the Problem N The approximate solutions of the integral equation (16) are obtained by using the standard iteration process x(pN1)   A2G11 GN1 ( x(pN ) )  f , p  0,1, 2, , x0( N )  f At the same time we will use “subsidiary” iteration We now construct the iterative algorithm to processes to invert the operators G1 , G2 , , GN at find approximate solution of the operator equation each step of this iteration process when calculating (1) Firstly, we construct the iterative algorithm to find approximate solution of the Problem The the value of G11G21 approximate solutions of the equation (10) are obtained by using the standard iteration process (1) 1 (1) x(1) j 1   A2G1 ( x j )  f , j  0,1, 2, , x0  f At the same time at each step of above iteration process when calculating the value G11 ( x(1) j ) we will again use the standard iteration process (1) xi 1   A1 ( xi )  x(1) j , i  0,1, 2, , x0  x j As a result, the approximate solutions of the equation (8) can be found by the following iteration processes xi 1   A1 ( xi )  x(1) j , i  0,1, 2, , x(1) j 1   A2G11 ( x(j1) )  f, j  0,1, 2, , x0(1)  17a  f 17b  GN1 ( x(pN ) ) Hence the approximate solutions of the equation (14) can be found by iteration processes 19a  (2) 1 (1) x(1) 19b  j 1   A1G1 ( x j )  xl , j  0,1, 2, , , 19c  x(pN1)   A2G11 GN1 ( x(pN ) )  f , p  0,1, 2, , x0( N )  f 19d  xi 1   A1 ( xi )  x(1) j , i  0,1, 2, , The iteration processes (19a)-(19d) can be written as the following symbolic notation xi 1   1 A1 ( xi )  A1 ( x j )   A1 ( xh )  A2 ( x p )  f , N N N N terms  20  i, j, , p  0,1, Assume that the number of steps in each Next, we construct the iterative algorithm to find approximate solution of the Problem The approximate solutions of the integral equation (13) iteration scheme of iteration processes (19a)-(19d) are obtained by using the standard iteration process on N Hence we denote x(n, N )  xn We have the xl(2) 1   A2G11G21 ( xl(2) )  f,l  0,1, 2, , x0(2)  f is the same and equals n Let xn be approximate solutions of the equation (1) Note that xn depends following theorem At the same time we will use “subsidiary” iteration Theorem 2.2 Let the assumptions of Theorem processes to invert the operators G1 , G2 at each 2.1 be satisfied Then the sequence of approximate step of this iteration process when calculating the solutions value of G11G21 ( xl(2) ) Hence the approximate iteration processes (19a)-(19d) converges to the {x(n, N )}, n  1, 2, constructed by exact solution x  X of the operator equation (1) 161 N.T.Binh/ No.21_Jun 2021|p.157-165 Moreover, the x(n, N )  x  following estimates Consequently, the values G11 ( x(1) j ) are calculated hold   q1n 1  q2 n 1 eq1N   q2n 1  f ,  q1  q2 1  q1  q2 e   with the error 1 (n)  1 (n)  xn  x*   (n), where (21)  ( n)  where N is the smallest natural number such that q1  L  , L is Lipschitz coefficient of the N q1n 1  q2n 1 f  q1  q2 (22) Since A2 is a contractive operator with contraction coefficient equal to q2  , the error 1 (n) in operator A1 , q2 is a contraction coefficient of the specifying the argument of the operator A2 is operator A2 , n  1, 2, equivalent to the error q2 1 (n) in specifying the Proof Without loss of generality, we assume that right - hand side f of the integral equation (8) On A1 (0)  0, A2 (0)  Let us consider successive problems 1, 2, , N The approximate solutions the other hand, the operator P11 is Lipschitz - of Problem assumes are obtained by iteration continuous with Lipschitz coefficient equal to processes (17a)-(17b) The values G11 ( x(1) j ) are Indeed, for any f , f  X , we have  q2 calculated by using the iteration process (17a) with P11 ( f )  P11 ( f )  x(1)  x (1) the error  x(1)  x (1)  A2G11 ( x(1) )  A2G11 ( x (1) )   A2G11 ( x(1) )  A2G11 ( x (1) )  q n 1 (1)  xj  q1 xn  x *  x(1)  x (1)  A2G11 ( x(1) )  A2G11 ( x (1) )  A2G11 ( x(1) )  A2G11 ( x (1) ) For any k  1, 2, , n , we have xk(1)  xk(1)1  A2G11 ( xk(1)1 )  A2G11 ( xk(1) )  q2 xk(1)1   xk(1)  q2k 1 x1(1)  x0(1) ,  f  f  q2 x (1)  x (1) , so that so that (1) x(1)  x(1) j j  x j 1    q2j 1  q2j     P1 ( x (1) )  P1 ( x (1) )  q2 x (1)  x (1) P11 ( f )  P11 ( f )   x1(1)  x0(1)  x0(1)   q2  x1(1)  x0(1)  x0(1)  q2 x(1)  x0(1)  x0(1)  q2 j Since A1 (0)  , we have G1 (0)    A1 (0)  Hence the substitution of the error q2 1 (n) into the right – hand side of the integral equation (8) causes an error of not more than q2 1 (n)  q2 in the corresponding solution x(1) The error of Hence x1(1)  x0(1)  A2G11 ( x0(1) )  f  x0(1)  A2G11 ( f )  A2G11 (0)  q2 f an iteration process in the calculation of x(1) equals q2 n 1 f Therefore we have  q2 Then from above inequality it follows that x(1)  j ff  q2  q2j (1)  q2j x1  x0(1)  x0(1)  q2 f  f  q2  q2  q2n  q2n 1  q2 f  f  f  q2  q2 xn(1)  x(1)  q2 q n 1 1 (n)  f  q2  q2 The inverse substitution, i.e., the transition from the variable x(1) to the variable x again introduces the error 1 (n) Then the error of approximate solutions xn of Problem gives the estimate 162 N.T.Binh/ No.21_Jun 2021|p.157-165 xn  x( )  q2 q n 1 1 (n)  1 (n)  f  q2  q2 x (1)  j  q2 j  q2 x1(2)  x0(2)  x0(2)  q2  q2 j  q2 f + f The approximate solutions of Problem assumes are obtained by iteration processes (18a)-(18c)  q2  q2 n  q2 f + f The values G11G21 ( xl(2) ) are calculated by using  q n 1  1 (n)  f  q2  q2 iteration processes G11 ( x(1) j ) (18a)-(18b) The values f Therefore the values G11 ( x(1) j ) are calculated with are calculated by using the iteration the error process (18a) with the error q n 1 (1) xn  x  xj  q1 xn  x*  * q1n 1  q2 n 1 f   (n)  q1  q2 Since  A1 is a contractive operator We have (1) x(1)  x(1) j j  x j 1   q2 n 1  q2 with contraction coefficient equal to q1  , the error  x1(1)  x0(1)  x0(1)  (n) in specifying the argument of the operator Since the operator G21 is Lipschitz - continuous  A1 is equivalent to the error q1 (n) with Lipschitz coefficient equal to , it follows that in the right - hand side x(2) of the specifying equation (12b): x(1)   A1G11 ( x(1) )  x(2) Since xk(1)  xk(1)1  G21 ( xk(2) )  G21 ( xk(2) 1 ) the operator G21 is Lipschitz – continuous with  xk(2)  xk(2) 1 , k  1, 2, , n Lipschitz coefficient equal to , the substitution of Hence (2) x(1)  x(2) j j  x j 1  the error q1 (n) into the right –hand side of the  x1(2)  x0(2)  x0(2) integral equation (12b) causes an error of not more than q1 (n) in the corresponding solution x(1) For any k  1, 2, , n , we have The error of an iteration process in the calculation 1 1 (2) 1 1 (2) xk(2)  xk(2) 1  F1G1 G2 ( xk 1 )  F1G1 G2 ( xk  )  q2 xk(2) 1  xk(2) 2   q2k 1 x1(2)  x0(2) of x(1) equals q1n 1 (2) x For any l  1, 2, , n ,  q1 l we have Thus  x(1) j  q2 j 1  q2 j 2  xl(2)  xl(2)  xl(2) 1    q2  x1(2)  x0(2)  x0(2)   q2l 1  q2l    q2 j (2) (2)  x1  x0  x0(2)  q2  Since A1 (0)  0, it follows that G1 (0)    A1 (0)  and G2 (0)    A1G11 (0)  Therefore x1(2)  x0(2)   A2G11G21 ( f ) f  f A2G11G21 ( f  q2 f ) A2G11G21 (0)  x1(2)  x0(2)  x0(2) Then  q2l x1(2)  x0(2)  x0(2)  q2  q2  q2l  q2  q2  q2 n  q2    q2  x1(2)  x0(2)  x0(2)  q2 n 1  q2 f  f f  f f the error of an iteration process in the calculation of x(1) equals  (n) Hence  (n)  xn(1)  x(1)  q1 (n)   (n)  q11 (n)   (n) Then we have 163 ... introduced the PCM for solving operator equations of the second kind x  A( x)  f , (1) where A is a Lipschitz-continuous and monotone operator, which operates in an arbitrary Banach 158 x1... coefficient equal to After changing the variables (1 2a) and (12b), the equation (11) will take the following form After changing the variable (9), the equation (8) will take the following form P2 (... monotone operator, A2 is Proof We take a minimal natural number that q1   L  1,   a N such , where L is the Lipschitz N coefficient of the operator A1 The equation (6) can be written as the