We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate connections between firm nonexpansiveness, cocoerciveness averagedness. The convergence of associated fixed-point iterations is discussed particular focus on case degenerate metric, since degeneracy often encountered when reformulating many existing first-order operator splitting algorithms as a res...

Abstract Answering a question of Garbulińska-Wȩgrzyn and Kubiś, we prove that Gurariĭ operators form dense $$G_\delta $$ G δ -set in the space $$\mathcal B(\mathbb G)$$ B ( ) all nonexpansive on $...

In this paper we investigate the following problem (see I.A. Rus, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca, 2006, pp. 193-194): Let X be a set with structure and (X,S(X),M) be a large fixed point structure on X. Let U ∈ S(X). An operator p : U → U is by definition coincidence producing operator on (X,S(X),M) if for each f ∈M(U) there exists u ∈ U such that f(u) = p(u). The pr...

we introduce an implicit method for nding a common element of the set of solutions of systems of equilibrium problems and the set of common xed points of a sequence of nonexpansive mappings and a representation of nonexpansive mappings. then we prove the strong convergence of the proposed implicit schemes to the unique solution of a variational inequality, which is the optimality condition fo...

A new monotonicity, M-monotonicity, is introduced, and the resolvant operator of an M-monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI (S+,F +G) is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is construc...

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi’s question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-...

Abstract In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function f such that its image under a bounded linear operator A minimizes another convex function g. Based on an idea introduced in [9], we propose a split proximal algorithm with a way of s...

To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century when Carl Friedrich Gauss developed the method of least squares of a system of linear equations — its solutions can be viewed as fixed points of averaged projections onto hyperplanes. A powerful ge...

with kn ≥ 0, rn ≥ 0, kn → 1, rn → 0 n → ∞ , for each x, y ∈ C, n 0, 1, 2, . . . . If kn 1 and rn 0, 1.1 reduces to nonexpansive mapping; if rn 0, 1.1 reduces to asymptotically nonexpansive mapping; if kn 1, 1.1 reduces to asymptotically nonexpansive-type mapping. So, a generalized asymptotically nonexpansive mapping is much more general than many other mappings. Browder 1 introduced the followi...