Dislocated Soft Metric Space with Soft Fixed Point Theorems

DOI: 10.4236/ojdm.2017.73012   PDF   HTML   XML   1,071 Downloads   1,989 Views   Citations

Abstract

In the present paper, we define Dislocated Soft Metric Space and discuss about the existence and uniqueness of soft fixed point of a cyclic mapping in soft dislocated metric space. We also prove the unique soft fixed point theorems of a cyclic mapping in the context of dislocated soft metric space. Examples are given for support of the results.

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Wadkar, B. , Mishra, V. , Bhardwaj, R. and Singh, B. (2017) Dislocated Soft Metric Space with Soft Fixed Point Theorems. Open Journal of Discrete Mathematics, 7, 108-133. doi: 10.4236/ojdm.2017.73012.

1. Introduction

The soft set theory is one of the branches of mathematics, which aims to describe phenomena and concepts of an ambiguous, undefined vague and imprecise meaning, which was initiated by Molodtsov [1] . This theory is applicable where there are no clearly defined mathematical models. Recently many papers concerning soft sets have been published (see [2] - [8] ). In many aspects of Mathematics, fixed point theory has wonderful applications. Shabir and Naz [9] presented soft topological spaces and they investigated some properties of soft topological spaces. Later many researchers were studied about soft topological spaces. In these studies, concepts of soft fixed point are expressed by different approaches. Das and Samanta [10] [11] introduced a different notion of soft metric space by using different concept of soft point and investigated some basic properties of these. In 2000, Hitzler and Seda [12] introduced the notion of dislocated metric space, in which self distance of a point need not be equal to zero. Aage and Salunke [13] established some important fixed point theorem in single and pair of mappings in dislocated metric space. Later Karapnar and Salimi [14] discussed the existence and uniqueness of fixed point of a cyclic mapping in the context of metric like space. The study of common fixed point of mapping in dislocated metric space satisfying certain contractive condition has been at the Centre of vigorous research activity (see [15] [16] [17] [18] ). Dislocated metric space plays very important role in topology, logical programming and in electronic engineering. Recently Wadkar et al. [19] , Mishra et al. [20] [21] [22] [23] , Deepmala and pathak [24] , Wadkar et al. [25] , [26] discussed and proved fixed point theorems by employing different concepts.

In the present paper, we discuss about the investigations concerning the existence and uniqueness of soft fixed point of a cyclic mapping in soft dislocated metric space. We also prove the unique soft fixed point theorems of a cyclic mapping in the context of dislocated metric space. To check the validity of the result we give the examples. Before starting to prove main result, some basic definitions are required.

Definition 1.1: Let X and E are respectively an initial inverse set and a parameter set. A soft set over X is pair denoted by (Y, E) if and only if Y is a mapping from E into the set of all subsets of the set X, i.e. Y: E → P(x), where P(x) is the power set of X.

Example of Soft Set: Let ( C , D ) be soft set, which describes the Nature of workers at Industry. Suppose that U = { u 1 , u 2 , u 3 , u 4 } , i.e. universe of four workers. Let a set of decision parameters be D = { s 1 , s 2 , s 3 , s 4 , s 5 } . Now s i ( i = 1 , 2 , 3 , 4 , 5 ) stand for the parameters: high working speed, slow working speed, average working speed, work delay working speed and no working speed properly respectively.

Now consider C ˜ ( s 1 ) = { u 1 , u 3 } , C ˜ ( s 2 ) = { u 1 , u 3 , u 4 } , C ˜ ( s 3 ) = U , C ˜ ( s 4 ) = { u 2 , u 4 } and C ˜ ( s 5 ) = empty . By consisting of the following collection of approximations, the soft set ( C , D ) can be viewed.

( C ˜ , D ˜ ) = { ( highworkingspeed, { u 1 , u 3 } ) , ( slowworkingspeed, { u 1 , u 3 , u 4 } ) , ( averageworkingspeed, U ) , ( workdelayworkingspeed, { u 2 , u 4 } ) , ( noworkingspeed,empty ) } .

Definition 1.2: The intersection of two soft sets ( Y , A ) and ( Z , B ) over X is a soft set over X denoted by ( I , C ) and is given by ( Y , A ) ˜ ( Z , B ) = ( I , C ) , where C = A B and ε C , I ( ε ) = Y ( ε ) Z ( ε ) .

Definition 1.3: The union of two soft sets ( Y , A ) and ( Z , B ) over X is the soft set ( I , C ) , where C = A B and for all k in C,

I ( k ) = { Y ( k ) , if k isanelementof A B , Z ( k ) , if k isanelementof B A , Y ( k ) Z ( k ) , if k isanelementof A B .

This relationship is denoted by ( Y , A ) ˜ ( Z , B ) = ( I , C ) .

Definition 1.4: A soft set (Y, A) over X is said to be a null soft set if Y ( k ) = empty , for all k in A and is denoted by Φ .

Definition 1.5: For all k A , if Y ( k ) = X then ( Y , A ) is called an absolute soft set over X.

Definition 1.6: The difference of two soft sets ( F , E ) and ( G , E ) over X is a soft set ( H , E ) over X, denoted by ( F , E ) \ ( G , E ) and is defined as H ( x ) = F ( x ) \ G ( x ) , x E .

Definition 1.7: The complement of soft set ( Y , A ) is denoted by ( Y , A ) c and is defined as ( Y , A ) c = ( Y c , A ) , where Y c : A P ( X ) is a mapping given by Y c ( β ) = X Y ( β ) , for all β .

Definition 1.8: Let B(R) be the collection of all non-empty bounded subsets of R and E taken as a set of parameters. Then the mapping Y : E B ( R ) is called a soft real set. It is denoted by (Y, E).

Definition 1.9: For two soft real numbers m ˜ and n ˜ the following conditions holds:

(i) m ˜ n ˜ if m ˜ ( s ) n ˜ ( s ) , for all s E ;

(ii) m ˜ n ˜ if m ˜ ( s ) n ˜ ( s ) , for all s E ;

(iii) m ˜ < n ˜ if m ˜ ( s ) < n ˜ ( s ) , for all s E ;

(iv) m ˜ > n ˜ if m ˜ ( s ) > n ˜ ( s ) , for all s E .

Definition 1.10: A soft set (P, E) over X is said to have a soft point if there is exactly one s E such that P ( s ) = { x } , for some x X , also P ( s ) = φ , s E / { s } . It will be denoted by x ˜ s .

Definition 1.11: Two soft points x ˜ i , y ˜ j are said to be equal if i = j and P ( i ) = P ( j ) i.e. x = y. Hence x ˜ i y ˜ j x y or i j .

Definition 1.12: A mapping ρ ˜ : S P ( X ˜ ) × S P ( X ˜ ) R ( E ) * is soft metric on soft set X ˜ with following properties.

SM1. for all x ˜ s 1 , y ˜ s 2 X ˜ , ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) ˜ 0 ¯ ;

SM2. ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) = 0 ¯ , if and only if x ˜ s 1 = y ˜ s 2 ;

SM3. for all x ˜ s 1 , y ˜ s 2 X ˜ , ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) = ρ ˜ ( y ˜ s 2 , x ˜ s 1 ) ;

SM4. for all x ˜ s 1 , y ˜ s 2 , z ˜ s 3 X ˜ , ρ ˜ ( x ˜ s 1 , z ˜ s 3 ) ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) + ρ ˜ ( y ˜ s 2 , z ˜ s 3 ) .

The soft set X ˜ with a soft metric ρ ˜ defined on X ˜ is called a soft metric space and denoted by ( X ˜ , ρ ˜ , E ) .

Definition 1.13: Let us consider a soft metric ( X ˜ , ρ ˜ , E ) and α ˜ be a non-negative soft real number. The soft open ball with center at x ˜ s and radius α ˜ is given by

B ( x ˜ s , α ˜ ) = { y ˜ s ' X ˜ : ρ ˜ ( x ˜ s , y ˜ s ) α ˜ } S P ( X ˜ ) ,

and the soft closed ball with center at x ˜ s and radius α ˜ is given by

B ( x ˜ s , α ˜ ) = { x ˜ s X ˜ : ρ ˜ ( x ˜ s , y ˜ s ) α ˜ } S P ( X ˜ ) .

Definition 1.14: A sequence { x ˜ λ n n } of soft points in soft metric space ( X ˜ , ρ ˜ , E ) is said to be convergent in ( X ˜ , ρ ˜ , E ) if there is a soft point y ˜ μ ˜ X ˜ such that ρ ˜ ( x ˜ λ n n , y ˜ μ ) 0 ˜ as n , that is for every ε ˜ > 0 ¯ , there is a natural number N = N ( ε ˜ ) such that 0 ¯ ρ ˜ ( x ˜ λ n n , y ˜ μ ) < ˜ ε ˜ , whenever n > N .

Definition 1.15: Let ( X ˜ , ρ ˜ , E ) be a soft metric space, then the sequence { x ˜ λ n n } of soft points in ( X ˜ , ρ ˜ , E ) is said to be a Cauchy sequence in X ˜ , if corresponding to every ε ˜ ˜ 0 ¯ , there exist m N such that ρ ˜ ( x ˜ λ i i , y ˜ λ j j ) ˜ ε ˜ , i , j m , i.e. ρ ˜ ( x ˜ λ i i , y ˜ λ j j ) 0 ¯ as i , j .

Definition 1.16: The soft metric space ( X ˜ , ρ ˜ , E ) is called complete, if every Cauchy sequence in X ˜ converges to some point of X ˜ .

Definition 1.17: Let ( X ˜ , ρ ˜ , E ) be a soft metric space. A function ( f , φ ) : ( X ˜ , ρ ˜ , E ) ( X ˜ , ρ ˜ , E ) is called a soft contraction mapping if there is a soft real number α R , 0 ¯ α < 1 ¯ such that for every point x ˜ λ , y ˜ μ S P ( X ) , we have

ρ ˜ ( ( f , φ ) ( x ˜ λ ) , ( f , φ ) ( y ˜ μ ) ) α ρ ˜ ( x ˜ λ , y ˜ μ ) .

Definition 1.18: A mapping ρ ˜ : S P ( X ˜ ) × S P ( X ˜ ) R ( E ) * is said to be dislocated soft metric on the soft set X ˜ if ρ ˜ satisfies the following conditions:

(d1) ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) = 0 ¯ then x ˜ s 1 = y ˜ s 2 ,

(d2) ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) = ρ ˜ ( y ˜ s 2 , x ˜ s 1 ) , for all x ˜ s 1 , y ˜ s 2 X ˜ ,

(d3) ρ ˜ ( x ˜ s 1 , z ˜ s 3 ) ρ ˜ ( x ˜ s 1 , y ˜ s 2 ) + ρ ˜ ( y ˜ s 2 , z ˜ s 3 ) , for all x ˜ s 1 , y ˜ s 2 , z ˜ s 3 X ˜ .

The soft set X ˜ with soft dislocated metric ρ ˜ defined on X ˜ is called a dislocated soft metric space and denoted by ( X ˜ , ρ ˜ , E ) .

2. Main Results

Theorem 2.1: Let A and B be two non-empty closed subsets of a complete dislocated soft metric space ( X ˜ , ρ ˜ , E ) . Suppose ( f , φ ) : A B A B is cyclic and satisfy the following:

(c) There exist a constant k ( 0 , 1 ) such that ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) k ρ ˜ ( x λ , y μ ) for all x λ A , y μ B .

Then ( f , φ ) has a unique soft fixed point that belongs to A B .

Definition 2.2: Let ( X ˜ , ρ ˜ , E ) be dislocated soft metric space and U be a subset of X ˜ . We say that U is ρ ˜ -open subset of X if x X ˜ , there exists r > 0 such that B ρ ( x , r ) U . Also, V X ˜ is a ρ ˜ -closed subset of X ˜ if ( X ˜ / V ) is ρ ˜ -open subset of X ˜ .

Lemma 2.3: Let ( X ˜ , ρ ˜ , E ) be dislocated soft metric space and V be a ρ ˜ - closed subset of X. Let { x λ n n } be a sequence in V. If x λ n n x λ as n , then x λ V .

Proof: Let x λ V , by definition 2.2, X \ V is a ρ ˜ -open set then there exist r > 0 such that B ρ ˜ ( x λ , r ) X \ V . On the other hand, we have lim n | ρ ˜ ( x λ n n , x λ ) ρ ˜ ( x λ , x λ ) | = 0 , Since x λ n n x λ as n . So there exist n 0 N such that

| ρ ˜ ( x λ n n , x λ ) ρ ˜ ( x λ , x λ ) | < r ,

for all n > n 0 . Hence we conclude that { x λ n n } B ρ ( x λ , r ) X \ V , for all n > n 0 . This is contradiction since { x λ n n } V , for all n in N.

Lemma 2.4: Let ( X ˜ , ρ ˜ , E ) be dislocated soft metric space and { x λ n n } be a sequence in ( X ˜ , ρ ˜ , E ) such that { x λ n n } x λ as n and ρ ˜ ( x λ , x λ ) = 0. Then

lim n ρ ˜ ( x λ n n , y μ ) = ρ ˜ ( x λ , y μ ) , for all y μ ( X ˜ , ρ ˜ , E ) .

Lemma 2.5: If ( X ˜ , ρ ˜ , E ) be soft dislocated metric space, then the following conditions holds

A. If ρ ˜ ( x λ n n , x λ n n ) = 0 , then ρ ˜ ( x λ x λ ) = ρ ˜ ( y μ , y μ ) = 0 ;

B. If { x λ n n } be a sequence such that lim n ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) = 0 , then we have lim n ρ ˜ ( x λ n n , x λ n n ) = lim n ρ ˜ ( x λ n + 1 n + 1 , x λ n + 1 n + 1 ) = 0 ;

C. If x λ y μ , then ρ ˜ ( x λ , y μ ) > 0 ;

ρ ˜ ( x λ x λ ) = 2 n i = 1 n ρ ˜ ( x λ , x λ i i ) , holds for all x λ , x λ i i X ˜ , where 1 i n .

D. At first we define the class of Ф and Ψ by the following ways:

Ψ = { ψ : [ 0 , ) [ 0 , ) suchthat ψ isnondecreasingandcontinuous } and Φ = { ϕ : [ 0 , ) [ 0 , ) suchthat ϕ islowersemicontinuous } .

Definition 2.6: Let ( X ˜ , ρ ˜ , E ) be a soft dislocated metric space, m N , let A 1 , A 2 , , A m be ρ ˜ -closed non empty subsets of X ˜ and let Y ˜ = i = 1 i = m A i . We say that ( f , φ ) is a cyclic generalized ϕ ψ contractive mapping if

i. Y ˜ = i = 1 i = m A i is a cyclic representation of Y ˜ with respect to ( f , φ ) ,

ii. ψ ( t ) ψ ( s ) + ϕ ( s ) > 0 for all t > 0 and s = t or s = 0 and

ψ ( ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ) ψ ( M ρ ˜ ( x λ , y μ ) ) ϕ ( M ρ ˜ ( x λ , y μ ) ) , (1)

for any x λ A i , y μ A i + 1 , i = 1 , 2 , 3 , , m , where A i + 1 = A 1 , ϕ Φ , ψ Ψ and

M ρ ˜ ( x λ , y μ ) = max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } .

Let X ˜ be a non-empty set and ( f , φ ) : ( X ˜ , ρ ˜ , E ) ( X ˜ , ρ ˜ , E ) be given map. The set of all soft fixed points of ( f , φ ) will be denoted by F i x ( ( f , φ ) ) i.e. F i x ( ( f , φ ) ) = { x λ X ˜ : x λ = ( f , φ ) x λ } .

Theorem 2.7: Let ( X ˜ , ρ ˜ , E ) be a complete dislocated soft metric space, m N , let A 1 , A 2 , , A m be non-empty ρ ˜ -closed subsets of ( X ˜ , ρ ˜ , E ) and

let Y ˜ = i = 1 i = m A i . Suppose that ( f , φ ) : Y ˜ Y ˜ is a cyclic generalized ϕ ψ con-

tractive mapping. Then ( f , φ ) has fixed point in i = 1 n A i . Moreover if ρ ˜ ( x λ , y μ ) ρ ˜ ( x λ , x λ ) for all x , y F i x ( f , φ ) , then ( f , φ ) has a unique fixed point in i = 1 n A i .

Proof: Let x λ 0 0 be an arbitrary point of Y ˜ , so there exists some i 0 such

that x λ 0 0 A i 0 . We know that ( f , φ ) ( A i 0 ) A i 0 + 1 , we conclude that ( f , φ ) ( x λ 0 0 ) A i 0 + 1 . Thus there exist x λ 1 1 in A i 0 + 1 such that ( f , φ ) x λ 0 0 = x λ 1 1 . Recursive ( f , φ ) x λ n n = x λ n + 1 n + 1 , where x λ n n A i n . Hence for n 0 there exist i n { 1 , 2 , 3 , , m } such that x λ n n A i n . In case x λ n 0 n 0 = x λ n 0 + 1 n 0 + 1 for some n 0 = 0 , 1 , 2 , , then it is clear that x λ n 0 n 0 is a soft fixed point of ( f , φ ) . Now assume that x λ n n x λ n + 1 n + 1 for all n. Hence by lemma 2.5(c) we have ρ ˜ ( x λ n 1 n 1 , x λ n n ) > 0 for all n. We shall show that the sequence { d n } is non-in- creasing, where d n = ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) . Assume that there exists some n 0 N such that

ρ ˜ ( x λ n 0 1 n 0 1 , x n 0 n 0 ) ρ ˜ ( x λ n 0 n 0 , x n 0 + 1 n 0 + 1 ) .

Hence we get

ψ ( ρ ˜ ( x n 0 1 n 0 1 , x n 0 n 0 ) ) ψ ( ρ ˜ ( x n 0 n 0 , x n 0 + 1 n 0 + 1 ) ) . (2)

Set x ˜ λ 1 1 = ( f , φ ) ( x ˜ λ 0 ) = ( f ( x ˜ λ 0 ) ) φ ( λ ) , x ˜ λ 2 2 = ( f , φ ) ( x ˜ λ 1 1 ) = ( f 2 ( x ˜ λ 0 ) ) φ 2 ( λ ) , ,

x ˜ λ n + 1 n + 1 = ( f , φ ) ( x ˜ λ n n ) = ( f n + 1 ( x ˜ λ 0 ) ) φ n + 1 ( λ ) , .

Using conditions (1) together with (2), we get

(3)

On the other hand, from lemma 2.5(D) we have ρ ˜ ( x λ n n , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) and ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n 1 n 1 , x λ n n ) 6 ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) 3 .

That is max { ρ ˜ ( x λ n 1 n 1 , x λ n n ) , ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) , ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) 3 } max { ρ ˜ ( x λ n 1 n 1 , x λ n n ) , ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) } .

Therefore from (3) we get

ψ ( ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ) ψ ( max { ρ ˜ ( x λ n 1 n 1 , x λ n n ) , ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) } ) ϕ ( max { ρ ˜ ( x λ n 1 n 1 , x λ n n ) , ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) } ) .

Now, if max { ρ ˜ ( x λ n 1 n 1 , x λ n n ) , ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) } = ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) , then

ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) α ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) β ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) .

This is contradiction. Hence we have

ψ ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ψ { ρ ˜ ( x λ n 1 n 1 , x λ n n ) } ϕ { ρ ˜ ( x λ n 1 n 1 , x λ n n ) } , (4)

for all n N . By taking x λ = x λ n 0 1 n 0 1 and y μ = x λ n 0 n 0 in (4) and keeping (2) in

mind, we deduce that

ψ ρ ˜ ( x λ n 0 1 n 0 1 , x λ n 0 n 0 ) ψ { ρ ˜ ( x λ n 0 1 n 0 1 , x λ n 0 n 0 ) } ϕ { ρ ˜ ( x λ n 0 1 n 0 1 , x λ n 0 n 0 ) } .

This is a contradiction. Hence we conclude that d n < d n 1 i.e.

ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) hold for all n N . Thus there exist r 0 such that lim n d n = lim n ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) = r . We shall show that r = 0 by the method of

reductio ad absurdum. For this purpose, we assume that r > 0 . By (4) together with the property of ϕ and ψ we have

ψ ( r ) = lim n sup ψ ( d n ) lim n sup [ ψ ( d n 1 ) ϕ ( d n 1 ) ] ψ ( r ) ϕ ( r ) .

This yields that φ ( r ) 0 . This is contradiction. Hence we obtain that

lim n d n lim n ( x λ n n , x λ n + 1 n + 1 ) = 0 . (5)

We shall show that { x λ n n } is a ρ ˜ -Cauchy sequence. To reach this goal, first we prove the followings claim:

(k) For every ϵ > 0 , there exists n N such that if r , q n with

r q 1 ( m ) , then ρ ˜ ( x λ r n r n , x λ q n q n ) < ϵ .

Suppose, on the contrary that there exist ϵ > 0 such that for any n N , we can find r n > q n n with r n q n 1 ( m ) satisfying

ρ ˜ ( x λ q n q n , x λ r n r n ) ϵ . (6)

Now we consider n > 2 m . Then, corresponding to q n n , we can choose r n in such a way that it is the smallest integer with r n > q n satisfying

r n q n 1 ( m ) and ρ ˜ ( x λ q n q n , x λ r n r n ) ϵ . Therefore ρ ˜ ( x λ q n q n , x λ r n m r n m ) ϵ . By using

triangular inequality, we obtain

ϵ ρ ˜ ( x λ q n q n , x λ r n r n ) ρ ˜ ( x λ q n q n , x λ q n m q n m ) + i = 1 m ρ ˜ ( x λ r n i r n i , x λ r n i + 1 r n i + 1 ) ϵ + i = 1 m ρ ˜ ( x λ r n i r n i , x λ r n i + 1 r n i + 1 ) .

Passing to the limit as n in the last inequality and taking (5) into account, we obtain that

lim n ρ ˜ ( x λ q n q n , x λ r n r n ) = ϵ . (7)

Again by (d3) we derive that

ϵ ρ ˜ ( x λ q n q n , x λ r n r n ) ρ ˜ ( x λ q n q n , x λ q n + 1 q n + 1 ) + ρ ˜ ( x λ q n + 1 q n + 1 , x λ r n + 1 r n + 1 ) + ρ ˜ ( x λ r n + 1 r n + 1 , x λ r n r n ) ρ ˜ ( x λ q n q n , x λ q n + 1 q n + 1 ) + ρ ˜ ( x λ q n + 1 q n + 1 , x λ q n q n ) + ρ ˜ ( x λ q n q n , x λ r n r n ) + ρ ˜ ( x λ r n r n , x λ r n + 1 r n + 1 ) + ρ ˜ ( x λ r n + 1 r n + 1 , x λ r n r n ) 2 ρ ˜ ( x λ q n q n , x λ q n + 1 q n + 1 ) + ρ ˜ ( x λ q n q n , x λ r n r n ) + 2 ρ ˜ ( x λ r n + 1 r n + 1 , x λ r n r n ) .

Taking (5) and (7) in account we get

lim n ρ ˜ ( x λ q n + 1 q n + 1 , x λ r n + 1 r n + 1 ) = ϵ . (8)

By ( d 3 ) we have the following inequality

ρ ˜ ( x λ q n q n , x λ r n + 1 r n + 1 ) ρ ˜ ( x λ q n q n , x λ r n r n ) + ρ ˜ ( x λ r n r n , x λ r n + 1 r n + 1 ) (9)

and ρ ˜ ( x λ q n q n , x λ r n r n ) ρ ˜ ( x λ q n q n , x λ r n + 1 r n + 1 ) + ρ ˜ ( x λ r n r n , x λ r n + 1 r n + 1 ) . (10)

Letting lim n in (9) and (10), we have

lim n ρ ˜ ( x λ q n q n , x λ r n + 1 r n + 1 ) = ϵ . (11)

Again by ( d 3 ) we have

ρ ˜ ( x λ r n r n , x λ q n + 1 q n + 1 ) ρ ˜ ( x λ r n r n , x λ r n + 1 r n + 1 ) + ρ ˜ ( x λ r n + 1 r n + 1 , x λ q n + 1 q n + 1 ) (12)

and ρ ˜ ( x λ r n + 1 r n + 1 , x λ q n + 1 q n + 1 ) ρ ˜ ( x λ r n + 1 r n + 1 , x λ r n r n ) + ρ ˜ ( x λ r n r n , x λ q n + 1 q n + 1 ) . (13)

Letting n in (12) and (13), we derive that

lim n ρ ˜ ( x λ r n r n , x λ q n + 1 q n + 1 ) = ϵ . (14)

Since x λ q n q n and x λ r n r n lie in different adjacently labeled sets and for certain 1 i m . By using (5), (7), (8), (11) and (14) together with the fact that ( f , φ ) is a generalized cyclic ϕ ψ contractive mappings, we find that

Regarding the property of ϕ and ψ in the last inequality, we obtain that ψ ( ϵ ) ψ ( ϵ ) ϕ ( ϵ ) , which is a contradiction. Hence the condition (k) is a satisfied. Fix ϵ > 0. By the claim we find n 0 N such that if r , q n 0 with r q 1 ( m ) ,

ρ ˜ ( x λ r r , x λ q q ) ϵ 2 . (15)

Since lim n d ˜ ( x λ n n , x λ n + 1 n + 1 ) = 0 , we also find n 1 N such that

ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ϵ 2 m , (16)

for any n n 1 . Suppose that r , s max { n 0 , n 1 } and s > r . There exist k { 1 , 2 , 3 , , m } such that s r k ( m ) . Therefore s r + ϕ 1 ( m ) , for ϕ = m k + 1.

So we have for j { 1 , 2 , 3 , , m } and s + j r 1 ( m ) ,

ρ ˜ ( x λ r r , x λ s s ) ρ ˜ ( x λ r r , x λ s + j s + j ) + ρ ˜ ( x λ s + j s + j , x λ s + j 1 s + j 1 ) + + ρ ˜ ( x λ s + 1 s + 1 , x λ s s ) .

By (15) & (16) and from the last inequality, we get

ρ ˜ ( x λ r r , x λ s s ) ϵ 2 + j × ϵ 2 m ϵ 2 + m × ϵ 2 m = ϵ .

This proves that { x λ n n } is a ρ ˜ -Cauchy sequence. Since ϵ is arbitrary,

{ x λ n n } is a Cauchy sequence. Since Y is ρ ˜ -closed in ( X ˜ , ρ ˜ , E ) , then ( Y ˜ , ρ ˜ , E ) is also complete, there exists x λ Y ˜ = i = 1 m A i such that lim n x λ n n = x λ in ( Y ˜ , ρ ˜ , E ) ; equivalently

ρ ˜ ( x λ , x λ ) = lim n ρ ˜ ( x λ , x λ n n ) = lim m , n ρ ˜ ( x λ n n , x λ m m ) = 0 . (17)

In what follows, we prove that x λ is a soft fixed point of ( f , φ ) . In fact,

since lim n x λ n n = x λ and Y ˜ = i = 1 m A i is a cyclic representation of Y with respect to

( f , φ ) . The sequence { x λ n n } has infinite terms in each A i , for i { 1 , 2 , 3 , , m } . Suppose that x λ A i , ( f , φ ) x λ A i + 1 and we take a subsequence { x λ n k n k } of { x λ n n } with x λ n k n k A i 1 (the existence of this subsequence is guaranteed by above- mentioned comment). By using the contractive condition we can obtain

ψ ( ρ ˜ ( ( f , φ ) x λ , ( f , φ ) x λ n k n k ) ) ψ ( max { ρ ˜ ( x λ , x λ n k n k ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) , ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) + ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } ) ϕ ( max { ρ ˜ ( x λ , x λ n k n k ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) , ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) + ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } ) .

Passing to the limit as n and using x λ n k n k x λ , lower semi-continuity of ϕ , we have

ψ ( ρ ˜ ( x λ , ( f , φ ) x λ ) ) ψ ( ρ ˜ ( x λ , ( f , φ ) x λ ) ) ϕ ( ρ ˜ ( x λ , ( f , φ ) x λ ) ) .

So, ρ ˜ ( x λ , ( f , φ ) x λ ) = 0. Therefore x λ is a soft fixed point of ( f , φ ) . Finally to prove the uniqueness of soft fixed point, suppose that y μ , z w ( X ˜ , ρ ˜ , E ) are two distinct soft fixed points of ( f , φ ) . The cyclic character of ( f , φ ) and the fact that y μ , z w ( X ˜ , ρ ˜ , E ) are soft fixed points of ( f , φ ) implies that

x λ , y μ i = 1 m A i . Suppose that x λ y μ and for all p λ , q θ F i x ( ( f , φ ) ) ,

ρ ˜ ( p λ , q θ ) ρ ˜ ( p λ , p λ ) . Using the contractive condition, we obtain

ψ ( ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ) ψ ( max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } ) ϕ ( max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } )

Then we have ψ ( ρ ˜ ( x λ , y μ ) ) ψ ( ρ ˜ ( x λ , y μ ) ) ϕ ( ρ ˜ ( x λ , y μ ) ) .

This is a contradiction. Thus we derive that ρ ˜ ( y μ , z w ) = 0 y μ = z w . Hence proved.

In the theorem 2.7, if we take ( X ˜ , ρ ˜ , E ) = A i , for all 0 i m , then we deduce the following theorem.

Theorem 2.8: Let ( X ˜ , ρ ˜ , E ) be a complete soft dislocated metric space and ( f , φ ) be self map on ( X ˜ , ρ ˜ , E ) . Assume that there exist φ Φ , ψ Ψ such that

ψ ( ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ) ψ ( M ρ ˜ ( x λ , y μ ) ) ϕ ( M ρ ˜ ( x λ , y μ ) ) , for all x λ , y μ ( X ˜ , ρ ˜ , E ) , where

M ρ ˜ ( x λ , y μ ) = max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } .

Then ( f , φ ) has a soft fixed point. Moreover if ρ ˜ ( x λ , y μ ) ρ ˜ ( x λ , x λ ) for all x λ , y μ F i x ( ( f , φ ) ) , then ( f , φ ) has a unique soft fixed point.

If in theorem (2.7) we take ψ ( t ) = t and ϕ ( t ) = ( 1 r ) t , where r [ 0 , 1 ) then we deduce the following corollary.

Corollary 2.9: Let ( X ˜ , ρ ˜ , E ) be a complete soft dislocated metric space, m N , let A 1 , A 2 , , A m be non empty ρ ˜ -closed subsets of ( X ˜ , ρ ˜ , E ) and let Y = i = 1 i = m A i . Suppose that ( f , φ ) : Y Y is an operator such that

i. Y = i = 1 i = m A i is a cyclic representation of ( X ˜ , ρ ˜ , E ) with respect to ( f , φ ) ,

ii. there exist r [ 0 , 1 ) such that

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = r max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } ,

for any x A i and y A i + 1 , i = 1 , 2 , 3 , , m . Where A m + 1 = A 1 , then ( f , φ ) has a soft fixed point z w i = 1 m A i . Moreover if ρ ˜ ( x λ , y μ ) ρ ˜ ( x λ , x λ ) for all x λ F i x ( f , φ ) , then ( f , φ ) has a unique soft fixed point.

Example 2.10: Let X = R with soft dislocated metric

ρ ˜ ( x λ , y μ ) = max { | x λ 2 | , | y μ 2 | } , for all x λ , y μ X ¯ . Suppose A 1 = [ 2 , 0 ] , A 2 = [ 0 , 2 ] & Y = i = 1 i = 2 A i . Define ( f , φ ) : Y Y by

( f , φ ) = { ( x λ ) 2 8 ,if x λ [ 2 , 0 ] x λ 5 ,if x λ [ 0 , 2 ]

It is clear that i = 1 i = 2 A i is a cyclic representation of Y with respect to ( f , φ ) .

Let x λ A 1 = [ 2 , 0 ] and y μ A 2 = [ 0 , 2 ] then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = ρ ˜ ( x λ 2 8 , y μ 5 ) = max { | x λ 2 / 8 2 | , | y μ / 5 2 | } = max { | x λ 2 16 | , | y μ 10 | } max { x λ 4 , y μ 4 } 1 2 max { x λ 2 , y μ 2 } 1 2 ρ ˜ ( x λ , y μ ) .

and so

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = r max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } .

Hence the condition of corollary (2.9) (theorem 2.7) holds and ( f , φ ) has a fixed point in A 1 A 2 . Here x λ = 0 is a fixed point of ( f , φ ) .

Example 2.11: Let X = R with soft dislocated metric

ρ ˜ ( x λ , y μ ) = max { | x λ | , | y μ | } , for all x λ , y μ X . Suppose A 1 = [ 1 , 0 ] and A 2 = [ 0 , 1 ] & Y = i = 1 i = 2 A i we define ( f , φ ) : Y Y by

( f , φ ) = { ( x λ ) 2 2 ,if x λ [ 1 , 0 ] , x λ 4 ,if x λ [ 0 , 1 ] .

It is clear that i = 1 i = 2 A i is a cyclic representation of Y with respect to ( f , φ ) . Let x λ A 1 = [ 1 , 0 ] and y μ A 2 = [ 0 , 1 ] , then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = ρ ˜ ( x λ 2 2 , y μ 2 ) = max { | x λ 2 2 | , | y μ 4 | } max { x λ 2 , y μ 2 } 1 2 max { x λ , y μ } 1 2 max { | x λ | , | y μ | } 1 2 ρ ˜ ( x λ , y μ ) ,

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = r max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } .

Hence the condition of corollary (2.9) (theorem 2.7) holds and ( f , φ ) has a soft fixed point in A 1 A 2 . Here x λ = 0 is a soft fixed point of ( f , φ ) .

In the above corollary we take A i = ( X ˜ , ρ ˜ , E ) for all 0 i m , then we deduce the following corollary.

Corollary 2.12: Let ( X ˜ , ρ ˜ , E ) be a complete soft dislocated metric space and let ( f , φ ) be a self map on X. Assume that there exist r [ 0 , 1 ) such that

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) r max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } ,

holds for all x λ , y μ ( X , ρ ˜ , E ) . Then ( f , φ ) has a soft fixed point. More- over if ρ ˜ ( x λ , y μ ) ρ ˜ ( x λ , x λ ) for all x λ , y μ F i x ( ( f , φ ) ) , then ( f , φ ) has a unique soft fixed point.

Example 2.13: Let X = R with soft dislocated metric space and ρ ˜ ( x λ , y μ ) = max { x λ , y μ } . For any x λ , let ( f , φ ) : ( X , ρ ˜ , E ) ( X , ρ ˜ , E ) be defined by

( f , φ ) x λ = { x λ 6 , if0 x λ < 1 2 x λ 2 5 ,if 1 2 x λ 1 x λ 7 ,if x > 1.

Proof: To show that the existence and uniqueness of soft point of ( f , φ ) , we need to consider the following cases

Let 0 x λ , y μ < 1 2 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = 1 6 max { x λ , y μ } 1 2 max { x λ , y μ } = 1 2 ρ ˜ ( x λ , y μ ) .

Let 1 2 x λ , y μ 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = 1 5 max { x λ 2 , y μ 2 } 1 5 max { x λ , y μ } 1 2 max { x λ , y μ } = 1 2 ρ ˜ ( x λ , y μ ) .

Let x λ , y μ > 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = 1 7 max { x λ , y μ } 1 2 max { x λ , y μ } = 1 2 ρ ˜ ( x λ , y μ ) .

Let 0 x λ < 1 2 and 1 2 y μ 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = max { x λ 6 , y μ 2 5 } 1 2 max { x λ , y μ } = 1 2 ρ ˜ ( x λ , y μ ) .

Let 0 x λ < 1 2 and y μ > 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = max { x λ 6 , y μ 7 } 1 2 max { x λ , y μ } = 1 2 ρ ˜ ( x λ , y μ ) .

Let 1 2 x λ 1 and 0 y μ 1 2 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = max { x λ 5 , y μ 6 } 1 2 max { x λ , y μ } = 1 2 ρ ˜ ( x λ , y μ ) .

and so

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) r max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } .

Hence we conclude that all the condition of corollary (2.12) (theorem 2.7) holds and ( f , φ ) has a soft fixed point in [ 0 , ) . By corollary 2.9 we deduce the following result.

Corollary 2.14: Let ( X ˜ , ρ ˜ , E ) be a complete soft dislocated metric, m N , let A 1 , A 2 , , A m be non empty ρ ˜ -closed subsets of X and Y = i = 1 i = m A i . Suppose that ( f , φ ) : Y Y is an operator such that

i. Y = i = 1 i = m A i is a cyclic representation of ( X ˜ , ρ ˜ , E ) with respect to ( f , φ ) .

ii. there exist r [ 0 , 1 ) such that

0 ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ρ ( t ) d t r 0 m ρ ( t ) d t ,

where m = max { ρ ˜ ( x λ , y μ ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) 6 } , for any x A i

y A i + 1 , i = 1 , 2 , 3 , , m , where A m + 1 = A 1 and ρ : [ 0 , ) [ 0 , ) be Lebesgue-

integrable mapping satisfying 0 ϵ ρ ( t ) d t > 0 , for ϵ > 0 . Then ( f , φ ) has a fixed point z ω i = 1 m A i . Moreover if ρ ˜ ( x λ , y μ ) ρ ˜ ( x λ , x λ ) for all

x λ , y μ F i x ( ( f , φ ) ) , then ( f , φ ) has a unique soft fixed point.

Definition 2.15: Let ( f , φ ) : ( X , ρ ˜ , E ) ( X , ρ ˜ , E ) with ψ : ( X , ρ ˜ , E ) [ 0 , ) and γ [ 0 , 1 ] . A mapping ( f , φ ) is said to be a γ ψ sub admissible soft mapping if ψ ( x λ ) γ ψ ( ( f , φ ) x λ ) γ , for x λ ( X , ρ ˜ , E ) .

Example 2.16: Let ( f , φ ) : [ π , π ] [ π , π ] and ψ : [ π , π ] R + be defined by ( f , φ ) x λ = π 4 ( tan x λ ) and ψ ( x λ ) = | x λ 1 4 π | + 1 2 then ( f , φ ) is a γ ψ sub admissible mapping, where γ = 1 2 , indeed if ψ ( x λ ) = | x λ 1 4 π | + 1 2 1 2 then x λ = 1 4 π .

Hence ( f , φ ) ( x λ ) = 1 4 π and ψ ( ( f , φ ) x λ ) = 1 2 .

Let Λ be the class of all the functions ϕ : [ 0 , ) 3 [ 0 , ) that are a continuous with the property: ϕ ( x , y , z ) = 0 if and only if x = y = z = 0 .

Definition 2.17: Let ( X , ρ ˜ , E ) be a soft dislocated metric space, m N , let

A 1 , A 2 , , A m be ρ ˜ -closed non empty subsets of ( X , ρ ˜ , E ) and let Y = i = 1 i = m A i . Assume that ( f , φ ) : Y Y is g-ψ-sub admissible mapping, where γ = 1 8 . Then ( f , φ ) is called ψ-cyclic generalized weakly C-contraction if

i. Y = i = 1 i = m A i is a cyclic representation of Y with respect to ( f , φ ) ,

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ψ ( x λ ) ρ ˜ ( y μ , ( f , φ ) x λ ) + ψ ( ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) + ψ ( ( f , φ ) 2 x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) + ψ ( ( f , φ ) 3 x λ ) ρ ˜ ( x λ , ( f , φ ) x λ ) + ψ ( ( f , φ ) 4 x λ ) ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ψ ( ( f , φ ) 5 x λ ) ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) ϕ { ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , 1 2 [ ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) x λ ) ] } (18)

for any x A i , y A i + 1 , i = 1 , 2 , 3 , , m and A m + 1 = A 1 and ϕ Λ .

Theorem 1.18: Let ( X , ρ ˜ , E ) be a complete soft dislocated metric space, m N , let A 1 , A 2 , , A m , be ρ ˜ -closed non-empty subsets of ( X , ρ ˜ , E ) and let

Y = i = 1 i = m A i . Assume that ( f , φ ) : Y Y is a ψ-cyclic generalized weakly C-

contraction. If there exists x λ 0 0 Y such that ψ ( x λ 0 ) 1 8 , then ( f , φ ) has a soft fixed point z ω i = 1 n A i . Moreover if ψ ( z w 0 0 ) 1 8 , then z w 0 0 is unique.

Proof: Let x λ 0 0 Y be such that ψ ( x λ 0 0 ) 1 8 . Since ( f , φ ) is a sub ψ-ad- missible mapping with respect to 1 8 , then ψ ( ( f , φ ) x λ 0 0 ) 1 8 . ψ ( ( f , φ ) n x λ 0 0 ) 1 8 for all n N 0 . Also, there exist some i0 such that x λ 0 0 A i 0 . Now ( f , φ ) ( A i 0 ) A i 0 + 1 implies that ( f , φ ) ( x λ 0 0 ) A i 0 + 1 . Thus there exist x λ 1 1 in A i 0 + 1 such that ( f , φ ) x λ 0 0 = x λ 1 1 . Similarly ( f , φ ) x λ n n = x λ n + 1 n + 1 , where x λ n n A i n . Hence for n 0 there exist i n { 1 , 2 , 3 , , m } such that x λ n n A i n and x λ n + 1 n + 1 A i n + 1 . In case x λ n 0 n 0 = x λ n 0 + 1 n 0 + 1 , for some n 0 = 0 , 1 , 2 , , then it is clear that x λ n 0 n 0 is a fixed point of ( f , φ ) . Now assume that x λ n n x λ n + 1 n + 1 , for all n.

Since ( f , φ ) : Y Y is a cyclic generalized weak C-contraction, we have for all n N we have

And so

ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) 1 8 { ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n 1 n 1 , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) 1 + ρ ˜ ( x λ n n , x λ n n ) } . (19)

On the other hand from (d3) we have

ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) .

And by lemma (2.5D) we have

ρ ˜ ( x λ n n , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) .

Also

ρ ˜ ( x λ n 1 n 1 , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) 1 + ρ ˜ ( x λ n n , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) ρ ˜ ( x λ n n , x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) [ ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ] ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 )

From (19) we have

ρ ˜ ( ( f , φ ) x λ n 1 n 1 , ( f , φ ) x λ n n ) 1 8 { ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n 1 n 1 , x λ n n ) } 1 8 { 4 ρ ˜ ( x λ n 1 n 1 , x λ n n ) + 4 ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) } 1 2 { ρ ˜ ( x λ n 1 n 1 , x λ n n ) + ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) } .

Therefore we have

ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ρ ˜ ( x λ n 1 n 1 , x λ n n ) . (20)

For any n 1 , set t n = ρ ˜ ( x λ n n , x λ n 1 n 1 ) . On the occasion of the facts above { t n }

is a non-increasing sequence of non negative real numbers. Consequently, there exist L 0 such that

lim n ρ ˜ ( x λ n n , x λ n 1 n 1 ) = L . (21)

We shall prove that L = 0 . Since ρ ˜ ( x λ n n , x λ n n ) = 2 ϕ ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) then we get lim n d ˜ ( x λ n n , x λ n n ) = 2 L . Similarly, lim n ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) = 2 L . Then

lim n [ ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) ] 4 L .

On the other hand, by taking limit as n in (19), we have,

L 1 8 [ 4 L + { ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) } ] ,

This implies 4 L lim n { ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) } .

Hence lim n { ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) } = 4 L .

Now from (18) we have

t n + 1 ψ ( x λ n 1 n 1 ) ρ ˜ ( x λ n n , x λ n n ) + ψ ( x λ n n ) ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) + ψ ( x λ n + 1 n + 1 ) t n + 1 + ψ ( x λ n + 2 n + 2 ) t n + ψ ( x λ n + 3 n + 3 ) t n + 1 + ψ ( x λ n + 4 n + 4 ) t n 1 ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) 1 + ρ ˜ ( x λ n n , x λ n n ) ϕ { t n , t n + 1 , 1 2 [ ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n n , x λ n n ) ] } .

t n + 1 1 8 { ρ ˜ ( x λ n n , x λ n n ) + ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) + t n + 1 + t n + t n + 1 + t n 1 ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) 1 + ρ ˜ ( x λ n n , x λ n n ) } ϕ { t n , t n + 1 , 1 2 [ ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) + ρ ˜ ( x λ n n , x λ n n ) ] }

By taking limit as n in the above inequality, we deduce that, L L ϕ ( L , L , 2 L ) .

So ϕ ( L , L , 2 L ) = 0 , since ϕ ( x λ , y μ , z ω ) = 0 x λ = y μ = z ω = 0 , we get L = 0 .

Due to lim n ρ ˜ ( x λ n n , x λ n n ) 2 L and lim n ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) 2 L we have

lim n ρ ˜ ( x λ n n , x λ n n ) = lim n ρ ˜ ( x λ n 1 n 1 , x λ n + 1 n + 1 ) n = lim n ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) = 0 . (22)

We shall show that { x λ n n } is a ρ ˜ -Cauchy sequence. At first, we prove the following fact:

(K) for every ϵ > 0 , there exist n N such that if r , q n with

r q 1 ( m ) , then ρ ˜ ( x λ r r , x λ q q ) < ϵ .

Suppose to the contrary that there exist ϵ > 0 , such that for any n in N, we can find r n > q n n with r n q n 1 ( m ) satisfying

ρ ˜ ( x λ r n r n , x λ q n q n ) ϵ . (23)

Following the related lines of the proof of theorem (1.8) we have

lim n ρ ˜ ( x λ q n q n , x λ r n r n ) = ϵ ,

lim n ρ ˜ ( x λ q n + 1 q n + 1 , x λ r n + 1 r n + 1 ) = ϵ ,

lim n ρ ˜ ( x λ q n q n , x λ r n + 1 r n + 1 ) = ϵ and lim n ρ ˜ ( x λ r n r n , x λ q n + 1 q n + 1 ) = ϵ . (24)

Since x λ q n q n & x λ r n r n lie in different adjacently labeled sets Ai and Ai+1, for a certain 1 i m . Using the fact that ( f , φ ) is ψ-cyclic generalized weakly C- contraction, we have

Now, by taking limit as n in the above inequality, we derive that

ϵ 1 8 [ ϵ + ϵ + 0 + 0 + 0 + 0 ] ϕ [ 0 , 0 , ϵ ] 1 4 ϵ .

This is a contradiction. Hence condition (k) holds. We are ready to show that the sequence { x λ n n } is a Cauchy. Fix ϵ > 0. By the claim, we find n 0 N such that if r , q n 0 with r q 1 ( m ) then

ρ ˜ ( x λ r n r n , x λ q n q n ) ϵ 4 ϵ 2 . (25)

Since lim n ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) = 0 , we find n 1 N such that

ρ ˜ ( x λ n n , x λ n + 1 n + 1 ) ϵ 2 m , (26)

for any n n 1 . Suppose that r , s max { n 0 , n 1 } with s > r . Then there exist k { 1 , 2 , , m } such that s r k ( m ) . Therefore, s r + ϕ 1 ( m ) , for ϕ = m k + 1 .

So we have, for j { 1 , 2 , 3 , , m } , s + j r 1 ( m ) ,

ρ ˜ ( x λ r r , x λ s s ) ρ ˜ ( x λ r r , x λ s + j s + j ) + ρ ˜ ( x λ r r , x λ s + j s + j ) + ρ ˜ ( x λ s + j s + j , x λ s + j 1 s + j 1 ) + + ρ ˜ ( x λ s + 1 s + 1 , x λ s s ) .

By (25) and (26) and from the last inequality, we get

ρ ˜ ( x λ r r , x λ s s ) ϵ 2 + j × ϵ 2 m ϵ 2 + m × ϵ 2 m = ϵ .

This proves that { x λ n n } is a ρ ˜ -Cauchy sequence.

Since Y is ρ ˜ -closed in ( X , ρ ˜ , E ) , then ( Y , ρ ˜ , E ) is also complete, there

exists z ω Y = i = 1 m A i such that lim n x λ n n = z ω in ( Y , ρ ˜ , E ) , equivalently

ρ ˜ ( z ω , z ω ) = lim n ρ ˜ ( z ω , x λ n n ) = lim m , n ρ ˜ ( x λ n n , x λ m m ) = 0 . (27)

In what follows, we prove that x λ is a soft fixed point of ( f , φ ) . In fact,

since lim n x λ n n = z w and as Y = i = 1 m A i is cyclic representation of Y with respect to ( f , φ ) , the sequence { x λ n n } has infinite terms in each A i for

i { 1 , 2 , 3 , , m } . Suppose that x λ A i , ( f , φ ) x λ A i + 1 and we take a subsequence { x λ n k n k } of { x λ n n } with x λ n k n k A i 1 (the existence of this subsequence is

guaranteed by the above mentioned comment). By using the contractive condition, we can obtain

ρ ˜ ( x λ n k + 1 n k + 1 , ( f , φ ) x λ ) x λ n k n k = ρ ˜ ( ( f , φ ) x λ n k n k , ( f , φ ) x λ ) ψ ( x λ n k n k ) ρ ˜ ( x λ , ( f , φ ) x λ n k n k ) + ψ ( ( f , φ ) x λ n k n k ) ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) + ψ ( ( f , φ ) 2 x λ n k n k ) ρ ˜ ( x λ , ( f , φ ) x λ ) + ψ ( ( f , φ ) 3 x λ n k n k ) ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) + ψ ( ( f , φ ) 4 x λ n k n k ) ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) ρ ˜ ( x λ , ( f , φ ) x λ ) 1 + ρ ˜ ( x λ n k n k , x λ )

+ ψ ( ( f , φ ) 5 x λ n k n k ) ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) 1 + ρ ˜ ( x λ , ( f , φ ) x λ n k n k ) ϕ { ρ ˜ ( x λ n k n k , ( f , φ ) x λ n k n k ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , 1 2 [ ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ n k n k ) ] } , 1 8 { ρ ˜ ( x λ , x λ n k + 1 n k + 1 ) + ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ n k n k , x λ n k + 1 n k + 1 ) + ρ ˜ ( x λ n k n k , x λ n k + 1 n k + 1 ) ρ ˜ ( x λ , ( f , φ ) x λ ) 1 + ρ ˜ ( x λ n k n k , x λ ) + ρ ˜ ( x λ n k n k , x λ n k + 1 n k + 1 ) ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) 1 + ρ ˜ ( x λ , x λ n k + 1 n k + 1 ) } ϕ { ρ ˜ ( x λ n k n k , x λ n k + 1 n k + 1 ) , ρ ˜ ( x λ , ( f , φ ) x λ ) , 1 2 [ ρ ˜ ( x λ n k n k , ( f , φ ) x λ ) + ρ ˜ ( x λ , x λ n k + 1 n k + 1 ) ] } .

Passing to the limit as lim n and using x λ n k n k x λ , lower semi-conti- nuity of ϕ , we have

ρ ˜ ( x λ , ( f , φ ) x λ ) 1 8 { 0 + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + 0 + 0 + 0 } ϕ { 0 , ρ ˜ ( x λ , ( f , φ ) x λ ) , 1 2 [ ρ ˜ ( x λ , ( f , φ ) x λ ) ] } 1 4 ρ ˜ ( x λ , ( f , φ ) x λ ) .

So ρ ˜ ( x λ , ( f , φ ) x λ ) = 0 and therefore x λ is fixed point of ( f , φ ) .

Finally to prove the uniqueness of soft fixed point theorem, suppose that y μ , z ω ( X , ρ ˜ , E ) are soft fixed points of ( f , φ ) . The cyclic character of

( f , φ ) and the fact that y μ , z ω ( X , ρ ˜ , E ) are soft fixed points of ( f , φ ) implies that y μ , z ω i = 1 m A i . Also suppose that ψ ( y μ ) 1 8 . By using contrac-

tive condition we derive that

ρ ˜ ( y μ , z ω ) = ρ ˜ ( ( f , φ ) y μ , ( f , φ ) z ω ) ψ ( y μ ) ρ ˜ ( z ω , ( f , φ ) y μ ) + ψ ( ( f , φ ) y μ ) ρ ˜ ( y μ , ( f , φ ) z ω ) + ψ ( ( f , φ ) 2 y μ ) ρ ˜ ( z ω , ( f , φ ) z ω ) + ψ ( ( f , φ ) 3 y μ ) ρ ˜ ( y μ , ( f , φ ) y μ ) + ψ ( ( f , φ ) 4 y μ ) ρ ˜ ( y μ , ( f , φ ) y μ ) ρ ˜ ( z w , ( f , φ ) z w ) 1 + ρ ˜ ( y μ , z w ) + ψ ( ( f , φ ) 5 y μ ) ρ ˜ ( y μ , ( f , φ ) y μ ) ρ ˜ ( y μ , ( f , φ ) z w ) 1 + ρ ˜ ( z w , ( f , φ ) y μ ) ϕ ( ρ ˜ ( y μ , ( f , φ ) y μ ) , ρ ˜ ( z w , ( f , φ ) z w ) , 1 2 [ ρ ˜ ( y μ , ( f , φ ) z w ) + ρ ˜ ( z w , ( f , φ ) y μ ) ] )

1 8 { ρ ˜ ( z ω , y μ ) + ρ ˜ ( y μ , z ω ) + ρ ˜ ( z ω , z ω ) + ρ ˜ ( y μ , y μ ) + ρ ˜ ( y μ , y μ ) ρ ˜ ( z w , z w ) 1 + ρ ˜ ( y μ , z w ) + ρ ˜ ( y μ , y μ ) ρ ˜ ( y μ , z w ) 1 + ρ ˜ ( z w , y μ ) } ϕ ρ ( ρ ˜ ( y μ , y μ ) , ρ ˜ ( z w , z w ) , 1 2 [ ρ ˜ ( y μ , z w ) + ρ ˜ ( z w , y μ ) ] ) 1 8 { 2 ρ ˜ ( z ω , y μ ) } ϕ ( 0 , ρ ˜ ( z w , z w ) , 1 2 [ ρ ˜ ( y μ , z w ) + ρ ˜ ( z w , y μ ) ] ) 1 4 ρ ˜ ( z ω , y μ ) .

This gives us ρ ˜ ( z ω , y μ ) = 0 , that is z ω = y μ . This finishes the proof.

Corollary 2.19: Let ( X , ρ ˜ , E ) be a complete soft dislocated metric space, m N , let A 1 , A 2 , , A m be non empty ρ ˜ -closed subsets of ( X , d , E ) and

let Y = i = 1 i = m A i . Suppose that ( f , φ ) : Y Y is an operator such that

i) Y = i = 1 i = m A i is cyclic representation of ( X , ρ ˜ , E ) with respect to ( f , φ ) ;

ii) there exist β [ 0 , 1 8 ) such that

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) β { ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) } , (28)

for any x λ A i , y μ A i + 1 , i = 1 , 2 , 3 , , m , where A m + 1 = A 1 . Then ( f , φ )

has fixed point z w i = 1 i = m A i .

Example 2.20: Let ( X , ρ ˜ , E ) = R with soft dislocates metric space

ρ ˜ ( x λ , y μ ) = max { | x λ | , | y μ | } , for any x λ , y μ ( X , ρ ˜ , E ) , suppose A 1 = [ 2 , 0 ]

and A 2 = [ 0 , 2 ] & Y = i = 1 i = 2 A i , we define ( f , φ ) : Y Y by

( f , φ ) x λ = { x λ 32 , if x λ [ 2 , 0 ] , x λ 16 , if x λ [ 0 , 2 ] ,

It clear that i = 1 i = 2 A i is cyclic representation of Y with respect to ( f , φ ) .

Proof: Let x λ 1 1 A 1 = [ 2 , 0 ] and x λ 2 2 A 2 = [ 0 , 2 ]

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = max { | x λ 32 | , | y μ 16 | } max { | x λ 32 | , y μ 16 } max { x λ 16 , y μ 16 } 1 16 max { | x λ | , | y μ | } = 1 16 ρ ˜ ( x λ , y μ ) .

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) 1 16 { ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) } .

Hence the condition of corollary (2.19) (theorem 2.18) holds and ( f , φ ) has a soft fixed point in A 1 A 2 . Here x λ = 0 is a soft fixed point of ( f , φ ) .

In the above theorem 2.18, if we take A i = ( X , ρ ˜ , E ) , for all 0 i m then we deduce the following theorem.

Theorem 2.21: Let ( X , ρ ˜ , E ) be a complete soft dislocated metric space and ( f , φ ) : ( X , ρ ˜ , E ) ( X , ρ ˜ , E ) be a sub ψ-admissible mapping such that

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ψ ( x λ ) ρ ˜ ( y μ , ( f , φ ) x λ ) + ψ ( ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) + ψ ( ( f , φ ) 2 x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) + ψ ( ( f , φ ) 3 x λ ) ρ ˜ ( x λ , ( f , φ ) x λ ) + ψ ( ( f , φ ) 4 x λ ) ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ψ ( ( f , φ ) 5 x λ ) ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) ϕ { ρ ˜ ( x λ , ( f , φ ) x λ ) , ρ ˜ ( y μ , ( f , φ ) y μ ) , 1 2 [ ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) x λ ) ] } ;

for any x λ , y μ ( X , ρ ˜ , E ) , where ϕ , ψ Ψ . Then ( f , φ ) has unique soft fixed point in ( X , ρ ˜ , E ) .

Corollary 2.22: Let ( X , ρ ˜ , E ) be a complete soft dislocated metric space and ( f , φ ) : ( X , ρ ˜ , E ) ( X , ρ ˜ , E ) be a sub ψ-admissible mapping such that

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) β { ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) } ;

for any x λ , y μ ( X , ρ ˜ , E ) , where β [ 0 , 1 8 ) . Then ( f , φ ) has unique soft

fixed point in ( X , ρ ˜ , E ) .

Example 2.23: Let ( X , ρ ˜ , E ) = R with soft dislocated metric space ρ ˜ ( x λ , y μ ) = max { x λ , y μ } , for any x λ , y μ ( X , ρ ˜ , E ) . Let ( f , φ ) : ( X , ρ ˜ , E ) ( X , ρ ˜ , E ) be defined by

( f , φ ) x λ = { ( x λ ) 2 + x λ 18 ,if0 x λ 1 , x λ 12 , if x λ 1 ,

Proof: To show the existence and uniqueness soft point of ( f , φ ) , we investigate the following cases

Let 0 x λ , y μ < 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = max { x λ 2 + x λ 18 , y μ 2 + y μ 18 } 1 10 max { x λ , y μ } = 1 10 ρ ˜ ( x λ , y μ ) .

Let x λ , y μ 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = 1 12 max { x λ , y μ } 1 10 max { x λ , y μ } = 1 10 ρ ˜ ( x λ , y μ )

Let 0 x λ < 1 and y μ 1 then

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) = max { x λ 2 + x λ 18 , y μ 12 } 1 12 max { x λ , y μ } = 1 10 ρ ˜ ( x λ , y μ ) .

Hence

ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) 1 10 { ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) } .

Hence all the condition of corollary (2.22) (theorem 2.21) are satisfied. Thus ( f , φ ) has a unique soft fixed point in ( X , d , E ) indeed 0 is the unique soft fixed point of ( f , φ ) .

Corollary 2.24: Let ( X , d , E ) be a complete soft dislocated metric space, m N , let A 1 , A 2 , , A m be non empty ρ ˜ -closed subsets of ( X , ρ ˜ , E ) and let Y = i = 1 i = m A i . Suppose that ( f , φ ) : Y Y is an operator such that

i) Y = i = 1 i = m A i is cyclic representation of ( X , ρ ˜ , E ) with respect to ( f , φ ) ,

ii) 0 ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ρ ( t ) d t β 0 m ρ ( t ) d t .

where

m = ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ )

for any x λ A i , y μ A i + 1 , i = 1 , 2 , , m . Where A m + 1 = A 1 and ρ : [ 0 , ) [ 0 , ) is

Lebsegue-integrable mapping satisfying 0 ϵ ρ ( t ) d t , for ϵ > 0 and the constant β [ 0 , 1 8 ) . Then ( f , φ ) has unique soft fixed point z ω i = 1 i = m A i .

In corollary 2.24, if we take A i = ( X , ρ ˜ , E ) , for i = 1 , 2 , , m . We obtain the following result.

Theorem 2.25: Let ( X , ρ ˜ , E ) be a complete soft dislocated metric space and ( f , φ ) : ( X , ρ ˜ , E ) ( X , ρ ˜ , E ) be a mapping such that for any x λ , y μ ( X , ρ ˜ , E ) then

0 ρ ˜ ( ( f , φ ) x λ , ( f , φ ) y μ ) ρ ( t ) d t β 0 m ρ ( t ) d t .

where

m = ρ ˜ ( y μ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) y μ ) + ρ ˜ ( y μ , ( f , φ ) y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( y μ , ( f , φ ) y μ ) 1 + ρ ˜ ( x λ , y μ ) + ρ ˜ ( x λ , ( f , φ ) x λ ) ρ ˜ ( x λ , ( f , φ ) y μ ) 1 + ρ ˜ ( y μ , ( f , φ ) x λ ) ,

Let ρ : [ 0 , ) [ 0 , ) is Lebsegue-integrable mapping satisfying 0 ϵ ρ ( t ) d t for ϵ > 0 and the constant β [ 0 , 1 8 ) . Then ( f , φ ) has unique soft fixed point.

3. Conclusion

In this paper, the investigations concerning the existence and uniqueness of soft fixed point of a cyclic mapping in soft dislocated metric space are established. Examples are given in the support of established results. These results can be extended to any directions, and can also be extended to fixed point theory of non-expansive multivalued mappings. These proved results lead to different directions and aspect of soft metric fixed point theory.

Acknowledgements

The authors would like their sincere thanks to the editor and the anonymous referees for their valuable comments and useful suggestions in improving the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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