Convergence theorems of solutions of a generalized variational inequality

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2011; no. 1; pp. 1 - 10
• Main Authors: Yu, Li, Liang, Ma
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.01.2011; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Analysis; Applications of Mathematics; Classification; Constrictions; Convergence; Differential Geometry; Feasibility; fixed point; Inequalities; Integers; Mapping; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; nonexpansive mapping; relaxed cocoercive mapping; S. Park's Contribution to the Development of Fixed Point Theory and KKM Theory; Topology; variational inequality; nonexpansive mapping; variational inequality; fixed point; relaxed cocoercive mapping
• ISSN: 1687-1812; 1687-1820; 1687-1812; 2730-5422
• DOI: 10.1186/1687-1812-2011-19

The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where r ≥ 1 is an integer and each C m is assumed to be the solution set of a generalized variational inequality. Let C be a nonempty closed and convex subset of a real Hilbert space H . Let A m , B m : C → H be relaxed cocoercive mappings for each 1 ≤ m ≤ r . It is proved that the sequence { x n } generated in the following algorithm: where u ∈ C is a fixed point, { α n }, { β n }, { γ n }, { δ (1, n ) }, ..., and { δ ( r , n ) } are sequences in (0, 1) and , are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions. 2000 AMS Subject Classification : 47H05; 47H09; 47H10.