The results of this thesis are: Propose and prove the strong convergence of a new modification of the Newton-Kantorovich iterative regularization method (0.6) to solve the problem (0.1) with A is a monotone mapping from Banach space E into the dual space E ∗ , which overcomes the drawbacks of method (0.6).
MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY *** NGUYEN DUONG NGUYEN NEWTON-KANTOROVICH ITERATIVE REGULARIZATION AND THE PROXIMAL POINT METHODS FOR NONLINEAR ILL-POSED EQUATIONS INVOLVING MONOTONE OPERATORS Major: Applied Mathematics Code: 46 01 12 SUMMARY OF MATHEMATICS DOCTORAL THESIS Hanoi - 2018 This thesis is completed at: Graduate University of Science and Technology - Vietnam Academy of Science and Technology Supervisor 1: Prof Dr Nguyen Buong Supervisor 2: Assoc Prof Dr Do Van Luu First referee 1: Second referee 2: Third referee 3: The thesis is to be presented to the Defense Committee of the Graduate University of Science and Technology - Vietnam Academy of Science and Technology on 2018, at o’clock The thesis can be found at: - Library of Graduate University of Science and Technology - Vietnam National Library Introduction Many issues in science, technology, economics and ecology such as image processing, computerized tomography, seismic tomography in engineering geophysics, acoustic sounding in wave approximation, problems of linear programming lead to solve problems having the following operator equation type (A Bakushinsky and A Goncharsky, 1994; F Natterer, 2001; F Natterer and F Wă ubbeling, 2001): A(x) = f, (0.1) where A is an operator (mapping) from metric space E into metric space E and f ∈ E However, there exists a class of problems among these problems that their solutions are unstable according to the initial data, i.e., a small change in the data can lead to a very large difference of the solution It is said that these problems are ill-posed Therefore, the requirement is that there must be methods to solve ill-posed problems such that the smaller the error of the data is, the closer the approximate solution is to the correct solution of the derived problem If E is Banach space with the norm then in some cases of the mapping A, the problem (0.1) can be regularized by minimizing Tikhonov’s functional: Fαδ (x) = A(x) − fδ + α x − x+ , (0.2) with selection suitable regularization parameter α = α(δ) > 0, where fδ is the approximation of f satisfying fδ − f ≤ δ and x+ is the element selected in E to help us find a solution of (0.1) at will If A is a nonlinear mapping then the functional Fαδ (x) is generally not convex Therefore, it is impossible to apply results obtained in minimizing a convex functional to find the minimum component of Fαδ (x) Thus, to solve the problem (0.1) with A is a monotone nonlinear mapping, a new type of Tikhonov regularization method was proposed, called the Browder-Tikhonov regularization method In 1975, Ya.I Alber constructed Browder-Tikhonov regularization method to solve the problem (0.1) when A is a monotone nonlinear mapping as follows: A(x) + αJ s (x − x+ ) = fδ (0.3) We see that, in the case E is not Hilbert space, J s is the nonlinear mapping, and therefore, (0.3) is the nonlinear problem, even if A is the linear mapping This is a difficult problem class to solve in practice In addition, some information of the exact solution, such as smoothness, may not be retained in the regularized solution because the domain of the mapping J s is the whole space, so we can’t know the regularized solution exists where in E Thus, in 1991, Ng Buong replaced the mapping J s by a linear and strongly monotone mapping B to give the following method: A(x) + αB(x − x+ ) = fδ (0.4) The case E ≡ H is Hilbert space, the method (0.3) has the simplest form with s = Then, the method (0.3) becomes: A(x) + α(x − x+ ) = fδ (0.5) In 2006, Ya.I Alber and I.P Ryazantseva proposed the convergence of the method (0.5) in the case A is an accretive mapping in Banach space E under the condition that the normalized duality mapping J of E is sequentially weakly continuous Unfortunately, the class of infinite-dimensional Banach space has the normalized duality mapping that satisfies sequentially weakly continuous is too small (only the space lp ) In 2013, Ng Buong and Ng.T.H Phuong proved the convergence of the method (0.5) without requiring the sequentially weakly continuity of the normalized duality mapping J However, we see that if A is a nonlinear mapping then (0.3), (0.4) and (0.5) are nonlinear problems For that reason, another stable method to solve the problem (0.1), called the Newton-Kantorovich iterative regularization method, has been studied This method is proposed by A.B Bakushinskii in 1976 to solve the variational inequality problem involving monotone nonlinear mappings This is the regularization method built on the well-known method of numerical analysis which is the NewtonKantorovich method In 1987, based on A.B Bakushinskii’s the method, to find the solution of the problem (0.1) in the case A is a monotone mapping from Banach space E into the dual space E ∗ , I.P Ryazantseva proposed Newton-Kantorovich iterative regularization method: A(zn ) + A (zn )(zn+1 − zn ) + αn J s (zn+1 ) = fδn (0.6) However, since the method (0.6) uses the duality mapping J s as a regularization component, it has the same limitations as the Browder-Tikhonov regularization method (0.3) The case A is an accretive mapping on Banach space E, to find the solution of the problem (0.1), also based on A.B Bakushinskii’s the method, in 2005, Ng Buong and V.Q Hung studied the convergence of the Newton-Kantorovich iterative regularization method: A(zn ) + A (zn )(zn+1 − zn ) + αn (zn+1 − x+ ) = fδ , (0.7) under conditions A(x) − A(x∗ ) − J ∗ A (x∗ )∗ J(x − x∗ ) ≤ τ A(x) − A(x∗ ) , ∀x ∈ E (0.8) and A (x∗ )v = x+ − x∗ , (0.9) where τ > 0, x∗ is a solution of the problem (0.1), A (x∗ ) is the Fréchet derivative of the mapping A at x∗ , J ∗ is the normalized duality mapping of E ∗ and v is some element of E We see that conditions (0.8) and (0.9) use the Fréchet derivative of the mapping A at the unknown solution x∗ , so they are very strict In 2007, A.B Bakushinskii and A Smirnova proved the convergence of the method (0.7) to the solution of the problem (0.1) when A is a monotone mapping from Hilbert space H into H (in Hilbert space, the accretive concept coincides with the monotone concept) under the condition A (x) ≤ 1, A (x) − A (y) ≤ L x − y , ∀x, y ∈ H, L > (0.10) The first content of this thesis presents new results of the NewtonKantorovich iterative regularization method for nonlinear equations involving monotone type operators (the monotone operator and the accretive operator) in Banach spaces that we achieve, which has overcome limitations of results as are mentioned above Next, we consider the problem: Find an element p∗ ∈ H such that ∈ A(p∗ ), (0.11) where H is Hilbert space, A : H → 2H is the set-valued and maximal monotone mapping One of the first methods to find the solution of the problem (0.11) is the proximal point method introduced by B Martinet in 1970 to find the minimum of a convex functional and generalized by R.T Rockafellar in 1976 as follows: xk+1 = Jk xk + ek , k ≥ 1, (0.12) where Jk = (I + rk A)−1 is called the resolvent of A with the parameter rk > 0, ek is the error vector and I is the identity mapping in H Since A is the maximal monotone mapping, Jk is the single-valued mapping (F Wang and H Cui, 2015) Thus, the prominent advantage of the proximal point method is that it varies from the set-valued problem to the singlevalued problem to solve R.T Rockafellar proved that the method (0.12) converges weakly to a zero of the mapping A under hypotheses are the k < ∞ and rk ≥ ε > 0, zero set of the mapping A is nonempty, ∞ k=1 e for all k In 1991, O Gă uler pointed out that the proximal point method only achieves weak convergence without strong convergence in infinite-dimensional space In order to obtain strong convergence, some modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space (OA Boikanyo and G Morosanu, 2010, 2012; S Kamimura and W Takahashi, 2000; N Lehdili and A Moudafi, 1996; G Marino and H.K Xu, 2004; Ch.A Tian and Y Song, 2013; F Wang and H Cui, 2015; H.K Xu, 2006; Y Yao and M.A Noor, 2008) as well as of an accretive mapping in Banach space (L.C Ceng et al., 2008; S Kamimura and W Takahashi, 2000; X Qin and Y Su, 2007; Y Song, 2009) were investigated The strong convergence of these modifications is given under conditions leading to the parameter sequence of the resol∞ vent of the mapping A is nonsummable, i.e k=1 rk = +∞ Thus, one question arises: is there a modification of the proximal point method that its strong convergence is given under the condition is that the parameter ∞ sequence of the resolvent is summable, i.e k=1 rk < +∞? In order to answer this question, the second content of the thesis introduces new modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space in which the strong convergence of methods is given under the assumption is that the parameter sequence of the resolvent is summable The results of this thesis are: 1) Propose and prove the strong convergence of a new modification of the Newton-Kantorovich iterative regularization method (0.6) to solve the problem (0.1) with A is a monotone mapping from Banach space E into the dual space E ∗ , which overcomes the drawbacks of method (0.6) 2) Propose and prove the strong convergence of the Newton-Kantorovich iterative regularization method (0.7) to find the solution of the problem (0.1) for the case A is an accretive mapping on Banach space E, with the removal of conditions (0.8), (0.9), (0.10) and does not require the sequentially weakly continuity of the normalized duality mapping J 3) Introduce two new modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space, in which the strong convergence of these methods are proved under the assumption that the parameter sequence of the resolvent is summable Apart from the introduction, conclusion and reference, the thesis is composed of three chapters Chapter is complementary, presents a number of concepts and properties in Banach space, the concept of the ill-posed problem and the regularization method This chapter also presents the NewtonKantorovich method and some modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space Chapter presents the Newton-Kantorovich iterative regularization method for solving nonlinear ill-posed equations involving monotone type operators in Banach spaces, includes: introducing methods and theorems about the convergence of these methods At the end of the chapter give a numerical example to illustrate the obtained research result Chapter presents modifications of the proximal point method that we achieve to find a zero of a maximal monotone mapping in Hilbert spaces, including the introduction of methods as well as results of the convergence of these methods A numerical example is given at the end of this chapter to illustrate the obtained research results Chapter Some knowledge of preparing This chapter presents the needed knowledge to serve the presentation of the main research results of the thesis in the following chapters 1.1 1.1.1 Banach space and related issues Some properties in Banach space This section presents some concepts and properties in Banach space 1.1.2 The ill-posed problem and the regularization method • This section mentions the concept of the ill-posed problem and the regularization method • Consider the problem of finding a solution of the equation A(x) = f, (1.1) where A is a mapping from Banach space E into Banach space E If (1.1) is an ill-posed problem then the requirement is that we must be used the solution method (1.1) such that when δ 0, the approximative solution is closer to the exact solution of (1.1) As presented in the Introduction, in the case where A is the monotone mapping from Banach space E into the dual space E ∗ , the problem (1.1) can be solved by Browder-Tikhonov type regularization method (0.3) (see page 2) or (0.4) (see page 2) The case A is an accretive mapping on Banach space E, one of widely used methods for solving the problem (1.1) is the Browder-Tikhonov type regularization method (0.5) (see page 2) Ng Buong and Ng.T.H Phuong (2013) proved the following result for the strong convergence of the method (0.5): Theorem 1.17 Let E be real, reflexive and strictly convex Banach space with the uniformly Gâteaux differentiable norm and let A be an m-accretive mapping in E Then, for each α > and fδ ∈ E, the equation (0.5) has a unique solution xδα Moreover, if δ/α → as α → then the sequence {xδα } converges strongly to x∗ ∈ E that is the unique solution of the following variational inequality x∗ ∈ S ∗ : x∗ − x+ , j(x∗ − y) ≤ 0, ∀y ∈ S ∗ , (1.2) where S ∗ is the solution set of (1.1) and S ∗ is nonempty We see, Theorem 1.17 gives the strong convergence of the regularization solution sequence {xδα } generated by the Browder-Tikhonov regularization method (0.5) to the solution x∗ of the problem (1.1) that does not require the sequentially weakly continuity of the normalized duality mapping J This result is a significant improvement compare with the result of Ya.I Alber and I.P Ryazantseva (2006) (see the Introduction) Since A is the nonlinear mapping, (0.3), (0.4) and (0.5) are nonlinear problems, in Chapter 2, we will present an another regularization method, called the Newton-Kantorovich iteration regularization method This is the regularization method built on the well-known method of the numerical analysis, that is the Newton-Kantorovich method, which is presented in Section 1.2 1.2 The Kantorovich-Newton method This section presents the Kantorovich-Newton method and the convergence theorem of this method 1.3 The proximal point method and some modifications In this section, we consider the problem: Find an element p∗ ∈ H such that ∈ A(p∗ ), (1.3) where H is Hilbert space and A : H → 2H is a maximal monotone mapping Denote Jk = (I + rk A)−1 is the resolvent of A with the parameter rk > 0, where I is the identity mapping in H 1.3.1 The proximal point method This section presents the proximal point method investigated by R.T Rockafellar (1976) to find the solution of the problem (1.3) and the assertion proposed by O Gă uler (1991) that this method only achieves weak convergence without strong convergence in the infinite-dimensional space 1.3.2 Some modifications of the proximal point method This section presents some modifications of the proximal point method with the strong convergence of them to find the solution of the problem (1.3) including the results of N Lehdili and A Moudafi (1996), H.K Xu (2006), O.A Boikanyo and G Morosanu (2010; 2012), Ch.A Tian and Y Song (2013), S Kamimura and W Takahashi (2000), G Marino and H.K Xu (2004), Y Yao and M.A Noor (2008), F Wang and H Cui (2015) Comment 1.6 The strong convergence of modifications of the proximal point method mentioned above uses one of the conditions (C0) exists constant ε > such that rk ≥ ε for every k ≥ (C0’) lim inf k→∞ rk > (C0”) rk ∈ (0; ∞) for every k ≥ and limk→∞ rk = ∞ These conditions lead to the parameter {rk } of the resolvent is nonsummable, ∞ rk = +∞ In Chapter 3, we introduce two new modifications of i.e k=1 the proximal point method that the strong convergence of these methods is given under the condition of the parameter sequence of the resolvent that is completely different from results we know Specifically, we use the condition that the parameter sequence of the resolvent is summable, i.e ∞ rk < +∞ k=1 10 the following variational inequality problem in Hilbert space H: Find an element x∗ ∈ Q ⊆ H such that A(x∗ ), x∗ − w ≤ 0, ∀w ∈ Q, (2.3) where A : H → H is a monotone mapping, Q is a closed and convex set in H A.B Bakushinskii introduced the iterative method to solve the problem (2.3) as follows: z ∈ H, A(z ) + A (z )(z n n n+1 − zn ) + αn zn+1 , zn+1 − w ≤ 0, ∀w ∈ Q (2.4) Based on the method (2.4), to find the solution of the equation (2.1) when A is a monotone mapping from Hilbert space H into H, A.B Bakushinskii and A Smirnova (2007) proved the strong convergence of the NewtonKantortovich type iterative regularization method: z0 = x+ ∈ H, A(zn ) + A (zn )(zn+1 − zn ) + αn (zn+1 − x+ ) = fδ , (2.5) with using the generalized discrepancy principle ≤ τ δ < A(zn ) − fδ , ≤ n < N = N (δ), (2.6) A (x) ≤ 1, A (x) − A (y) ≤ L x − y , ∀x, y ∈ H (2.7) A(zN ) − fδ and the condition Comment 2.1 The advantage of the method (2.5) is its linearity This method is an important tool for solving the problem (2.1) in the case A is a monotone mapping in Hilbert space However, we see that the condition (2.7) is fairly strict and should overcome such that the method (2.5) can be applied to the wider mapping class When E is Banach space, to solve the equation (2.1) in the case instead of f , we only know its approximation fδn ∈ E ∗ satisfying (2.2), in which δ is replaced by δn , I.P Ryazantseva (1987, 2006) also developed the method (2.4) to propose the iteration: z0 ∈ E, A(zn ) + A (zn )(zn+1 − zn ) + αn J s (zn+1 ) = fδn (2.8) The convergence of the method (2.8) was provided by I.P Ryazantseva under the assumption that E is Banach space having the ES property, the 11 dual space E ∗ is strictly convex and the mapping A satisfies the condition A (x) ≤ ϕ( x ), ∀x ∈ E, (2.9) where ϕ(t) is a nonnegative and nondecreasing function Comment 2.2 We see that lp and Lp (Ω) (1 < p < +∞) are Banach spaces having the ES property and the dual space is strictly convex However, since the method (2.8) uses the duality mapping J s as a regularization component, it has the same disadvantages as the Browder-Tikhonov regularization method (0.3) mentioned above To overcome these drawbacks, in [3 ], we propose the new NewtonKantorovich iterative regularization method as follows: z0 ∈ E, A(zn ) + A (zn )(zn+1 − zn ) + αn B(zn+1 − x+ ) = fδn , (2.10) where B is a linear and strongly monotone mapping Firstly, to find the solution of the equation (2.1) in the case without perturbation for f , we have the following iterative method: z0 ∈ E, A(zn ) + A (zn )(zn+1 − zn ) + αn B(zn+1 − x+ ) = f (2.11) The convergence of the method (2.11) is given by the following theorem: Theorem 2.4 Let E be a real and reflexive Banach space, B be a linear, mB -strongly monotone mapping with D(B) = E, R(B) = E ∗ and A be a monotone, L-Lipschitz continuous and twice Fréchet differentiable mapping on E with condition (2.9) Assume that the sequence {αn } and the initial point z0 in (2.11) satisfy the following conditions: a) {αn } is a monotone decreasing sequence with < αn < and there exists σ > such that αn+1 ≥ σαn for all n = 0, 1, ; b) ϕ0 z0 − x0 ≤ q < 1, ϕ0 = ϕ(d + γ), 2mB σα0 d ≥ max (2.12) B(x+ − x∗ ) /mB + x∗ , L B(x+ − x∗ ) /m2B , a positive number γ is found from the estimate 2mB σα0 /ϕ0 ≤ γ, where x∗ is the unique solution of the variational inequality x∗ ∈ S, B(x+ − x∗ ), x∗ − y ≥ 0, ∀y ∈ S (2.13) 12 and x0 is the solution of the following equation with n = 0: A(x) + αn B(x − x+ ) = f c) αn − αn+1 2mB σ ≤ c(q − q ), c = αn3 ϕ0 d Then, zn → x∗ , where zn is defined by (2.11) Now, we have the following result for the convergence of (2.10): Theorem 2.5 Let E, A and B be as in Theorem 2.4 and let fδn be elements in E ∗ satisfies (2.2), in which δ replaced by δn Assume that the sequence {αn }, the real number d and the initial point z0 in (2.10) satisfy conditions a), b) in Theorem 2.4 and c) mB σ δn m2B σ αn − αn+1 2 (2.14) ≤ c (q − q ), c = , ≤ c (q − q ), c = 1 2 αn3 ϕ0 d αn2 ϕ0 Then, zn → x∗ , where zn is defined by (2.10) Comment 2.3 We see, (2.10) and (2.11) are linear problems The introduction of these methods overcomes the "nonlinear" property of previous methods for finding a solution of nonlinear ill-posed equations involving monotone mappings in Banach spaces For strong convergence, the method (2.8) only applies to Banach space E having the ES property and the dual space E ∗ is strictly convex, while methods (2.10) and (2.11) can be used in any real and reflexive Banach space However, methods (2.10) and (2.11) require A is the Lipschitz continuous mapping 2.2 Newton-Kantorovich iterative regularization for nonlinear equations involving accretive operators in Banach spaces Consider the problem of finding a solution of the nonlinear equation A(x) = f, f ∈ E, (2.15) where A is an accretive mapping on Banach space E Assume that the solution set of (2.15), denote by S ∗ , is nonempty and instead of f , we only know its approximation fδ satisfies (2.2) If A does not have additional property as strongly accretive or uniformly accretive then the problem (2.15), in general, is the ill-posed problem 13 One of widely used methods to solve the problem (2.15) is the BrowderTikhonov regularization method (0.5) (see page 2) However, if A is the nonlinear mapping then (0.5) is the nonlinear problem To overcome this drawback, Ng Buong and V.Q Hung (2005) investigated the convergence of the following Newton-Kantorovich type iterative regularization method to find the solution of the problem (2.15) in the case instead of f we only know its approximation fδn ∈ E which satisfies the condition (2.2), where δ is replaced by δn : z0 ∈ E, A(zn ) + A (zn )(zn+1 − zn ) + αn (zn+1 − x+ ) = fδn (2.16) The strong convergence of the method (2.16) proposed by Ng Buong and V.Q Hung under hypotheses that E with the dual space E ∗ are uniformly convex spaces, E possesses the approximation and the mapping A satisfies conditions A(x) − A(x∗ ) − J ∗ A (x∗ )∗ J(x − x∗ ) ≤ τ A(x) − A(x∗ ) , ∀x ∈ E (2.17) and A (x∗ )v = x+ − x∗ , (2.18) wher τ is some positive constant, x∗ ∈ S ∗ is uniquely determined by (2.17), J ∗ is the normalized duality mapping of E ∗ and v is some element of E Comment 2.4 We see, (2.16) has the advantage that is the linear problem However, the strong convergence of this Newton-Kantorovich type iterative regularization method require conditions (2.17) and (2.18) These conditions are relatively strict because they use the Fréchet derivative of the mapping A at the unknown solution x∗ Recently, in [2 ], in order to solve the equation (2.15), we prove the convergence of the following Newton-Kantorovich type iterative regularization method: z0 ∈ E, A(zn ) + A (zn )(zn+1 − zn ) + αn (zn+1 − x+ ) = fδ , (2.19) without using conditions (2.7), (2.17) and (2.18) Our results are proved based on Theorem 1.17 Therefore, the strong convergence of the method (2.19) also must not utilize the assumption that the normalized duality mapping J is sequentially weakly continuous 14 Firstly, consider the Newton-Kantorovich type iterative regularization method in the case without perturbation for f : x0 ∈ E, A(xn ) + A (xn )(xn+1 − xn ) + αn (xn+1 − x+ ) = f (2.20) Theorem 2.7 Let E be a real, reflexive and strictly convex Banach space with the uniformly Gâteaux differentiable norm and let A be an m-accretive and twice Fréchet differentiable mapping on E with condition (2.9) Assume that the sequence {αn }, the real number d and the initial point x0 satisfy the following conditions: a) {αn } is a monotone decreasing sequence with < αn < and there exists σ > such that αn+1 ≥ σαn for all n = 0, 1, ; b) ϕ0 x0 − xα0 ≤ q < 1, ϕ0 = ϕ(d + γ), 2σα0 a positive number γ is found from the estimate 2σα0 /ϕ0 ≤ γ, d ≥ x∗ − x+ + x+ , (2.21) where x∗ ∈ S ∗ is the unique solution of the variational inequality (1.2) and xα0 is the solution of the following equation with n = 0: A(x) + αn (x − x+ ) = f (2.22) c) αn − αn+1 ϕ0 d q − q2 ≤ ; c = αn2 c1 2σ Then, xn → x∗ , where xn is defined by (2.20) Theorem 2.8 Let E and A be as in Theorem 2.7 The mapping A satisfies additional property that is L-Lipschitz continuous Suppose the sequence {αn } satisfies the condition a) in Theorem 2.7 Let a number τ > in (2.6) be chosen such that 3dL ϕ˜ z0 − xα0 ≤ q, < q < − , 2σα0 τ˜σ (2.23) √ 2 ϕ˜ = ϕ0 + 2L /˜ τ , τ˜ = ( τ − 1) , where d is defined as in Theorem 2.7, a positive number γ is taken from estimate (2.21) with ϕ0 replaced by ϕ˜ and αn − αn+1 d 2Lσ + ≤ q αn2 τ˜ ϕ˜ ˜τ (2.24) 15 Then, For n = 0, 1, , N (δ), ϕ˜ zn − xαn ≤ q, 2σαn (2.25) where zn is a solution of (2.19) and N (δ) is chosen by (2.6) limδ→0 zN (δ) − y = 0, where y ∈ S ∗ If N (δ) → ∞ as δ → then y = x∗ , where x∗ ∈ S ∗ is the unique solution of the variational inequality (1.2) Comment 2.5 Beside eliminating conditions (2.7), (2.17), (2.18) and the sequentially weakly continuity of the normalized duality mapping J, the hypothesis about Banach space E in our results is also lighter than one in Ng Buong and V.Q Hung’s result (2005) Specifically, in Theorem 2.7 and Theorem 2.8, we only need to assume that E is the reflexive and strictly convex space with the uniformly Gâteaux differentiable norm, instead of E with the dual space E ∗ are uniformly spaces and E possesses the approximation as in Ng Buong and V.Q Hung’s result However, the strong convergence of the Newton-Kantorovich type iterative regularization method (2.19) requires the additional condition that is the Lipschitz continuity of the mapping A 2.3 Numerical example of the finite-dimensional approximation for the Newton-Kantorovich iterative regularization method To solve the equation (2.1), we can use the Browder-Tikhonov type regularization method (0.4) or the Newton-Kantorovich type iterative regularization method (2.10) However, to be able to use (0.4) and (2.10) for solving problems in practice by computer, the first task is to approximate (0.4) and (2.10) by corresponding equations in finite-dimensional spaces Ng Buong (1996; 2001) introduced the finite-dimensional approximation method for the solution xδα of (0.4) as follows: An (x) + αBn (x − x+ ) = fnδ , (2.26) where An = Pn∗ APn , Bn = Pn∗ BPn , fnδ = Pn∗ fδ , Pn is a linear projection from E onto the subspace En of E satisfies En ⊂ En+1 , for all n, Pn ≤ c, 16 where c is a constant and Pn∗ is the conjugate mapping of Pn We have the following result for the convergence of the solution sequence {xδαn } of (2.26) to the solution xδα of (0.4): Theorem 2.9 (Ng Buong, 2001) Assume that Pn∗ BPn x → Bx, for all x ∈ D(B) The necessary and sufficient condition for every α > and fδ ∈ E ∗ , the solution sequence {xδαn } of (2.26) converges strongly to the solution xδα of (0.4) is Pn x → x as n → ∞ for every x ∈ E Apply (2.26) to (2.10) for αn = α that is a fixed number, we have An (zn ) + An (zn )(zn+1 − zn ) + αBn (zn+1 − x+ ) = fnδn (2.27) Let k(t, s) is a real, two variables, continuous, nondegenerate and nonnegative function on the square [a, b]×[a, b] satisfies: there exists a constant q = 2, < q < ∞ such that b b |k(t, s)|q dsdt < +∞ a (2.28) a Then, the mapping A is defined by b k(t, s)x(s)ds, x(s) ∈ Lp [a, b], (Ax)(t) = (2.29) a is the mapping from the space Lp [a, b] into the space Lq [a, b] with + p = and continuous (S Banach, 1932) Since k(t, s) is continuous and q nonnegative on [a, b] × [a, b], A is a monotone mapping on Lp [a, b] Hereafter, we will present the application of the method (2.27) to solve the following Hammerstein type integral equation: b (Ax)(t) = k(t, s)x(s)ds = f (t), (2.30) a where f (t) ∈ Lq [a, b] Assume the solution x(t) of (2.30) is twice Fréchet differentiable and satisfies the boundary condition x(a) = x(b) = Taking Bx(t) = x(t) − x (t), where x(t) ∈ D(B) is the closure of all functions in C [a, b] in the metric of Wq2 [a, b], satisfies x(a) = x(b) = Let {tn0 = a < tn1 < · · · < tnn = b} be an uniformly partition of [a, b] Approximating E by the sequence of linear subspaces En = L{ψ1 , ψ2 , · · · , ψn }, where ψi (t) = 1, if t ∈ (tni−1 , tni ] 0, if t ∈ / (tni−1 , tni ] 17 Selecting projection n x(tni )ψi (t) Pn x(t) = (2.31) i=1 Consider the equation (2.30) with a = 0, b = 1, k(t, s) = |t − s| Clearly, 1 3/2 |k(t, s)| 0 |t − s|3/2 dsdt < +∞ dsdt = (2.32) So, we consider q = 3/2 and p = With the exact solution x∗ (s) = s(1−s), we have f (t) = −(1/6)t4 +(1/3)t3 −(1/6)t+1/12 Hereafter is the calculated result with x+ (t) = 2.22 and fδn = f +δn , where δn = 1/(1+n)2 : Table 2.1 Calculated results with α = 0.5 n zn+1 − x∗ n zn+1 − x∗ 0.2689666069 64 0.0424663883 0.1620043546 128 0.0298464819 16 0.1003942097 256 0.0230577881 32 0.0640826159 1024 0.0203963532 Table 2.2 Calculated results with α = 0.1 n zn+1 − x∗ n zn+1 − x∗ 0.2600031372 64 0.0388129413 0.1534801504 128 0.0269295563 16 0.0936525099 256 0.0204623013 32 0.0591546836 1024 0.0176684288 Table 2.3 Calculated results with α = 0.01 n zn+1 − x∗ n zn+1 − x∗ 0.1948813288 64 0.0295640389 0.1176798737 128 0.0196621910 16 0.0739066898 256 0.0138863679 32 0.0461148835 1024 0.0099425015 Observing above results, we see, the application of the method (2.27) brings results in the convergence to the solution of the equation (2.30) is quite good Especially, with α getting smaller and going to 0, zn+1 is closer to the exact solution x∗ Chapter Iterative method for finding a zero of a monotone mapping in Hilbert space This chapter presents modifications of the proximal point method that we achieve to find a zero of a maximal monotone mapping in Hilbert space The content of this chapter is presented based on works [1 ] and [4 ] in list of works has been published 3.1 The problem finding a zero of a maximal monotone mapping • This section introduces the problem: Find an element p∗ ∈ H such that ∈ A(p∗ ), (3.1) where H is Hilbert space and A : H → 2H is a maximal monotone mapping • One of the first methods for finding a solution of the problem (3.1) is the proximal point method (0.12) However, the proximal point method (0.12) only converges weakly without converging strongly in the infinitedimensional space In order to achieve strong convergence, some modifications of the proximal point method were proposed (see Section 1.3.2) As Comment 1.6 in Section 1.3.2, the strong convergence of these modifications is proved under conditions leading to the parameter sequence of the resolvent of the mapping A is nonsummable, i.e ∞ k=1 rk = +∞ • To find p∗ ∈ H that is the solution of the variational inequality problem p∗ ∈ C : F p∗ , p∗ − p ≤ 0, ∀p ∈ C, (3.2) where C = ZerA is the set of zeros of the mapping A, F is a L-Lipschitz continuous and η-strongly monotone mapping on H, S Wang (2012) pro- 19 posed the iterative method: xk+1 = Jk [(I − tk F )xk + ek ], k ≥ 1, (3.3) where Jk is the resolvent of A and ek is the error vector The author proved the convergence of the method (3.3) under the condition (C0’) lim inf k→∞ rk > that is mentioned in Section 1.3.2 This is the condition leading to the parameter sequence {rk } of the resolvent of the mapping A is nonsummable • To answer the question in the Introduction (see page 4), in the next section, we will introduce two new modifications of the proximal point method to find a zero of the maximal monotone mapping A in Hilbert space H with the strong convergence given under the condition that the parameter ∞ sequence of the resolvent is summable, i.e k=1 rk < +∞ The obtained new modifications are individual cases of an extension of the method (3.3) to find the solution of the variational inequality problem (3.2) 3.2 Modifications of the proximal point method with the parameter sequence of the resolvent is summable In [1 ], we introduce two new modifications of the proximal point method corresponding to sequences {xk } and {z k } defined by: xk+1 = J k (tk u + (1 − tk )xk + ek ), k ≥ 1, (3.4) z k+1 = tk u + (1 − tk )J k z k + ek , k ≥ 1, (3.5) and where J k = J1 J2 · · · Jk is the product of k resolvents Ji = (I + ri A)−1 , i = 1, 2, , k of the mapping A First of all, we propose the iterative method: xk+1 = J k [(I − tk F )xk + ek ], k ≥ 1, (3.6) to find a solution p∗ ∈ H of the variational inequality problem (3.2), where C = ZerA, F : H → H is an η-strongly monotone and γ-strictly pseudocontractive mapping Then, from (3.6), by selecting the appropriate mapping F , we obtain methods (3.4) and (3.5) Denote |Ax| = inf{ y : y ∈ Ax}, x ∈ D(A) Let A0 is the mapping defined by A0 x = {y ∈ Ax : 20 y = |Ax|}, x ∈ D(A) Since A is the maximal monotone mapping, A0 is a single-valued mapping (Ya.I Alber and I.P Ryazantseva, 2006) Theorem 3.2 Let A be a maximal monotone mapping in Hilbert space H such that D(A) = H, C := ZerA = ∅ and the mapping A0 be bounded, F be an η-strongly monotone and γ-strictly pseudocontractive mapping with η + γ > Assume that tk , ri and ek satisfy conditions (C1), (C5’) and (C0” ’) ri > for all i ≥ and ∞ i=1 ri < +∞ Then, the sequence {xk }, defined by the method (3.6), converges strongly to the element p∗ as k → ∞, where p∗ is the unique solution of (3.2) Remark 3.1 This remark presents how to change and selection the mapping F in the method (3.6) to obtain methods (3.4) and (3.5) Indeed, in (3.6), setting z k = (I − tk F )xk + ek , then rewriting tk := tk+1 and ek := ek+1 , we obtain z k+1 = (I − tk F )J k z k + ek (3.7) Next, take F = I − f , where f = aI + (1 − a)u, with a is a fixed number in (0; 1) and u is a fixed point of H With the mapping F selected as above, (3.6) and (3.7) respectively become xk+1 = J k (tk (1 − a)u + (1 − tk (1 − a))xk + ek ), (3.8) z k+1 = tk (1 − a)u + (1 − tk (1 − a))J k z k + ek (3.9) Then, in (3.8) and (3.9), denoting tk := (1 − a)tk , we obtain methods (3.4) and (3.5) corresponding Remark 3.2 The condition that the parameter sequence of the resolvent is summable, i.e the condition (C0” ’) is satisfied, which leads to limk→∞ rk = The result in this section is a suggestion for the research of the strong convergence of modifications of the proximal point method under the condition that the parameter sequence of the resolvent satisfies limk→∞ rk = 3.3 Numerical example Consider the following convex optimization problem: find an element p∗ ∈ R2 such that f (p∗ ) = inf2 f (x) x∈R (3.10) 21 We know that if f (x) is a lower-semicontinuous, proper and convex functional then its sub-differential ∂f is a maximal monotone mapping and the problem (3.10) is equivalent to the problem of finding a zero of ∂f (H.H Bauschke and P.L Combettes, 2017; R.T Rockafellar, 1966) Hereafter, we will apply methods (3.4) and (3.5) to find a solution of the problem (3.10) with the function f (x) given as follows: 0, if x2 ≤ 1, f (x) = x − 1, if x > (3.11) For r > 0, we have (x , x ), if x2 ≤ 1, −1 (I + r∂f ) (x) = (x , x /(1 + r)), if x > 1 2 (3.12) Taking a = 1/2 and u = (0; 2) Then, the solution of the problem (3.10) satisfies the variational inequality (3.2) with A = ∂f is p∗ = (0; 1) Using tk = 1/(k + 1), ri = 1/(i(i + 1) and ek = (0; 0), we obtain the following result tables: a) The case of the initial point is (2,0; 6,0): Table 3.1 Calculated result when applying the method (3.4) k xk+1 with calculated time is 2.745 seconds xk+1 k xk+1 xk+1 1.5000000000 3.3333333333 2000 0.0504468881 0.9999996953 10 0.6727523804 0.9982716570 5000 0.0319113937 0.9999999357 20 0.4895426850 0.9997219609 8000 0.0252293542 0.9999999775 50 0.3152358030 0.9998464349 10000 0.0225661730 0.9999999803 100 0.2242781046 0.9999270505 12000 0.0206002179 0.9999999883 500 0.1007993740 0.9999960565 15000 0.0184255869 0.9999999946 1000 0.0713204022 0.9999986543 20000 0.0159571926 0.9999999954 Table 3.2 Calculated result when applying the method (3.5) with calculated time is 2.730 seconds 22 z1k+1 k z2k+1 k z1k+1 z2k+1 1.5000000000 3.5000000000 2000 0.0504468881 1.0002497795 10 0.6727523804 1.0367234670 5000 0.0319113937 1.0000999281 20 0.4895426850 1.0225868918 8000 0.0252293542 1.0000624662 50 0.3152358030 1.0092381210 10000 0.0225661730 1.0000499833 100 0.2242781046 1.0048024836 12000 0.0206002179 1.0000416476 500 0.1007993740 1.0009925330 15000 0.0184255869 1.0000333306 1000 0.0713204022 1.0004981392 20000 0.0159571926 1.0000249980 b) The case of the initial point is (10; 20): Table 3.3 Calculated result when applying the method (3.4) k xk+1 with calculated time is 2.699 seconds xk+1 k xk+1 xk+1 7.5000000000 10.3333333333 2000 0.2522344403 0.9999996174 10 3.3637619019 0.9837468002 5000 0.1595569687 0.9999999232 20 2.4477134250 0.9999068009 8000 0.1261467712 0.9999999725 50 1.5761790149 0.9997667170 10000 0.1128308650 0.9999999996 100 1.1213905229 0.9998979723 12000 0.1030010894 0.9999999861 500 0.5039968702 0.9999947597 15000 0.0921279346 0.9999999932 1000 0.3566020110 0.9999983305 20000 0.0797859628 0.9999999946 Table 3.4 Calculated result when applying the method (3.5) with calculated time is 2.683 seconds k+1 k z1 z2k+1 k z1k+1 z2k+1 7.5000000000 10.5000000000 2000 0.2522344403 1.0002497442 10 3.3637619019 1.0343656695 5000 0.1595569687 1.0000999239 20 2.4477134250 1.0224448516 8000 0.1261467712 1.0000624636 50 1.5761790149 1.0091958993 10000 0.1128308650 1.0000499805 100 1.1213905229 1.0047946859 12000 0.1030010894 1.0000416627 500 0.5039968702 1.0009920670 15000 0.0921279346 1.0000333302 1000 0.3566020110 1.0004979605 20000 0.0797859628 1.0000249977 Observing above results, we see, the application of methods (3.4) and (3.5) brings results in a good convergence to the solution of the problem (3.10) 23 GENERAL CONCLUSION The thesis achieves the following results: - Propose and prove theorems of the strong convergence of the NewtonKantorovich type iterative regularization method for finding a solution of nonlinear equations involving monotone mappings in Banach spaces - Propose and prove theorems of the strong convergence of the NewtonKantorovich type iterative regularization method for finding a solution of nonlinear equations involving accretive mappings in Banach spaces - Propose and prove the theorem of the strong convergence of new modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space, with a different approach about the condition of the parameter sequence of the resolvent, that is the convergence of previous modifications was given under the assumption that the parameter sequence of the resolvent is nonsummable, while the strong convergence of these new modifications is proved under the assumption that the parameter sequence of the resolvent is summable Recommendations for future research: • Continue the study of the finite-dimensional approximation with the regularization parameter sequence {αn } and evaluating the convergence rate to the solution of Newton-Kantorovich iterative regularization methods given in Chapter for solving equations involving monotone type operators • Develope Newton-Kantorovich iterative regularization methods proposed in Chapter to build regularization methods for solving equation systems involving monotone type operators • Evaluate the convergence rate to the solution of iterative methods to find a zero of a maximal monotone mapping in Hilbert space given in Chapter • Propose and investigate the convergence of new iterative methods to find zeros of monotone type mappings in Hilbert spaces and Banach spaces 24 LIST OF WORKS HAS BEEN PUBLISHED [1 ] Ng Buong, P.T.T Hoai, Ng.D Nguyen, Iterative methods for a class of variational inequalities in Hilbert spaces, J Fixed Point Theory Appl., 2017, 19 (4), 2383-2395 [2 ] Ng Buong, Ng.D Nguyen, Ng.T.T Thuy, Newton-Kantorovich iterative regularization and generalized discrepancy principle for nonlinear ill-posed equations involving accretive mappings, Russian Math (Iz VUZ), 2015, 59 (5), 32-37 [3 ] Ng.D Nguyen, Ng Buong, Regularization Newton-Kantorovich iterative method for nonlinear monotone ill-posed equations on Banach spaces, Proceedings of the 18th national workshop: "Selected issues of information technology and communication", Ho Chi Minh city, November 5-6, 2015, Science and Technics Pushlishing House, 2015, 278-281 [4 ] Ng.D Nguyen, Numerical results for iteration methods of the NewtonKantorovich type and the proximal point type for solving equations involving monotone mappings, Journal of Science and Technology-TNU, 2018, 178 (2), 145-150 ... the concept of the ill- posed problem and the regularization method This chapter also presents the NewtonKantorovich method and some modifications of the proximal point method to find a zero of. .. The ill- posed problem and the regularization method • This section mentions the concept of the ill- posed problem and the regularization method • Consider the problem of finding a solution of the. .. difference of the solution It is said that these problems are ill- posed Therefore, the requirement is that there must be methods to solve ill- posed problems such that the smaller the error of the data