The concept of a cone b-metric space has been introduced recently as a generalization of a b-metric space and a cone metric space in 2011. The aim of this paper is to establish some fixed point and common fixed point theorems on ordered cone b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Some supporting examples are given.

Fixed point theory has attracted many researchers since 1922 with the admired Banach fixed point theorem. This theorem supplies a method for solving a variety of applied dilemma in mathematical sciences and engineering. A large literature on this subject exists, and this is a very active area of research at present. Banach contraction principle has been generalized in dissimilar directions in different spaces by mathematicians over the years; for more details on this and related topics, we refer to [

In contemporary time, fixed point theory has evolved speedily in partially ordered cone metric spaces; that is, cone metric spaces equipped with a partial ordering, for some new results in ordered metric spaces see [

In 2011, Hussain and Shah [

In [

Let

there exists

there exists

In [

Let

there exist

there exists

Let

there exist

there exists

if an increasing sequence

In the same paper, they also presented the following two common fixed point results in ordered cone metric spaces.

Let

there exist

Let

there exist

if an increasing sequence

In this paper, we prove some fixed point and common fixed point theorems on ordered cone b-metric spaces. Our results extend and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Throughout this paper, we do not impose the normality condition for the cones, but the only assumption is that the cone

The following definitions and results shall be needed in the sequel.

Let

if

Let

Observe that if

Let

Let

Let

For all

For all

A cone metric space

The following lemma is useful in our work.

If

If

If

If

In this section, we prove some fixed point theorems on ordered cone b-metric space. We begin with a simple but a useful lemma.

Let

Let

Let

there exist

there exists

If

If we use condition (iii) instead of the continuity of

Let

there exist

there exists

if an increasing sequence

As in the Theorem

Now, we give two common fixed point theorems on ordered cone b-metric spaces. We need the following definition.

Let

Let

there exist

Let

On the other hand and by symmetry we have

Let

there exist

if an increasing sequence

As in Theorem

Now, we present two examples to illustrate our results. In the first example (the case of a normal cone), the conditions of Theorem

Let

Let

The authors would like to acknowledge the financial support received from Ministry of Higher Education, Malaysia, under the Research Grant no. ERGS/1/2011/STG/UKM/01/13. The authors thank the referee for his/her careful reading of the paper and useful suggestions.