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In this paper, we obtain unique common fixed point theorems for two mappings satisfying the variable coefficient linear contraction of integral type and the implicit contraction of integral type respectively in metric spaces.

Throughout this paper, we assume that

integral, summable on each compact subset of

The famous Banach’s contraction principle is as follows:

Theorem 1.1 ([

where

It is known that the Banach contraction principle has a lot of generalizations and various applications in many directions; see, for examples, [

Theorem 1.2 ( [

where

In 2002, Branciari [

Theorem 1.3 ( [

where

In 2011, Liu and Li [

Theorem 1.4 ([

where

Then f has a unique fixed point

Here, we will use the methods in [

Lemma 2.1 ([

Lemma 2.2([

Now, we will give the first main result in this paper.

Theorem 2.1. Let

where

Then f and g have a unique common fixed point u, and the sequence

Proof.

Let

For

hence by (2.3),

Similarly, by (2.1),

hence by (2.3),

Combining (2.4) and (2.5), we have

Now, we prove that

Otherwise, there exists

Obviously,

which is a contradiction. Similarly, if

which is also a contradiction. Hence (2.7) holds. Therefore there exists

which is a contradiction. Therefore

We claim that

For k, let

hence

Let

hence we obtain

If

which is a contradiction. Similarly, we obtain the same contradiction for the case that

If

which is a contradiction, hence

If

which is a contradiction, hence

From Theorem 2.1, we obtain the next more general common fixed point theorem.

Theorem 2.2. Let

where

Proof.

Let

which is a contradiction, hence

From now on, we will discuss the second common fixed point problem for two mappings with implicit contraction of integral type.

Let

(i) There exists

(ii) There exists

(iii)

Example 2.1. Define

where

The function

Example 2.2. Let

Hence

Theorem 2.3. Let

where

Proof.

We take any element

Since

So by (i),

Similarly,

So by (ii),

Combining (2.14) and (2.15), we have

Obviously,

which is a contradiction. Similarly, if

which is also a contradiction. Hence we have

Therefore there exists

which is a contradiction. Therefore,

We claim that

If

This is a contradiction. Similarly, we obtain the same contradiction for the case that

If

which is a contradiction, hence

If

This is a contradiction. Hence

Using Theorem 2.3 and the Example 2.2, we have the next result.

Theorem 2.4. Let

where

Combining Theorem 2.4 and Example 2.1, we obtain the following result.

Theorem 2.5. Let

where

The research is partially supported by the National Natural Science of Foundation of China (No. 11361064).