"Approximating fixed points of demicontractive mappings via the quasi-nonexpansive case"

Abstract

"We prove that the convergence theorems for Mann iteration used for approximation of the fixed points of demicontractive mappings in Hilbert spaces can be derived from the corresponding convergence theorems in the class of quasi-nonexpansive mappings. Our derivation is based on an important auxiliary lemma (Lemma \ref{lem3}), which shows that if $T$ is $k$-demicontractive, then for any $\lambda\in (0,1-k)$, $T_{\lambda}$ is quasi-nonexpansive. In this way we obtain a unifying technique of proof for various well known results in the fixed point theory of demicontractive mappings. We illustrate this reduction technique for the case of two classical convergence results in the class of demicontractive mappings: [M\u aru\c ster, \c St. The solution by iteration of nonlinear equations in Hilbert spaces. {\em Proc. Amer. Math. Soc.} {\bf 63} (1977), no. 1, 69--73] and [Hicks, T. L.; Kubicek, J. D. On the Mann iteration process in a Hilbert space. {\em J. Math. Anal. Appl.} {\bf 59} (1977), no. 3, 498--504]."

An inertial self-adaptive algorithm for solving split feasibility problems and fixed point problems in the class of demicontractive mappings

“…Proof. Since S is k-demicontractive, by Lemma 5 we deduce that the averaged mapping S λ = (1 − λ)I + λS is also quasi-nonexpansive, for any λ ∈ (0, 1 − k), and that F ix (S) = F ix (S λ ), for any λ ∈ (0, 1] (see for example [3]).…”

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

“…one can see that the two parameter sequences {d n } and (9) satisfy exactly the same boundedness condition, which is due to the corresponding boundedness condition in the case of quasi-nonexpansive mappings; see Theorem 4.3 in [1].…”

“…Now we state the weak convergence result from Hicks and Kubicek [4] and note that, despite the fact Theorems 1 and 3 were discovered independently, their statements are quite similar. In our opinion, the explanation is that both results were obtained using an important convergence result for quasi-nonexpansive mappings;see [1] for more details.…”

“…Demicontractive mappings are ubiquitous objects in several important contexts of metric fixed-point theory with relevant applications in solving nonlinear problems, such as equilibrium problems, split problems, split common fixed-point problems, split feasibility problems with multiple output sets, split variational inequality problems, split common null point problems, etc. (see [1,2], among others). The main theoretical and applicative interest in studying this class of mappings lies in the fact that demicontractive mappings form one of the largest classes of discontinuous nonexpansive type mappings for which the fixed points can be approximated by means of iterative algorithms.…”

“…It is easily seen that any nonexpansive mapping is k-strictly pseudocontractive, any nonexpansive mapping with Fix (T) = ∅ is quasi-nonexpansive, any quasi-nonexpansive mapping is demicontractive, and a k-strictly pseudocontractive mapping with Fix (T) = ∅ is demicontractive, but the reverses may not be true, as illustrated by the next example, which collects Examples 1.1 and 1.2 in [1]. The natural ambient spaces for studying the iterative approximation of fixed points of demicontractive mappings were, for a long time, the linear settings, i.e., Hilbert and Banach spaces, but very recently, authors started to study this problem in nonlinear settings, as well.…”