Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities

Abstract

We introduce a new iteration method and prove strong convergence theorems for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of monotone and Lipschitz-type continuous Ky Fan inequality. Under certain conditions on parameters, we show that the iteration sequences generated by this method converge strongly to the common element in a real Hilbert space. Some preliminary computational experiences are reported.

Hybrid Proximal Point and Extragradient Algorithms for Solving Equilibrium Problems

“…The authors showed that under certain conditions on {α k } and {r k }, {x k } and {u k } strongly converge to z = P r Sol(f,C)∩F ix(S) (g(z)). Recently, iterative methods for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonexpansive mapping have further developed by many authors (see [2,8,10,24,25] and the references therein).…”

Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

“…The method has been modified and extended in various ways by many authors in Hilbert spaces for finding a common point of the solution set of variational inequalities and the fixed point set of nonexpansive mappings. [25][26][27][28] Recently, it was also extended to the lexicographic variational inequality problem in the space  s by Anh et al 2 The iteration algorithm is defined as…”

A New Iterative Method for Equilibrium Problems and Fixed Point Problems

“…During last few years, iterative algorithms for finding a common element of the set of solutions of Fan inequality and the set of fixed points of nonexpansive mappings in a real Hilbert space have been studied by many authors (see, e.g., [2,4,[22][23][24][25][26][27][28]). Recently, Anh [22] studied the existence of a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of Fan inequality for monotone and Lipschitz-type continuous bifunctions. He introduced the following new iterative process:…”