The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of weak reciprocal continuity. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. We demonstrate that weak reciprocal continuity ensures the existence of common fixed points under contractive conditions, which otherwise do not ensure the existence of fixed points. Our results generalize and extend Banach contraction principle and Meir-Keeler-type fixed point theorem.

In his earlier works, Pant [

In 1986, Jungck [

Two self-maps

The definition of compatibility implies that the mappings

Two self-maps

Two self-maps

Two self-maps

A pair

It is well known now that pointwise

In a recent work, Al-Thagafi and Shahzad [

A pair

Two self-mappings

From the definition itself, it is clear that if two maps are weakly compatible or owc then they are necessarily conditionally commuting; however, the conditionally commuting mappings are not necessarily weakly compatible or owc [

Let

It is well known that the absorbing maps are neither a subclass of compatible maps nor a subclass of noncompatible maps [

Two self-mappings

If

Two self-mappings

We now give examples of compatible and weakly reciprocally continuous mappings with or without common fixed points.

Let

Let

If

Let

If

Let

Suppose that

Next suppose that

Finally suppose that

Next suppose that

Uniqueness of the common fixed point theorem follows easily in each of the two cases.

We now give an example to illustrate the above theorem.

Let

Then

Putting

We now establish a common fixed point theorem for a pair of mappings satisfying an

Let

Then

In view of the above example, the next theorem demonstrates the usefulness of weak reciprocal continuity and shows that the new notion ensures the existence of a common fixed point under an

Let

given

Let

Let

Suppose that

Next suppose that

When

We now give an example to illustrate Theorem

Let

Theorem

It may be observed that the mappings

In the area of fixed point theory, Lipschitz type mappings constitute a very important class of mappings and include contraction mappings, contractive mappings and, nonexpansive mappings as subclasses. The next theorem provides a good illustration of the applicability of recently introduced notions of conditional commutativity and weak reciprocal continuity to establish a situation in which a pair of mappings may possess common fixed points as well as coincidence points, which may not be common fixed points.

Let

If

Since

Next suppose that

Now suppose that

Next suppose that

We now give examples to illustrate Theorem

Let

Then

In Example

Let

Putting

Let

If

The author is thankful to the learned referee for his deep observations and pertinent suggestions, which improved the exposition of the paper.

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