Common Fixed Point Theorems for a Pair of Set-Valued Maps and Two Pairs of Single-Valued Maps

YU Jing，GU Feng

（College of Science，Hangzhou Normal University，Hangzhou 310036，China）

Common Fixed Point Theorems for a Pair of Set-Valued Maps and Two Pairs of Single-Valued Maps

YU Jing，GU Feng

（College of Science，Hangzhou Normal University，Hangzhou 310036，China）

Under strict contractive conditions，the paper established some common fixed point theorems for a pair of set-valued mappings and two pairs of single-valued mappings with no compacity and continuity.The theorems extended and improved the corresponding results of some existing literatures.

weakly compatible maps；D-mappings；single and set-valued maps；common fixed points

1 Introduction and preliminaries

Throughout this paper，we assume that（X，d）is a metric space，B（X）is the set of all nonempty bounded subsets of X.As in［1，2］，we define the functionsδ（A，B）and D（A，B）as follows： for all A，B∈B（X）.If Acontains a single point a，we writeδ（A，B）＝δ（a，B）.Also，if Bcontains a single point b，it yieldsδ（A，B）＝d（a，b）.

The definition of the functionδ（A，B）yields the following：

Definition 1［1，3-5］A sequence｛An｝of subsets of Xis said to be convergent to a subset Aof Xif

（i）Given a∈A，there is a sequence｛an｝in Xsuch that an∈Anfor n＝1，2，3，…，and｛an｝converges to a.

（ii）Givenε＞0，there exists a positive integer Nsuch that An⊂Aεfor n＞Nwhere Aεis the unionof all open spheres with centers in Aand radiusε.

Lemma 1［1-2］If｛An｝and｛Bn｝are sequences in B（X）converging to Aand Bin B（X），respectively，then the sequence｛δ（An，Bn）｝converges toδ（A，B）.

Lemma 2［2-3］Let｛An｝be a sequence in B（X）and ybe a point in Xsuch thatδ（An，y）→0as n→∞.Then，the sequence｛An｝converges to the set｛y｝in B（X）.

Definition 2［2］A set-valued mapping Fof Xinto B（X）is said to be continuous at x∈Xif the sequence｛Fxn｝in B（X）converges to Fx whenever｛xn｝is a sequence in Xconverging to xin X.Fis said to be continuous on Xif it is continuous at every point in X.

Lemma 3［2］Let｛An｝be a sequence of nonempty subsets of Xand zin Xsuch thatindependent of the particular choice of each an∈An.If a self-map I of Xis continuous，then｛Iz｝is the limit of the sequence｛IAn｝.

Definition 3［6］The mappings F：X→B（X）and f：X→Xareδ-compatible if whenever｛xn｝is a sequence in Xsuch that fFxn∈B（X），Fxn→｛t｝and fxn→t for some t∈X.

Definition 4［7］The mappings F：X→B（X）and f：X→Xare weakly compatible if they commute at coincidence points，that is

It can be seen thatδ-compatible maps are weakly compatible but the converse is not true.Examples supporting this fact can be found in［7］.

Definition 5［8］The mappings F：X→B（X）and I：X→Xare said to be D-mappings if there exists a

sequence｛xn｝in Xsuch that＝｛t｝for some t∈X.Examples supporting this fact can be found in［8］.

In［9］，Fisher proved the following theorem：

Theorem1［9］Let F，Gbe mappings of Xinto B（X）and I，Jbe mappings of Xinto itself satisfying

for all x，y∈X，where 0≤c＜1.If Fcommutes with Iand Gcommutes with J，GX⊂IX，FX⊂JXand I or Jis continuous，then F，G，I and Jhave a unique common fixed point uin X.

On the other hand，Fisher［9］proved the following fixed point theorem on compact metric spaces：

Theorem2［9］Let F，Gbe continuous mappings of a compact metric space（X，d）into B（X）and I，Jare continuous mappings of Xinto itself satisfying the inequality

for all x，y∈Xfor which the righthand side of the above inequality is positive.If the mappings Fand I commute and Gand Jcommute and GX⊂IX，FX⊂JX，then there is a unique point uin Xsuch that

In［10］，Ahmed extended Theorem 1and Theorem 2，he proved the following theorem：

Theorem3［10］Let I，Jbe function of a compact metric space（X，d）into itself and F，G：X→B（X）two set-valued functions with＜1-（a＋b），holds whenever the right hand side of

（2）is positive.

If the pair｛F，I｝and｛G，J｝are weakly compatible，and if the functions Fand I are continuous，then there is a unique point uin Xsuch that

Recently，Bouhadjera and Djoudi［11］extended and improved the above results，proved the following theorem：

Theorem4［11］Let（X，d）be a metric space，let F，G：X→B（X）and I，J：X→Xbe set and singlevalued mappings，respectively satisfying the conditions： for all x，y∈X，where 0≤α＜1，a≥0，b≥0，a＋b＜1，whenever the right hand side of（2）is positive.If either

（3）F，I are weakly compatible D-mappings；G，Jare weakly compatible and FXor JXis closed or

（3’）G，Jare weakly compatible D-mappings；F，I are weakly compatible and GXor IXis closed.

Then there is a unique common fixed point t in Xsuch that

Inspired by above works，in this paper，we prove some new common fixed point theorem for a pair of set-valued mappings and two pair of single-valued mappings under strict contractive conditions.These theorems use minimal type commutativity with no continuity and compacity requirement.Our results presented improve and extend some recent results in Fisher［9］，Ahmed［10］and Bouhadjera and Djoudi［11］.

（3）0≤α＜1，a≥0，b≥0，a≤

2 Main results

Theorem5 Let（X，d）be a metric space，let F，G：X→B（X）be two set-valued mappings，and I，J，S，T：X→Xbe four single-valued mappings，respectively satisfying the conditions：

for all x，y∈X，where 0≤α＜1，a≥0，b≥0，a＋b＜1，whenever the right hand side of（iii）is positive.If one of the following conditions is satisfied，then F，G，I，J，Sand Thave a unique common fixed point t in Xsuch that

（1）F，ISare weakly compatible D-mappings；G，JTare weakly compatible and FXor JTXis closed；

（2）G，JTare weakly compatible D-mappings；F，ISare weakly compatible of and GXor ISXis closed.

Proof （1）Suppose that F，ISare weakly compatible D-mappings，G，JTare weakly compatible and FXor JTXis closed.Then there exists a sequence｛xn｝in Xsuch that，for some t∈X.

If FXis closed，from the condition FX⊂JTX，there exists a point uin Xsuch that JTu＝t.Using

inequality（iii）we have

Taking the limit as ntends to infinity and using Lemma 1，it comes

It is obvious thatα＋（1-α）a＜1，and so from the above inequality that Gu＝｛JTu｝.Since Gand JTare weakly compatible，thus Gu＝｛JTu｝implies that GJTu＝JTGuand hence

Again using inequality（iii），we have

Letting n→∞and using Lemma 1，we obtain

N

ote thatα＋（1-α）（a＋b）＜1，then we have GGu＝｛JTu｝.Hence｛JTu｝＝GGu＝JTGu，further we obtain Gu＝GGu＝JTGu，hence Guis a common fixed point of Gand JT.Since GX⊂ISX，then there is apoint v∈Xsuch that｛ISv｝＝Gu.Moreover，using inequality（iii），we get

It is easy to see thatα＋（1-α）b＜1，the above inequality implies that Fv＝Gu＝｛ISv｝.Since Fv＝｛ISv｝，by the weak compatibility of Fand IS，we get FISv＝ISFv，hence we have

Next we will show that FFv＝Gu.In fact，if FFv≠Gu，by the condition（iii），we obtain which is a contradiction，thus FFv＝Gu.Further we get FGu＝Gu＝ISGu，and so Guis also a common fixed point of Fand IS.

Now we show that SGu＝Gu.In fact，from the condition（iii），we have

Since IS＝SI，FS＝SF，so FSGu＝SFGu＝SGu，ISSGu＝SISGu＝SGu，hence，the above inequality implies that

Sinceα＋（1-α）（a＋b）＜1so the above inequality implies that SGu＝Gu.Therefore，we have ISGu＝IGu＝Gu.

Next we will show that TGu＝Gu.In fact，from the condition（iii），we have

Since JT＝TJ，GT＝TG，so JTTu＝TJTu＝TGu，hence，the above inequality implies that

Sinceα＋（1-α）（a＋b）＜1，so the above inequality implies that TGu＝Gu.Therefore，we have JTGu＝JGu＝Gu.

Since Gu＝｛t｝，then we have

If JTXis closed，we can similarly prove（＊）hold.

（2）If G，JTare weakly compatible D-mappings；F，ISare weakly compatible of and GXor ISXis closed，similar to（1）for the same reason we can prove that（＊）hold.

Finally，we prove that t is unique.In fact，let t′be another common fixed point of the maps F，G，I，J，Sand Tsuch that t′≠t.Then，using the condition（iii），we get

which is a contradiction，this implies that t′＝t.Hence，tis the unique common fixed point of F，G，I，J，Sand T.

Remark 1 Truly，our result generalizes the result of Fisher［9］and Ahmed［10］，since we have not assuming compacity but only the so-called D-mappings and the minimal condition of the closedness.

Remark 2 If we take S＝T＝E（Eis identity mapping）in Theorem 5，then we can obtain corresponding results of Bouhadjera and Djoudi［11］，is omitted in here.

In Theorem 5let S＝Twe have the following Theorem 6.

Theorem6 Let（X，d）be a metric space，let F，G：X→B（X）be two set-valued mappings，and I，J，T：X→Xbe three single-valued mappings，respectively satisfying the conditions：

for all x，y∈X，where 0≤α＜1，a≥0，b≥0，a＋b＜1，whenever the right hand side of（iii）is positive.If one of the following conditions is satisfied，then F，G，I，Jand Thave a unique common fixed point t in Xsuch that

（1）F，ITare weakly compatible D-mappings；G，JTare weakly compatible and FXor JTXis closed；

（2）G，JTare weakly compatible D-mappings；F，ITare weakly compatible of and GXor ITXis closed.

In Theorem 5let I＝Jand S＝Twe have the following Theorem 7.

Theorem7 Let（X，d）be a metric space，let F，G：X→B（X）be two set-valued mappings，and I，T：X→Xbe two single-valued mappings，respectively satisfying the conditions：

for all x，y∈X，where 0≤α＜1，a≥0，b≥0，a＋b＜1，whenever the right hand side of（iii）is positive.If one of the following conditions is satisfied，then F，G，I and Thave a unique common fixed point t in X such that

（1）F，ITare weakly compatible D-mappings；G，ITare weakly compatible and FXor ITXis closed；

（2）G，ITare weakly compatible D-mappings；F，ITare weakly compatible of and GXor ITXis closed.

Remark 3 If we take 1）F＝G；2）F＝Gand S＝T＝E（Eis identity mapping）；3）F＝G，I＝Jand S＝T；4）F＝G，I＝S，J＝T；5）I＝Sand J＝Tin Theorem 5，several new result can be obtain.

［1］Fisher B.Common fixed points of mappings and set-valued mappings［J］.Rostock Math Kolloq，1981，18：69-77.

［2］Fisher B，Sessa S.Two common fixed point theorems for weakly commuting mappings［J］.Period Math Hungar，1989，20：207-218.

［3］Imdad M，Khan M S，Sessa S.On some weak conditions of commutativity in common fixed point theorems［J］.Internat J Math Math Sci，1988，11（2）：289-296.

［4］Sessa S，Fisher B.On common fixed points of weakly commuting mappings［J］.Internat J Math Math Sci，1986，9（2）：323-329.

［5］Naidu S N R，Prasad J R.Fixed point theorems for a pair of set-valued maps on a metric space［J］.Nonlinear Anal，1986，10（12）：1421-1426.

［6］Liu Lishan.Common fixed points of a pair of single-valued mapping and a pair of set-valued mappings［J］.Qufu Shifan Daxue Xuebao：Ziran Kexue Ban，1992，18（1）：6-10.

［7］Jungck G，Rhoades B E.Fixed points for set valued functions without continuity［J］.Indian J Pure Appl Math，1998，29（3）：227-238.

［8］Djoudi A，Khemis R.Fixed points for set and single valued maps without continuity［J］.Demonstratio Mathematica，2005，38（3）：739-751.

［9］Fisher B.Common fixed points of mappings and set-valued mappings on a metric spaces［J］.Kyungpook Math J，1985，25：35-42.

［10］Ahmed M A.Common fixed point theorems for weakly compatible mappings［J］.Rocky Mountain J Math，2003，33（4）：1189-1203.

［11］Bouhadjera H，Djoudi A.Common fixed point theorems for single and set-valued maps without continuity［J］.An St Univ Ovidius Constanta，2008，16（1）：49-58.

关于一对集值映象和两对单值映象的公共不动点定理

余 静，谷 峰

（杭州师范大学理学院，浙江杭州310036）

该文的主要目的是，对于一类严格压缩条件，在不具有紧性和不使用连续性的条件下，建立了一对集值映象和两对单值映象的公共不动点定理.定理推广和改进了一些现有文献的相应结果.

弱相容映象；D-映象；单值和集值映象；公共不动点

10.3969／j.issn.1674-232X.2012.02.012

O177.91 MSC2010：47H10；54H25 Article character：A

1674-232X（2012）02-0151-06

Received date：2011-03-07

Supported by the National Natural Science Foundation of China（10771141）；the Natural Science Foundation of Zhejiang Province（Y6110287）and Teaching Reformation Foundation of Graduate Student of Hangzhou Normal University.

GU Feng（1960—），male，professor，engaged in nonlinear functional analysis and its application.E-mail：gufeng99＠sohu.com