A new algorithm for variational inequality problems with a generalized phi-strongly monotone map over the set of common fixed points of a finite family of quasi-phi-nonexpansive maps, with applications

• Published in: Journal of fixed point theory and applications 2018-03, Vol.20 (1), p.1-15, Article 29
• Main Authors: Chidume, C. E., Nnyaba, U. V., Romanus, O. M.
• Format: Article
• Language: English
• Subjects: Analysis; Banach spaces; Computational geometry; Convergence; Convexity; Iterative algorithms; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Optimization; Theorems
• ISSN: 1661-7738; 1661-7746
• DOI: 10.1007/s11784-018-0502-0

Let E be a uniformly convex and uniformly smooth real Banach space with dual space E ∗ and C be a nonempty, closed and convex subset of E . Let A : E → E ∗ be a generalized Φ -strongly monotone and bounded map and let T i : C → E , i = 1 , 2 , 3 , … , N be a finite family of quasi- ϕ -nonexpansive maps such that ∩ i = 1 N F ( T i ) ≠ ∅ . Suppose V I ( A , ∩ i = 1 N F ( T i ) ) ≠ ∅ . A new iterative algorithm that converges strongly to a point in V I ( A , ∩ i = 1 N F ( T i ) ) is constructed. Results obtained are applied to a convex optimization problem. Furthermore, the theorems proved complement, improve and unify several recent important results. Finally, we consider a family { T i } i = 1 N of maps where for each i , T i maps E into its dual space E ∗ and prove a strong convergence theorem for V I ( A , ∩ i = 1 N F ( T i ) ) , where F J ( T i ) is the set of J -fixed points introduced by Chidume and Idu.