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We introduce a new class of mappings satisfying the “common limit range property” in symmetric spaces and utilize the same to establish common fixed point theorems for such mappings in symmetric spaces. Our results generalize and improve some recent results contained in the literature of metric fixed point theory. Some illustrative examples to highlight the realized improvements are also furnished.

In 1986, Jungck [

In this paper, we introduce a new notion called the common limit range property and show that this new notion buys a typically required condition up to a pair of mappings along with the notion of absorbing property in proving common fixed point theorems for Lipschitz type mappings in symmetric spaces. Consequently, the relevant recent fixed point theorems due to Soliman et al. [

A symmetric

Since symmetric spaces are not necessarily Hausdorff and the symmetric

(

(

(

(

(

Clearly,

Recall that a sequence

Let

A pair

compatible (cf. [

noncompatible (cf. [

tangential (or satisfying the property (E.A)) (cf. [

Let

(cf. [

the common limit range property with respect to the map

let

a pair

let

let

On similar lines, we can define pointwise

For the sake of completeness, we state below some theorems contained in Soliman et al. [

Let

the pairs

Then there exist

Moreover, if

the pairs

Let

the pairs

Moreover, if

Let

the pair

Moreover, if

In this paper, we provide a unified approach to certain theorems in symmetric (semimetric) spaces using a blend of common limit range property along with absorbing pair property and obtain generalizations of various results due to Gopal et al. [

We start to section with the following definition.

Let

If

Consider

For sequences

In view of the preceeding example, the following proposition is predictable.

If the pairs

We now prove our first result employing

Let

the pairs

Since the pairs

On using

As the pairs

With a view to demonstrate the utility of Theorem

Consider

Consider sequences

Notice that at

By restricting

Let

the pairs

The following example illustrates the preceding corollary involving a pair of two self-mappings.

Consider

By routine calculations, one can easily verify that the maps in the pair

Our next theorem is essentially inspired by Theorem

Let

the pairs

Since the pairs

On using condition (ii), we have

Next, we show that

On using pointwise absorbing property of the pairs

The following example demonstrates Theorem

Consider

Then, by a routine calculation, it can be easily verified that

Observe that

Choosing

By restricting

Suppose that (in the setting of Theorem

the pairs

Let

the pairs

Moreover, if

Proof follows from Theorem

Our next theorem is essentially inspired by a Lipschitzian condition utilized by Cho et al. [

Theorem

The proof can be completed on the lines of proof of Theorem

By restricting

Suppose that (in the setting of Theorem

the pairs

Let

the mappings satisfy the

The proof can be completed on the lines of proof of Theorem

The authors are grateful to two anonymous referees for their helpful comments and suggestions.