Một vài phương pháp tìm điểm bất động chung của họ hữu hạn các ánh xạ không giãn

I express my sincere gratitude to my thesis advisor Prof. Pham Ky Anh, who has introduced me to the field of Numerical Analysis. I am especially grateful for his patience and ability of making abstract mathematics so easily to be perceived. I also want to thank my family since they always motivate, encourage and create favorable conditions for my study and research. Finally, I want to thank my friends in K53 Advanced Maths. They always stay by my side to encourage and to help me.

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Nguyen Ly Vinh Hanh SOME METHODS FOR COMMON FIXED POINTS OF A FAMILY OF NONEXPANSIVE MAPPINGS Undergraduate Thesis Advanced Undergraduate Program in Mathematics Hanoi - 2012 VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Nguyen Ly Vinh Hanh SOME METHODS FOR COMMON FIXED POINTS OF A FAMILY OF NONEXPANSIVE MAPPINGS Undergraduate Thesis Advanced Undergraduate Program in Mathematics Thesis Advisor: Prof. Dr. Sc. Pham Ky Anh Hanoi - 2012 Acknowledgments I express my sincere gratitude to my thesis advisor Prof. Pham Ky Anh, who has intro- duced me to the field of Numerical Analysis. I am especially grateful for his patience and ability of making abstract mathematics so easily to be perceived. I also want to thank my family since they always motivate, encourage and create favor- able conditions for my study and research. Finally, I want to thank my friends in K53 Advanced Maths. They always stay by my side to encourage and to help me. Ha Noi, May, 2012. Nguyen Ly Vinh Hanh 2 Contents Introduction 5 1 Preliminary 6 2 Iterations for nonexpansive mappings 14 2.1 Krasnoselskij iteration for nonexpansive mappings . . . . . . . . . . . . . 14 2.2 Mann iterations for nonexpansive mappings . . . . . . . . . . . . . . . . . 20 2.2.1 Strongly Pseudocontractive Operators . . . . . . . . . . . . . . . . 23 2.2.2 Nonexpansive and quasi-nonexpansive operators . . . . . . . . . . 26 3 Iterations for relatively nonexpansive mappings 34 3.1 A hybrid method for relatively nonexpansive mappings . . . . . . . . . . . 34 3.2 Strong Convergence theorems for a Finite Family of Relatively Nonexpan- sive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Parallel hybrid methods for a finite family of relatively nonexpansive mappings 44 4 Applications 52 4.1 Some basic facts about projections onto hyperplanes . . . . . . . . . . . . 52 4.2 The algebraic reconstruction technique . . . . . . . . . . . . . . . . . . . . 54 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.2 The convergence of the ART . . . . . . . . . . . . . . . . . . . . . 55 4.3 Reconstruction by successive approximation . . . . . . . . . . . . . . . . . 57 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 4.3.2 The convergence of the reconstruction by successive approximation 58 4.4 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.1 Applying the ART method . . . . . . . . . . . . . . . . . . . . . . 62 4.4.2 Applying Liu’s method . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Introduction In this thesis, we deal with iteration methods for finding fixed points of nonexpansive map- pings on normed spaces. The origin of these methods dates back to 1920, when Stefan Ba- nach (1892 −1945) formulated his famous contraction mapping principle. Banach proved that if (X,d) is a complete metric space and T : X → X is a given contraction, then T has a unique fixed point p, i.e., T(p) = p and T n (x) → p (as n → ∞). Many scientific problems in game theory, theory of phase transitions, optimization theory, differential equations, differential geometry, image processing, etc. lead to a problem of finding fixed points of nonexpansive mappings. This minor thesis will attempt to highlight some achievements in the theory of nonexpansive mappings and concentrate to the following problems. • The existence of fixed points; • The approximation of fixed points; • Applications of the fixed point theory. The thesis consists of four chapters. The first chapter is devoted to some minimal functional analysis background. The second chapter is devoted to Krasnoselskij and Mann iterations for nonexpansive mappings and the third chapter deals with hybrid methods for relatively nonexpansive mappings. The last chapter provides a crucial application of nonexpansive mappings in image processing. 5 Chapter 1 Preliminary In this chapter we collect some facts on nonlinear operators and geometry of Banach spaces. Definition 1.1. Let C be a nonempty set of a metric space and T be a mapping from C into itseft. An element x ∗ ∈ C is called a fixed point of T if Tx = x and the set of all fixed points of T is denoted by F(T). Example 1.1. 1) Let X = R Tx = x 2 − 3x . Then F(T) = {0,4}. 2) Let X = R Tx = x+ 5 . Then F(T) = { /0}. Definition 1.2. The sequence {x n } ∞ n=0 in a normed space X is said to be strongly convergent to a if ||x n − a|| → 0, as n → ∞. Further, {x n } ∞ n=0 ⊂ X coverges weakly to a if for any f ∈ X ∗  f,x n  →  f,a, as n → ∞. Definition 1.3. A subset C of a real normed space is called bounded if there exists M > 0 such that ||x|| ≤ M, for all x ∈ C. Definition 1.4. A subset C of a real vector space X is called covex if, for any pair of points x,y in C, the closed segment with the endpoints x, y, i.e., the set { λ x+ (1− λ )y : λ ∈ [0,1)} is contained in C. Definition 1.5. A Banach space (X,||.||) is called strictly convex if, for all x,y ∈ X satisfying ||x|| ≤ 1, ||y|| ≤ 1 and x = y, we have ||x+ y|| < 2. Example 1.2. 6 • All inner product spaces are strictly convex. • Let X = R 2 . Then (X,||.|| 2 ) is strictly covex, (X,||.|| 1 ) and (X,||.|| ∞ ) are not strictly convex. Example 1.3. C[a,b] is not stritly convex. Proof. Choose, x ≡ 1 and y = t−a b−a . Clearly, x = y. On the other hand, ||x|| = max t∈[a,b] x(t) = 1, ||y|| = max t∈[a,b] t −a b− a = 1, and (x+ y)(t) = 1+ t −a b− a = t +b− 2a b− a . Thus, ||x+ y|| = max t∈[a,b] (x+ y)(t) = b+ b− 2a b− a = 2(b− a) b− a = 2 = x + y. Definition 1.6. A Banach space (X, ||.||) is called uniformly convex if, given any ε > 0, there exists δ > 0 such that for all x,y ∈ X satisfying ||x|| ≤ 1,||y|| ≤ 1, and ||x− y|| ≥ ε , we have 1 2 ||x+ y|| < 1− δ . Definition 1.7. Let X be a real Banach space. The space X ∗ of all linear continous functionals on X is called the dual space of X. For f ∈ X ∗ and x ∈ X the value of f at x is denoted by  f,x and is called the duality pairing. • The dual X ∗ is a Banach space with respect to the norm || f || ∗ = sup{ f, x : ||x|| ≤ 1} usually denoted by ||.||; • The dual space of X ∗ is X ∗∗ , the bidual space of X. Since, in general, X ⊆ X ∗∗ , we say X is reflexive if X = X ∗∗ . Definition 1.8. Let X ∗ be the dual space of a real Banach space. The multivalued mapping J defined by Jx = { f ∗ ∈ X ∗ :< f ∗ ,x >= ||x|| 2 = || f ∗ || 2 } is called the normalized duality mapping of E. 7