 Applied Mathematics, 2011, 2, 1114-1118 doi:10.4236/am.2011.29153 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM New Common Fixed Point Results for Four Maps on Cone Metric Spaces* Yan Han, Shaoyuan Xu School of Mathematics and Statistics, Hubei Normal University, Huangshi, China E-mail: hanyan702@126.com Received June 6, 2011; revised July 2, 2011; accepted July 9, 2011 Abstract In this paper, some new existence and uniqueness of common fixed points for four mappings are obtained, which do not satisfy continuity and commutation on non-normal cone metric spaces. These results improve and generalize several well-known comparable results in the literature. Keywords: Common Fixed Point, Cone Metric Space, Non-Normal Cone 1. Introduction and Preliminaries Since Huang and Zhang  introduced the concept of cone metric space, the study of common fixed points of mappings satisfying certain contractive conditions on cone metric spaces has been at the center of strong research activity, because it has not only important theoretical meaning but also wide applications. Recently, some authors obtained a number of meaningful consult- ing fixed point theorems for one or two mappings on cone metric spaces(see [1-3,5-8]). The aim of this paper is to present coincidence points results for four mappings without satisfying the notion of continuity and commuta- tion on non-normal cone metric spaces. Common fixed point theorems are obtained under weakly compatible maps. Our results generalized and unified these main results in [1-5]. We recall some definitions and properties of cone metric spaces in . Let be a real Banach space and be a subset of , EP E denotes the zero element of and int denotes the interior of . The subset is called a cone if and only if EPP P1) is closed, nonempty and PPax by, 2) , ,, ,0, ,abx yPP úab3) xP=xP x and  . Given a cone , we define a partial ordering PE with respect to by Pxy if and only if yxP. We shall write x < y if xy and xy, while xy will stand for . A cone is called normal if there is a number such that for all intyxNP>0P,xyP, .xyimpliesxN y The least positive number satisfying the above inequa- lity is called the normal constant of . PDefinition 1.1. () Let X be a nonempty set. Suppose that the mapping satisfies: :X EdX(d1) ,dxy for all ,xyX and ,=dxy if and only if =xy; (d2) ,= ,dxydyx for all ,xyX; (d3) dzy,,,dxydxz for all ,,xyz X. Then is called a cone metric on dX and ,Xd is called a cone metric space. It is clear that the cone metric space is more general than metric space. Definition 1.2 () Let ,Xd be a cone metric space. Then we say that nx is: 1) a Cauchy sequence if for every with cEc, there is such that for all ; N,>nm xccE, ,N dxnm2) a convergent sequence if for every  with c, there is such that for all Nx>,mN ,dx cm for some fixed x in X. A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X. Definition 1.3. () Let f and g be self maps of a set X. If for some ==wfxgxx in X, then x is called a coincidence point of f and g, and is called a point of coincidence of wf and g. Definition 1.4. () The mappings ,:fgX X are weakly compatible if, for every xX, holds =fgxggx whenever =fxgx. Lemma 1.5. () Let f and g be weakly compati- ble self maps of a set X. If f and g have a unique point of coincidence , then is the uni- que common fixed point of ==xwf gxwf and g. Remark 1.6. Let ,Xd be a cone metric space with a cone . If Pdx dxy,y h*This paper is supported by the Foundation of Education Ministry, Hubei Province, China (No: D20102502). , for all Y. HAN ET AL.1115 1,, 0,xyX h , then ,=dxy, which implies that =xy. 2. Main Results In this section, we give some common fixed point theo- rems for four mappings defined on a cone metric space. Normality of the cone is not assumed. Theorem 2.1. Let ,Xd, :J XX1234,,adIxSxadIxTy be a cone metric space. Suppose mappings satisfy ,,SITdSxTy aa d,  5,,,dIxJyJy Tya dJySx, for all xyX, where , 4,5i0= 1, 2,3ai satisfying 451232,2,maxaaaa aIX12 aaor aSX JX ,,SX IXJX345nm 2121212221222 1110=1 =11010,,, ,,,11=1 ,.1nmnnnnm mnniiim immmmdyydyydy ydyyMNNMNd yyMNN MNdyyMN MNNMNMdyyMN In analogous way, we gain 221 10,11mnmNMNdy yMdyyMN ,, Copyright © 2011 SciRes. AM Y. HAN ET AL. 1116 122 10,11mnmMNdy yNdyyMN , and 121 21 0,11mnmMNdy yNdyyMN ,. Thus, for >>0nm11010,max1,1,11=,,nmmmmdyyMNN MNNM dMN MNdyyyy  where 0mc as . mFor each , choose >0 such that cxintP, where , ..,xiex c. For this , we can choose a natural number 1 such that N10, for . Thus, we get 0>mN 10 0,,, >nm md yydyycforallnmN. Therefore ny is a Cauchy sequence in ,Xd. Suppose JX is complete, there exists qJX such that 2. So we can find a such that 22ySX1== nnnxJx qasn=pJpqSX qSX JXn. (If is com- plete, there exists , then the conclu- sions remain the same.) Letting , and by 221=,nnSx Jxq as , we can choose a natural nnumber such that 1N3421,4naacdy q and 34221nny1,2aacdy ,,nn. for 1n. Now we show Nshow that . By (2.1), we have =Tp q  2212222 3425 221212212 34215 221212 22212 3421,, ,,,,,=,, ,,, ,,,,,,nnnnnnnnnnnnnn nnnndTpqdSx TpdSxqa dIxJpadIxSxadJpTpad IxTpadJpSxd Sxqad yqad yyadqTpad yTpadqyd yqadyydyqadyyad qTpady 2252,,1,.nnnydyqdqTpadqy Taking , we get n 34124 212145 21, ,1,nnnaadqTp aaady yaaa dyq Then, we get ,=dqTp qJp . =qTpTXIX22ccc, i.e. At the same time, as ==Tp, therexe ists u in X such that =Iuq. From (2.1),    1234525,,,,,=,.d qadIuJpadIuSua dJp Tpad IuTpadJpSuaadSuq = ,Su dSuTp,Hence, from Remark 1.6, we know Therefore, Next if we assume=Suq.==,==JpqSuIuq . Tp IX is complete, there exists IX such that 2221n nx q21nyuX==IxT asn q. So we can find a  such that =Iuq. (If TX is complete, there exists Xn cqTX, thenonclusions remain the samhoose a natural number 2N such that I the c.) Then, we cae251,aacdy q and 21 4n2521 21,2nnaacdyy for . Now we show that . By (2.1), w 2nN=Su qe get  21 211212321214 21521 2112 2322142152 21121 21223, ,,,,,,,=, ,,,,,,,nnnnn nnnnnnnnnnnq dqadIuJxa dIuSua dJxTxa dIu TxadJxSud Txqad yqadqSuadyyadqyadySudyqadyq dyyad qSuady 21 242152 2121,1,,,nnnnnnyadqyady ydyqdqSu,dSuSuTx dTx,n,. Letting , by Lemma 1.6 we get n25135 212145 211,.nnnadyaa adyq 1,aSuq aaady ,Thus, we have ,=22ccdqSu c, i.e. SuJ X=qS X ists pX, then there ex such that =Jpq. In the samq. e method we canobtain =Tp also Finally, we show that and TJ, S and I have a unique point of coinciden in ceX. Assume there exists another point zX such that ==Jxz , then Tx12345,,,,dSuJx adad JxTxadIuTxad JxSu ,= ,,dqzTx adIuIuSu,Copyright © 2011 SciRes. AM Y. HAN ET AL.1117 which gives ,,. That is to say, 145,dqzaaa dqz thus , =dqz, .., =====ieTxTp Jx Jpq zq is a unique point of coincidence in X of T and J. Similarly, we also have q is a uniqueoincidence of S and point of cI by induction. So, according to Lemma 1.5, q is the unique comon ed point of ,TJ and mfix,SI. Terefore, q is the unique com- mon fix point the four mappings ,,TJS and hed ofI. The proof ofe theorem is completed. As a variant of Theoremwe get the following Corollary. ollary 2.2. Let , th 2.1, CorXd be a cone metric space. Suppose mappings ,, ,:SITJXX satisfy  12,,(,),,,mn mmnn nmdSxTy adIadIxSxadJyTy dJyS (2.4) 345mnmnxJyadIxTy axfor all ,xyX, where 0=1, 2,3, 4,5iai satisfying 1 45123452max,<1 2max,<1.aa aar aaaaa23ao If and onef , SXJX TXIX ,,IXJX and oSXTX space of is a comlete sub- pX, then the four mappings ,,SIT and J have a unique point of coincidence in X. Moreover, if ,TJ and ,SI are weakly compatible, respectively, all of the mappings ,,SIT and J have a unique common fixed point. Proof. It follows from Theorem 2.1, that the four mappings ,,mm nSIT and nJ have a unique common fixed point q. Now, 1===mm mSq  S S qSqSSqand 1===mm mIqIIqIqI Iq implies that Sq and Iq are d poinfd malso fixets or mS anI. Hence, ==Sq Iq qng ==TqReark 2.3. Compared with corresponding results in e [4, Theorem 2.8 and Corollary 2.9], Theo- w that our hy. By usiq. ame method, we havethe sJqmthe literaturrem es metric spaces, which improve and ge2.1 shopotheses are greatly weaker than those conditions. On the one hand, we ensure the existence and uniqueness of a common fixed point of four mappings without continuity and commutation; on the other hand, they are special cases of Theorem 2.1 and Corollary 2. 2 with 45=aa. Hence, Theorem 2.1 present a more general format of common fixed point for four mappings without continuity and commutation on non- normal cone metric spaces, which extends the main results in . Now we use Theorem 2.1 to obtain a series of new common fixed point theorems for four mappings in non-normal conneralize several known results in [1-3]. Corollary 2.4. Let ,Xd be a cone metric space. Suppose mappings ,,,:SITJ XX satisfy ,, ,,,d SxTyd IxJydIxSxdJyTydIxTy d (2.5) ,,JySx,xyX,for all where ,, 0 and 122<. If TX IX, SX JX and one of X and ,,JTX is a comSX Ie of Xplete subspacX,then theings four mapp,,SIT and J have a unique point ofn coincidence iX. Moreover, if ,TJ and  ,SIpiare patibweakly comle, respectively, all of the mapngs ,,SIT and J haveue com- mon fixed point. of. Let2345, ==, ==aaaa a uniqPro 1=a in Theo- rem 2.1. Corollary 2.5. Let ,Xd ,, ,:SITJbe a cone metric space. Sugs satisfyppose mappin X X d,, ,,,Sx TydIxJyIxSxdJyTy (2.6) for all ,dxyX, where0 , and 2<1. If IX , JX TSXXand one of SXX and ,,IX J TX is a complete sub- space ofX, then the four mings app, ,SIT andJ ence in have a nt of coiunique poincidX. Moreover, if ,TJ and  ,SIappings are weampatible, respectively, all of the m,,SIT and kly coJ haveue com- mon fixed point. ollary 2Let a uniqCor.6. ,Xd be a cone metric space. Suppose mappings :SIJ XX satisfy ,,,T kdIxSx ldJyTy ,,dSx ,,Ty (2.7) for all ,xyX, and klwhere ,0,1kl<1. If IX and one of , JX TSXX,,IX J SXX and TX is a complete sub- space ofX, then the four ms ,SI apping ,T andJ ence in have a nt of coiunique poincidX. Moreover, if ,TJ and ,SIappings are weampatible, respectively, all of the m,,SIT and kly coJ haveue com- mon fixed point. of. Let45 23==0, =, =a aa kal in Theo- rem 2.1, the conclusi true. Corollary 2.7. a uniqProLet 1=aons are ,Xd ,, ,:SITJbe a cone metric space. Sugs satisfyppose mappin X X  ,,,,dSxTyTyldJySx (2.8) for all ,kdIxxyX, and klwhere ,0,1kl<1 . If IX and one of , JX TSXXSXX and ,,IX J TX is a complete sub- space ofX, then the four ms ,SI apping ,T andJ ence in have a nt of coiunique poincidX. Moreover, if ,TJ and  ,SI are weampatible, respectively, kly coCopyright © 2011 SciRes. AM Y. HAN ET AL. Copyright © 2011 SciRes. AM 1118 aall of the mppings ,,SIT and J haveque com mon fixed point. ollary () Let a uni-Cor2.8. ,Xd be a cone metric space. Suppose mapp,, ,:SITX X satisfy dSings xTy J,,Ix,kdJy for all ,xyX, where k. If 0,1, SXJX TXIX and one of  SX X,,IX J and TX is a complete sub- space of X, then ,TJ and ,SI have a unique point of coincidence in X. Moreover, if ,TJ and ,SIly are weakSc id ompat spectively, all of the mappings ,,IT anble, reJ llais a norhave a ue common fixed point. ark 2.9. Compared to Theorem 2.1 and Corollary 2.2-2.8 in [r Corory 2.2, 2.4-2.8 do not require that condnconeiquRem3], ouition mal ”. Moreover, when w“P e further restrict ==xIJI in Corollary 2.4, which xI is the identity map on X, we get Theorem 2.1 in , and we get Corollary 2.2-2.8 in  when we even define =ST in Corollary .8. Hence, Corollary 2.2, 4-2.8 improve and geeralize Theorem 2.1 and Corollary 2.2-2.8 in . ollary 2.10. Let 2.2, 2.n5-22.Cor,Xd be a cone metric space. Suppose mappings ,:fgX X satisfy  ,, ,,,fx fydgxgygx fxdgyfyd fx(2.9) ,,gyd ddgxfyfor all ,xyX, where ,, 0 and 122< . If the range of g contains the range of f and gX or fX is a complete subspace of X, then f and g ha point ofve a unique coincidence in X. Moreover, if f and g are weakly compatible, f d ang ha uniqumon fixed point. P f. Let =ST fnd ==ve oa=e acomroIJg in Corollary 2.4. Corollary.11. Let, 2 Xd be a cone metric space. Suppose mappings ,:fgX X satisfy 345,,,,adfygyayadgyfx for all ,12,,dfxfygy a,a dfxgxfdgxd gx (2.10) xyX, where 0a=i1, 2,3, 4,5i and . If 5=1 <1iiafXgX and gX or fX is a complete subspace of X, then f and g have aoint of unique pcoincidence in X. Moreover, if f and g are compatible, weakly f and g hamon fipointgi lvexed a unique com. Proof. In (2.10) interchanng the ro es of x and y, and adding the new inequality to.10), yield (2.9ith ) w(223 451=, =, =2aa aaa . 2Remark 2.12. We note that in our Corolla 2.10 and Corollary 2.11 rygX or fX is complete, and the results remae same, weorem 2.1 in  requires that in thhile ThgX is co addition, Theorem 2. i [mplete. In1 in  and Theorem 2.1 in  generalize the corre- sponding resultsn1-2]. Therefore, above all, our results improve and unify all of these main results in [1-5]. Remark 2.13. In Theorem 2.1 we do not require mappings ,,ST I and J to be compact or continuous. Moreover, we delete the condition “P is a normal cone” in this papers. In addition, when we choose =, =0,RPE in the above theorems and corollaries, similar conns will be gained in simple metric spaces. 3. Acknowements The authors thank the referee for his/her careful reading nd usefuclusioledgl suggestions of the manuscript. ppings,” Journal al Analysis and Applications, Vol. 332, No. 68-1476. doi:10.1016/j.jmaa.2005.03.087a . References 4  L. G. Huang and X. Zhang, “Cone Metric Space and Fixed Point Theorems of Contractive Maof Mathematic2, 2007, pp. 14  M. Abbas and G. Jungck, “Common Fixed Point Results for Noncommuting Mappings without Continuity in Cone Metric Spaces,” Journal of Mathematical Analysis and Applications, Vol. 341, No. 1, 2008, pp. 416-420. doi:10.1016/j.jmaa.2007.09.070  M. Abbas and B. E. Rhoades, “Fixed and Periodic Point Results in Cone Metric Spaces,” Applied Mathematics Letters, Vol. 22, No. 4, 2009, pp. 511-515. doi:10.1016/j.aml.2008.07.001  M. Abbas, B. E. Rhoades and T. Nazir, “Common Fixed Points for Four Maps in Cone Metric Spaces,” Applied Mathematics and Computation, Vol. 216, No. 1, 2010, pp. 80-86. doi:10.1016/j.amc.2010.01.003  G. X. Song, X. Y. Sun, Y. Zhao and G. T. Wang, “New Common Fixed Point Theorems for Maps on Cone Met-ric Spaces,” Applied Mathematics Letters, Vol. 23, No. 9, 2010, pp. 1033-1037. doi:10.1016/j.aml.2010.04.032  C. D. Bari and P. Vetro, “φ-Pairs and Common Fixed Points in Cone Metric Spaces,” Rendiconti del Circolo Matematico Palermo, Vol. 57, 2008, pp. 279-285.  S. Radenovic, “Common Fixed Points under Contractive Conditions in Cone Metric Spaces,” Computers and Mathematics with Applications, Vol. 58, No. 6, 2009, pp. 1273-1278. doi:10.1016/j.camwa.2009.07.035  Sh. Rezapour and R. Hamlbarani, “Some Note on the Paper ‘Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analy-sis and Applications, Vol. 345, No. 2, 2008, pp. 719-724. doi:10.1016/j.jmaa.2008.04.049