Convergence theorems of fixed points for <textbox>κtextbox>-strict pseudo-contractions in Hilbert spaces

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• 作者：Haiyun Zhou
• 关键词：style=""text-decoration ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml24&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=813d75916ab9f2d0b089282eed7b871f"" ti
• 刊名：Nonlinear Analysis
• 出版年：2008
• 出版时间：15 July 2008
• 年：2008
• 卷：69
• 期：2
• 页码：456-462
• 全文大小：228 K

文摘

Let style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml2&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=9073b60498c38b2a3bb029dc094eefad"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">C be a closed convex subset of a real Hilbert space style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml3&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=767eb2bf328bb32f6a1d69d9aca87d16"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">H and assume that style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml4&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=53b311f0e5c0f51198c0e564ea594bf5"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">T:C→H is a style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml5&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=45993647185f393d02ff9c1ed318e067"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">κ-strict pseudo-contraction such that style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml6&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=a5d1ef0836242dc6417dc68bb65d5f9c"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">F(T)={xC:x=Tx}≠0/. Consider the normal Mann’s iterative algorithm given by

where style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml8&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=acf77e3b116431cce31b1174fc47ccfa"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">S:C→H is defined by style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml9&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=e915dfa5791d42f53892ec47b3af0c72"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">Sx=κx+(1−κ)Tx, style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml10&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=bccfb0172f29f13ec4050cbc852ab446"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">P_C is the metric projection of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml11&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=4a4526d8c57b7c47621ae1e530ba4b46"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">H onto style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml12&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=7f27e9075e2f48fe618c843a971ef299"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">C and for all style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml14&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=e0f71ca2a6351c837e8fc3bca7e4b8fc"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">n≥1. It is proved that if the control parameter sequence style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml15&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=9f74ffc6e13c88a4b68b1cac5d26127e"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">{_n} is chosen so that style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml16&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=d5f38117b7fa713c3d0adbf297760d42"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">κ≤_n≤1 and , then style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml18&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=0dc881e2197766fe117677e89633a39a"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">{x_n} converges weakly to a fixed point of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml19&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=15c6411e9baa35ac005d402ae4a9fa56"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">T. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml20&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=5a0be91dce5cf767ba6bac73ec5f44f0"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">C. The results presented in this article respectively improve and extend the recent results of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml21&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=ca3eb30ff74834d8a410b843ad47f2d6"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">κ-strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349] from style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml22&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=dcaabfd3ad2195dbf460f9655d9b4fa8"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">κ-strictly pseudo-contractive self-mappings to nonself-mappings and of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] from nonexpansive mappings to style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4NWCGSN-7&_mathId=mml23&_user=1067359&_cdi=5659&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=6ce11a3ac06100dcad3d2aa1df6dc87a"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">κ-strict pseudo-contractions.