Some Notes on the Paper “New Common Fixed Point Theorems for Maps on Cone Metric Spaces”

Abstract

In this paper, we show that Theorem 2.1 [1] (resp. Theorem 2.2 [1]) is a consequence of Corollary 2.1 [1] ( resp. Corollary 2.2 [1]).

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M. Ahmed, "Some Notes on the Paper “New Common Fixed Point Theorems for Maps on Cone Metric Spaces”," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 2, 2013, pp. 150-151. doi: 10.4236/ijmnta.2013.22019.

1. Introduction

In 2007, Huang and Zhang [2] initiated fixed point theory in cone metric spaces. On the other hand, in 2011, Haghi, Rezapour and Shahzad [3] gave a lemma and showed that some fixed point generalizations are not real generalizations. In this note, we show that Theorem 2.1 [1] and Theorem 2.2 [1] are so.

Following [2], let be a real Banach space and be the zero vector in, and. is called cone iff 1) is closed, nonempty and2) for all and nonnegative real numbers3).

For a given cone, we define a partial ordering with respect to by iff. (resp.) stands for and (resp.), where denotes the interior of. In the paper we always assume that is solid, i.e.,. It is clear that leads to but the reverse need not to be true.

The cone is called normal if there exists a number such that for all, implies.

The least positive number satisfying above is called the normal constant of.

Definition 1.1 [2]. Let be a nonempty set. A function is called cone metric iff

(M1)(M2) iff,

(M3),

(M4)for all. is said to be a cone metric space.

Lemma 1.1 [3]. Let be a nonempty and. Then there exists a subset such that and is one-to-one.

Definition 1.2 [4]. Let be a cone metric space and be mappings. Then, is called a coincidence point of and iff.

Definition 1.3 [4]. Let be a cone metric space. The mappings are weakly compatible iff for every coincidence point of and,.

Theorem 1.1 (Theorem 2.1 [1]). Let be a cone metric space and let be constants with. Suppose that the mappings satisfy the condition

for all.

If the range of contains the range of and is a complete subspace, then and have a unique point of coincidence in. Moreover, if and are weakly compatible, then and have a unique fixed point.

Theorem 1.2 (Corollary 2.1 [1]). Let be a complete cone metric space and let i = (1,2,3,4,5) be constants with. Suppose that the mapping satisfies the condition

for all.

Then has a unique fixed point in.

Theorem 1.3 (Theorem 2.2 [1]). Let be a cone metric space and let the mappings satisfy the condition

, for allwhere

,.

If the range of contains the range of and is a complete subspace, then and have a unique point of coincidence in. Moreover, if and are weakly compatible, then and have a unique fixed point.

Theorem 1.4 (Corollary 2.2 [1]). Let be a complete cone metric space and let the mapping satisfies the condition

, for allwhere

,.

Then has a unique fixed point in.

2. Main Result

In this section, we show that that Theorem 1.1 (resp. Theorem 1.3) is a consequence of Theorem 1.2 (resp. Theorem 1.4).

Theorem 2.1. Theorem 1.1 is a consequence of Theorem 1.2.

Proof. By Lemma 1.1, there exists such that and is one-to-one. Define a map by for each. Since is one-to-one on, then is well-defined. Also, for arbitrary,

where are constants with

.

From the completeness of, there exists such that

by Theorem 1.2. Hence, and have a point of coincidence which is also unique. Since and are weakly compatible, then and have a unique common fixed point.

Theorem 2.2. Theorem 1.3 is a consequence of Theorem 1.4.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] G. Song, X. Sun, Y. Zhao and G. Wang, “New Common Fixed Point Theorems for Maps on Cone Metric Spaces,” Applied Mathematics Letters, Vol. 23, No. 9, 2010, pp. 1033-1037. doi:10.1016/j.aml.2010.04.032 [2] 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 [3] R. H. Haghi, Sh. Rezapour and N. Shahzad, “Some Fixed Point Generalizations Are Not Real Generalizations,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 74, 2011, pp. 1799-1803. [4] C. Di 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

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