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 be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure, a nonempty bounded closed convex subset of , a finite family of asymptotically nonexpansive self-mappings on with common sequence , , be two sequences in such that and be a contraction on . Under suitable conditions on the sequences , , we show the existence of a sequence satisfying the relation where for some unique integers and . Further we prove that converges strongly to a common fixed point of , which solves some variational inequality, provided as for . As an application, we prove that the iterative process defined by , , converges strongly to the same common fixed point of .
|49J40||Variational methods including variational inequalities|
|47J20||Inequalities involving nonlinear operators|
|47H10||Fixed point theorems for nonlinear operators on topological linear spaces|
|49L25||Viscosity solutions (infinite-dimensional problems)|
|47H09||Mappings defined by “shrinking” properties|