Erratum to "Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Space"

We regret making following mistake in the above-mentioned paper [1]. We would like to correct it and explain some notations.

(1) In [1] we introduced a new concept of integral type contraction in cone metric spaces and generalized Brancieri and Meir-Keeler theorems in such spaces. [1, Theorem  2.9] is an extension of Brancieri's theorem, and [1, Theorem  3.2] is an extension of Brancieri and Meir-Keeler's results. We asserted the following in [1, Theorem  2.9].

(i)"Let be a complete cone metric space and be a normal cone. Suppose is a non-vanishing map and a sub-additive cone integrable on each such that for each , . If is a map such that for all

(1)

for some , then has a unique fixed point in ."

Also, we asserted in [1, Theorem  3.2] the following.

(ii)"Let be a complete regular cone metric space and be a mapping on . Assume that there exists a function from into itself satisfying the following:

(B1) and for all .

(B2) is nondecreasing and continuous function. Moreover, its inverse is continuous.

(B3)For all , there exists such that for all

(2)

(B4)For all

(3)

Then has a unique fixed point."

After this theorem, we asserted the following in [1, Remark  3.3] that:

(iii)"If is a non-vanishing map and a sub-additive cone integrable on each such that for each , and , then is satisfies in all conditions of [1, Theorem  3.2]. Equivalently [1, Theorem  2.9] is concluded from [1, Theorem  3.2]."

Note that, in (B2) of [1, Theorem  3.2] and [1, Remark  3.3], we have emphasized that the map must have the continuous inverse, but unfortunately this assumption has been forgotten mistakenly in [1, Theorem  2.9]. Note that this assumption is a necessary condition to prove [1, Theorem  2.9].

(2) To prove [1, Theorem  3.2] and [1, Theorem  2.9], it is sufficient that satisfy the following: for each sequence

(4)

On the other hand, (4) is equivalent to continuity of at zero.

(3) In [2] the authors gave a counterexample on [1, Theorem  2.9] only for our misprint that we have asserted it in the above as you have seen. They also gave a comment for us at the end of their paper to correct such misprint and emphasized that must have the continuous inverse. As you have seen, we have asserted and emphasized such note in (B2) of [1, Theorem  3.2] and [1, Remark  3.3] before the authors in [2] mentioned it.

Nevertheless, we do apologize to the readers for this mistake.