### Journal Menu >> Advances in Pure Mathematics, 2012, 2, 6-7 http://dx.doi.org/10.4236/apm.2012.21002 Published Online January 2012 (http://www.SciRP.org/journal/apm) A Note on the Paper “Generalized -Contraction for a Pair of Mappings on Cone Metric Spaces” Mohamed Abd El-Rahman Ahmed Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt Email: mahmed68@yahoo.com Received August 10, 2011; revised September 20, 2011; accepted September 28, 2011 ABSTRACT We note that Theorem 2.3  is a consequence of the same theorem for one map. Keywords: Generalized -Contraction; Weakly Compatible; Solid Cone 1. Introduction Huang and Zhang [2 ] initiated fixed po int theory in cone metric spaces. On the other hand, the authors  gave a lemma and show ed that some fixed point generalizations are not real generalizations. In this note, we show that Theorem 2.3 [1 ] is so. Following , let E be a real Banach space and  be the zero vector in E, and . P is called cone iff PEP1) P is closed, nonempty and by P ,, 2) for all axxyP and nonnegative real numbers a, b, 3) PP . For a given cone P, we define a partial ordering  with respect to P by xy iff yxP . xy (resp. xy) stands for xy and xy (resp. yxin ), where denotes the interior of P. In the paper we always assume that P is solid, i.e., tP int Pint P. It is clear that xy leads to xy>0K but the reverse need not to be true. The cone P is called normal if there exists a number such that for all ,xyE, xy implies xKyE. The least positive number satisfying above is called the normal constant of P. Definition 1.1 . Let X be a nonempty set. A func-tion is called cone metric iff :dX X1M , ,ydx2M dxy,yx,d iff xy, 3M ,  ,,y dyxdx4M , dxy d,,,xz dzyfor all ,,xyz X, d: is said to be a cone metric space. In , the authors gave the following important lem- ma. Lemma 1.1 Let X be a nonempty and fXXX. Then there exists a subset Y such that fY fX: and fYX is one-to-one. ,Definition 1.2 . Let Xd,: be a cone metric space and fgX XzX  be mappings. Then, is called a coincidence point of f and g iff fzgz. Definition 1.3 . Let X,d,: be a cone metric space. The mappings fgX XzX are weakly compatible iff for every coincid ence point of f and g, fgx gfx:PP. Definition 1.4 (see ). Let P be a solid cone in a real Banach space E. A nondecreasing function  is called a compari son function iff xPxx for 1) ,X and  ; int P int implies xxxPlimn; 2) 0nxxP for all 3)  . In , the authors established the following fixed point theorem. ,Theorem 1.1 Let Xd,: be a cone metric space, P a solid cone and ,fgX X. Assume that fg is a generalized  -contraction; i.e., ,dfx fyu, for all xyX and some u where      ,, ,,,,,, .2udgxgydfxgxdgxf ydgyfxdfygy XgX, ffX or gSuppose that X is a complete subspace of X, and f and g are weakly compati-Copyright © 2012 SciRes. APM M. ABD EL-R. AHMED 7ble. Then the mappings f and g have a unique common fixed poi n t in X. fX or g2. Main Result In Theorem 1.1, if we choose XgI (XI:= the identity map on X), then we have the following theorem. Theorem 2.1 Let ,Xd: be a cone metric space, P a solid cone and fXX. Assume that f is a general-ized -contraction; i.e., fy u,dfx for all ,xyX and some u where  ,, ,,,,2udxydxfxdxfx dyf, ,.dyfyy Suppose that fXYX or X is a complete subspace of X. Then the mapping f has a unique fixed point in X. Now, we state and prove our main result in the fol-lowing way. Theorem 2.2 Theorem 1.1 is a consequence of Theo-rem 2.1. Proof. By Lemma 1.1, there exists such that  gYgX and :gYX is one-to-one. Define a map :hgYgY by xf xhg for each xgY. Since g is one-to-one on Y, then h is well- defined. u,dfx fy for all ,xyX and some u where     Since YgX is complete, by us-ing Theorem 2.1, there exists 0xX such that hgxgxf x000. Hence, f and g have a point of coincidence which is also unique. Since f and g are weakly compatible, then f and g have a unique common fixed poi nt. Remark 2.1 Since Theorem 1  is a special case of Theorem 1.1, then it is a consequence of Theorem 2.1, too. REFERENCES  A. Razani, V. Rakocevic and Z. Goodarzi, “Generalized -Contraction for a Pair of Mappings on Cone Metric Spaces,” Applied Mathematics and Computation, Vol. 217, No. 22, 2011, pp. 8899-8906.  L.-G. Huang and X. Zhang, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp. 1468-1476. doi:10.1016/j.jmaa.2005.03.087  R. H. Haghi, Sh. Rezapour and N. Shahzad, “Some Fixed Point Generalizations Are Not Real Generalizations,” Nonlinear Analysi s, Vol. 74, 2011, pp. 1799-1803.  C. D. Bari and P. Vetro, “-Pairs and Common Fixed Points in Cone Metric Spaces,” Rendiconti del Circolo Matematico di Palermo, Vol. 57, No. 2, 2008, pp. 279- 285. doi:10.1007/s12215-008-0020-9  ,, ,,,, 2udgxgydfxgxdgxf ydgdfygy ,,.yfx  Copyright © 2012 SciRes. APM