strong convergence implies weak convergence proof

... ∈ C, x = T x}. We use “ → ” to stand for strong convergence and “  ” for weak convergence. For a given sequence {x n } ⊂ C, let ω w (x n ) denote the weak ω−limit set. Recall that T is said to ... the sequence {x n } converges strongly to a common fixed point of the semigroup T := {T (t) : t ≥ 0}. The purpose of this article is to prove the strong and weak convergence of implicit iterative process x n = ... semigroups in Hilbert spaces. Xu [7] gave strong convergence theorem for contraction semigroups in Banach spaces. Chang et al. [8] proved the strong convergence the- orem for nonexpansive semi-groups...

Ngày tải lên: 21/06/2014, 17:20

... x 0 ∈ C. (5.3.4) The mapping x → f  (x) for x ∈ X is weak- to -weak ∗ continuous. Then for any x ∈ C, {T n x} n∈N is weakly convergent to an element v ∈ F(T ). Proof. Let v ∈ F (T ) and x ∈ C. If {T n x} n∈N is ... x, T y)−D f (x, y) ≤ 0 for all x, y ∈ [0, 0.85].  5 Weak convergence theorems In this section, we discuss the demiclosedness and the weak convergence prob- 24 The modulus of uniform convexity ... Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967) 32 which in conjunction with (3), (4), and (6) implies...

Ngày tải lên: 18/06/2014, 22:20

... common fixed point. Proof. Since C is a nonempty bounded closed convex subset of the reflexive Banach space X, it is weakly compact. By Proposition 3.5, each F (T i ) is a nonempty weakly compact subset ... f  (x) denotes the Gˆateaux derivative of f at x. In this article, we study the fixed point and weak convergence problem for 4 mappings satisfying (1.5). This article is organized in the following ... Aoyama–Iemoto–Kohsaka–Takahashi fixed point theorem properly in Section 4. Section 5 is devoting to studying the weak convergence problem for this new class of nonlinear mappings. 2 Preliminaries In what follows,...

Ngày tải lên: 20/06/2014, 04:20

... Zegeye, H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. J Math Anal Appl. 280, 364–374 (2003). doi:10.1016/S0022-247X(03)00061-1 17. Wang, C, Zhu, J: Convergence ... ||f - g|| 2 ≤ ||f - g||(h(l)+ε). Therefore ||f - g|| ≤ h(l). This completes the proof. Now we can prove the weak convergence theorem of the iterative sequence (3.1). Zhu et al. Fixed Point Theory ... 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 2 of 13 8. Cho, Y, Zhou, H, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive...

Ngày tải lên: 20/06/2014, 21:20

... Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc. 73, 591–597 (1967). doi:10.1090/S0002-9904-1967-11761-0 9. Reich, S: Weak convergence ... Then,{Q C (I - l n A)} is a strongly nonexpansive sequence. Proof. Notice first that Q C is a strongly nonexpansive mapping (see [16,17]). Next, we prove that {I - l n A} is a strongly nonexpansive ... Science, Khon kaen University, Khon kaen 40002, Thailand Abstract We present two weak convergence the orems for inverse strongly accretive mappings in Banach spaces, which are supplements to the recen...

Ngày tải lên: 21/06/2014, 01:20

... whose inner product and norm are denoted by ·, · and ·. → and  are denoted by strong convergence and weak convergence, respectively. Let C be a nonempty closed convex subset of H and T : C ... Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967. 12 G. Marino and H K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical ... of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991. 17 J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian...

Ngày tải lên: 21/06/2014, 05:20

... aspects. We will use the following notation: i  for weak convergence and → for strong convergence. ii ω ω x n {x : x n i x} denotes the weak ω-limit set of {x n }. 2. Preliminaries A Banach ... Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. 8 H. Zhou, Convergence ... Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. 7 G. Marino and H K. Xu, “Weak...

Ngày tải lên: 21/06/2014, 11:20

... Zegeye, H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. J Math Anal Appl. 280, 364–374 (2003). doi:10.1016/S0022-247X(03)00061-1 17. Wang, C, Zhu, J: Convergence ... ||f - g|| 2 ≤ ||f - g||(h(l)+ε). Therefore ||f - g|| ≤ h(l). This completes the proof. Now we can prove the weak convergence theorem of the iterative sequence (3.1). Zhu et al. Fixed Point Theory ... 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 7 of 13 8. Cho, Y, Zhou, H, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive...

Ngày tải lên: 21/06/2014, 17:20

Ngày tải lên: 22/06/2014, 22:20

... có sự so sánh trực quan sinh động về các âm theo dạng mạnh và dạng yếu: Strong form Weak form Strong form Weak form u Butter / 'btə/ Autumn / ':təm/ o Potato / pə'teitou/ ... Chúc các bạn học tốt và ngày càng “nói hay”! Strong form và weak form trong tiếng Anh Phát âm dạng mạnh (strong form) và d ạng yếu (weak form) không phải là m ột khái niệm xa lạ đối ... Bên cạnh những nguyên tắc chung như đã đề cập ở trên v ề phát âm dạng mạnh và dạng yếu (strong form and weak form), phát âm dạng yếu nhẹ, không được nhấn mạnh mà thường biến âm thành /ə/. Mời...

Ngày tải lên: 20/03/2014, 23:21

... âm dạng mạnh và dạng yếu (strong form and weak form), phát âm dạng yếu nhẹ, không được nhấn mạnh mà thường biến âm thành /ə/. Strong form Weak form Strong form Weak form u Butter / 'btə/ ... công việc.) At /æt/ /ət/ I’ll meet you at the office. Phát âm dạng mạnh (strong form) và dạng yếu (weak form) không phải là một khái niệm xa lạ đối với những người học tiếng Anh, thế ... (auxiliary verb). Một số liên từ, giới từ hoặc đại từ có dạng phát âm mạnh và yếu như sau: WordStrong form Weak form The /ði/ - Đứng trước nguyên âm /ðə/ - Đứng trước phụ âm What are you looking...

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Ngày tải lên: 08/08/2014, 09:21

... conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong convergence theorem ... Takahashi, W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 37–47 (2004). 2004 21. Matsushita, S, Takahashi, W: A strong convergence ... we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition (C). Next, we consider strong convergence theorem...

Ngày tải lên: 20/06/2014, 21:20

... conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong convergence theorem ... Takahashi, W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 37–47 (2004). 2004 21. Matsushita, S, Takahashi, W: A strong convergence ... we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition (C). Next, we consider strong convergence theorem...

Ngày tải lên: 20/06/2014, 21:20

... C. 1.13 They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was ... pseudocontractions was introduced by Qihou 4 in 1996 see also 5.KimandXu6 studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically ... Applications, vol. 70, no. 10, pp. 3502–3511, 2009. 8 M. O. Osilike and S. C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,” Mathematical...

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