A Fixed Point Theorems for Soft Contractive Mapping by Using Altering Distance Function

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 8, Issue 3, March 2018)

436

A Fixed Point Theorems for Soft Contractive Mapping by Using

Altering Distance Function

Mahesh Tiwari1, Ramakant Bhardwaj 2, Basant Kumar Singh3, Sunil Garge4

1,3_{Department Of Mathematics, RNT University, Bhopal (M.P.)} 2_{Technocrats Institute of Technology , Bhopal, M.P., India}

4_{Senier Scintific Officer, MPCST Bhopal}

ABSTRACT

In the present paper, we prove some fixed point theorem in complete soft metric spaces by using altering distance function.

Keywords: - Soft metric space, soft contractive mapping, fixed point, altering distance function.

Mathematics Subject Classification: - 47H10, 54H25.

1. INTRODUCTION &PRELIMINARIES

A new category of contractive fixed point problem was introduced by M. S. Khan, M. Swalech and S. Sessa [9]. In this work, they introduced the concept of altering distance function which is a control function that alters distance between two points in a metric space.

In the year 1999, Molodtsov [13] initiated a novel concept of soft sets theory as a new mathematical tool for dealing with uncertainties. A soft set is a collection of approximate descriptions of an object. Soft systems provide a very general framework with the involvement of parameters. Since soft set theory has a rich potential, applications of soft set theory in other disciplines and real life problems are progressing rapidly.

Maji et al. [10,11] worked on soft set theory and presented an application of soft sets in decision making problems. Chen [3] introduced a new definition of soft set parameterization reduction and a comparison of it with attribute reduction in rough set theory. Many researchers contributed towards many structure on soft set theory. [1, 3].

M. Shabir and M. Naz [14] presented soft topological spaces and they investigated some properties of soft topological spaces. Later, many researches about soft topological spaces were studied in [2,6,7,12,14]. In these studies, the concept of soft point is expressed by different approaches. In the study we use the concept of soft point which was given in [2,4].

Definition 2.1: Let be an initial universe set and be a set of parameters. A pair is called a soft set over if and only if is a mapping from into the set of all subsets of the set where is the power set of

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437

Definition 2.3: The union of two soft sets and over is the soft set where and

{

This relationship is denoted by

Definition 2.4: The soft set over is said to be a null soft set denoted by if for all

Definition 2.5: A soft set over is said to be an absolute soft set, if for all

Definition 2.6: The difference of two soft sets and over denoted by , is defined as for all

Definition 2.7: The complement of a soft set is denoted by and is defined by where is mapping given by

Definition 2.8: Let be the set of real numbers and be the collection of all nonempty bounded subsets of and taken as a set of parameters. Then a mapping is called a soft real set. It is denoted by If specifically is a singleton soft set, then identifying with the corresponding soft element, it will be called a soft real number and denoted ̃ ̃ ̃ etc.

are the soft real numbers where for all respectively.

Definition 2.9: For two soft real numbers

(i) ̃ ̃ if ̃ ̃ for all

(ii) ̃ ̃ if ̃ ̃ for all

(iii) ̃ ̃ if ̃ ̃ for all

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438

Definition 2.10: A soft set over X is said to be a soft point if there is exactly one such that for some and It will be denoted by ̃

Definition 2.11: Two soft points ̃ ̃ are said to be equal if and i.e. Thus ̃ ̃ or

Definition 2.12: A mapping ̃ ( ̃) ( ̃) is said to be a soft metric on the soft set ̃ if satisfies the following conditions:

(M1) ̃( ̃ ̃ ) ̃ for all ̃ ̃ ̃ ̃

(M2) ̃( ̃ ̃ ) if and only if ̃ ̃

(M3) ̃( ̃ ̃ ) ̃ ̃( ̃ ̃ ) for all ̃ ̃ ̃ ̃

(M4) ̃( ̃ ̃ ) ̃ ̃( ̃ ̃ ) ̃( ̃ ̃ ) for all ̃ ̃ ̃ ̃ ̃

The soft set ̃ with a soft metric ̃ on ̃ is called a soft metric space and denoted by ( ̃ ̃ )

Definition 2.13 (Cauchy Sequence): A sequence { ̃} of soft points in ( ̃ ̃ ) is considered as a Cauchy sequence in ̃ if corresponding to every ̃ ̃ such that ( ̃ ̃) ̃ ̃ i.e. ( ̃ ̃) as

Definition 2.14 (Soft Complete Metric Space): A soft metric space ( ̃ ̃ )is called complete,

if every Cauchy Sequence in ̃ converges to some point of ̃

Definition 2.15: Let( ̃ ̃ )be a soft metric space. A function ( ̃ ̃ ) ( ̃ ̃ )is called a soft contractive mapping if there exist a soft real number such that for every point ̃ ̃ we have

̃ ( ̃ ( ̃ )) ̃( ̃ ̃ )

Definition 2.16[9]: The function is called an altering distance function if the following properties are satisfied:

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439

3.MAIN RESULTS

Theorem 3.1: Let ( ̃ ̃ ) be a soft complete metric space. Suppose the soft mapping

( ̃ ̃ ) ( ̃ ̃ )satisfies the soft contractive condition:

* ̃ ( ̃ ( ̃ ))+ [ ̃ ( ̃ ̃ )] [ ̃ ( ̃ ̃ )] …(3.1.1)

For each ̃ ̃ ̃ ̃ ̃ where are altering distance functions, and

Where ̃ ( ̃ ̃ ) { ̃ ( ̃ ̃ ) ̃ ( ̃ ( ̃ ))

̃ ( ̃ ̃ ) ̃ ( ̃ ( ̃ ))}

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )

̃( ̃ ( ̃ )) ̃( ̃ ̃ )} { ̃( ̃ ̃ )} …(3.1.2)

Where and is a soft constant. Then has a unique fixed point in ̃

Proof: Let ̃ be any soft point in

Set ̃ ( ̃ ) ( ( ̃ ))

̃ ( ̃ ) ( ( ̃ ))

Now consider, from (3.1.2) we have

̃ ( ̃ ̃ ) { ̃ ( ̃ _{( ̃}

_{)) ̃} _{( ̃}

( ̃ ))

̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃))}

{ ̃ ( ̃ _{( ̃}

)) ̃ ( ̃ ( ̃ ))

̃( ̃ ( ̃)) ̃( ̃ ( ̃ ))} { ̃( ̃

_̃ _)}

{ ̃ ( ̃ _̃

) ̃ ( ̃ ̃ )

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440

{ ̃ ( ̃ _̃

) ̃ ( ̃ ̃ )

̃( ̃ ̃ ) ̃( ̃ ̃)} { ̃( ̃

_̃ _)}

{( ̃( ̃ _̃

) ̃( ̃ ̃ )) ( ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ))

̃ ( ̃ ̃) ̃ ( ̃ ̃ ) }

{ ̃ ( ̃ _̃

₎

̃ ̃ ̃ } { ̃( ̃

_̃ _)}

{ ̃( ̃ ̃ ) ̃( ̃ ̃ )} { ̃( ̃ ̃ )} { ̃( ̃ ̃ )}

̃( ̃ ̃ ) ̃( ̃ ̃ )

From (3.1.1) we have

[ ̃( ̃ ̃ )] * ̃ ( ( ̃ ) ( ̃ ))+

[ ̃ ( ̃ ̃ )] [ ̃ ( ̃ ̃ )]

[ ̃( ̃ ̃ ) ̃( ̃ ̃ )] [ ̃ ( ̃ ̃ )]

[ ̃( ̃ ̃ )] [ ̃( ̃ ̃ ) ̃( ̃ ̃ )]

Since is non-decreasing, we have

̃( ̃ ̃ ) ̃( ̃ ̃ ) ̃( ̃ ̃ )

̃( ̃ ̃ )

̃( ̃

_̃ ₎

̃( ̃ ̃ ) ̃( ̃ ̃ ) Where

Thus ̃( ̃ ̃ ) ̃( ̃ ̃ )

Taking we have

̃( ̃ ̃

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441

Now, we will show that { ̃ }is a soft Cauchy sequence. Suppose that { ̃ } is not a Soft

Cauchy sequence, which means that there is a constant such that for each positive integer k, there are positive integer and with such that

̃ ( ̃ ̃ ) ̃ ( ̃ _̃ )

By triangle inequality

̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ )

Letting and using (3.1.3), we have

̃ ( ̃ ̃

) …(3.1.4)

Similarly, we have

̃ (

̃

)

̃ (

̃

)

̃ (

̃

)

_}

…(3.1.5)

Putting ̃ ̃

and ̃ ̃

in (3.1.2) we have

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442

{

̃ ( ̃

̃

) ̃ ( ̃

̃

) ̃( ̃

̃

_{) ̃( ̃}

̃

₎} , ̃ ( ̃

̃

)-Letting and using (3.1.3),(3.1.4) and (3.1.5), we have

̃ ( ̃

̃

) …(3.1.6)

* ̃ ( ̃

̃

)+ * ̃ ( ( ̃

) ( ̃

))+

* ̃ ( ̃

̃

)+ * ̃ ( ̃

̃

)+

Taking using (3.1.5), (3.1.6) and the continuity of and ,we have

] ] ]

] ]

This leads to ] , and property of we get .

This is a contradiction. Thus{ ̃ }is a soft Cauchy sequence in ̃, which is complete. Thus,

there is ̃ ̃such that ̃ ̃ … (3.1.6)

Putting ̃ ̃ and ̃ ̃ in (3.1.2) we have

̃ ( ̃ ̃ ) { ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))

̃ ( ̃ ( ̃)) ̃ ( ̃ ( ̃ ))}

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))

̃( ̃ ( ̃ )) ̃( ̃ ( ̃))} { ̃( ̃ ̃ )}

Taking and using (3.1.3) and (3.1.6) we have

̃ ( ̃ ̃ ) { ̃( ̃ ̃ )}

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443

[ ̃ ( ̃ ̃ )] [ ̃ ( ̃ ̃ )]

[ ̃( ̃ ̃ )] [ { ̃( ̃ ̃ )}] [ { ̃( ̃ ̃ )}]

[ ̃( ̃ ̃ )] [ ̃( ̃ ̃ )] [ { ̃( ̃ ̃ )}]

Which implies [ { ̃( ̃ ̃ )}]

So ̃( ̃ ̃ ) that is ̃ ̃

Uniqueness: Let ̃ is another fixed point of in ̃such that ̃ ̃ , then we have

̃ ̃ ̃ { ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))

̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))}

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))

̃( ̃ ( ̃ )) ̃( ̃ ( ̃ ))} { ̃ ̃ ̃ }

̃ ̃ ̃ ̃ ̃ ̃

[ ̃ ̃ ̃ ] [ ̃( ̃ ̃ )]

[ ̃ ̃ ̃ ] [ ̃ ̃ ̃ ]

[ ̃ ̃ ̃ ] [ ̃ ̃ ̃ ] [ ̃ ̃ ̃ ]

So [ ̃ ̃ ̃ ] thus ̃ ̃ ̃ that is ̃ ̃

Hence fixed point of is unique.

Corollary 3.2: Let ( ̃ ̃ ) be a soft complete metric space. Suppose the soft mapping

( ̃ ̃ ) ( ̃ ̃ )satisfies the following condition:

* ̃ ( ̃ ( ̃ ))+ [ ̃ ( ̃ ̃ )] [ ̃ ( ̃ ̃ )] …(3.2.1)

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444

Where ̃ ( ̃ ̃ ) { ̃ ( ̃ ̃ ) ̃ ( ̃ ( ̃ ))

̃ ( ̃ ̃ ) ̃ ( ̃ ( ̃ ))}

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )

̃( ̃ ( ̃ )) ̃( ̃ ̃ )} { ̃( ̃ ̃ )}

, ̃( ̃ ̃ ) ̃ ( ̃ ( ̃

, ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )- … (3.2.2)

Proof: It can be proved easily.

Theorem 3.3: Let ( ̃ ̃ ) be a soft complete metric space. Suppose the soft mapping

( ̃ ̃ ) ( ̃ ̃ )satisfies the following condition:

* ̃ ( ̃ ( ̃ ))+ [ ̃ ( ̃ ̃ )] [ ̃ ( ̃ ̃ )] …(3.3.1)

For each ̃ ̃ ̃ ̃ ̃ where are altering distance functions, and

Where ̃ ( ̃ ̃ ) { ̃ ( ̃ ̃ ) ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )

̃( ̃ ̃ ) ̃( ̃ ( ̃ )) ̃( ̃ ̃ ) }

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ )

̃( ̃ ( ̃ )) ̃( ̃ ̃ ) ̃( ̃ ̃ ) } …(3.3.2)

Proof: Let ̃ be any soft point in

Set ̃ ( ̃ ) ( ( ̃ ))

̃ ( ̃ ) ( ( ̃ ))

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445

Now consider from(3.3.1) we have

̃ ( ̃ ̃ ) { ̃ ( ̃ _{( ̃}

_{)) ̃} _{( ̃}

( ̃ )) ̃ ( ̃ ( ̃ ))

̃( ̃ ( ̃ )) ̃( ̃ ( ̃)) ̃( ̃ ( ̃ )) }

{ ̃ ( ̃ _{( ̃}

)) ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )

̃( ̃ ( ̃)) ̃( ̃ ( ̃ )) ̃( ̃ ̃) }

{ ̃ ( ̃ _̃

) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ )

̃( ̃ ̃) ̃( ̃ ̃ ) ̃( ̃ ̃) }

{ ̃ ( ̃ _̃

) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ )

̃( ̃ ̃ ) ̃( ̃ ̃) ̃( ̃ ̃) }

{( ̃( ̃ _̃

) ̃( ̃ ̃ ))

̃( ̃ ̃) ̃( ̃ ̃ ) } {

( ̃( ̃ ̃ ) ̃( ̃ ̃)) ̃( ̃ ̃ ) ̃( ̃ ̃) }

{ ̃( ̃ ̃ ) ̃( ̃ ̃ )} { ̃( ̃ ̃ ) ̃( ̃ ̃ ) }

{ ̃( ̃ ̃ )} { ̃( ̃ ̃ ) }

From (3.3.1), we have

[ ̃( ̃ ̃ )] * ̃ ( ( ̃ ) ( ̃ ))+

[ ̃ ( ̃ ̃ )] [ ̃ ( ̃ ̃ )]

[ { ̃( ̃ ̃ )} { ̃( ̃ ̃ ) }] [ ̃ ( ̃ ̃ )]

[ ̃ ( ̃ ̃ )]

[ ̃( ̃ ̃ )] [ { ̃( ̃ ̃ )} { ̃( ̃ ̃ ) }]

Since is non-decreasing, we have

̃( ̃ ̃ ) { ̃( ̃ ̃ )} { ̃( ̃ ̃ ) }

̃( ̃ ̃ )

̃( ̃

_̃ ₎

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446

̃( ̃ ̃ ) ̃( ̃ ̃ ) Where

Thus ̃( ̃ ̃ ) ̃( ̃ ̃ )

Taking we have

̃( ̃ ̃

₎_…(3.3.3)

Now, we will show that { ̃ }is a soft Cauchy sequence. Suppose that { ̃ } is not a Soft

Cauchy sequence, which means that there is a constant such that for each positive integer k, there are positive integer and with such that

̃ ( ̃ ̃ ) ̃ ( ̃ ̃ )

By triangle inequality

̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ )

Letting and using (3.3.3), we have

̃ ( ̃ ̃

) …(3.3.4)

Similarly, we have

̃ (

̃

)

̃ (

̃

)

̃ (

̃

)

_}

…(3.3.5)

Putting ̃ ̃

and ̃ ̃

in (3.3.2) we have

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447 { ̃ . ̃ ( ̃ )/ ̃ . ̃ ( ̃ )/ ̃ ( ̃ ̃ ) ̃( ̃ ( ̃ )) ̃( ̃ ( ̃ )) ̃( ̃ ̃ ) } { ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃( ̃ ̃ _{) ̃( ̃} ̃ _{) ̃( ̃} ̃ ₎ } { ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃( ̃ ̃ _{) ̃( ̃} ̃ _{) ̃( ̃} ̃ ) }

Letting and using (3.3.3) and (3.3.5), we have

̃ ( ̃ ̃

) …(3.3.6)

* ̃ ( ̃ ̃ )+ * ̃ ( ( ̃ ) ( ̃ ))+ * ̃ ( ̃ ̃ )+ * ̃ ( ̃ ̃ )+

Taking and from (3.3.5), (3.3.6), we have

] ] ]

] ]

This leads to ] , and property of we get .

This is a contradiction. Thus{ ̃ }is a soft Cauchy sequence, which is complete. Thus, there is ̃ ̃such that ̃ ̃ … (3.3.7)

̃ ( ̃ ̃ ) { ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))

̃( ̃ ( ̃)) ̃( ̃ ( ̃ )) ̃( ̃ ( ̃)) }

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )

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448

Taking and using (3.3.3), (3.3.7) we have

̃ ( ̃ ̃ ) { ̃( ̃ ̃ )}

* ̃ ( ̃ ̃ )+ * ̃ ( ( ̃ ) ̃ )+

[ ̃ ( ̃ ̃ )] [ ̃( ̃ ̃ )]

[ ̃( ̃ ̃ )] [ { ̃( ̃ ̃ )}] [ { ̃( ̃ ̃ )}]

[ ̃( ̃ ̃ )] [ ̃( ̃ ̃ )] [ { ̃( ̃ ̃ )}]

Which implies [ { ̃( ̃ ̃ )}]

So ̃( ̃ ̃ ) that is ̃ ̃

Uniqueness: Let ̃ is another fixed point of in ̃such that ̃ ̃ , then we have

̃ ̃ ̃ { ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ ))

̃( ̃ ( ̃ )) ̃( ̃ ( ̃ )) ̃( ̃ ( ̃ )) }

{ ̃ ( ̃ ( ̃ )) ̃ ( ̃ ( ̃ )) ̃ ( ̃ ̃ )

̃( ̃ ( ̃ )) ̃( ̃ ( ̃ )) ̃( ̃ ̃ ) }

̃ ̃ ̃ { ̃ ̃ ̃ }

[ ̃ ̃ ̃ ] [ ̃( ̃ ̃ )]

[ ̃ ̃ ̃ ] [ ̃ ̃ ̃ ]

[ ̃ ̃ ̃ ] [ ̃ ̃ ̃ ] [ ̃ ̃ ̃ ]

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449

Hence fixed point of is unique.

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