A variational inequality in complete CAT(0) spaces. (English) Zbl 1323.47066
Summary: In this paper, first, existence of solutions for a variational inequality associated with a nonexpansive mapping in Hadamard spaces is studied. Then, the inexact proximal point algorithm for approximation of a solution of the variational inequality, which is also a fixed point of the nonexpansive mapping, is proposed. We prove the \(\Delta\)-convergence of the generated sequence by the algorithm as well as the strong convergence of a Halpern-type regularization one to a fixed point of the nonexpansive mapping. Our motivation is to give a step toward investigation of variational inequalities in CAT(0) spaces.
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 54E50 | Complete metric spaces |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47J25 | Iterative procedures involving nonlinear operators |
Keywords:
Hadamard space; fixed point; variational inequality; nonexpansive mapping; \(\Delta\)-convergence; strong convergenceReferences:
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