weak convergence and norm convergence implies strong convergence

... Banach space and let E* b e the dual space of E. For a sequence {x n }ofE and a point x ẻ E, the weak convergence of {x n }tox and the strong convergence of {x n }tox are denoted by x n x and x n đ ... Lin et al.: Weak and strong convergence theorems of implicit iteration process on Banach spaces. Fixed Point Theory and Applications 2011 2011:96. Submit your manuscript to a journal and benefi ... J: Weak and strong convergence to fixed points of asymptotically non-expansive mappings. Bull Austral Math Soc. 43, 153–159 (1991). doi:10.1017/S0004972700028884 Lin et al. Fixed Point Theory and...

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... Banach space and let E* b e the dual space of E. For a sequence {x n }ofE and a point x ẻ E, the weak convergence of {x n }tox and the strong convergence of {x n }tox are denoted by x n x and x n đ ... Lin et al.: Weak and strong convergence theorems of implicit iteration process on Banach spaces. Fixed Point Theory and Applications 2011 2011:96. Submit your manuscript to a journal and benefi ... two nonexpansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the...

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... 3502–3511, 2009. 8 M. O. Osilike and S. C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 32, ... κ-strict 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 ... 2007. 11 G. Marino and H K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1,...

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... spaces, Fixed Point Theory and Appl., vol. 2010, Article ID 262691, 12 (2010) [20] Takahashi, W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. ... appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Strong convergence theorems and rate of convergence of multi-step iterative methods for ... problems and fixed point problems with applications. Nonlinear Anal. 72, 99–112 (2010) [18] Shimoji, K, Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications....

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... al.: Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems. Fixed Point Theory and Applications 2011 2011:53. Submit your manuscript to a journal and ... {z n } converge weakly to thesameelementof  N i =1 Fix(S i ) ∩  if and only if lim inf n 〈Ax n , x - x n 0, x ẻ C. Note that Ceng, Teboulle and Yao’s method has only weak convergence. Very ... approximation method are well defined and converge strongly to P F(S)∩Ω . Moti vated and inspired by the works of Nadezhkina and Takaha shi [23], Ceng et al. [25], and Ceng et al. [27], in this paper...

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... 1 n 0 T(t)y n dt, n ≥ 0 . Then {x n } and {y n } convergence strongly to x* which is the unique solution of the sys- tems of variational inequalities: Piri and Badali Fixed Point Theory and Applications 2011, ... δ n+1 Bx n )  . Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 9 of 16 RESEARC H Open Access Strong convergence ... iterative process, strong convergence, variational inequality 1 Introduction Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by 〈.,.〉 and || . ||,...

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... product 〈·,·〉 andnorm||·||,andletC be a closed convex subset of H.Wewritex n ⇀ x to indicate that the sequence {x n }con- verges weakly to x,andx n đ x implies that {x n } converges strongly to ... is maximal if and only if for (x, f ) ẻ H × H, 〈x- y, f-g〉 ≥ 0 for every (y, g) Î G(T) implies f Î Tx.LetA be an inverse strongly mono- tone mapping of C into H,andletN C v be t he normal cone ... article as: Bunyawat and Suantai: Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings. Fixed Point Theory and Applications 2011...

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... Banach space, and {x n }beasequenceinE.Wedenotebyx n đ x and x n x the strong convergence and weak convergence of {x n }, respectively. The normal- ized duality mapping J : E ® 2 E* is defined by J x ... and relatively nonexpansive mappings. Fixed Point Theory Appl 2008, 2008:11, (Article ID 528476). 9. Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and ... x ), which implies that ¯ u = x and f 0 = J ¯ u . It follows from Remark 2.1(iv) and (v) that E* has the Kadec-Klee property, and so J y n k → J ¯ u by (2.4) and (2.5). By Remar k 2.1(vii) and (ii),...

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... is bounded, closed and con- vex and T is a nonexpansive mapping of C into itself, then F(T) is nonem pty; for instance, see [5]. Recently, Tada and Takahashi [6] and Takahashi and Takahashi [7] considered ... Kangtunyakarn: Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions. Fixed Point Theory and Applications ... b < 1 , and j 1 , j 2 , α j 3 ∈ (κ,1 ) for all j = 1, 2, . For every n ẻ N, let S n and S are S-mapping gener- ated by T n , ,T 1 and r n , r n-1 , , r 1 and T n , T n-1 , , and r n , r n-1 ,...

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... closed convex subset of H and T : K đ H be a mapping. In this paper, the set of fixed points of T is denoted by F(T). We use symbols đ and to denote strong and weak convergence, respectively. For ... article as: He and Du: Strong convergence theorems for equilibrium problems and fixed point problems: A new iterative method, some comments and applications. Fixed Point Theory and Applications ... n ≥ 0 . If He and Du Fixed Point Theory and Applications 2011, 2011:33 http://www.fixedpointtheoryandapplications.com/content/2011/1/33 Page 2 of 15 RESEARC H Open Access Strong convergence theorems...

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... notations: 1. for the weak convergence and đ for the strong convergence. 2. ω w (x n )={x : ∃x n j  x } denotes the weak ω-limit set of {x n }. 2. Preliminaries We need some facts and tools in a real ... (2009). doi:10.1016/j.na.2008.02.090 14. Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mappingand a equilibrium problem. J Optim Theory Appl. 133, 359–370 (2007). ... [0, 1/2] and C 2 : = (1/2, 1]. Hence, | T n x − T n y | = k n |x − y |≤|x − y | for all x, y ∈ C 1 and n ∈ N and | T n x − T n y | =0≤|x − y | for all x, y C 2 and n N . For x ẻ C 1 and y ẻ...

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... 14 Fixed Point Theory and Applications 19 W. Takahashi and Kei Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings ... Accretive and Monotone Type,vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. 2 S Y. Matsushita and W. Takahashi, Weak and strong convergence ... 20–30, 2009. 15 A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133,...

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... Su and Qin 6 modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems. In 2008, Takahashi et al. 7 proved strong ... asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in 3, 5–8 and many authors. 2. ... with inner product ·, · and norm ·, and we write x n → x to indicate that the sequence {x n } converges strongly to x.LetC be a nonempty closed convex subset of H,andletT : C → C be a mapping....

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... Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. 24 W. Takahashi, Weak and strong convergence ... A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008. 15 S. Plubtieng and R. Punpaeng, ... quasi-φ- nonexpansive mappings and equilibrium problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 750–760, 2010. 18 S. H. Wang, G. Marino, and F. H. Wang, Strong convergence theorems...

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