In this section, we will present some fixed point theorems for contractive mappings in the setting of cone b-metric spaces. Furthermore, we will give examples to support our main results.
Theorem 2.1 Let be a complete cone b-metric space with the coefficient . Suppose the mapping satisfies the contractive condition
where is a constant. Then T has a unique fixed point in X. Furthermore, the iterative sequence converges to the fixed point.
Proof Choose . We construct the iterative sequence , where , , i.e., . We have
For any , , it follows that
Let be given. Notice that as for any k. Making full use of Lemma 1.8, we find such that
for each . Thus,
for all and any p. So, by Lemma 1.9, is a Cauchy sequence in . Since is a complete cone b-metric space, there exists such that . Take such that for all . Hence,
for each . Then, by Lemma 1.10, we deduce that , i.e., . That is, is a fixed point of T.
Now we show that the fixed point is unique. If there is another fixed point , by the given condition,
By Lemma 1.11, . The proof is completed. □
Example 2.2 Let , and be a constant. Take . We define as
Then is a complete cone b-metric space. Let us define as
Here is the unique fixed point of T.
Theorem 2.3 Let be a complete cone b-metric space with the coefficient . Suppose the mapping satisfies the contractive condition
where the constant and , . Then T has a unique fixed point in X. Moreover, the iterative sequence converges to the fixed point.
Proof Fix and set and . Firstly, we see
It follows that
This establishes that
Adding up (2.1) and (2.2) yields
Put , it is easy to see that . Thus,
Following an argument similar to that given in Theorem 2.1, there exists such that . Let be arbitrary. Since , there exists N such that
Next we claim that is a fixed point of T. Actually, on the one hand,
which implies that
On the other hand,
which means that
Combining (2.3) and (2.4) yields
Simple calculations ensure that
It is easy to see from Lemma 1.10 that , i.e., is a fixed point of T. Finally, we show the uniqueness of the fixed point. Indeed, if there is another fixed point , then
Owing to , we deduce from Lemma 1.11 that . Therefore, we complete the proof. □
Remark 2.4 Theorem 2.1 extends the famous Banach contraction principle to that in the setting of cone b-metric spaces.
Remark 2.5 Any fixed point theorem in the setting of a metric space, a b-metric space or a cone metric space cannot cope with Example 2.2. So, Example 2.2 shows that the fixed point theory of cone b-metric spaces offers independently a strong tool for studying the positive fixed points of some nonlinear operators and the positive solutions of some operator equations.
Remark 2.6 The main results are some valuable additions to the available references regarding cone b-metric spaces since we have known few fixed point theorems of contractive mappings in the setting of cone b-metric spaces.