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=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),ifkisanelementofA−B,Z(k),ifkisanelementofB−A,Y(k)∪Z(k),ifkisanelementofA∩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=(Yc,A) , where
Yc:A→P(X) is a mapping given by
Yc(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
˜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≠˜yj⇔x≠y or
i≠j .
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
m∈N such that
˜ρ(˜xiλi,˜yjλj)˜≤˜ε,∀i,j≥m, 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
α∈R,ˉ0≤α<ˉ1 such that for every point
˜xλ,˜yμ∈SP(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):A∪B→A∪B 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
xλ∈A ,
yμ∈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
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
xnλn→xλ as
n→∞, then
xλ∈V.
Proof: Let
xλ∉V, 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
xnλn→xλ as
n→∞. So there exist
n0∈N 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
xλ≠yμ, then
˜ρ(xλ,yμ)>0;
˜ρ(xλxλ)=2nn∑i=1˜ρ(xλ,xiλi), holds for all
xλ,xiλi∈˜X, where
1≤i≤n .
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,
m∈N , 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
ϕ−ψ 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
xλ∈Ai,yμ∈Ai+1,
i=1,2,3,⋯,m , where
Ai+1=A1,ϕ∈Φ,ψ∈Ψ 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,
m∈N , 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
ϕ−ψ con-
tractive mapping. Then
(f,φ) has fixed point in
∩ni=1Ai . Moreover if
˜ρ(xλ,yμ)≥˜ρ(xλ,xλ) for all
x,y∈Fix(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
x0λ0∈Ai0. 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
xnλn∈Ain. Hence for
n≥0 there exist
in∈{1,2,3,⋯,m} such that
xnλn∈Ain . 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
xnλn≠xn+1λn+1 for all n. Hence by lemma 2.5(c) we have
˜ρ(xn−1λn−1,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
n0∈N such that
˜ρ(xn0−1λn0−1,xn0n0)≤˜ρ(xn0λn0,xn0+1n0+1).
Hence we get
ψ(˜ρ(xn0−1n0−1,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(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1) and
˜ρ(xnλn,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn−1λn−1,xnλn)6≤˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)3.
That is
max{˜ρ(xn−1λn−1,xnλn),˜ρ(xnλn,xn+1λn+1),˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)3}≤max{˜ρ(xn−1λn−1,xnλn),˜ρ(xnλn,xn+1λn+1)}.
Therefore from (3) we get
ψ(˜ρ(xnλn,xn+1λn+1))≤ψ(max{˜ρ(xn−1λn−1,xnλn),˜ρ(xnλn,xn+1λn+1)}) −ϕ(max{˜ρ(xn−1λn−1,xnλn),˜ρ(xnλn,xn+1λn+1)}).
Now, if
max{˜ρ(xn−1λn−1,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(xn−1λn−1,xnλn)}−ϕ{˜ρ(xn−1λn−1,xnλn)}, (4)
for all
n∈N . By taking
xλ=xn0−1λn0−1 and
yμ=xn0λn0 in (4) and keeping (2) in
mind, we deduce that
ψ˜ρ(xn0−1λn0−1,xn0λn0)≤ψ{˜ρ(xn0−1λn0−1,xn0λn0)}−ϕ{˜ρ(xn0−1λn0−1,xn0λn0)} .
This is a contradiction. Hence we conclude that
dn<dn−1 i.e.
˜ρ(xnλn,xn+1λn+1)≤˜ρ(xn−1λn−1,xnλn) hold for all
n∈N . Thus there exist
r≥0 such that
limn→∞dn=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)=limn→∞supψ(dn)≤limn→∞sup[ψ(dn−1)−ϕ(dn−1)]≤ψ(r)−ϕ(r) .
This yields that
φ(r)≤0 . This is contradiction. Hence we obtain that
limn→∞dn≤limn→∞(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
n∈N such that if
r,q≥n with
r−q≡1(m), then
˜ρ(xrnλrn,xqnλqn)<ϵ .
Suppose, on the contrary that there exist
ϵ>0 such that for any
n∈N , we can find
rn>qn≥n with
rn−qn≡1(m) satisfying
˜ρ(xqnλqn,xrnλrn)≥ϵ . (6)
Now we consider
n>2m. Then, corresponding to
qn≥n , we can choose
rn in such a way that it is the smallest integer with
rn>qn satisfying
rn−qn≡1(m) and
˜ρ(xqnλqn,xrnλrn)≥ϵ. Therefore
˜ρ(xqnλqn,xrn−mλrn−m)≤ϵ . By using
triangular inequality, we obtain
ϵ≤˜ρ(xqnλqn,xrnλrn)≤˜ρ(xqnλqn,xqn−mλqn−m)+m∑i=1˜ρ(xrn−iλrn−i,xrn−i+1λrn−i+1)≤ϵ+m∑i=1˜ρ(xrn−iλrn−i,xrn−i+1λrn−i+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
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
n0∈N such that if
r,q≥n0 with
r−q≡1(m) ,
˜ρ(xrλr,xqλq)≤ϵ2 . (15)
Since
limn→∞˜d(xnλn,xn+1λn+1)=0 , we also find
n1∈N such that
˜ρ(xnλn,xn+1λn+1)≤ϵ2m, (16)
for any
n≥n1 . Suppose that
r,s≥max{n0,n1} 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(xrλr,xsλs)≤˜ρ(xrλr,xs+jλs+j)+˜ρ(xs+jλs+j,xs+j−1λs+j−1)+⋯+˜ρ(xs+1λs+1,xsλs) .
By (15) & (16) and from the last inequality, we get
˜ρ(xrλr,xsλs)≤ϵ2+j×ϵ2m≤ϵ2+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
limn→∞xnλn=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
limn→∞xnλn=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
xλ∈Ai,
(f,φ)xλ∈Ai+1 and we take a subsequence
{xnkλnk} of
{xnλn} with
xnkλnk∈Ai−1 (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
xnkλnk→xλ , 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,yμ∈∩mi=1Ai . Suppose that
xλ≠yμ and for all
pλ,qθ∈Fix((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)=0⇔yμ=zw. Hence proved.
In the theorem 2.7, if we take
(˜X,˜ρ,E)=Ai , 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)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,yμ∈Fix((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
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 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
x∈Ai and
y∈Ai+1 ,
i=1,2,3,⋯,m . Where
Am+1=A1 , then
(f,φ) has a soft fixed point
zw∈∩mi=1Ai . Moreover if
˜ρ(xλ,yμ)≥˜ρ(xλ,xλ) for all
xλ∈Fix(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,yμ∈ˉX . Suppose
A1=[−2,0] ,
A2=[0,2] &
Y=∪i=2i=1Ai . Define
(f,φ):Y→Y 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
xλ∈A1=[−2,0] and
yμ∈A2=[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
A1∩A2 . 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,yμ∈X . Suppose
A1=[−1,0] and
A2=[0,1] &
Y=∪i=2i=1Ai we define
(f,φ):Y→Y 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
xλ∈A1=[−1,0] and
yμ∈A2=[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
A1∩A2 . Here
xλ=0 is a soft fixed point of
(f,φ) .
In the above corollary we take
Ai=(˜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)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,yμ∈Fix((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, if0≤xλ<12x2λ5,if12≤xλ≤1xλ7,ifx>1.
Proof: To show that the existence and uniqueness of soft point of
(f,φ) , we need to consider the following cases
Let
0≤xλ,yμ<12 then
˜ρ((f,φ)xλ,(f,φ)yμ)=16max{xλ,yμ}≤12max{xλ,yμ}=12˜ρ(xλ,yμ) .
Let
12≤xλ,yμ≤1 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
0≤xλ<12 and
12≤yμ≤1 then
˜ρ((f,φ)xλ,(f,φ)yμ)=max{xλ6,y2μ5}≤12max{xλ,yμ}=12˜ρ(xλ,yμ) .
Let
0≤xλ<12 and
yμ>1 then
˜ρ((f,φ)xλ,(f,φ)yμ)=max{xλ6,yμ7}≤12max{xλ,yμ}=12˜ρ(xλ,yμ) .
Let
12≤xλ≤1 and
0≤yμ≤12 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,
m∈N , let
A1,A2,⋯,Am be non empty
˜ρ -closed subsets of X and
Y=∪i=mi=1Ai . Suppose that
(f,φ):Y→Y 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)dt≤r∫m0ρ(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
x∈Ai
y∈Ai+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
zω∈∩mi=1Ai . Moreover if
˜ρ(xλ,yμ)≥˜ρ(xλ,xλ) for all
xλ,yμ∈Fix((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(xλ)≤γ⇒ψ((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)=|xλ−14π|+12 then
(f,φ) is a
γ−ψ sub admissible mapping, where
γ=12 , indeed if
ψ(xλ)=|xλ−14π|+12≤12 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,
m∈N , let
A1,A2,⋯,Am be
˜ρ -closed non empty subsets of
(X,˜ρ,E) and let
Y=∪i=mi=1Ai . Assume that
(f,φ):Y→Y 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
x∈Ai ,
y∈Ai+1 ,
i=1,2,3,⋯,m and
Am+1=A1 and
ϕ∈Λ .
Theorem 1.18: Let
(X,˜ρ,E) be a complete soft dislocated metric space,
m∈N , let
A1,A2,⋯,Am , be
˜ρ -closed non-empty subsets of
(X,˜ρ,E) and let
Y=∪i=mi=1Ai . Assume that
(f,φ):Y→Y is a ψ-cyclic generalized weakly C-
contraction. If there exists
x0λ0∈Y such that
ψ(x0λ)≤18, then
(f,φ) has a soft fixed point
zω∈∩ni=1Ai. Moreover if
ψ(z0w0)≤18, then
z0w0 is unique.
Proof: Let
x0λ0∈Y 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
n∈N∪0 . Also, there exist some i0 such that
x0λ0∈Ai0. 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
xnλn∈Ain. Hence for
n≥0 there exist
in∈{1,2,3,⋯,m} such that
xnλn∈Ain and
xn+1λn+1∈Ain+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
xnλn≠xn+1λn+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(xnλn,xn+1λn+1)≤18{˜ρ(xnλn,xnλn)+˜ρ(xn−1λn−1,xn+1λn+1)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn−1λn−1,xnλn) +˜ρ(xnλn,xn+1λn+1)+˜ρ(xn−1λn−1,xnλn)˜ρ(xn−1λn−1,xn+1λn+1)1+˜ρ(xnλn,xnλn)}. (19)
On the other hand from (d3) we have
˜ρ(xn−1λn−1,xn+1λn+1)≤˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1) .
And by lemma (2.5D) we have
˜ρ(xnλn,xnλn)≤˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1) .
Also
˜ρ(xn−1λn−1,xnλn)˜ρ(xn−1λn−1,xn+1λn+1)1+˜ρ(xnλn,xnλn)≤˜ρ(xn−1λn−1,xnλn)˜ρ(xn−1λn−1,xn+1λn+1)˜ρ(xnλn,xnλn)≤˜ρ(xn−1λn−1,xnλn)[˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)]˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)≤˜ρ(xn−1λn−1,xn+1λn+1)
From (19) we have
˜ρ((f,φ)xn−1λn−1,(f,φ)xnλn)≤18{˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1) +˜ρ(xnλn,xn+1λn+1)+˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)+˜ρ(xn−1λn−1,xnλn)}≤18{4˜ρ(xn−1λn−1,xnλn)+4˜ρ(xnλn,xn+1λn+1)}≤12{˜ρ(xn−1λn−1,xnλn)+˜ρ(xnλn,xn+1λn+1)}.
Therefore we have
˜ρ(xnλn,xn+1λn+1)≤˜ρ(xn−1λn−1,xnλn) . (20)
For any
n≥1 , set
tn=˜ρ(xnλn,xn−1λn−1) . On the occasion of the facts above
{tn}
is a non-increasing sequence of non negative real numbers. Consequently, there exist
L≥0 such that
limn→∞˜ρ(xnλn,xn−1λn−1)=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(xn−1λn−1,xn+1λn+1)=2L. Then
limn→∞[˜ρ(xnλn,xnλn)+˜ρ(xn−1λn−1,xn+1λn+1)]≤4L .
On the other hand, by taking limit as
n→∞ in (19), we have,
L≤18[4L+{˜ρ(xnλn,xnλn)+˜ρ(xn−1λn−1,xn+1λn+1)}] ,
This implies
4L≤limn→∞{˜ρ(xnλn,xnλn)+˜ρ(xn−1λn−1,xn+1λn+1)} .
Hence
limn→∞{˜ρ(xnλn,xnλn)+˜ρ(xn−1λn−1,xn+1λn+1)}=4L.
Now from (18) we have
tn+1≤ψ(xn−1λn−1)˜ρ(xnλn,xnλn)+ψ(xnλn)˜ρ(xn−1λn−1,xn+1λn+1)+ψ(xn+1λn+1)tn+1 +ψ(xn+2λn+2)tn+ψ(xn+3λn+3)tn+1+ψ(xn+4λn+4)tn−1˜ρ(xn−1λn−1,xn+1λn+1)1+˜ρ(xnλn,xnλn) −ϕ{tn,tn+1,12[˜ρ(xn−1λn−1,xn+1λn+1)+˜ρ(xnλn,xnλn)]}.
tn+1≤18{˜ρ(xnλn,xnλn)+˜ρ(xn−1λn−1,xn+1λn+1)+tn+1+tn+tn+1+tn−1˜ρ(xn−1λn−1,xn+1λn+1)1+˜ρ(xnλn,xnλn)} −ϕ{tn,tn+1,12[˜ρ(xn−1λn−1,xn+1λn+1)+˜ρ(xnλn,xnλn)]}
By taking limit as
n→∞ in the above inequality, we deduce that,
L≤L−ϕ(L,L,2L) .
So
ϕ(L,L,2L)=0, since
ϕ(xλ,yμ,zω)=0⇔xλ=yμ=zω=0, we get
L=0 .
Due to
limn→∞˜ρ(xnλn,xnλn)≤2L and
limn→∞˜ρ(xn−1λn−1,xn+1λn+1)≤2L we have
limn→∞˜ρ(xnλn,xnλn)=limn→∞˜ρ(xn−1λn−1,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
n∈N such that if
r,q≥n with
r−q≡1(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>qn≥n with
rn−qn≡1(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
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≤18[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
n0∈N such that if
r,q≥n0 with
r−q≡1(m) then
˜ρ(xrnλrn,xqnλqn)≤ϵ4≤ϵ2 . (25)
Since
limn→∞˜ρ(xnλn,xn+1λn+1)=0 , we find
n1∈N such that
˜ρ(xnλn,xn+1λn+1)≤ϵ2m, (26)
for any
n≥n1. Suppose that
r,s≥max{n0,n1} 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(xrλr,xsλs)≤˜ρ(xrλr,xs+jλs+j)+˜ρ(xrλr,xs+jλs+j) +˜ρ(xs+jλs+j,xs+j−1λs+j−1)+⋯+˜ρ(xs+1λs+1,xsλs) .
By (25) and (26) and from the last inequality, we get
˜ρ(xrλr,xsλs)≤ϵ2+j×ϵ2m≤ϵ2+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
zω∈Y=∪mi=1Ai such that
limn→∞xnλn=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
limn→∞xnλn=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
xλ∈Ai,
(f,φ)xλ∈Ai+1 and we take a subsequence
{xnkλnk} of
{xnλn} with
xnkλnk∈Ai−1 (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
xnkλnk→xλ, 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,zω∈∩mi=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,
m∈N , let
A1,A2,⋯,Am be non empty
˜ρ -closed subsets of
(X,d,E) and
let
Y=∪i=mi=1Ai . Suppose that
(f,φ):Y→Y 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
xλ∈Ai ,
yμ∈Ai+1 ,
i=1,2,3,⋯,m , where
Am+1=A1 . Then
(f,φ)
has fixed point
zw∈∩i=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):Y→Y 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
x1λ1∈A1=[−2,0] and
x2λ2∈A2=[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
A1∩A2 . 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
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)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∈∧ ,
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)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,if0≤xλ≤1,xλ12, ifxλ≥1,
Proof: To show the existence and uniqueness soft point of
(f,φ) , we investigate the following cases
Let
0≤xλ,yμ<1 then
˜ρ((f,φ)xλ,(f,φ)yμ)=max{x2λ+xλ18,y2μ+yμ18}≤110max{xλ,yμ}=110˜ρ(xλ,yμ).
Let
xλ,yμ≥1 then
˜ρ((f,φ)xλ,(f,φ)yμ)=112max{xλ,yμ}≤110max{xλ,yμ}=110˜ρ(xλ,yμ)
Let
0≤xλ<1 and
yμ≥1 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,
m∈N , 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 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)dt≤β∫m0ρ(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
xλ∈Ai,yμ∈Ai+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
zω∈∩i=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)dt≤β∫m0ρ(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.