481601

7172019 481601

httpslidepdfcomreaderfull481601 14

Hindawi Publishing CorporationTe Scienti1047297c World JournalVolume 983090983088983089983091 Article ID 983092983096983089983094983088983089 983091 pageshttpdxdoiorg983089983088983089983089983093983093983090983088983089983091983092983096983089983094983088983089

Research ArticleOn Multivalued Contractions in Cone Metric Spaces

without Normality

Muhammad Arshad1 and Jamshaid Ahmad2

983089 Department of Mathematics International Islamic University H-983089983088 Islamabad 983092983092983088983088983088 Pakistan983090 Department of Mathematics COMSAS Institute of Information echnology Chak Shahzad Islamabad 983092983092983088983088983088 Pakistan

Correspondence should be addressed to Jamshaid Ahmad jamshaid jasimyahoocom

Received 983092 April 983090983088983089983091 Accepted 983089983093 May 983090983088983089983091

Academic Editors A Agouzal A Ibeas F Khani and F Kittaneh

Copyright copy 983090983088983089983091 M Arshad and J Ahmad Tis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wardowski (983090983088983089983089) in this paper for a normal cone metric space ( 1038389) and for the family A of subsets of X established a new conemetric 1103925 A times A rarr 907317 and obtained 1047297xed point of set-valued contraction of Nadler type Further it is noticed in the work of Jankovic et al 983090983088983089983089 that the 1047297xed-point problem in the setting of cone metric spaces is appropriate only in the case when theunderlying cone is nonnormal In the present paper we improve Wardowskirsquos result by proving the same without the assumptionof normality on cones

1 Introduction and Preliminaries

Huang and Zhang [983089] generalized the notion of metric spaceby replacing the set of real numbers by ordered Banachspace and de1047297ned cone metric space and extended Banachtype 1047297xed-point theorems for contractive type mappingsSubsequently some other authors (eg see [983090ndash983089983093] and ref-erences therein) studied properties of cone metric spacesand 1047297xed points results of mappings satisfying contractivetype condition in cone metric spaces Recently Choa et al[983097] Kadelburg and Radenovic [983089983094] Klim and Wardowski

[983089983095] Latif and Shaddad [983089983096] Radenovic and Kadelburg [983089983097]Rezapour and Haghi [983090983088] and Wardowski [983089983092 983090983089] obtained1047297xed points of set-valued mappings in normal cone metricspaces On the other hand it is shown in [983089983089] that most of the 1047297xed points results of mappings satisfying contractivetype condition in cone metric spaces with a normal cone canbe reduced to the corresponding results from metric spacetheory Te 1047297xed-point problem in the setting of cone metricspaces is appropriate only in the case when the underlyingcone is nonnormal because the results concerning 1047297xedpoints and other results in the case of cone metric spaceswith nonnormal solid cones cannot be proved by reducing tometric spaces In this paper we prove the result of Wardowski

[983089983092] without the assumption of normality of cones We needthe following de1047297nitions and resultsconsistent with [983089 983089983089 983089983092]

Let 907317 be a Banach Space and a subset of 907317 Ten iscalled a cone whenever

(i) is closed nonempty and =(ii) + isin for all isin and nonnegative real

numbers

(iii) cap (minus) =

Each cone induces a partial ordering ≼ on 907317 by ≼ if and only if

minus isin So

lt will stand for

≼ and

= while ≪ will stand for minus isin int where int denotes the interior of Te cone is called normal if thereis a number gt 0 such that for all isin 907317

≼ ≼ rArr le (983089)

Te least positive number satisfying(983089) is called the normal constant of

De1047297nition 983089 Let be a nonempty set Suppose the mapping1038389 times rarr 907317 satis1047297es

(d1) ≼ 1038389( ) for all isin and1038389( ) = if and only if

=

7172019 481601


983090 Te Scienti1047297c World Journal

(d2) 1038389( ) = 1038389() for all isin

(d3) 1038389() ≼ 1038389( ) + 1038389() for all isin

Ten 1038389 is called a cone metric on and ( 1038389) is called acone metric space

Let ( 1038389) be a cone metric space isin and ge1 asequence in Ten ge1 converges to whenever forevery isin 907317 with ≪ there is a natural number such that 1038389( ) ≪ for all ge We denote this by limrarrinfin = or rarr ge1 is a Cauchy sequencewhenever for every isin 907317with ≪ there is a natural number such that 1038389( 1038389) ≪ for all ge ( 1038389) is called acomplete cone metric space if every Cauchy sequence in isconvergent

A set sub ( 1038389) is called closed if for any sequence sub convergent to we have isin Denote by ()the collection of all nonempty subsets of and by () acollection of all nonempty closed subsets of Denote by

Fix a set of all 1047297xed points of a mapping In the presentpaper we assume that 907317 is a real Banach space is a cone in907317 with nonempty interior (such cones are called solid) and≼ is a partial ordering with respect to In accordance with[983089983092 De1047297nition 983091983089 and Lemma 983091983089] we minutely modify theidea of 1103925-cone metric to make it more comparable with astandard metric

De1047297nition 983090 Let ( 1038389) be a cone metric space and A bea collection of nonempty subsets of A map 1103925 A timesA rarr 907317 is called an 1103925-cone metric on A induced by 1038389 if the following conditions hold

(H1

) ≼ 1103925() for all

isinA and

1103925() = if and

only if = (H2) 1103925( ) = 1103925() for all isin A

(H3) 1103925() ≼ 1103925( ) + 1103925() for all isin A

(H4) If isin A ≺ isin 907317 with 1103925() ≺ then foreach isin there exists isin such that 1038389( ) ≺

Examples can be seen in [983089983092 examples 983091983089 and 983091983090]

2 Main Result

Teorem983091 Let (1038389) be a complete cone metric space Let Abe a nonempty collection of nonempty closed subsets of

and

let 1103925 A times A rarr 907317 be an 1103925-cone metric induced by 1038389 If for a map rarr A there exists isin (0 1) such that for all isin

1103925983080983081 ≼ 1038389 983080 983081 (983090)

then Fix = 983088

Proof Let 0 be an arbitrary but 1047297xed element of and 1 isin0 If 0 = 1 then 0 isin Fix and if 0 = 1 using the factthat

11039259830800 1983081 ≼ 1038389 9830800 1983081 ≺ radic 10383899830800 1983081 (983091)

we may choose 2 isin such that 2 isin 1 and

1038389 9830801 2983081 ≺ radic 10383899830800 1983081 (983092)

Similarly in case 1 = 2 we may choose 3 isin such that3 isin 2 and

1038389 9830802 3983081 ≺ radic 10383899830801 2983081 ≺ 1048616radic 104861721038389 9830800 1983081 (983093)

We can continue this process to 1047297nd a sequence of pointsof such that

+1 isin = 01 2 1038389 983080 +1983081 ≺ radic 1038389983080minus1 983081

≺ 1048616radic 104861721038389 983080minus2 minus1983081 ≺ sdot sdot sdot ≺ 1048616radic 10486171038389 9830800 1983081 (983094)

Now for any gt

1038389 9830801038389 983081 ≼ 1038389 983080 +1983081 + 1038389 983080+1 +2983081+sdot sdot sdot+10383899830801038389minus1 1038389983081≼ 2 + (+1)2 + sdotsdot sdot + (1038389minus1)210383899830800 1983081≼ 983131 2

1 minus 1298313310383899830800 1983081 (983095)

Let ≪ be given Choose a symmetric neighborhood of such that + sube int Also choose a natural number 1

such that

2

(1minus12

)1038389(0

1

) isin for all

ge 1 Ten

(2

(1 minus 12

))1038389(1 0) ≪ for all ge 1 Tus

1038389 9830801038389 983081 ≼ 983131 2

1 minus 1298313310383899830800 1983081 ≪ (983096)

for all gt Terefore ge1 is a Cauchy sequence Since is complete there exists isin such that rarr Since

1103925983080 983081 ≼ 1038389 983080 983081 ≺ radic 1038389 983080 983081 (983097)

for each +1 isin we have isin such that

1038389(+1 ) ≺ radic 1038389() Now choose a natural number

2 such that

1038389 983080 983081 ≪ 2 forall ge 2 (983089983088)

Ten for all ge 2

1038389983080983081 ≼ 1038389 983080 +1983081 + 1038389 983080+1 983081≼ 1038389 983080 +1983081 + radic 1038389983080 983081≼ 1038389 983080 +1983081 + 1038389 983080 983081 ≪

2 + 2 =

(983089983089)

It follows that rarr and it implies that isin

7172019 481601


Te Scienti1047297c World Journal 983091

Example 983092 Suppose = 0 1 907317 = 2110392501 with the norm

= infin + infin = isin 907317 ge 0 () = and() = 2907317 Ten 0 le le = 2 and = 1 + 2For all ge 1 since lt Terefore is non-normalDe1047297ne 1038389 times rarr 907317 as follows

9830801038389983080983081983081 () = 1039252103925210392521039252 minus 1039252103925210392521039252

(983089983090)

Let A be a family of subsets of of the formA = 0 isin cup isin and de1047297ne 1103925 A timesA rarr 907317 as follows

1103925( )

=9831639831639831631048699983163983163983163852091

1039252103925210392521039252 minus 1039252103925210392521039252 for = 0 = 10486670 1048669 1039252103925210392521039252 minus 1039252103925210392521039252 for = = 983165 Max 1039252103925210392521039252 minus 1039252103925210392521039252983165 for = 0 = 983165Max 1039252103925210392521039252 minus 1039252103925210392521039252983165 for = = 10486670 1048669

(983089983091)

It iseasyto observethat1103925 satis1047297es (H1)ndash(H4) of De1047297nition 983090De1047297ne rarr A as

= 98316310486999831638520910 for isin 8520590 12852061 [0 12852008 minus

12852009

2] for isin 852008 12 1852061 (983089983092)

Note that satis1047297es the conditions of Teorem 983091 with = 12and 0 isin Fix

References

[983089] L-G Huang andX Zhang ldquoCone metric spacesand 1047297xed pointtheorems of contractive mappingsrdquo Journal of Mathematical Analysis and Applications vol 983091983091983090 no 983090 pp 983089983092983094983096ndash983089983092983095983094 983090983088983088983095

[983090] Abdeljawad P Murthy and K as ldquoA Gregus type common1047297xed point theorem of set-valued mappings in cone metricspacesrdquo Journalof Computational Analysis and Applications vol983089983091 no 983092 pp 983094983090983090ndash983094983090983096 983090983088983089983089

[983091] J Ahmad M Arshad and C Vetro ldquoOn a theorem of khan in ageneralized metric spacerdquo International Journal of Analysis vol983090983088983089983091 Article ID 983096983093983090983095983090983095 983094 pages 983090983088983089983091

[983092] M Arshad J Ahmad and E Karapınar ldquoSome common 1047297xedpoint results in rectangular metric spacesrdquo Journal of Analysis vol 983090983088983089983091 Article ID 983091983088983095983090983091983092 983095 pages 983090983088983089983091

[983093] A G Ahmad Z M Fadail M Abbas Z Kadelburg and SRadenovic ldquoSome 1047297xed and periodic points in abstract metricspacesrdquo Abstract and Applied Analysis vol 983090983088983089983090 Article ID983097983088983096983092983090983091 983089983093 pages 983090983088983089983090

[983094] M Arshad A Azam and P Vetro ldquoSome common 1047297xedpoint results in cone metric spacesrdquo Fixed Point Teory and Applications vol 983090983088983088983097 Article ID 983092983097983091983097983094983093 983090983088983088983097

[983095] S H Cho J S Bae and K S Na ldquoFixed point theoremsfor multivalued contractive mappings and multivalued Caristitype mappings in cone metric spacesrdquo Fixed Point Teory and Applications vol 983090983088983089983090 article 983089983091983091 983089983088 pages 983090983088983089983090

[983096] S H Cho and J S Bae ldquoFixed point theorems for multivalued maps in cone metric spacesrdquo Fixed Point Teory and Applications vol 983090983088983089983089 article 983096983095 983090983088983089983089

[983097] Y J Choa S Hirunworakit and N Petrot ldquoSet-valued 1047297xed-point theorems for generalized contractive mappings withoutthe Hausdorff metricrdquo Applied Mathematics Letters vol 983090983092 no983089983089 pp 983089983097983093983097ndash983089983097983094983095 983090983088983089983089

[983089983088] L Gajic and V Rakocevic ldquoQusai-contractions on a nonnormalcone metric spacerdquo Fund Analyzer Application vol 983092983094 pp 983095983093ndash983095983097 983090983088983089983090

[983089983089] S Jankovic Z Kadelburg and S Radenovic ldquoOn cone metricspaces a survey surveyrdquo Nonlinear Analysis Teory Methods amp Applications vol 983095983092 pp 983090983093983097983089ndash983090983094983088983089 983090983088983089983089

[983089983090] G Jungck S Radenovic S Radojevic and V RakocevicldquoCommon 1047297xed point theorems for weakly compatible pairs oncone metric spacesrdquo Fixed Point Teory and Applications vol983090983088983088983097 Article ID 983094983092983091983096983092983088 983090983088983088983097

[983089983091] S R Kumar ldquoCommon 1047297xed pointtheoremsfor subcompatibleand sub sequentially continuous maps in 983090 metric spacesrdquoInternational MatheMatical Forum vol 983095 no 983090983092 pp 983089983089983096983095ndash983089983090983088983088983090983088983089983090

[983089983092] D Wardowski ldquoOn set-valued contractions of Nadler type incone metric spacesrdquo Applied Mathematics Letters vol 983090983092 no 983091

pp 983090983095983093ndash983090983095983096 983090983088983089983089[983089983093] Z Kadelburg S Radenovic and V Rakocevic ldquoA note on the

equivalence of some metric and conemetric 1047297xed point resultsrdquo Applied Mathematics Letters vol 983090983092 pp 983091983095983088ndash983091983095983092 983090983088983089983089

[983089983094] Z Kadelburg and S Radenovic ldquoSome results on set-valuedcontractions in abstract metric spacesrdquo Computers and Math-ematics with Applications vol 983094983090 no 983089 pp 983091983092983090ndash983091983093983088 983090983088983089983089

[983089983095] D Klim and D Wardowski ldquoDynamic processes and 1047297xedpoints of set-valued nonlinear contractions in cone metricspacesrdquo Nonlinear Analysis Teory Methods and Applications vol 983095983089 no 983089983089 pp 983093983089983095983088ndash983093983089983095983093 983090983088983088983097

[983089983096] A Latif and F Y Shaddad ldquoFixed point results for multivaluedmaps in cone metric spacesrdquo Fixed Point Teory and Applica-tions vol 983090983088983089983088 article 983096983095 Article ID 983097983092983089983091983095983089 983090983088983089983088

[983089983097] S Radenovic and Z Kadelburg ldquoSome results on 1047297xed pointsof multifunctions on abstract metric spacesrdquo Mathematical and Computer Modelling vol 983093983091 no 983093-983094 pp 983095983092983094ndash983095983093983092 983090983088983089983089

[983090983088] Sh Rezapour and R H Haghi ldquoFixed point of multifunctionson cone metric spacesrdquo Numerical Functional Analysis and Optimization vol 983091983088 no 983095-983096 pp 983089ndash983096 983090983088983088983097

[983090983089] D Wardowski ldquoEndpoints and 1047297xed points of set-valuedcontractions in cone metric spacesrdquo Nonlinear Analysis Teory Methods and Applications vol 983095983089 no 983089-983090 pp 983093983089983090ndash983093983089983094 983090983088983088983097

7172019 481601


Submit your manuscripts at

httpwwwhindawicom

7172019 481601


983090 Te Scienti1047297c World Journal

(d2) 1038389( ) = 1038389() for all isin

(d3) 1038389() ≼ 1038389( ) + 1038389() for all isin

Ten 1038389 is called a cone metric on and ( 1038389) is called acone metric space

Let ( 1038389) be a cone metric space isin and ge1 asequence in Ten ge1 converges to whenever forevery isin 907317 with ≪ there is a natural number such that 1038389( ) ≪ for all ge We denote this by limrarrinfin = or rarr ge1 is a Cauchy sequencewhenever for every isin 907317with ≪ there is a natural number such that 1038389( 1038389) ≪ for all ge ( 1038389) is called acomplete cone metric space if every Cauchy sequence in isconvergent

A set sub ( 1038389) is called closed if for any sequence sub convergent to we have isin Denote by ()the collection of all nonempty subsets of and by () acollection of all nonempty closed subsets of Denote by

Fix a set of all 1047297xed points of a mapping In the presentpaper we assume that 907317 is a real Banach space is a cone in907317 with nonempty interior (such cones are called solid) and≼ is a partial ordering with respect to In accordance with[983089983092 De1047297nition 983091983089 and Lemma 983091983089] we minutely modify theidea of 1103925-cone metric to make it more comparable with astandard metric

De1047297nition 983090 Let ( 1038389) be a cone metric space and A bea collection of nonempty subsets of A map 1103925 A timesA rarr 907317 is called an 1103925-cone metric on A induced by 1038389 if the following conditions hold

(H1

) ≼ 1103925() for all

isinA and

1103925() = if and

only if = (H2) 1103925( ) = 1103925() for all isin A

(H3) 1103925() ≼ 1103925( ) + 1103925() for all isin A

(H4) If isin A ≺ isin 907317 with 1103925() ≺ then foreach isin there exists isin such that 1038389( ) ≺

Examples can be seen in [983089983092 examples 983091983089 and 983091983090]

2 Main Result

Teorem983091 Let (1038389) be a complete cone metric space Let Abe a nonempty collection of nonempty closed subsets of

and

let 1103925 A times A rarr 907317 be an 1103925-cone metric induced by 1038389 If for a map rarr A there exists isin (0 1) such that for all isin

1103925983080983081 ≼ 1038389 983080 983081 (983090)

then Fix = 983088

Proof Let 0 be an arbitrary but 1047297xed element of and 1 isin0 If 0 = 1 then 0 isin Fix and if 0 = 1 using the factthat

11039259830800 1983081 ≼ 1038389 9830800 1983081 ≺ radic 10383899830800 1983081 (983091)

we may choose 2 isin such that 2 isin 1 and

1038389 9830801 2983081 ≺ radic 10383899830800 1983081 (983092)

Similarly in case 1 = 2 we may choose 3 isin such that3 isin 2 and

1038389 9830802 3983081 ≺ radic 10383899830801 2983081 ≺ 1048616radic 104861721038389 9830800 1983081 (983093)

We can continue this process to 1047297nd a sequence of pointsof such that

+1 isin = 01 2 1038389 983080 +1983081 ≺ radic 1038389983080minus1 983081

≺ 1048616radic 104861721038389 983080minus2 minus1983081 ≺ sdot sdot sdot ≺ 1048616radic 10486171038389 9830800 1983081 (983094)

Now for any gt

1038389 9830801038389 983081 ≼ 1038389 983080 +1983081 + 1038389 983080+1 +2983081+sdot sdot sdot+10383899830801038389minus1 1038389983081≼ 2 + (+1)2 + sdotsdot sdot + (1038389minus1)210383899830800 1983081≼ 983131 2

1 minus 1298313310383899830800 1983081 (983095)

Let ≪ be given Choose a symmetric neighborhood of such that + sube int Also choose a natural number 1

such that

2

(1minus12

)1038389(0

1

) isin for all

ge 1 Ten

(2

(1 minus 12

))1038389(1 0) ≪ for all ge 1 Tus

1038389 9830801038389 983081 ≼ 983131 2

1 minus 1298313310383899830800 1983081 ≪ (983096)

for all gt Terefore ge1 is a Cauchy sequence Since is complete there exists isin such that rarr Since

1103925983080 983081 ≼ 1038389 983080 983081 ≺ radic 1038389 983080 983081 (983097)

for each +1 isin we have isin such that

1038389(+1 ) ≺ radic 1038389() Now choose a natural number

2 such that

1038389 983080 983081 ≪ 2 forall ge 2 (983089983088)

Ten for all ge 2

1038389983080983081 ≼ 1038389 983080 +1983081 + 1038389 983080+1 983081≼ 1038389 983080 +1983081 + radic 1038389983080 983081≼ 1038389 983080 +1983081 + 1038389 983080 983081 ≪

2 + 2 =

(983089983089)

It follows that rarr and it implies that isin

7172019 481601





9830801038389983080983081983081 () = 1039252103925210392521039252 minus 1039252103925210392521039252

(983089983090)


1103925( )

=9831639831639831631048699983163983163983163852091


(983089983091)


= 98316310486999831638520910 for isin 8520590 12852061 [0 12852008 minus

12852009

2] for isin 852008 12 1852061 (983089983092)


References























7172019 481601



httpwwwhindawicom

7172019 481601





9830801038389983080983081983081 () = 1039252103925210392521039252 minus 1039252103925210392521039252

(983089983090)


1103925( )

=9831639831639831631048699983163983163983163852091


(983089983091)


= 98316310486999831638520910 for isin 8520590 12852061 [0 12852008 minus

12852009

2] for isin 852008 12 1852061 (983089983092)


References























7172019 481601



httpwwwhindawicom

7172019 481601



httpwwwhindawicom

481601

Documents

Transcript of 481601