Dislocated Soft Metric Space with Soft Fixed Point Theorems

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={u1,u2,u3,u4} , i.e. universe of four workers. Let a set of decision parameters be D={s1,s2,s3,s4,s5} . Now si(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(s1)={u1,u3} , ˜C(s2)={u1,u3,u4} , ˜C(s3)=U , ˜C(s4)={u2,u4} and ˜C(s5)=empty . By consisting of the following collection of approximations, the soft set (C,D) can be viewed.

(˜C,˜D)={(highworkingspeed,{u1,u3}),(slowworkingspeed,{u1,u3,u4}),(averageworkingspeed,U),(workdelayworkingspeed,{u2,u4}),(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=AB and εC , 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=AB and for all k in C,

I(k)={Y(k),ifkisanelementofAB,Z(k),ifkisanelementofBA,Y(k)Z(k),ifkisanelementofAB.

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 kA , 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) , xE .

Definition 1.7: The complement of soft set (Y,A) is denoted by (Y,A)c and is defined as (Y,A)c=(Yc,A) , where Yc:AP(X) is a mapping given by Yc(β)=XY(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:EB(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 sE ;

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

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

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

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

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

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

SM1. for all ˜xs1,˜ys2˜X, ˜ρ(˜xs1,˜ys2)˜ˉ0;

SM2. ˜ρ(˜xs1,˜ys2)=ˉ0 , if and only if ˜xs1=˜ys2;

SM3. for all ˜xs1,˜ys2˜X, ˜ρ(˜xs1,˜ys2)=˜ρ(˜ys2,˜xs1);

SM4. for all ˜xs1,˜ys2,˜zs3˜X, ˜ρ(˜xs1,˜zs3)˜ρ(˜xs1,˜ys2)+˜ρ(˜ys2,˜zs3).

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 ˜xs and radius ˜α is given by

B(˜xs,˜α)={˜ys'˜X:˜ρ(˜xs,˜ys)˜α}SP(˜X),

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

B(˜xs,˜α)={˜xs˜X:˜ρ(˜xs,˜ys)˜α}SP(˜X).

Definition 1.14: A sequence {˜xnλn} 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 ˜yμ˜˜X such that ˜ρ(˜xnλn,˜yμ)˜0 as n, that is for every ˜ε>ˉ0, there is a natural number N=N(˜ε) such that ˉ0˜ρ(˜xnλn,˜yμ)˜<˜ε , whenever n>N.

Definition 1.15: Let (˜X,˜ρ,E) be a soft metric space, then the sequence {˜xnλn} of soft points in (˜X,˜ρ,E) is said to be a Cauchy sequence in ˜X, if corresponding to every ˜ε˜ˉ0, there exist mN such that ˜ρ(˜xiλi,˜yjλj)˜˜ε,i,jm, i.e. ˜ρ(˜xiλi,˜yjλj)ˉ0 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 αR,ˉ0α<ˉ1 such that for every point ˜xλ,˜yμSP(X), we have

˜ρ((f,φ)(˜xλ),(f,φ)(˜yμ))α˜ρ(˜xλ,˜yμ) .

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

(d1) ˜ρ(˜xs1,˜ys2)=ˉ0 then ˜xs1=˜ys2,

(d2) ˜ρ(˜xs1,˜ys2)=˜ρ(˜ys2,˜xs1) , for all ˜xs1,˜ys2˜X,

(d3) ˜ρ(˜xs1,˜zs3)˜ρ(˜xs1,˜ys2)+˜ρ(˜ys2,˜zs3) , for all ˜xs1,˜ys2,˜zs3˜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):ABAB is cyclic and satisfy the following:

(c) There exist a constant k(0,1) such that ˜ρ((f,φ)xλ,(f,φ)yμ)k˜ρ(xλ,yμ) for all xλA , yμB.

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

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 Bρ(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 {xnλn} be a sequence in V. If xnλnxλ as n, then xλV.

Proof: Let xλV, by definition 2.2, X\V is a ˜ρ -open set then there exist r > 0 such that B˜ρ(xλ,r)X\V. On the other hand, we have limn|˜ρ(xnλn,xλ)˜ρ(xλ,xλ)|=0 , Since xnλnxλ as n. So there exist n0N such that

|˜ρ(xnλn,xλ)˜ρ(xλ,xλ)|<r,

for all n>n0 . Hence we conclude that {xnλn}Bρ(xλ,r)X\V, for all n>n0 . This is contradiction since {xnλn}V, for all n in N.

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

limn˜ρ(xnλn,yμ)=˜ρ(xλ,yμ) , for all yμ(˜X,˜ρ,E) .

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

A. If ˜ρ(xnλn,xnλn)=0, then ˜ρ(xλxλ)=˜ρ(yμ,yμ)=0;

B. If {xnλn} be a sequence such that limn˜ρ(xnλn,xn+1λn+1)=0, then we have limn˜ρ(xnλn,xnλn)=limn˜ρ(xn+1λn+1,xn+1λn+1)=0;

C. If xλyμ, then ˜ρ(xλ,yμ)>0;

˜ρ(xλxλ)=2nni=1˜ρ(xλ,xiλi), holds for all xλ,xiλi˜X, where 1in .

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

Ψ={ψ:[0,)[0,)suchthatψisnondecreasingandcontinuous} and Φ={ϕ:[0,)[0,)suchthatϕislowersemicontinuous} .

Definition 2.6: Let (˜X,˜ρ,E) be a soft dislocated metric space, mN , let A1,A2,,Am be ˜ρ -closed non empty subsets of ˜X and let ˜Y=i=mi=1Ai . We say that (f,φ) is a cyclic generalized ϕψ contractive mapping if

i. ˜Y=i=mi=1Ai 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)xλ,(f,φ)yμ))ψ(M˜ρ(xλ,yμ))ϕ(M˜ρ(xλ,yμ)), (1)

for any xλAi,yμAi+1, i=1,2,3,,m , where Ai+1=A1,ϕΦ,ψΨ and

M˜ρ(xλ,yμ)=max{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)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 Fix((f,φ)) i.e. Fix((f,φ))={xλ˜X:xλ=(f,φ)xλ} .

Theorem 2.7: Let (˜X,˜ρ,E) be a complete dislocated soft metric space, mN , let A1,A2,,Am be non-empty ˜ρ -closed subsets of (˜X,˜ρ,E) and

let ˜Y=i=mi=1Ai . Suppose that (f,φ):˜Y˜Y is a cyclic generalized ϕψ con-

tractive mapping. Then (f,φ) has fixed point in ni=1Ai . Moreover if ˜ρ(xλ,yμ)˜ρ(xλ,xλ) for all x,yFix(f,φ) , then (f,φ) has a unique fixed point in ni=1Ai.

Proof: Let x0λ0 be an arbitrary point of ˜Y, so there exists some i0 such

that x0λ0Ai0. We know that (f,φ)(Ai0)Ai0+1, we conclude that (f,φ)(x0λ0)Ai0+1. Thus there exist x1λ1 in Ai0+1 such that (f,φ)x0λ0=x1λ1. Recursive (f,φ)xnλn=xn+1λn+1, where xnλnAin. Hence for n0 there exist in{1,2,3,,m} such that xnλnAin . In case xn0λn0=xn0+1λn0+1 for some n0=0,1,2, , then it is clear that xn0λn0 is a soft fixed point of (f,φ) . Now assume that xnλnxn+1λn+1 for all n. Hence by lemma 2.5(c) we have ˜ρ(xn1λn1,xnλn)>0 for all n. We shall show that the sequence {dn} is non-in- creasing, where dn=˜ρ(xnλn,xn+1λn+1) . Assume that there exists some n0N such that

˜ρ(xn01λn01,xn0n0)˜ρ(xn0λn0,xn0+1n0+1).

Hence we get

ψ(˜ρ(xn01n01,xn0n0))ψ(˜ρ(xn0n0,xn0+1n0+1)). (2)

Set ˜x1λ1=(f,φ)(˜x0λ)=(f(˜x0λ))φ(λ),˜x2λ2=(f,φ)(˜x1λ1)=(f2(˜x0λ))φ2(λ),,

˜xn+1λn+1=(f,φ)(˜xnλn)=(fn+1(˜x0λ))φn+1(λ),.

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

(3)

On the other hand, from lemma 2.5(D) we have ˜ρ(xnλn,xnλn)˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1) and ˜ρ(xnλn,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn1λn1,xnλn)6˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)3.

That is max{˜ρ(xn1λn1,xnλn),˜ρ(xnλn,xn+1λn+1),˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)3}max{˜ρ(xn1λn1,xnλn),˜ρ(xnλn,xn+1λn+1)}.

Therefore from (3) we get

ψ(˜ρ(xnλn,xn+1λn+1))ψ(max{˜ρ(xn1λn1,xnλn),˜ρ(xnλn,xn+1λn+1)})ϕ(max{˜ρ(xn1λn1,xnλn),˜ρ(xnλn,xn+1λn+1)}).

Now, if max{˜ρ(xn1λn1,xnλn),˜ρ(xnλn,xn+1λn+1)}=˜ρ(xnλn,xn+1λn+1), then

˜ρ(xnλn,xn+1λn+1)α˜ρ(xnλn,xn+1λn+1)β˜ρ(xnλn,xn+1λn+1) .

This is contradiction. Hence we have

ψ˜ρ(xnλn,xn+1λn+1)ψ{˜ρ(xn1λn1,xnλn)}ϕ{˜ρ(xn1λn1,xnλn)}, (4)

for all nN . By taking xλ=xn01λn01 and yμ=xn0λn0 in (4) and keeping (2) in

mind, we deduce that

ψ˜ρ(xn01λn01,xn0λn0)ψ{˜ρ(xn01λn01,xn0λn0)}ϕ{˜ρ(xn01λn01,xn0λn0)} .

This is a contradiction. Hence we conclude that dn<dn1 i.e.

˜ρ(xnλn,xn+1λn+1)˜ρ(xn1λn1,xnλn) hold for all nN . Thus there exist r0 such that limndn=limn˜ρ(xnλn,xn+1λn+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)=limnsupψ(dn)limnsup[ψ(dn1)ϕ(dn1)]ψ(r)ϕ(r) .

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

limndnlimn(xnλn,xn+1λn+1)=0 . (5)

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

(k) For every ϵ>0, there exists nN such that if r,qn with

rq1(m), then ˜ρ(xrnλrn,xqnλqn)<ϵ .

Suppose, on the contrary that there exist ϵ>0 such that for any nN , we can find rn>qnn with rnqn1(m) satisfying

˜ρ(xqnλqn,xrnλrn)ϵ . (6)

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

rnqn1(m) and ˜ρ(xqnλqn,xrnλrn)ϵ. Therefore ˜ρ(xqnλqn,xrnmλrnm)ϵ . By using

triangular inequality, we obtain

ϵ˜ρ(xqnλqn,xrnλrn)˜ρ(xqnλqn,xqnmλqnm)+mi=1˜ρ(xrniλrni,xrni+1λrni+1)ϵ+mi=1˜ρ(xrniλrni,xrni+1λrni+1) .

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

limn˜ρ(xqnλqn,xrnλrn)=ϵ . (7)

Again by (d3) we derive that

ϵ˜ρ(xqnλqn,xrnλrn)˜ρ(xqnλqn,xqn+1λqn+1)+˜ρ(xqn+1λqn+1,xrn+1λrn+1)+˜ρ(xrn+1λrn+1,xrnλrn)˜ρ(xqnλqn,xqn+1λqn+1)+˜ρ(xqn+1λqn+1,xqnλqn)+˜ρ(xqnλqn,xrnλrn)+˜ρ(xrnλrn,xrn+1λrn+1)+˜ρ(xrn+1λrn+1,xrnλrn)2˜ρ(xqnλqn,xqn+1λqn+1)+˜ρ(xqnλqn,xrnλrn)+2˜ρ(xrn+1λrn+1,xrnλrn).

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

limn˜ρ(xqn+1λqn+1,xrn+1λrn+1)=ϵ . (8)

By ( d3 ) we have the following inequality

˜ρ(xqnλqn,xrn+1λrn+1)˜ρ(xqnλqn,xrnλrn)+˜ρ(xrnλrn,xrn+1λrn+1) (9)

and ˜ρ(xqnλqn,xrnλrn)˜ρ(xqnλqn,xrn+1λrn+1)+˜ρ(xrnλrn,xrn+1λrn+1) . (10)

Letting limn in (9) and (10), we have

limn˜ρ(xqnλqn,xrn+1λrn+1)=ϵ . (11)

Again by ( d3 ) we have

˜ρ(xrnλrn,xqn+1λqn+1)˜ρ(xrnλrn,xrn+1λrn+1)+˜ρ(xrn+1λrn+1,xqn+1λqn+1) (12)

and ˜ρ(xrn+1λrn+1,xqn+1λqn+1)˜ρ(xrn+1λrn+1,xrnλrn)+˜ρ(xrnλrn,xqn+1λqn+1) . (13)

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

limn˜ρ(xrnλrn,xqn+1λqn+1)=ϵ . (14)

Since xqnλqn and xrnλrn lie in different adjacently labeled sets and for certain 1im . By using (5), (7), (8), (11) and (14) together with the fact that (f,φ) is a generalized cyclic ϕψ 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 n0N such that if r,qn0 with rq1(m) ,

˜ρ(xrλr,xqλq)ϵ2 . (15)

Since limn˜d(xnλn,xn+1λn+1)=0 , we also find n1N such that

˜ρ(xnλn,xn+1λn+1)ϵ2m, (16)

for any nn1 . Suppose that r,smax{n0,n1} and s>r . There exist k{1,2,3,,m} such that srk(m). Therefore sr+ϕ1(m) , for ϕ=mk+1.

So we have for j{1,2,3,,m} and s+jr1(m),

˜ρ(xrλr,xsλs)˜ρ(xrλr,xs+jλs+j)+˜ρ(xs+jλs+j,xs+j1λs+j1)++˜ρ(xs+1λs+1,xsλs) .

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

˜ρ(xrλr,xsλs)ϵ2+j×ϵ2mϵ2+m×ϵ2m=ϵ .

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

{xnλn} is a Cauchy sequence. Since Y is ˜ρ -closed in (˜X,˜ρ,E) , then (˜Y,˜ρ,E) is also complete, there exists xλ˜Y=mi=1Ai such that limnxnλn=xλ in (˜Y,˜ρ,E) ; equivalently

˜ρ(xλ,xλ)=limn˜ρ(xλ,xnλn)=limm,n˜ρ(xnλn,xmλm)=0 . (17)

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

since limnxnλn=xλ and ˜Y=mi=1Ai is a cyclic representation of Y with respect to

(f,φ) . The sequence {xnλn} has infinite terms in each Ai, for i{1,2,3,,m} . Suppose that xλAi, (f,φ)xλAi+1 and we take a subsequence {xnkλnk} of {xnλn} with xnkλnkAi1 (the existence of this subsequence is guaranteed by above- mentioned comment). By using the contractive condition we can obtain

ψ(˜ρ((f,φ)xλ,(f,φ)xnkλnk))ψ(max{˜ρ(xλ,xnkλnk),˜ρ(xλ,(f,φ)xλ),˜ρ(xnkλnk,(f,φ)xnkλnk),˜ρ(xnkλnk,(f,φ)xλ)+˜ρ(xnkλnk,(f,φ)xnkλnk)+˜ρ(xλ,(f,φ)xλ)6})ϕ(max{˜ρ(xλ,xnkλnk),˜ρ(xλ,(f,φ)xλ),˜ρ(xnkλnk,(f,φ)xnkλnk),˜ρ(xnkλnk,(f,φ)xλ)+˜ρ(xnkλnk,(f,φ)xnkλnk)+˜ρ(xλ,(f,φ)xλ)6}).

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

ψ(˜ρ(xλ,(f,φ)xλ))ψ(˜ρ(xλ,(f,φ)xλ))ϕ(˜ρ(xλ,(f,φ)xλ)) .

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

xλ,yμmi=1Ai . Suppose that xλyμ and for all pλ,qθFix((f,φ)),

˜ρ(pλ,qθ)˜ρ(pλ,pλ) . Using the contractive condition, we obtain

ψ(˜ρ((f,φ)xλ,(f,φ)yμ))ψ(max{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6})ϕ(max{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6})

Then we have ψ(˜ρ(xλ,yμ))ψ(˜ρ(xλ,yμ))ϕ(˜ρ(xλ,yμ)) .

This is a contradiction. Thus we derive that ˜ρ(yμ,zw)=0yμ=zw. Hence proved.

In the theorem 2.7, if we take (˜X,˜ρ,E)=Ai , for all 0im, 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 φΦ , ψΨ such that

ψ(˜ρ((f,φ)xλ,(f,φ)yμ))ψ(M˜ρ(xλ,yμ))ϕ(M˜ρ(xλ,yμ)) , for all xλ,yμ(˜X,˜ρ,E) , where

M˜ρ(xλ,yμ)=max{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6}.

Then (f,φ) has a soft fixed point. Moreover if ˜ρ(xλ,yμ)˜ρ(xλ,xλ) for all xλ,yμFix((f,φ)), then (f,φ) has a unique soft fixed point.

If in theorem (2.7) we take ψ(t)=t and ϕ(t)=(1r)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, mN , let A1,A2,,Am be non empty ˜ρ -closed subsets of (˜X,˜ρ,E) and let Y=i=mi=1Ai . Suppose that (f,φ):YY is an operator such that

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

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

˜ρ((f,φ)xλ,(f,φ)yμ)=rmax{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6},

for any xAi and yAi+1 , i=1,2,3,,m . Where Am+1=A1 , then (f,φ) has a soft fixed point zwmi=1Ai . Moreover if ˜ρ(xλ,yμ)˜ρ(xλ,xλ) for all xλFix(f,φ), then (f,φ) has a unique soft fixed point.

Example 2.10: Let X = R with soft dislocated metric

˜ρ(xλ,yμ)=max{|xλ2|,|yμ2|} , for all xλ,yμˉX . Suppose A1=[2,0] , A2=[0,2] & Y=i=2i=1Ai . Define (f,φ):YY by

(f,φ)={(xλ)28,ifxλ[2,0]xλ5,ifxλ[0,2]

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

Let xλA1=[2,0] and yμA2=[0,2] then

˜ρ((f,φ)xλ,(f,φ)yμ)=˜ρ(x2λ8,yμ5)=max{|x2λ/82|,|yμ/52|}=max{|x2λ16|,|yμ10|}max{xλ4,yμ4}12max{xλ2,yμ2}12˜ρ(xλ,yμ).

and so

˜ρ((f,φ)xλ,(f,φ)yμ)=rmax{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6}.

Hence the condition of corollary (2.9) (theorem 2.7) holds and (f,φ) has a fixed point in A1A2 . Here xλ=0 is a fixed point of (f,φ) .

Example 2.11: Let X = R with soft dislocated metric

˜ρ(xλ,yμ)=max{|xλ|,|yμ|} , for all xλ,yμX . Suppose A1=[1,0] and A2=[0,1] & Y=i=2i=1Ai we define (f,φ):YY by

(f,φ)={(xλ)22,ifxλ[1,0],xλ4,ifxλ[0,1].

It is clear that i=2i=1Ai is a cyclic representation of Y with respect to (f,φ) . Let xλA1=[1,0] and yμA2=[0,1] , then

˜ρ((f,φ)xλ,(f,φ)yμ)=˜ρ(x2λ2,yμ2)=max{|x2λ2|,|yμ4|}max{xλ2,yμ2}12max{xλ,yμ}12max{|xλ|,|yμ|}12˜ρ(xλ,yμ),

˜ρ((f,φ)xλ,(f,φ)yμ)=rmax{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6}.

Hence the condition of corollary (2.9) (theorem 2.7) holds and (f,φ) has a soft fixed point in A1A2 . Here xλ=0 is a soft fixed point of (f,φ) .

In the above corollary we take Ai=(˜X,˜ρ,E) for all 0im , 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)xλ,(f,φ)yμ)rmax{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6},

holds for all xλ,yμ(X,˜ρ,E) . Then (f,φ) has a soft fixed point. More- over if ˜ρ(xλ,yμ)˜ρ(xλ,xλ) for all xλ,yμFix((f,φ)) , then (f,φ) has a unique soft fixed point.

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

(f,φ)xλ={xλ6,if0xλ<12x2λ5,if12xλ1xλ7,ifx>1.

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

Let 0xλ,yμ<12 then

˜ρ((f,φ)xλ,(f,φ)yμ)=16max{xλ,yμ}12max{xλ,yμ}=12˜ρ(xλ,yμ) .

Let 12xλ,yμ1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=15max{x2λ,y2μ}15max{xλ,yμ}12max{xλ,yμ}=12˜ρ(xλ,yμ).

Let xλ,yμ>1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=17max{xλ,yμ}12max{xλ,yμ}=12˜ρ(xλ,yμ) .

Let 0xλ<12 and 12yμ1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=max{xλ6,y2μ5}12max{xλ,yμ}=12˜ρ(xλ,yμ) .

Let 0xλ<12 and yμ>1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=max{xλ6,yμ7}12max{xλ,yμ}=12˜ρ(xλ,yμ) .

Let 12xλ1 and 0yμ12 then

˜ρ((f,φ)xλ,(f,φ)yμ)=max{xλ5,yμ6}12max{xλ,yμ}=12˜ρ(xλ,yμ) .

and so

˜ρ((f,φ)xλ,(f,φ)yμ)rmax{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)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, mN , let A1,A2,,Am be non empty ˜ρ -closed subsets of X and Y=i=mi=1Ai . Suppose that (f,φ):YY is an operator such that

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

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

˜ρ((f,φ)xλ,(f,φ)yμ)0ρ(t)dtrm0ρ(t)dt,

where m=max{˜ρ(xλ,yμ),˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),˜ρ(yμ,(f,φ)xλ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)6}, for any xAi

yAi+1 , i=1,2,3,,m , where Am+1=A1 and ρ:[0,)[0,) be Lebesgue-

integrable mapping satisfying ϵ0ρ(t)dt>0, for ϵ>0 . Then (f,φ) has a fixed point zωmi=1Ai . Moreover if ˜ρ(xλ,yμ)˜ρ(xλ,xλ) for all

xλ,yμFix((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 γψ sub admissible soft mapping if ψ(xλ)γψ((f,φ)xλ)γ, for xλ(X,˜ρ,E) .

Example 2.16: Let (f,φ):[π,π][π,π] and ψ:[π,π]R+ be defined by (f,φ)xλ=π4(tanxλ) and ψ(xλ)=|xλ14π|+12 then (f,φ) is a γψ sub admissible mapping, where γ=12 , indeed if ψ(xλ)=|xλ14π|+1212 then xλ=14π .

Hence (f,φ)(xλ)=14π and ψ((f,φ)xλ)=12 .

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, mN , let

A1,A2,,Am be ˜ρ -closed non empty subsets of (X,˜ρ,E) and let Y=i=mi=1Ai . Assume that (f,φ):YY is g-ψ-sub admissible mapping, where γ=18. Then (f,φ) is called ψ-cyclic generalized weakly C-contraction if

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

˜ρ((f,φ)xλ,(f,φ)yμ)ψ(xλ)˜ρ(yμ,(f,φ)xλ)+ψ((f,φ)xλ)˜ρ(xλ,(f,φ)yμ)+ψ((f,φ)2xλ)˜ρ(yμ,(f,φ)yμ)+ψ((f,φ)3xλ)˜ρ(xλ,(f,φ)xλ)+ψ((f,φ)4xλ)˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+ψ((f,φ)5xλ)˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)ϕ{˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),12[˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)xλ)]} (18)

for any xAi , yAi+1 , i=1,2,3,,m and Am+1=A1 and ϕΛ .

Theorem 1.18: Let (X,˜ρ,E) be a complete soft dislocated metric space, mN , let A1,A2,,Am , be ˜ρ -closed non-empty subsets of (X,˜ρ,E) and let

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

contraction. If there exists x0λ0Y such that ψ(x0λ)18, then (f,φ) has a soft fixed point zωni=1Ai. Moreover if ψ(z0w0)18, then z0w0 is unique.

Proof: Let x0λ0Y be such that ψ(x0λ0)18 . Since (f,φ) is a sub ψ-ad- missible mapping with respect to 18 , then ψ((f,φ)x0λ0)18. ψ((f,φ)nx0λ0)18 for all nN0 . Also, there exist some i0 such that x0λ0Ai0. Now (f,φ)(Ai0)Ai0+1 implies that (f,φ)(x0λ0)Ai0+1. Thus there exist x1λ1 in Ai0+1 such that (f,φ)x0λ0=x1λ1 . Similarly (f,φ)xnλn=xn+1λn+1 , where xnλnAin. Hence for n0 there exist in{1,2,3,,m} such that xnλnAin and xn+1λn+1Ain+1 . In case xn0λn0=xn0+1λn0+1, for some n0=0,1,2, , then it is clear that xn0λn0 is a fixed point of (f,φ) . Now assume that xnλnxn+1λn+1 , for all n.

Since (f,φ):YY is a cyclic generalized weak C-contraction, we have for all nN we have

And so

˜ρ(xnλn,xn+1λn+1)18{˜ρ(xnλn,xnλn)+˜ρ(xn1λn1,xn+1λn+1)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn1λn1,xnλn)˜ρ(xn1λn1,xn+1λn+1)1+˜ρ(xnλn,xnλn)}. (19)

On the other hand from (d3) we have

˜ρ(xn1λn1,xn+1λn+1)˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1) .

And by lemma (2.5D) we have

˜ρ(xnλn,xnλn)˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1) .

Also

˜ρ(xn1λn1,xnλn)˜ρ(xn1λn1,xn+1λn+1)1+˜ρ(xnλn,xnλn)˜ρ(xn1λn1,xnλn)˜ρ(xn1λn1,xn+1λn+1)˜ρ(xnλn,xnλn)˜ρ(xn1λn1,xnλn)[˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)]˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)˜ρ(xn1λn1,xn+1λn+1)

From (19) we have

˜ρ((f,φ)xn1λn1,(f,φ)xnλn)18{˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn1λn1,xnλn)}18{4˜ρ(xn1λn1,xnλn)+4˜ρ(xnλn,xn+1λn+1)}12{˜ρ(xn1λn1,xnλn)+˜ρ(xnλn,xn+1λn+1)}.

Therefore we have

˜ρ(xnλn,xn+1λn+1)˜ρ(xn1λn1,xnλn) . (20)

For any n1 , set tn=˜ρ(xnλn,xn1λn1) . On the occasion of the facts above {tn}

is a non-increasing sequence of non negative real numbers. Consequently, there exist L0 such that

limn˜ρ(xnλn,xn1λn1)=L . (21)

We shall prove that L=0 . Since ˜ρ(xnλn,xnλn)=2ϕ˜ρ(xnλn,xn+1λn+1) then we get limn˜d(xnλn,xnλn)=2L . Similarly, limn˜ρ(xn1λn1,xn+1λn+1)=2L. Then

limn[˜ρ(xnλn,xnλn)+˜ρ(xn1λn1,xn+1λn+1)]4L .

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

L18[4L+{˜ρ(xnλn,xnλn)+˜ρ(xn1λn1,xn+1λn+1)}] ,

This implies 4Llimn{˜ρ(xnλn,xnλn)+˜ρ(xn1λn1,xn+1λn+1)} .

Hence limn{˜ρ(xnλn,xnλn)+˜ρ(xn1λn1,xn+1λn+1)}=4L.

Now from (18) we have

tn+1ψ(xn1λn1)˜ρ(xnλn,xnλn)+ψ(xnλn)˜ρ(xn1λn1,xn+1λn+1)+ψ(xn+1λn+1)tn+1+ψ(xn+2λn+2)tn+ψ(xn+3λn+3)tn+1+ψ(xn+4λn+4)tn1˜ρ(xn1λn1,xn+1λn+1)1+˜ρ(xnλn,xnλn)ϕ{tn,tn+1,12[˜ρ(xn1λn1,xn+1λn+1)+˜ρ(xnλn,xnλn)]}.

tn+118{˜ρ(xnλn,xnλn)+˜ρ(xn1λn1,xn+1λn+1)+tn+1+tn+tn+1+tn1˜ρ(xn1λn1,xn+1λn+1)1+˜ρ(xnλn,xnλn)}ϕ{tn,tn+1,12[˜ρ(xn1λn1,xn+1λn+1)+˜ρ(xnλn,xnλn)]}

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

So ϕ(L,L,2L)=0, since ϕ(xλ,yμ,zω)=0xλ=yμ=zω=0, we get L=0 .

Due to limn˜ρ(xnλn,xnλn)2L and limn˜ρ(xn1λn1,xn+1λn+1)2L we have

limn˜ρ(xnλn,xnλn)=limn˜ρ(xn1λn1,xn+1λn+1)n=limn˜ρ(xnλn,xn+1λn+1)=0 . (22)

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

(K) for every ϵ>0 , there exist nN such that if r,qn with

rq1(m), then ˜ρ(xrλr,xqλq)<ϵ.

Suppose to the contrary that there exist ϵ>0 , such that for any n in N, we can find rn>qnn with rnqn1(m) satisfying

˜ρ(xrnλrn,xqnλqn)ϵ. (23)

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

limn˜ρ(xqnλqn,xrnλrn)=ϵ,

limn˜ρ(xqn+1λqn+1,xrn+1λrn+1)=ϵ,

limn˜ρ(xqnλqn,xrn+1λrn+1)=ϵ and limn˜ρ(xrnλrn,xqn+1λqn+1)=ϵ. (24)

Since xqnλqn & xrnλrn lie in different adjacently labeled sets Ai and Ai+1, for a certain 1im . 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

ϵ18[ϵ+ϵ+0+0+0+0]ϕ[0,0,ϵ]14ϵ .

This is a contradiction. Hence condition (k) holds. We are ready to show that the sequence {xnλn} is a Cauchy. Fix ϵ>0. By the claim, we find n0N such that if r,qn0 with rq1(m) then

˜ρ(xrnλrn,xqnλqn)ϵ4ϵ2 . (25)

Since limn˜ρ(xnλn,xn+1λn+1)=0 , we find n1N such that

˜ρ(xnλn,xn+1λn+1)ϵ2m, (26)

for any nn1. Suppose that r,smax{n0,n1} with s>r . Then there exist k{1,2,,m} such that srk(m) . Therefore, sr+ϕ1(m) , for ϕ=mk+1 .

So we have, for j{1,2,3,,m} , s+jr1(m),

˜ρ(xrλr,xsλs)˜ρ(xrλr,xs+jλs+j)+˜ρ(xrλr,xs+jλs+j)+˜ρ(xs+jλs+j,xs+j1λs+j1)++˜ρ(xs+1λs+1,xsλs) .

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

˜ρ(xrλr,xsλs)ϵ2+j×ϵ2mϵ2+m×ϵ2m=ϵ .

This proves that {xnλn} is a ˜ρ -Cauchy sequence.

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

exists zωY=mi=1Ai such that limnxnλn=zω in (Y,˜ρ,E) , equivalently

˜ρ(zω,zω)=limn˜ρ(zω,xnλn)=limm,n˜ρ(xnλn,xmλm)=0 . (27)

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

since limnxnλn=zw and as Y=mi=1Ai is cyclic representation of Y with respect to (f,φ) , the sequence {xnλn} has infinite terms in each Ai for

i{1,2,3,,m} . Suppose that xλAi, (f,φ)xλAi+1 and we take a subsequence {xnkλnk} of {xnλn} with xnkλnkAi1 (the existence of this subsequence is

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

˜ρ(xnk+1λnk+1,(f,φ)xλ)xnkλnk=˜ρ((f,φ)xnkλnk,(f,φ)xλ)ψ(xnkλnk)˜ρ(xλ,(f,φ)xnkλnk)+ψ((f,φ)xnkλnk)˜ρ(xnkλnk,(f,φ)xλ)+ψ((f,φ)2xnkλnk)˜ρ(xλ,(f,φ)xλ)+ψ((f,φ)3xnkλnk)˜ρ(xnkλnk,(f,φ)xnkλnk)+ψ((f,φ)4xnkλnk)˜ρ(xnkλnk,(f,φ)xnkλnk)˜ρ(xλ,(f,φ)xλ)1+˜ρ(xnkλnk,xλ)

+ψ((f,φ)5xnkλnk)˜ρ(xnkλnk,(f,φ)xnkλnk)˜ρ(xnkλnk,(f,φ)xλ)1+˜ρ(xλ,(f,φ)xnkλnk)ϕ{˜ρ(xnkλnk,(f,φ)xnkλnk),˜ρ(xλ,(f,φ)xλ),12[˜ρ(xnkλnk,(f,φ)xλ)+˜ρ(xλ,(f,φ)xnkλnk)]},18{˜ρ(xλ,xnk+1λnk+1)+˜ρ(xnkλnk,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xnkλnk,xnk+1λnk+1)+˜ρ(xnkλnk,xnk+1λnk+1)˜ρ(xλ,(f,φ)xλ)1+˜ρ(xnkλnk,xλ)+˜ρ(xnkλnk,xnk+1λnk+1)˜ρ(xnkλnk,(f,φ)xλ)1+˜ρ(xλ,xnk+1λnk+1)}ϕ{˜ρ(xnkλnk,xnk+1λnk+1),˜ρ(xλ,(f,φ)xλ),12[˜ρ(xnkλnk,(f,φ)xλ)+˜ρ(xλ,xnk+1λnk+1)]}.

Passing to the limit as limn and using xnkλnkxλ, lower semi-conti- nuity of ϕ , we have

˜ρ(xλ,(f,φ)xλ)18{0+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)+0+0+0}ϕ{0,˜ρ(xλ,(f,φ)xλ),12[˜ρ(xλ,(f,φ)xλ)]}14˜ρ(xλ,(f,φ)xλ).

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

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

(f,φ) and the fact that yμ,zω(X,˜ρ,E) are soft fixed points of (f,φ) implies that yμ,zωmi=1Ai . Also suppose that ψ(yμ)18 . By using contrac-

tive condition we derive that

˜ρ(yμ,zω)=˜ρ((f,φ)yμ,(f,φ)zω)ψ(yμ)˜ρ(zω,(f,φ)yμ)+ψ((f,φ)yμ)˜ρ(yμ,(f,φ)zω)+ψ((f,φ)2yμ)˜ρ(zω,(f,φ)zω)+ψ((f,φ)3yμ)˜ρ(yμ,(f,φ)yμ)+ψ((f,φ)4yμ)˜ρ(yμ,(f,φ)yμ)˜ρ(zw,(f,φ)zw)1+˜ρ(yμ,zw)+ψ((f,φ)5yμ)˜ρ(yμ,(f,φ)yμ)˜ρ(yμ,(f,φ)zw)1+˜ρ(zw,(f,φ)yμ)ϕ(˜ρ(yμ,(f,φ)yμ),˜ρ(zw,(f,φ)zw),12[˜ρ(yμ,(f,φ)zw)+˜ρ(zw,(f,φ)yμ)])

18{˜ρ(zω,yμ)+˜ρ(yμ,zω)+˜ρ(zω,zω)+˜ρ(yμ,yμ)+˜ρ(yμ,yμ)˜ρ(zw,zw)1+˜ρ(yμ,zw)+˜ρ(yμ,yμ)˜ρ(yμ,zw)1+˜ρ(zw,yμ)}ϕρ(˜ρ(yμ,yμ),˜ρ(zw,zw),12[˜ρ(yμ,zw)+˜ρ(zw,yμ)])18{2˜ρ(zω,yμ)}ϕ(0,˜ρ(zw,zw),12[˜ρ(yμ,zw)+˜ρ(zw,yμ)])14˜ρ(zω,yμ).

This gives us ˜ρ(zω,yμ)=0, that is zω=yμ . This finishes the proof.

Corollary 2.19: Let (X,˜ρ,E) be a complete soft dislocated metric space, mN , let A1,A2,,Am be non empty ˜ρ -closed subsets of (X,d,E) and

let Y=i=mi=1Ai . Suppose that (f,φ):YY is an operator such that

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

ii) there exist β[0,18) such that

˜ρ((f,φ)xλ,(f,φ)yμ)β{˜ρ(yμ,(f,φ)xλ)+˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)}, (28)

for any xλAi , yμAi+1 , i=1,2,3,,m , where Am+1=A1 . Then (f,φ)

has fixed point zwi=mi=1Ai .

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

˜ρ(xλ,yμ)=max{|xλ|,|yμ|} , for any xλ,yμ(X,˜ρ,E), suppose A1=[2,0]

and A2=[0,2] & Y=i=2i=1Ai, we define (f,φ):YY by

(f,φ)xλ={xλ32,ifxλ[2,0],xλ16,ifxλ[0,2],

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

Proof: Let x1λ1A1=[2,0] and x2λ2A2=[0,2]

˜ρ((f,φ)xλ,(f,φ)yμ)=max{|xλ32|,|yμ16|}max{|xλ32|,yμ16}max{xλ16,yμ16}116max{|xλ|,|yμ|}=116˜ρ(xλ,yμ).

˜ρ((f,φ)xλ,(f,φ)yμ)116{˜ρ(yμ,(f,φ)xλ)+˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)}.

Hence the condition of corollary (2.19) (theorem 2.18) holds and (f,φ) has a soft fixed point in A1A2 . Here xλ=0 is a soft fixed point of (f,φ) .

In the above theorem 2.18, if we take Ai=(X,˜ρ,E), for all 0im 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)xλ,(f,φ)yμ)ψ(xλ)˜ρ(yμ,(f,φ)xλ)+ψ((f,φ)xλ)˜ρ(xλ,(f,φ)yμ)+ψ((f,φ)2xλ)˜ρ(yμ,(f,φ)yμ)+ψ((f,φ)3xλ)˜ρ(xλ,(f,φ)xλ)+ψ((f,φ)4xλ)˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+ψ((f,φ)5xλ)˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)ϕ{˜ρ(xλ,(f,φ)xλ),˜ρ(yμ,(f,φ)yμ),12[˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)xλ)]};

for any xλ,yμ(X,˜ρ,E) , where ϕ , ψΨ . 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)xλ,(f,φ)yμ)β{˜ρ(yμ,(f,φ)xλ)+˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)};

for any xλ,yμ(X,˜ρ,E) , where β[0,18) . 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(xλ,yμ)=max{xλ,yμ} , for any xλ,yμ(X,˜ρ,E) . Let (f,φ):(X,˜ρ,E)(X,˜ρ,E) be defined by

(f,φ)xλ={(xλ)2+xλ18,if0xλ1,xλ12,ifxλ1,

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

Let 0xλ,yμ<1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=max{x2λ+xλ18,y2μ+yμ18}110max{xλ,yμ}=110˜ρ(xλ,yμ).

Let xλ,yμ1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=112max{xλ,yμ}110max{xλ,yμ}=110˜ρ(xλ,yμ)

Let 0xλ<1 and yμ1 then

˜ρ((f,φ)xλ,(f,φ)yμ)=max{x2λ+xλ18,yμ12}112max{xλ,yμ}=110˜ρ(xλ,yμ) .

Hence

˜ρ((f,φ)xλ,(f,φ)yμ)110{˜ρ(yμ,(f,φ)xλ)+˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)}.

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, mN , let A1,A2,,Am be non empty ˜ρ -closed subsets of (X,˜ρ,E) and let Y=i=mi=1Ai . Suppose that (f,φ):YY is an operator such that

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

ii) ˜ρ((f,φ)xλ,(f,φ)yμ)0ρ(t)dtβm0ρ(t)dt.

where

m=˜ρ(yμ,(f,φ)xλ)+˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ)

for any xλAi,yμAi+1 , i=1,2,,m . Where Am+1=A1 and ρ:[0,)[0,) is

Lebsegue-integrable mapping satisfying ϵ0ρ(t)dt , for ϵ>0 and the constant β[0,18). Then (f,φ) has unique soft fixed point zωi=mi=1Ai.

In corollary 2.24, if we take Ai=(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 xλ,yμ(X,˜ρ,E) then

˜ρ((f,φ)xλ,(f,φ)yμ)0ρ(t)dtβm0ρ(t)dt.

where

m=˜ρ(yμ,(f,φ)xλ)+˜ρ(xλ,(f,φ)yμ)+˜ρ(yμ,(f,φ)yμ)+˜ρ(xλ,(f,φ)xλ)+˜ρ(xλ,(f,φ)xλ)˜ρ(yμ,(f,φ)yμ)1+˜ρ(xλ,yμ)+˜ρ(xλ,(f,φ)xλ)˜ρ(xλ,(f,φ)yμ)1+˜ρ(yμ,(f,φ)xλ),

Let ρ:[0,)[0,) is Lebsegue-integrable mapping satisfying ϵ0ρ(t)dt for ϵ>0 and the constant β[0,18). 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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