Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption

Lu-Chuan Ceng Ngai-Ching Wong Jen-Chih Yao

Functional Analysis mathscidoc:1910.43650

Computers & Mathematics with Applications, 56, (9), 2312-2322, 2008.11
Let E be a real Banach space with a uniformly Gteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E,{T i} i= 1 N a finite family of asymptotically nonexpansive self-mappings on K with common sequence {k n} n= 1[1,),{t n},{s n} be two sequences in (0, 1) such that s n+ t n= 1 (n 1) and f be a contraction on K. Under suitable conditions on the sequences {s n},{t n}, we show the existence of a sequence {x n} satisfying the relation x n=(1 1 k n) x n+ s n k n f (x n)+ t n k n T r n n x n where n= l n N+ r n for some unique integers l n 0 and 1 r n N. Further we prove that {x n} converges strongly to a common fixed point of {T i} i= 1 N, which solves some variational inequality, provided x n T i x n 0 as n for i= 1, 2,, N. As an application, we prove that the iterative process defined by z 0 K, z n+ 1=(1 1 k n) z n+ s n k n f (z n)+ t n k n
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@inproceedings{lu-chuan2008fixed,
  title={Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption},
  author={Lu-Chuan Ceng, Ngai-Ching Wong, and Jen-Chih Yao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020205502705744179},
  booktitle={Computers & Mathematics with Applications},
  volume={56},
  number={9},
  pages={2312-2322},
  year={2008},
}
Lu-Chuan Ceng, Ngai-Ching Wong, and Jen-Chih Yao. Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption. 2008. Vol. 56. In Computers & Mathematics with Applications. pp.2312-2322. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020205502705744179.
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