1. Introduction and Prelimnaries

Fixed point theory is one of the most important topic in development of nonlinear analysis. Also, fixed point theory has been used effectively in many other branch of science, such as chemistry, biology, economics, computer science, engineering, and many others.

It is well known that Banach’s contraction mapping theorem is one of the pivotal results of functional analysis. A mapping where (X, d) is a metric space, is said to be a contraction if there exists such that for all ;

(1.1)

If the metric space (X, d) is complete then the mapping satisfying (1.1) has a unique fixed point. Inequalty (1.1) implies continuity of T. A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity.

Kannan in ^[2] established the following result in which the above question has been answered in the affirmative.

If where (X, d) is a complete metric space, satisfies the inequlity

(1.2)

where and then T has a unique fixed point. The mappings satisfying (1.2) are called Kannan type mappings.

A similar contractive condition has been introduced by Chatterjea ^[3] as following.

If where (X, d) is a complete metric space, satisfies the inequality

(1.3)

where and then T has a unique fixed point. The mappings satisfying (1.3) are called Chatterjea type mappings.

Some problems, particularly the problem of the convergence of measurable functions with respect to a measure, lead to a generalization of notation of metric. Using this idea, Czerwik in ^[4] and ^[5] presented a generalization of the well known Banach’s fixed point theorem in so called b-metric spaces.

Our purpose is to show that some well-known fixed point theorems are valid in b-metric spaces. It is interesting that we see some properties in (usual) metric space is not valid in b-metric space.

Definition 1. ^[7] Let X be a nonempty set and let s ≥ 1 be a given real number. A function is called a b-metric provided that, for all ,

1. if and only if ,

2. ,

3. ,

A pair (X, d) is called a b-metric space.

It is clear that the definiton of b-metric space is a extension of usual metric space. Also, if we consider s = 1 in Definition 1, then we obtain definition of usual metric space. By this reason, our results are more general than the same results in (usual) metric space.

We can give some example of b-metric space as following:

Example 1. ^[6] The space ,

together with the function

where is a b-metric space. By an elementary calculation we obtain that

Example 2. ^[6] The space of all real functions such that

is b-metric space if we take

for each ,

Remark 1. Note that a (usual) metric space is evidently a b-metric space. However, Czerwik ^{[4, 5]} has shown that a b-metric on X need not be a metric on X.

Now, we illüstrate Remark 1 as follows:

Example 3. ^[7] Let X = {0, 1, 2}and d (2, 0) = d (0 ; 2) = m ≥ 2, d (0 1) = d (1, 2) = d (0, 1) = d (2, 1) = 1 and d (0, 0) = d (1, 1) = d (2, 2) = 0. Then,

for all . If m > 2, the ordinary triangle inequality does not hold.

We continue by presenting definition of Cauchy sequence, convergent sequence and complete b-metric space.

Definition 2. ^[6] Let (X, d) be a b-metric space. Then a sequence (x_n) in X is called Cauchy sequence if and only if for all there exists such that for each we have .

Definition 3. ^[6] Let (X, d) be a b-metric space. Then a sequence (x_n) in X is called convergent sequence if and only if there exists such that for all there exists such that for all we have . In this case we write .

Definition 4. ^[6] The b-metric space is complete if every Cauchy sequence convergent.

Definition 5. ^[8] Let E be a nonempty set and a selfmap. We say that is a fixed point of T if T (x) = x and denote by FT or Fix (T) the set of all fixed points of T.

Let E be any set and a selfmap. For any given , we define Tⁿ (x) inductively by T⁰ (x) = x and ; we recall T_n(x) the nth iterative of x under T.

For any , the sequence given by

(1.4)

is called the sequence of successive approximations with the initial value x₀. It is also known as the Picard iteration starting at x₀.

Definition 6. ^[8] Let (X, d) be a metric space. A mapping is called weak contraction if there exists a constant and some such that

(1.5)

fall all .

Remark 2. ^[8] Due to the symmetry of the distance, the weak contractive condition (1.5) implicitly includes the following dual one

(1.6)

fall all .

Consequently, in order to check the weak contractiveness of T; it is necessary to check both (1.5) and (1.6).

Remark 3. It is clear that any contraction mapping is also weak contraction mapping in a (usual) metric space.

2. Main Results

In this section, we give some fixed point theorems arising from b-metric spaces. Also, we find an intersting comparison between (usual) metric spaces and b-metric spaces. Our first theorem about Banach’s contraction princible in b-metric spaces.

Theorem 1. Let (X, d) be a complete b-metric space with constant s ≥ 1 and define the sequence by the recursion (1.4). Let be a contraction with the restrictions and . Then, there exists such that and is unique fixed point of T.

Proof. Let and be a sequence in X defined as following

Since T is a contraction with constant then, we obtain

Continuing this process, we easily arrive at

(2.1)

Now, we show that is a Cauchy sequence in X. Let m, n > 0 with m > n,

(2.2)

(2.3)

When we take in (2.3), we arrive at

Hence, is a Cauchy sequence in X. In view of completeness of X; we consider that convergent to .

Now, we show that is the unique fixed point of T. Indeed

Therefore, is a fixed point of T. T_o is complete the proof finaly, we have to show that the fixed point is unique. Assume that is an other fixed point of T. Then, .

(2.4)

The (2.4) implies that ; but this case is a contradiction to . So the fixed point is unique. This completes the proof.

Our next theorem about Kannan type fixed point theorem in b-metric spaces.

Theorem 2. Let (X, d) be a complete b-metric space with constant and define the sequence by the recursion (1.4). Let be a mapping for which there exists such that

(2.5)

for all .

Then, there exists such that and is unique fixed point of T.

Proof. Let and be a sequence in X defined as , . By using (2.5) and (1.4) we obtain that

and we obtain

(2.6)

Similarly, we have

(2.7)

Note that then . Thus, T is a contraction mapping. We deduce, in similar manner to that in the proof of Theorem 1 that is a Cauchy sequence and hence, a convergent sequence, too. We consider that convergent to then we have

and we arrive at

(2.8)

Also, thanks to (2.7), we obtain that

(2.9)

Letting in (2.9),

Therefore, and implies that is a fixed point of T. It is easy to see the fixed point is unique.

Remark 4. In proof of Theorem 2, it is clear that we have not heard of any restriction to actualize the Kannan’s fixed point theorem in b-metric spaces.

Proposition 1. Contrary to the known (usual) metric spaces, any mapping satisfying the contractive condition (2.5) need not be a weak contraction in a b-metric spaces unless under the terms .

Proof. Let be a mapping satisfying (2.5), we have

and we obtain

(2.10)

According to the inequalty (2.10), T is not a weak contraction unless the terms

Remark 5. In Theorem 2, we see that a mapping satisfying the contractive condition (2.5) is also a contraction mapping in b-metric space. Also, in Proposition 1, we see that a mapping satisfying the contractive condition (2.5) is not need to be a weak conraction. Consequently, it is interesting that contrary to the known (usual) metric spaces, any contraction mapping is not need to be a weak conraction mapping in b-metric spaces.

Our next theorem is about Chatterjea type fixed point theorem in b-metric spaces.

Theorem 3. (X, d) be a complete b-metric space and define the sequence by the Recursion (1.4). Let be a mapping under the terms such that

(2.11)

fall all .

Then, there exists such that and is unique fixed point of T.

Proof. Let and be a sequence in X defined as , . By using (2.11) and (1.4) we obtain that

(2.12)

(2.13)

thus, we have

(2.14)

Note that then.Thus, T is a contraction mapping. By using similar method in the proof of Theorem 1 and Theorem 2, we see that is a Cauchy sequence and hence a convergent sequence, too. we consider that convergent to .

Now, we show that is a fixed point of T. We have

(2.15)

Thus, we obtain that

(2.16)

Letting in (2.16),

(2.17)

The inequalty (2.17) is false unless . Thus, we arrive at .

Now, we show that is the unique fixed point of T. Assume that is an other fixed point of T. Then, we have .. and

(2.18)

From (2.18), we obtain . This implies that . This completes the proof.

Remark 6. The headlights from Theorem 2, we needed a restriction to valid Chatterjea’s fixed point theorem in b-metric space.

References

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