Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property

Agarwal, Sintunavarat Wutiphol, Kumam, Kumam Poom

Description

In this paper, we prove the coupled coincidence point theorems for a w ∗ -compatible mapping in partially ordered cone metric spaces over a solid cone without the mixed g -monotone property. In the case of a totally ordered space, these results are automatically obvious under the assumption given. Therefore, these results can be applied in a much wider class of problems. We also prove the uniqueness of a common coupled fixed point in this setup and give some example which is not applied to the existence of a common coupled fixed point by using the mixed g -monotone property but can be applied to our results. MSC: 47H10, 54H25.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2013
• Nombre de lectures: 2
• Langue: English

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Agarwaletal.FixedPointTheoryandApplications2013,2013:22
http://www.fixedpointtheoryandapplications.com/content/2013/1/22
RESEARCH OpenAccess
Coupledcoincidencepointandcommon
cofixedpointtheoremslackingthe
mixedmonotoneproperty
1,2 3* 3*RaviPAgarwal ,WutipholSintunavarat andPoomKumam
*Correspondence:
[e-mail]; Abstract
[e-mail] ∗
3 Inthispaper,weprovethecoupledcoincidencepointtheoremsforaw -compatibleDepartmentofMathematics,
FacultyofScience,KingMongkut’s mappinginpartiallyorderedconemetricspacesoverasolidconewithoutthemixed
UniversityofTechnologyThonburi g-monotoneproperty.Inthecaseofatotallyorderedspace,theseresultsare
(KMUTT),BangMod,ThrungKru,
automaticallyobviousundertheassumptiongiven.Therefore,theseresultscanbeBangkok,10140,Thailand
Fulllistofauthorinformationis appliedinamuchwiderclassofproblems.Wealsoprovetheuniquenessofa
availableattheendofthearticle commoncoupledfixedpointinthissetupandgivesomeexamplewhichisnot
appliedtotheexistenceofacommoncoupledfixedpointbyusingthemixed
g-monotonepropertybutcanbeappliedtoourresults.
MSC: 47H10;54H25
Keywords: conemetricspaces;commoncoupledfixedpoint;coupledcoincidence
∗point;w -compatiblemappings;mixedg-monotoneproperty
1 Introduction
The famous Banach contraction principle states that if (X,d)isacompletemetricspace
and T :X →X isacontractionmapping(i.e., d(Tx,Ty) ≤αd(x,y) forall x,y ∈X,whereα
isanon-negativenumbersuchthatα<),thenT hasauniquefixedpoint.Thisprinciple
isoneofthecornerstonesinthedevelopmentofnonlinearanalysis.Fixedpointtheorems
haveapplicationsnotonlyinthevariousbranchesofmathematics,butalsoineconomics,
chemistry, biology, computer science, engineering, and others. Due to the importance,
generalizationsofBanach’scontractionprinciplehavebeeninvestigatedheavilybyseveral
authors.
Following this trend, the problem of existence and uniqueness of fixed points in
partiallyorderedsetshasbeenstudiedthoroughlybecauseofitsinterestingnature.In,
Turinici [] presented the first result in this direction. Afterward, Ran and Reurings []
gavesomeapplicationsofTurinici’stheoremtomatrixequations.TheresultsofRanand
Reurings were further extended to ordered cone metric spaces in [–]. In ,
Nieto
andRodríguez-López[]extendedRanandReurings’stheoremsfornondecreasingmappings and obtained a uniquesolution for a first-order ordinarydifferential equationwith
periodicboundaryconditions.
The notion of coupled fixed points was introduced by Guo and Lakshmikantham [].
Since then, the concept has been of interest to many researchers in metrical fixed point
theory. In , Bhaskar and Lakshmikantham [] introduced the concept of a mixed
© 2013 Agarwal et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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inanymedium,providedtheoriginalworkisproperlycited.Agarwaletal.FixedPointTheoryandApplications2013,2013:22 Page2of20
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monotoneproperty(seefurtherDefinition.).Theyprovedclassicalcoupledfixedpoint
theorems for mappings satisfying the mixed monotone property and also discussed an
application of their result by investigating the existence and uniqueness of a solution of
theperiodicboundaryvalueproblem.Followingthisresult,Harjanietal.[](seealso[,
])studiedtheexistenceanduniquenessofsolutionsofanonlinearintegralequationas
anapplicationofcoupledfixedpoints.Veryrecently,motivatedbytheworkofCaballeroet
al.[],JleliandSamet[]discussedtheexistenceanduniquenessofapositivesolution
forthesingularnonlinearfractionaldifferentialequationboundaryvalueproblem

αD u(t)=f t,u(t),u(t),<t<,+
(.)
u(a)=u(b)=, a,b∈{,},
αwhereα ∈R such that <α ≤, D is the Riemann-Liouville fractional derivative and+
f :(,] ×[,∞) ×[,∞) →[,∞) is continuous, lim +f(t,·,·)=+∞ (f is singular att→
t = ) for all t ∈ (,], f(t,·,·) is nondecreasing with respect to the first component and
decreasingwithrespecttoitssecondandthirdcomponents.
Since their important role in the study of the existence and uniqueness of a solution of
theperiodicboundaryvalueproblem,anonlinearintegralequation,andtheexistenceand
uniquenessofapositivesolutionforthesingularnonlinearfractionaldifferentialequation
boundary value problem, a wide discussion on coupled fixed point theorems aimed the
interestofmanyscientists.
In , Lakshmikantham and Ćirić [] extended the concept of a mixed monotone
property to a mixed g-monotone mapping and proved coupled coincidence point and
commoncoupledfixedpointtheoremswhicharemoregeneralthantheresultofBhaskar
and Lakshmikantham in
[].Anumberofarticlesoncoupledfixedpoint,coupledcoincidence point, and common coupled fixed point theorems have been dedicated to the
improvement;see[–]andthereferencestherein.
Ontheotherhand,in,HuangandZhang[]havere-introducedtheconceptofa
conemetricspacewhichisreplacingthesetofrealnumbersbyanorderedBanachspaceE.
They went further and defined the convergence via interior points of the cone by
which
theorderinEisdefined.Thisapproachallowstheinvestigationofconespacesinthecase
whentheconeisnotnecessarilynormal.Theyalsocontinuedwithresultsconcernedwith
thenormalconesonly.Oneofthemainresultsfrom[]istheBanachcontractionprincipleinthesettingofnormalconespaces.Afterward,manyauthorsgeneralizedtheirfixed
pointtheoremsinconespaceswithnormalcones.Inotherwords,thefixedpointproblem
in the setting of cone metric spaces is appropriate only in the case when the underlying
cone is non-normal but just has interior that is nonempty. In this case only, proper
generalizationsofresultsfromtheordinarymetricspacescanbeobtained.In,Janković
etal.[]gavesomeexamplesshowingthattheoremsfromordinarymetricspacescannot
beappliedinthesettingofconemetricspaces,whentheconeisnon-normal.
Recently, Nashine et al. [] established common coupled fixed point theorems for
∗mixedg-monotoneandw
-compatiblemappingssatisfyingmoregeneralcontractiveconditions in ordered cone metric spaces over a cone that is only solid
(i.e.,hasanonempty
interior)whichimproveworksofKarapınar[]andShatanawi[].ThisresultisanorderedversionextensionoftheresultsofAbbasetal.[].