we prove a strong convergence result for a sequence generated by halpern's type iteration for approximating a common fixed point of a countable family of quasi-lipschitzian mappings in a real hilbert space. consequently, we apply our results to the problem of finding a common fixed point of asymptotically nonexpansive mappings, an equilibrium problem, and a variational inequality problem for co...

*Correspondence: [email protected] Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand Abstract We prove a convergence theorem of the Mann iteration scheme for a uniformly L-Lipschitzian asymptotically demicontractive mapping in a CAT(κ ) space with κ > 0. We also obtain a convergence theorem of the Ishikawa iteration scheme for a uniformly L-Lipsc...

In this paper, we show that for a polyhedral multifunction F : R n ! R m with convex range, the inverse function F ?1 is locally lower Lipschitzian at every point of the range of F (equivalently Lipschitzian on the range of F) if and only if the function F is open. As a consequence, we show that for a piecewise aane function f : R n ! R n , f is surjective and f ?1 is Lipschitzian if and only i...

We show that the variational inequality $VI(C,A)$ has aunique solution for a relaxed $(gamma , r)$-cocoercive,$mu$-Lipschitzian mapping $A: Cto H$ with $r>gamma mu^2$, where$C$ is a nonempty closed convex subset of a Hilbert space $H$. Fromthis result, it can be derived that, for example, the recentalgorithms given in the references of this paper, despite theirbecoming more complicated, are not...

We consider another class of generalized Lipschitzian type mappings and utilize the same to prove fixed point theorems for asymptotically regular and uniformly generalized Lipschitzian one-parameter semigroups of self-mappings defined on bounded metric spaces equipped with uniform normal structure which yield corresponding results in respect of semigroup of iterates of a self-mapping as corolla...

Let (X,‖ · ‖) be an infinite-dimensional Banach space with the unit ball B and the unit sphere S. Since the works of Nowak [11], Benyamini and Sternfeld [1], and Lin and Sternfeld [10], it is known that S is a Lipschitzian retract of B. It means that there exists a mapping (retraction) R : B → S satisfying Rx = x for all x ∈ S and also being Lipschitzian. If R satisfies the Lipschitz condition ...

Given an integer function f , the problem is to find its best uniform approximation from a set K of integervalued bounded functions. Under certain conditions on K , the best extremal (maximal or minimal) approximation is identified. Furthermore, the operator mapping f to its extremal best approximation is shown to be Lipschitzian with some constant C or optimal Lipschitzian having the smallest ...

Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → H be a boundedly Lipschitzian strong pseudo-contractionwith a nonempty fixed point set. Three iterative algorithms are proposed for approximating the unique fixed point of T ; one of them is for the selfmapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the d...

we show that the variational inequality $vi(c,a)$ has aunique solution for a relaxed $(gamma , r)$-cocoercive,$mu$-lipschitzian mapping $a: cto h$ with $r>gamma mu^2$, where$c$ is a nonempty closed convex subset of a hilbert space $h$. fromthis result, it can be derived that, for example, the recentalgorithms given in the references of this paper, despite theirbecoming more complicated, are not...