{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:45:27Z","timestamp":1753886727003,"version":"3.41.2"},"reference-count":21,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,6,10]],"date-time":"2012-06-10T00:00:00Z","timestamp":1339286400000},"content-version":"vor","delay-in-days":161,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11071279"],"award-info":[{"award-number":["11071279"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self\u2010mapping T<\/jats:italic> of a closed convex subset C<\/jats:italic> of a CAT(0) space X<\/jats:italic>. Suppose that the set Fix(T<\/jats:italic>) of fixed points of T<\/jats:italic> is nonempty. For a contraction f<\/jats:italic> on C<\/jats:italic> and t<\/jats:italic> \u2208 (0,1), let x<\/jats:italic>t<\/jats:italic><\/jats:sub> \u2208 C<\/jats:italic> be the unique fixed point of the contraction x<\/jats:italic> \u21a6 t<\/jats:italic>f<\/jats:italic>(x<\/jats:italic>)\u2295(1 \u2212 t<\/jats:italic>)T<\/jats:italic>x<\/jats:italic>. We will show that if X<\/jats:italic> is a CAT(0) space satisfying some property, then {x<\/jats:italic>t<\/jats:italic><\/jats:sub>} converge strongly to a fixed point of T<\/jats:italic> which solves some variational inequality. Consider also the iteration process {x<\/jats:italic>n<\/jats:italic><\/jats:sub>}, where x<\/jats:italic>0<\/jats:sub> \u2208 C<\/jats:italic> is arbitrary and x<\/jats:italic>n<\/jats:italic>+1<\/jats:sub> = \u03b1<\/jats:italic>n<\/jats:italic><\/jats:sub>f<\/jats:italic>(x<\/jats:italic>n<\/jats:italic><\/jats:sub>)\u2295(1 \u2212 \u03b1<\/jats:italic>n<\/jats:italic><\/jats:sub>)T<\/jats:italic>x<\/jats:italic>n<\/jats:italic><\/jats:sub> for n<\/jats:italic> \u2265 1, where {\u03b1<\/jats:italic>n<\/jats:italic><\/jats:sub>}\u2282(0,1). It is shown that under certain appropriate conditions on \u03b1<\/jats:italic>n<\/jats:italic><\/jats:sub>, {x<\/jats:italic>n<\/jats:italic><\/jats:sub>} converge strongly to a fixed point of T<\/jats:italic> which solves some variational inequality.<\/jats:p>","DOI":"10.1155\/2012\/421050","type":"journal-article","created":{"date-parts":[[2012,6,10]],"date-time":"2012-06-10T21:00:22Z","timestamp":1339362022000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces"],"prefix":"10.1155","volume":"2012","author":[{"given":"Luo Yi","family":"Shi","sequence":"first","affiliation":[]},{"given":"Ru Dong","family":"Chen","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,6,10]]},"reference":[{"key":"e_1_2_4_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12494-9"},{"key":"e_1_2_4_2_2","doi-asserted-by":"publisher","DOI":"10.1155\/S1687182004406081"},{"key":"e_1_2_4_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1019-1"},{"key":"e_1_2_4_4_2","series-title":"Monographs and Textbooks in Pure and Applied Mathematics","volume-title":"Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings","author":"Goebel K.","year":"1984"},{"key":"e_1_2_4_5_2","series-title":"Colecc. 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