Research Article
A New Faster Iterative Scheme for Numerical Fixed Points
Estimation of Suzuki’s Generalized Nonexpansive Mappings
Shanza Hassan,
1
Manuel De la Sen ,
2
Praveen Agarwal ,
3,4,5
Qasim Ali,
1
and Azhar Hussain
6,7
1
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
2
Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Leioa, Bizkaia,
P.O. Box 48490, Spain
3
Harish-Chandra Research Institute (HRI), Allahabad, UP, India
4
Anand International College of Engineering, Jaipur 303012, India
5
International Center for Basic and Applied Sciences, Jiapur 302029, India
6
Nonlinear Analysis Research Group, Ton Duc ang University, Ho Chi Minh City, Vietnam
7
Faculty of Mathematics and Statistics, Ton Duc ang University, Ho Chi Minh City, Vietnam
CorrespondenceshouldbeaddressedtoAzharHussain;azharhussain@tdtu.edu.vn
Received 5 January 2020; Accepted 13 April 2020; Published 3 July 2020
AcademicEditor:KishinSadarangani
Copyright©2020ShanzaHassanetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
epurposeofthispaperistointroduceanewfour-stepiterationschemeforapproximationoffixedpointofthenonexpansive
mappingsnamedas S
∗
-iterationschemewhichisfasterthanPicard,Mann,Ishikawa,Noor,Agarwal,Abbas,akur,andUllah
iterationschemes.Weshowthestabilityofourproposedscheme.Wepresentanumericalexampletoshowthatouriteration
scheme is faster than the aforementioned schemes. Moreover, we present some weak and strong convergence theorems for
Suzuki’sgeneralizednonexpansivemappingsintheframeworkofuniformlyconvexBanachspaces.Ourresultsextend,improve,
and unify many existing results in the literature.
1. Introduction
Mostofthenonlinearequationscanbetransformedintoa
fixed point problem as follows:
Fu � u, (1)
where F isaself-maponacertaindistancespace X andthe
solution of the aforementioned equation is considered as a
fixed point of the mapping F.Banach[1]provedthatifa
self-map F on a complete metric space is such that
d(Fu, Fv) ≤ qd(u, v), (2)
for 0 ≤ q < 1, then it possesses a unique fixed point u
∗
.
Moreover, the iterative process
Fu
n
� u
n+1
, (3)
called the Picard iteration process, converges to u
∗
. It is
worthmentioningthatPicarditerationprocessisusefulfor
the approximation of the fixed point of the contraction
mappingsbutthecasewhenonesdealingwithnonexpansive
mappingsitmayfailtoconvergetothefixedpointevenif F
has a unique fixed point. Krasnosel’skii [2] showed that
Mann[3]iterationprocesscanapproximatethefixedpoints
of a nonexpansive mapping. In this iteration scheme, the
sequence (u
n
) is generated by an arbitrary u
0
∈ C as
u
n+1
� 1 − α
n
( u
n
+ α
n
Fu
n
, ∀ n ≥ 0, (4)
where (α
n
) isin (0, 1).
In 1974, Ishikawa [4] developed an iterative scheme to
approximate the fixed point of nonexpansive mappings,
where (u
n
) is defined iteratively starting from u
0
∈ C by
u
n+1
� 1 − α
n
( u
n
+ α
n
Fv
n
v
n
� 1 − β
n
( u
n
+ β
n
Fu
n
, (5)
for all n ≥ 0, where (α
n
) and (β
n
) are in (0, 1).
Hindawi
Mathematical Problems in Engineering
Volume 2020, Article ID 3863819, 9 pages
https://doi.org/10.1155/2020/3863819