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Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption. (English) Zbl 1165.49007
Summary: Let E be a real Banach space with a uniformly Gâteaux 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 (n1) and f be a contraction on K. Under suitable conditions on the sequences {t n }, {s 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 1rnN. 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 T r n n z n , converges strongly to the same common fixed point of {T i } i=1 N .
MSC:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49L25Viscosity solutions (infinite-dimensional problems)
47H09Mappings defined by “shrinking” properties
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