_{1}

^{*}

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]).

In 2007, Huang and Zhang [

Following [

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 [

(M_{1})(M_{2}) iff,

(M_{3}),

(M_{4})for all. is said to be a cone metric space.

Lemma 1.1 [

Definition 1.2 [

Definition 1.3 [

Theorem 1.1 (Theorem 2.1 [

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 [

for all.

Then has a unique fixed point in.

Theorem 1.3 (Theorem 2.2 [

, 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 [

, for allwhere

,.

Then has a unique fixed point in.

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.