Research Article
Approximating Fixed Points of Reich–Suzuki Type Nonexpansive
Mappings in Hyperbolic Spaces
Kifayat Ullah,
1
Junaid Ahmad ,
1
Manuel De La Sen ,
2
and Muhammad Naveed Khan
1
1
Department of Mathematics, University of Science and Technology, Bannu 28100, Khyber Pakhtunkhwa, Pakistan
2
Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia),
P. O. Box 644-Bilbao, Barrio Sarriena, Leioa 48940, Spain
CorrespondenceshouldbeaddressedtoJunaidAhmad;ahmadjunaid436@gmail.com
Received 25 May 2020; Revised 17 June 2020; Accepted 24 June 2020; Published 22 July 2020
AcademicEditor:AliJaballah
Copyright©2020KifayatUllahetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Inthiswork,weprovesomestrongand Δ convergenceresultsforReich-Suzukitypenonexpansivemappingsthrough M iterative
process.Auniformlyconvexhyperbolicmetricspaceisusedasunderlyingsettingforourapproach.Wealsoprovideanillustrate
numerical example. Our results improve and extend some recently announced results of the metric fixed-point theory.
1. Introduction
A self-map S on a subset B of a metric space X �(X, p) is
calledcontractionifthereexistssomeconstant θ ∈ [0, 1) such
that for all a, b ∈ B it follows that p(Sa, Sb) ≤ θp(a, b). If
p(Sa, Sb) ≤ p(a, b) for all a, b ∈ B, then S is called non-
expansive.Apoint w ∈ B iscalledafixedpointof S whenever
w � Sw.Banach[1]theorem(1922)statesthatanycontraction
map S on a complete metric space has a unique fixed point
whichisthelimitofthesequence x
k
generatedbythePicard
iterates,thatis, x
k+1
� Sx
k
.In1965,Kirk[2],Browder[3],and
G¨ ohde [4] independently proved that any self nonexpansive
mapping S definedonaboundedclosedconvexsubset B ofa
uniformlyconvexBanachspacealwayshasafixedpoint.Now,a
natural question which comes in mind is that whether the
sequence x
k
ofPicarditeratesconvergestoafixedpointofa
self nonexpansive mapping. e answer of this question in
generalisnegative.erefore,thereisaneedtoconstructsome
new procedures to overcome such situations and to obtain a
betterrateofconvergence,forexample,Mann[5],Ishikawa[6],
Noor [7], S [8], Abbas and Nazir [9], and akur et al. [10]
iterativeprocessesareoftenusedtoapproximatefixedpointsof
nonexpansive mappings. In 2008, Suzuki [11] introduced a
weaker notion of nonexpansive mappings: a self-map S ona
subset B ofametricspaceissaidtobeSuzukitypenonexpansive
if for every two elements a, b in B, p(Sa, Sb) ≤ p(a, b) holds
whenever (1/2)p(a, Sa) ≤ p(a, b).Itiseasytoobservethatthe
classofSuzukitypenonexpansivemappingsproperlyincludes
the class of nonexpansive mappings.
e class of Suzuki type nonexpansive mappings was
studiedextensivelybymanyauthors(cf.[10,12–21]).Very
recently, Pant and Pandey [22] introduced Reich–Suzuki
typenonexpansivemappingswhichinturnincludetheclass
of Suzuki type nonexpansive mappings.
Definition 1 (see [22]). Let B be a nonempty subset of a
metricspace.Amap S: B ⟶ B issaidtobeReich–Suzuki
typenonexpansiveifforall a, b ∈ B,thereissomeconstant
t ∈ [0, 1) such that
1
2
p(a, Sa) ≤ p(a, b) ⟹ p(Sa, Sb)
≤ tp(a, Sa)+ tp(b, Sb)+(1 − 2t)p(a, b).
(1)
Approximation of fixed points of nonexpansive and
generalized nonexpansive mappings is an active area of
researchonitsown[23–27].Recently,PantandPandey[22]
usedakuretal.[10]iterativeprocesstoapproximatefixed
points of Reich–Suzuki type nonexpansive mappings. e
purposeofthispaperistoprovestrongand Δ convergence
resultsforReich–Suzukitypenonexpansivemappingsunder
M iterativeprocess[12],whichisknowntoconvergefaster
Hindawi
Journal of Mathematics
Volume 2020, Article ID 2169652, 6 pages
https://doi.org/10.1155/2020/2169652