In this paper, we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces. Furthermore, we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated with

Fixed point theory plays an important role in various branches of mathematics as well as in nonlinear functional analysis, and is very useful for solving many existence problems in nonlinear differential and integral equations with applications in engineering and behavioural sciences. Recently, many authors have provided the extended fixed point theorems for the different classes of contraction type mappings, such as Kannan, Reich, Chatterjea and Ćirić-Reich-Rus mappings (see [

Let

Kannan [

In 1980, Gregus [

In [

In 1971, Ćirić [

In addition,

Ćirić in [

Note that, 1-continuity is equivalent to continuity and for any

On the other hand, the concept of asymptotic regularity has been introduced by Browder et al. [

In [

Recently, Bisht [

This paper is organised as follows: First, we establish some fixed point theorems for Reich and Chatterjea nonexpansive mappings to include asymptotically regular or continuous mappings in complete metric spaces. After that, we prove some fixed point theorems and common fixed points for Reich and Chatterjea type nonexpansive mappings in Banach space using the Krasnoselskii-Ishikawa method associated with

In this section, we study fixed point and common fixed point theorems for Reich and Chatterjea type nonexpansive mappings in complete metric space.

To start with the following lemma, which is useful to prove the results of this section:

Then,

As

Using triangle inequality and asymptotic regularity of

Thereafter, suppose that

Let

Choosing

Thus, we have

As

Further, by asymptotic regularity of

Now, using

Taking limit as

For the uniqueness of the limit, we get

Similarly, let

In addition, since

Clearly, the two mappings

Therefore, in all the cases,

In the special case of our result, we can generate the Theorem 1.4 of Gòrnicki [

Choosing

Obviously, as all the assumptions of Corollary 2.1 hold,

(i) The mapping

(ii) For

Then,

Using the triangle inequality and asymptotic regularity in

Thus,

Hence,

This shows that

Next, we will consider condition (ii). Choose

As

Thus,

Since

Suppose that

The compatibility of

In the next theorem, we establish a common fixed point result on Chatterjea nonexpansive mapping.

As

By the asymptotic regularity of

Next, suppose that

Let

Assume that

Choosing

Now, we have that

As

Furthermore, by asymptotic regularity of

Then, using

Let

Since the limit is unique, it implies

Similarly, suppose that

If we choose

In fact, we have the following four cases:

Therefore,

Thus,

However,

Therefore, in all cases,

According to Theorem 2.7,

The mapping

For

Then

Using the triangle inequality and asymptotic regularity in

Then,

Hence,

This shows that

We next prove the condition (ii) holds. Suppose that

Finally, we show that

This implies

As

Thus,

Since

Assuming that

Then, compatibility of

Then

Similarly, if take

If

Otherwise, if

In this section, we present some fixed point and common fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space.

Consider a fixed point iteration, which is given by

Define

However, if

In the following, we prove basic lemmas for the Reich nonexpansive mapping which in turn are useful to proving the results of this section.

Since

Using

Now,

Since

Let

Then,

Let

Applying a triangle inequality and Lemma 3.1, we obtain

Therefore,

Since

Now, we state and prove our main results of this section:

Let

We shall show that the sequence of iterates

Again, by Lemma 3.1, we have

Moreover, by

Continuing this process, we obtain

Taking limit as

Hence,

Since

In the next theorem, we present a common fixed point result for Reich mappings.

Now, using the operator defined by

From

Similarly,

Continuing the process, we get the following:

Now, we show that

Since

Following similar lines of the proof of Theorem 3.2, we obtain

Let us choose

Similarly, we obtain

Continuing the process, we get the following:

Next, we show that

Since

Since

Take

Let

Therefore, in all the cases,

Let

We consider the following integral equations formulated as a common fixed point problem of the following nonlinear mappings:

The two mappings

We also have,

Suppose that

Therefore,

Now, since

By Theorem 2.1 there exists a common fixed point of

Fractional differential equations have applications in various fields of engineering and science including diffusive transport, electrical networks, fluid flow and electricity. Many researchers have studied this topic because it has many applications. Related to this matter, we suggest the recent literature [

The classical Caputo fractional derivative is defined by

Now, we consider the following fractional differential equation:

Assume that

Let

The exact solution of the above problem

The operator

By taking

On the other hand, we obtain that

By Theorem 2.7 there exists a common fixed point of

This paper contains the study of Reich and Chatterjea nonexpansive mappings on complete metric and Banach spaces. The existence of fixed points of these mappings which are asymptotically regular or continuous mappings in complete metric space is discussed. Furthermore, we provide some fixed point and common fixed point theorems for Reich and Chatterjea nonexpansive mappings by employing the Krasnoselskii-Ishikawa iteration method associated with

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

_{b}-metric spaces