http://www.uab.ro/auajournal/ doi: 10.17114/j.aua.2019.57.04
ON A GENERALIZED SOFT METRIC SPACE N. Tas¸ and N. Y. ¨Ozg¨ur
Abstract. In this paper our aim is to obtain new generalized fixed-point results.
To do this, we introduce a new generalized soft metric space called as a softS-metric space. We investigate some basic facts, relations and topological properties of this space. Also we define a soft S-contraction condition and study some fixed-point theorems on a complete softS-metric space with necessary examples.
2010Mathematics Subject Classification: 47H10, 54H25, 54A05.
Keywords: soft S-metric space, fixed point, topological properties.
1. Introduction and Background
Metric spaces and fixed-point theory have very important role in mathematics and lead to some applications. Some mathematicians have studied new generalizations of metric spaces using various ways. Recently, it has been introduced the notion of an S-metric space as a generalization of a metric space [14]. Then some basic fixed-point theorems and their generalizations were obtained in some studies (for more details see [9], [10], [11], [14], [15] and [16]). These fixed-point theorems were used in other mathematical areas such as complex valued metric spaces, differential equations etc. (see [12] and [13]).
There are some uncertain concepts in the areas of medical science, engineering, economics etc. Hence some set theories such as fuzzy set theory, rough set theory, intuitionistic fuzzy set theory etc. can be dealt with uncertainties. Unfortunately, they are not sufficient to cope with encountered problems. Therefore, Molodtsov introduced the soft set theory as a general mathematical tool for dealing with some complicated problems [8]. Maji et al. made a theoretical study of the soft set theory [7]. Shabir and Naz studied some soft topological concepts and investigated their basic properties [17].
Das and Samanta defined the notion of a soft real number and studied their properties [4]. Therefore they introduced the concept of a soft metric space and gave some fundamental properties of this space [5]. Then some fixed-point results
were obtained using various approaches on a soft metric space (see [1], [2] and [3]
for more details). G¨uler et al. defined the notion of a soft G-metric and proved a fixed-point theorem in soft G-metric spaces [6].
Motivated by the above studies, in this paper we introduce the concept of a soft S-metric space as a generalization of a soft metric space. We expect that our study will help to generate some new researches and applications. For example, various generalized soft contractive conditions can be given as generalizations of our results.
In Section 2, we define the notion of a softS-metric according to a soft point and determine the relationships between the other soft metrics. In Section 3, we describe basic topological concepts. In Section 4, we present the notion of a soft fixed point on a soft S-metric space and prove a fixed point-theorem of Banach contraction principle type. Also we generalize this theorem with a counter example.
On the other hand, Abbas et al. showed that a soft metric induces a compatible metric on the collection of all soft points of the absolute soft set when the set of parameters is a finite set [3]. Therefore a cardinality of a parameter set is to be significant. The results obtained in Section 4 can be also proved using this approach on a soft S-metric space.
Before stating our main results, we recall some definitions, a proposition and an example.
Definition 1. [8] Let U be an initial universe set andE be a set of parameters. A pair (F, E) is called a soft set over U if and only if F is a mapping fromE into the set of all subsets of the universe set U. That is,F :E →P(U), where P(U) is the set of all subsets of the set U.
Definition 2. [7] Let (F, E) be a soft set over a universe set U.
1. (F, E) is said to be a null soft set denoted bye∅ifF(e) =∅ for all e∈E.
2. (F, E) is said to be an absolute soft set denoted byUe ifF(e) =U for alle∈E.
Definition 3. [4] Let R be the set of real numbers, B(R) be the collection of all nonempty bounded subsets of R and E be a set of parameters. Then a mapping F :E →B(R) is called a soft real set. It is denoted by (F, E). If specifically (F, E) is a singleton soft set then identifying (F, E) with the corresponding soft element, it will be called a soft real number and denoted by er, s,e et etc.
0,1are the soft real numbers where0(e) = 0,1(e) = 1 for alle∈E, respectively.
Definition 4. [4] Let (F, E) and (G, E) be two soft real numbers.
1. (F, E) = (G, E) if F(e) =G(e) for each e∈E.
2. (F+G)(e) ={x+y :x∈F(e), y ∈G(e)} for each e∈E.
3. (F−G)(e) ={x−y :x∈F(e), y ∈G(e)} for each e∈E.
4. (F.G)(e) ={x.y:x∈F(e), y ∈G(e)} for each e∈E.
5. (F/G)(e) ={x/y:x∈F(e), y∈G(e)− {0}} for each e∈E.
Definition 5. [4] For two soft real numbers 1. re≤e esif er(e)≤es(e) for alle∈E, 2. re≥e esif er(e)≥es(e) for alle∈E, 3. re<e esif er(e)<es(e) for alle∈E, 4. re>e esif er(e)>es(e) for alle∈E.
Definition 6. [5] A soft set(P, E)overU is said to be a soft point if there is exactly one e∈E such that P(e) ={x} for somex∈U and P(e0) =∅ for all e0∈E− {e}.
It will be denoted by Pex.
Definition 7. [5] A soft point Pex is said to be belongs to a soft set(F, E) if e∈E and P(e) ={x} ⊂F(e). It is written by Pex ∈e (F, E).
Definition 8. [5] Two soft points Pex, Pey0 are said to be equal if e=e0 andP(e) = P(e0), that is, x=y. Thus,
Pex6=Pey0 ⇐⇒x6=y or e6=e0.
Proposition 1. [5] The union of any collection of soft points can be considered as a soft set and every soft set can be expressed as union of all soft points belonging to it, that is,
(F, E) = [
Pex∈(F,E)e
Pex.
Let SP(Ue) be the collection of all soft points of Ue and R(E)∗ be the set of all nonnegative soft real numbers.
Definition 9. [5] A mapping de: SP(Ue)×SP(Ue) → R(E)∗ is said to be a soft metric on the soft set Ue if desatisfies the following conditions:
(d1)e d(Pe ex1, Pey2) ≥e 0 for allPex1, Pey2 ∈SP(Ue).
(d2)e d(Pe ex1, Pey2) = 0 if and only if Pex1 =Pey2.
(d3)e d(Pe ex1, Pey2) =d(Pe ey2, Pex1) for all Pex1, Pey2 ∈SP(Ue).
(d4)e d(Pe ex1, Pez3) ≤e d(Pe ex1, Pey2) +d(Pe ey2, Pez3) for all Pex1, Pey2, Pez3 ∈SP(Ue).
The soft setUe with a soft metricdeonUe is called a soft metric space and denoted by
U ,e d, Ee
.
Example 1. [5] Let U ⊂R be a nonempty set and E ⊂R be the nonempty set of parameters. Let Ue be the absolute soft set and x denotes the soft real number such that
x(e) =x,
for all e∈E. Then the functionde:SP(Ue)×SP(Ue)→R(E)∗ defined by d(Pe ex1, Pey2) =|x−y|+|e1−e2|,
for all Pex1, Pey2 ∈ SP(Ue), where “|.|”denotes the modulus of soft real numbers, is a soft metric on Ue.
Definition 10. [5] Let
Pe,nx n be a sequence of soft points in a soft metric space
U ,e d, Ee
. The sequence Pe,nx
n is said to be convergent in
U ,e d, Ee
if there is a soft point Pαβ ∈SP(Ue) such that
d(Pe e,nx , Pαβ)→0 as n→ ∞.
That is, for everyεe≥e 0, chosen arbitrarily, there exists a natural numberN =N(eε) such that 0 ≤e d(Pe e,nx , Pαβ) <e ε, whenevere n > N.
Definition 11. [5] A sequence
Pe,nx nof soft points in
U ,e d, Ee
is called a Cauchy sequence if corresponding to every εe≥e 0, there exists m ∈N such that d(Pe e,ix , Pe,jx )
<e εefor everyi, j≥m, that is, d(Pe e,ix, Pe,jx )→0 as i, j→ ∞.
Definition 12. [5] A soft metric space
U ,e d, Ee
is called complete if every Cauchy sequence in Ue converges to some point of Ue.
Definition 13. [6] Let U be a nonempty set and E be the nonempty set of parame- ters. A mappingGe:SP(Ue)×SP(Ue)×SP(Ue)→R(E)∗ is said to be a softG-metric on the soft set Ue if Ge satisfies the following conditions:
(G1)e G Pe ex1, Pey2, Pez3
= 0 ifPex1 =Pey2 =Pez3. (G2) 0e <e G Pe ex1, Pex1, Pey2
for all Pex1, Pey2 ∈SP(Ue) with Pex1 6=Pey2. (G3)e G Pe ex1, Pex1, Pey2
≤e G Pe ex1, Pey2, Pez3
for all Pex1, Pey2, Pez3 ∈ SP(Ue) with Pey2 6=Pez3.
(G4)e G Pe ex1, Pey2, Pez3
=G Pe ex1, Pez3, Pey2
=G Pe ey2, Pez3, Pex1
=· · · for all possi- ble triples Pex1, Pey2, Pez3 ∈SP(Ue).
(G5)e G Pe ex1, Pey2, Pez3
≤e G Pe ex1, Pea, Pea
+G Pe ea, Pey2, Pez3
for all Pex1,Pey2,Pez3, Pea∈SP(Ue).
The triplet
U ,e G, Ee
is said to be a soft G-metric space.
2. Soft S-Metric spaces
In this section we define the notion of a softS-metric space and determine its basic properties. Also we investigate some relationships between a soft metric and a soft S-metric (resp. a soft G-metric and a soft S-metric).
LetU be an initial universe set and E be the nonempty set of parameters. Let SP(Ue) be a collection of all soft points ofUe andR(E)∗be the set of all nonnegative soft real numbers.
Definition 14. A mapping Se:SP(Ue)×SP(Ue)×SP(Ue)→R(E)∗ is said to be a soft S-metric on the soft set Ue if Se satisfies the following conditions for each Pex1, Pey2, Pez3, Pea∈SP(Ue):
(S1)e S Pe ex1, Pey2, Pez3
≥e 0.
(S2)e S Pe ex1, Pey2, Pez3
= 0 if and only if Pex1 =Pey2 =Pez3. (S3)e S Pe ex1, Pey2, Pez3
≤e S Pe ex1, Pex1, Pea
+Se(Pey2, Pey2, Pea) +S Pe ez3, Pez3, Pea . The soft set Ue with a soft S-metric Se on Ue is called a soft S-metric space and denoted by
U ,e S, Ee .
Now we give the following examples for a soft S-metric.
Example 2. Let U ⊂ R be a nonempty set and E ⊂ R be the nonempty set of parameters. Let Ue be the absolute soft set, that is, F(e) = U for all e∈ E, where (F, E) =Ue. Let x denote a soft real number such that x(e) =x for all e∈E. We define Se:SP(Ue)×SP(Ue)×SP(Ue)→R(E)∗ by
S Pe ex1, Pey2, Pez3
=|y+z−2x|+|y−z|+|e2+e3−2e1|+|e2−e3|,
for all Pex1,Pey2, Pez3 ∈SP(Ue), where “|.|”denotes the modulus of soft real numbers.
Then Se is a soft S-metric on Ue.
Example 3. Let U be a nonempty set, E be the nonempty set of parameters and de be a soft metric on Ue. Then the function Se :SP(Ue)×SP(Ue)×SP(Ue) → R(E)∗ defined as
S Pe ex1, Pey2, Pez3
=d(Pe ex1, Pey2) +d(Pe ey2, Pez3) +d(Pe ex1, Pez3),
for all Pex1, Pey2,Pez3 ∈SP(Ue), is a soft S-metric on Ue.
Example 4. Let U be a nonempty set and E be the nonempty set of parameters.
Let us define the function Se:SP(Ue)×SP(Ue)×SP(Ue)→R(E)∗ as follows:
S Pe ex1, Pey2, Pez3
=
0 if Pex1 =Pey2 =Pez3
1 if otherwise ,
for all Pex1, Pey2, Pez3 ∈SP(Ue). Then the function Seis a soft S-metric. We call this softS-metric is the soft discrete S-metric onUe. The triplet
U ,e S, Ee
is called soft discrete S-metric space.
Lemma 1. Let
U ,e S, Ee
be a soft S-metric space. Then we have S Pe ex1, Pex1, Pey2
=S Pe ey2, Pey2, Pex1 . Proof. By the condition (S3) we obtaine
S Pe ex1, Pex1, Pey2
≤e 2S Pe ex1, Pex1, Pex1
+S Pe ey2, Pey2, Pex1
=S Pe ey2, Pey2, Pex1 (1) and
S Pe ey2, Pey2, Pex1
≤e 2Se(Pey2, Pey2, Pey2) +S Pe ex1, Pex1, Pey2
=S Pe ex1, Pex1, Pey2
. (2)
Using the inequalities (1) and (2) we get S Pe ex1, Pex1, Pey2
=S Pe ey2, Pey2, Pex1 .
Proposition 2. Let U be a nonempty set, E be a nonempty set of the parameters and debe a soft metric onUe. Then
Sed(Pex1, Pey2, Pez3) =d(Pe ex1, Pez3) +d(Pe ey2, Pez3), for all Pex1, Pey2,Pez3 ∈SP(Ue), is a soft S-metric on Ue.
Proof. It is obvious from Definitions 9 and 14.
We call the soft metric Sedas the soft S-metric generated by d.e
Example 5. Let U and E be nonempty subsets of R. Let us define Se :SP(Ue)× SP(Ue)×SP(Ue)→R(E)∗ as
S Pe ex1, Pey2, Pez3
=|x−z|+|y−z|+|e1−e3|+|e2−e3|,
for all Pex1, Pey2, Pez3 ∈SP(Ue), where “|.|”denotes the modulus of soft real numbers and x,e1 are constant real numbers defined by
x(e) =x and e1(e) =e1,
for all e∈E, respectively. ThenSeis a softS-metric onUe and the triplet
U ,e S, Ee is a soft S-metric space. This soft S-metric is generated by soft metricdedefined in Example 1.
In the following example, we see that there exists a soft S-metric which is not generated by any soft metric.
Example 6. Let U ⊆R be a nonempty set and E ={ei : 1≤i≤n} ⊆R,
be the nonempty set of parameters. Let us define a function Se:SP(Ue)×SP(Ue)× SP(Ue)→R(E)∗ as
S Pe ex1, Pey2, Pez3
=|x−z|+|x+z−2y|+|e1−e3|+|e1+e3−2e2|, for all Pex1, Pey2, Pez3 ∈ SP(Ue). Then Se is a soft S-metric on Ue and the triplet
U ,e S, Ee
is a soft S-metric space.
Now we show that there does not exist any soft metric desuch that Se = Sed. Conversely, assume that there exists a soft metric desuch that
S(Pe ex1, Pey2, Pez3) =d(Pe ex1, Pez3) +d(Pe ey2, Pez3), for all Pex1, Pey2,Pez3 ∈SP(Ue). Therefore we find
S(Pe ex1, Pex1, Pez3) = 2d(Pe ex1, Pez3) = 2 (|x−z|+|e1−e3|) and
S(Pe ey2, Pey2, Pez3) = 2d(Pe ey2, Pez3) = 2 (|y−z|+|e2−e3|). Hence we obtain
d(Pe ex1, Pez3) =|x−z|+|e1−e3| and d(Pe ey2, Pez3) =|y−z|+|e2−e3|.
Therefore we have a contradiction since
|x−z|+|x+z−2y|+|e1−e3|+|e1+e3−2e2|=|x−z|+|e1−e3|+|y−z|+|e2−e3|. Consequently we get Se6=Sed.
Notice that the class of all soft G-metrics and the class of all soft S-metrics are distinct as seen in the following examples.
Example 7. Let
U ,e S, Ee
be the soft S-metric space defined in Example 5. For e= 0, x= 2, y= 1, z= 3, we obtain
S(Pe 02, P01, P03)(e) =|2−3|+|1−3|= 1 + 2 = 3 and
S(Pe 01, P03, P02)(e) =|1−2|+|3−2|= 1 + 1 = 2.
Then we get S(Pe 02, P01, P03) 6= S(Pe 01, P03, P02). Consequently, the condition (G4)e is not satisfied and Seis not a soft G-metric.
Example 8. Let U = {x, y}, E = {0} and the function Ge : SP(Ue)×SP(Ue)× SP(Ue)→R(E)∗ be defined by
G(Pe ex, Pex, Pex) =G(Pe ey, Pey, Pey) = 0,
G(Pe ex, Pex, Pey) =G(Pe ex, Pey, Pex) =G(Pe ey, Pex, Pex) = 2 and
G(Pe ex, Pey, Pey) =G(Pe ey, Pex, Pey) =G(Pe ey, Pey, Pex) = 3, for all Pex,Pey ∈SP(Ue). Then Ge is a soft G-metric onUe and the triplet
U ,e G, Ee is a soft G-metric space. But it is not soft S-metric space. Indeed, the condition (S3)e is not satisfied since
S(Pe ex, Pey, Pey)(e) = 3(e) = 3≤h
S(Pe ex, Pex, Pey) +S(Pe ey, Pey, Pey) +S(Pe ey, Pey, Pey)i (e)
= 2(e) + 0(e) + 0(e) = 2 + 0 + 0 = 2.
3. Some Topological Properties of Soft S-Metric Spaces
In this section we define some topological concepts on soft S-metric spaces and investigate some properties related to these notions.
Definition 15. Let
U ,e S, Ee
be a softS-metric space and(F, E)be a non-null soft subset of Ue. Then the diameter of (F, E) is denoted by δ(F, E) and
δ(F, E)(α) = sup{S(Pe ex1, Pex1, Pey2)(α) :Pex1, Pey2e∈(F, E)}, for all α∈E.
Definition 16. Let
U ,e S, Ee
be a soft S-metric space, Pex1 be a fixed soft point of Ue and(F, E) be a non-null soft subset ofUe. Then the distance of the soft pointPex1 from the soft set (F, E) is denoted by δ(Pex1,(F, E)) and
δ(Pex1,(F, E))(α) = inf{S(Pe ex1, Pex1, Pey2)(α) :Pey2e∈(F, E)}, for all α∈E.
Definition 17. Let
U ,e S, Ee
be a soft S-metric space and (F, E), (G, E) be two non-null soft subsets of U. The distance between the soft setse (F, E), (G, E) is denoted by δ((F, E),(G, E))and
δ((F, E),(G, E))(α) = inf{S(Pe ex1, Pex1, Pey2)(α) :Pex1∈(F, Ee ), Pey2∈(G, E)},e for all α∈E.
Definition 18. Let
U ,e S, Ee
be a soft S-metric space. If there exists a positive soft real number ek such that
S(Pe ex1, Pex1, Pey2) ≤e ek, for allPex1, Pey2∈eUe, then
U ,e S, Ee
is called a bounded softS-metric space. Otherwise it is called unbounded.
In the following definition we define the notion of soft openS-ball and soft closed S-ball, respectively.
Definition 19. Let
U ,e S, Ee
be a softS-metric space and er be a nonnegative soft real number.
1. The soft open S-ball is defined by
BS(Pex1,r) =e {Pey2∈eUe :S(Pe ex1, Pex1, Pey2) <e r},e with center Pex1 and radius er.
2. The soft closed S-ball is defined by
BS[Pex1,r] =e {Pey2∈eUe :S(Pe ex1, Pex1, Pey2) ≤e r},e with center Pex1 and radius er.
Proposition 3. Let
U ,e S, Ee
be a soft S-metric space,Pex ∈SP(Ue) and re≥0. Ife Pey ∈BS(Pex,er) then there exists a ρe≥0e such that
BS(Pey,ρ)e ⊆BS(Pex,er).
Proof. LetPey ∈BS(Pex,er). Then we get
S(Pe ey, Pey, Pex)<eer.
We show that
BS(Pey,ρ)e ⊆BS(Pex,er).
Let us choose
ρe= er−S(Pe ex, Pex, Pey)
2 .
If Pez∈BS(Pey,ρ), then we havee
S(Pe ez, Pez, Pey)<eρ.e Using the condition (S3) we finde
S(Pe ez, Pez, Pex)≤2e S(Pe ez, Pez, Pey) +S(Pe ex, Pex, Pey)<2e ρe+S(Pe ex, Pex, Pey) =er and so
BS(Pey,ρ)e ⊆BS(Pex,er).
Definition 20. Let
U ,e S, Ee
be a soft S-metric space having at least two soft points. Then
U ,e S, Ee
is said to poses soft S-Hausdorff property if for any two soft elements Pex1,Pey2 such that S(Pe ex1, Pex1, Pey2)>e 0, there are two soft openS-balls BS(Pex1,er) andBS(Pey2,er) with radiuser and centers Pex1, Pey2, respectively, such that
BS(Pex1,r)e ∩BS(Pey2,r) =e e∅.
Theorem 2. Every soft S-metric space is Hausdorff.
Proof. Let
U ,e S, Ee
be a softS-metric space having at least two soft elements. Let Pex1, Pey2 be two soft elements in Ue such that S(Pe ex1, Pex1, Pey2) >e 0. Let us consider any nonnegative soft real number er such that
0 <e er <e 1
3S(Pe ex1, Pex1, Pey2),
and the soft open S-balls BS(Pex1,r) ande BS(Pey2,r) with radiuse er and centers Pex1, Pey2, respectively.
Assume that there existsPez3 ∈SP(Ue) such that Pez3 ∈BS(Pex1,er)∩BS(Pey2,er).
Then we get
Pez3 ∈BS(Pex1,r)e ⇒S(Pe ex1, Pex1, Pez3) <e er (3) and
Pez3 ∈BS(Pey2,er)⇒S(Pe ey2, Pey2, Pez3)<e er. (4) Using the conditions (S3), (3) and (4) we havee
S(Pe ex1, Pex1, Pey2) ≤e 2S(Pe ex1, Pex1, Pez3) +S(Pe ey2, Pey2, Pez3) = 3r,e which is a contradiction since 0 <e er <e 13S(Pe ex1, Pex1, Pey2). Hence it should be
BS(Pex1,er)∩BS(Pey2,r) =e e∅.
Consequently, soft S-metric spaces satisfy the softS-Hausdorff property.
Definition 21. Let
U ,e S, Ee
be a soft S-metric space andPex∈SP(Ue). A collec- tionNS(Pea)of soft points containing the soft pointPeais called softS-neighbourhood of the soft point Pex if there exists a positive soft real number er such that
Pex ∈BS(Pex,er)⊂NS(Pea).
Theorem 3. Every soft open S-ball is a soft S-neighbourhood of each of its soft points.
Proof. Let
U ,e S, Ee
be a softS-metric space andBS(Pex1,r) be a soft opene S-ball with centerPex1 and radiuser. By Definition 21,BS(Pex1,r) is a softe S-neighbourhood of the soft pointPex1 in
U ,e S, Ee .
Let us consider any soft point Pey2 ∈ BS(Pex1,er) such that Pex1 6=Pey2. Then we have
06=S(Pe ey2, Pey2, Pex1) <e er.
If we choose ρesuch that
0 <e ρe<e er−S(Pe ey2, Pey2, Pex1)
2 ,
then ρeis a positive soft real number. ForPez3 ∈BS(Pey2,ρ), we havee S(Pe ez3, Pez3, Pey2)<e ρ.e
Using the condition (S3) and Lemma 1, we gete
S(Pe ez3, Pez3, Pex1)<e 2S(Pe ez3, Pez3, Pey2) +S(Pe ex1, Pex1, Pey2) <e er, that is
Pez3 ∈BS(Pey2,ρ)e ⊂BS(Pex1,r).e
Consequently, BS(Pex1,er) is a soft S-neighbourhood of its soft points.
Definition 22. Let
U ,e S, Ee
be a soft S-metric space and Pe,nx
n be a sequence of soft points in Ue. The sequence
Pe,nx
n is called soft S-convergent in Ue if there is a soft point Pαβ ∈SP(Ue) such that
S(Pe e,nx , Pe,nx , Pαβ)→0,
as n→ ∞. That is, for every εe≥e 0, there exists a natural number N =N(ε)e such that
0 ≤e S(Pe e,nx , Pe,nx , Pαβ) <e ε,e whenever n≥N. Then we get
Pe,nx ∈BS(Pαβ,ε).e We denote this by
n→∞limPe,nx =Pαβ or
Pe,nx →Pαβ as n→ ∞.
Lemma 4. Let
U ,e S, Ee
be a soft S-metric space. If the sequence
Pe,nx n con- verges to Pαβ thenPαβ is unique.
Proof. Let
Pe,nx n converges to Pαβ and Pµλ. There exist two natural numbers n1, n2 for everyεe≥e 0 such that
S(Pe e,nx , Pe,nx , Pαβ) <e εe 4 and
S(Pe e,nx , Pe,nx , Pµλ) <e eε 2,
where n≥n1,n2. If we choose n0 = max{n1, n2}, then for each n≥n0, using the condition (S3) and Lemma 1, we obtaine
S(Pe αβ, Pαβ, Pµλ) ≤e 2S(Pe αβ, Pαβ, Pe,nx ) +S(Pe µλ, Pµλ, Pe,nx ) <e ε.e Therefore we get
S(Pe αβ, Pαβ, Pµλ) = 0 and using the condition (S2),e
Pαβ =Pµλ. Consequently, the limit of
Pe,nx n is unique.
Lemma 5. Let
U ,e S, Ee
be a softS-metric space. If there exist sequences Pe,nx n and {Pe,ny }n such that
n→∞limPe,nx =Pαβ and
n→∞limPe,ny =Pµλ, then we get
n→∞limS(Pe e,nx , Pe,nx , Pe,ny ) =S(Pe αβ, Pαβ, Pµλ).
Proof. Using the hypothesis, for eachεe≥e 0, there exist two natural numbersn1,n2 such that
S(Pe e,nx , Pe,nx , Pαβ) <e εe 4 and
S(Pe e,ny , Pe,ny , Pµλ) <e eε 4,
where n≥n1,n2. If we choose n0 = max{n1, n2}, then for each n≥n0, using the condition (S3) we havee
S(Pe e,nx , Pe,nx , Pe,ny ) ≤e 2S(Pe e,nx , Pe,nx , Pαβ) +S(Pe e,ny , Pe,ny , Pαβ)
≤e 2S(Pe e,nx , Pe,nx , Pαβ) + 2S(Pe e,ny , Pe,ny , Pµλ) +S(Pe αβ, Pαβ, Pµλ)
<e ε2e+eε2 +S(Pe αβ, Pαβ, Pµλ).
Therefore we get
S(Pe e,nx , Pe,nx , Pe,ny )−S(Pe αβ, Pαβ, Pµλ) <e ε.e (5) On the other hand, using the condition (S3) and Lemma 1, we finde
S(Pe αβ, Pαβ, Pµλ)≤e 2S(Pe αβ, Pαβ, Pe,nx ) +S(Pe µλ, Pµλ, Pe,nx )
≤e 2S(Pe αβ, Pαβ, Pe,nx ) + 2S(Pe µλ, Pµλ, Pe,ny ) +S(Pe e,nx , Pe,nx , Pe,ny )
<e ε2e+ε2e+S(Pe e,nx , Pe,nx , Pe,ny ).
Hence we obtain
S(Pe αβ, Pαβ, Pµλ)−S(Pe e,nx , Pe,nx , Pe,ny ) <e ε.e (6) Using the inequalities (5) and (6) we have
S(Pe e,nx , Pe,nx , Pe,ny )−S(Pe αβ, Pαβ, Pµλ) <e εe and so
n→∞limS(Pe e,nx , Pe,nx , Pe,ny ) =S(Pe αβ, Pαβ, Pµλ).
Definition 23. Let
U ,e S, Ee
be a soft S-metric space and
Pe,nx n be a sequence of soft points in Ue. The sequence
Pe,nx n is called soft S-bounded if there exists a positive soft real number Re >e 0 such that
S(Pe e,nx , Pe,nx , Pe,mx ) ≤e R,e for each m, n∈N.
Definition 24. Let
U ,e S, Ee
be a soft S-metric space and
Pe,nx n be a sequence of soft points in Ue. The sequence
Pe,nx n is called soft S-Cauchy sequence in Ue if S(Pe e,nx , Pe,nx , Pe,mx )→0 as m, n→ ∞,
that is, for every εe≥e 0, there exists a natural number n0 such that S(Pe e,nx , Pe,nx , Pe,mx ) <e ε,e
whenever n, m≥n0.
Definition 25. A soft S-metric space
U ,e S, Ee
is called complete if every soft S-Cauchy sequence in Ue converges to some soft points of Ue.
Lemma 6. Let
U ,e S, Ee
be a soft S-metric space and
Pe,nx n be a sequence of soft points inUe. If the sequence
Pe,nx nconverges toPαβ, then it is a softS-Cauchy sequence.
Proof. Using the hypothesis, for eachεe≥e 0, there exist two natural numbersn1,n2
such that
S(Pe e,nx , Pe,nx , Pαβ) <e εe 4 and
S(Pe e,mx , Pe,mx , Pαβ) <e εe 2,
where n≥ n1, m ≥ n2. If we choose n0 = max{n1, n2}, then for each n, m ≥n0, using the condition (S3) we finde
S(Pe e,nx , Pe,nx , Pe,mx )≤e 2S(Pe e,nx , Pe,nx , Pαβ) +S(Pe e,mx , Pe,mx , Pαβ) <e ε.e Therefore
Pe,nx n is a soft S-Cauchy sequence.
Corollary 7. Every soft S-Cauchy sequence is soft S-bounded.
Corollary 8. Let
U ,e d, Ee
be a soft metric space and
U ,e fSd, E
be a softS-metric space which is generated by soft metric d. Then we havee
1.
Pe,nx n→Pαβ in
U ,e d, Ee
if and only if
Pe,nx n→Pαβ in
U ,e Sfd, E
. 2.
Pe,nx n is Cauchy in
U ,e d, Ee
if and only if
Pe,nx n is soft S-Cauchy in
U ,e Sfd, E . 3.
U ,e d, Ee
is complete if and only if
U ,e fSd, E
is complete.
4. Some Fixed-Point Results In this section we study some fixed-point results.
Definition 26. Let
U ,e S, Ee
be a soft S-metric space and T : Ue → Ue be a soft mapping. If there exists a soft point Pαβ ∈SP(Ue) such that
T(Pαβ) =Pαβ, then Pαβ is called a soft fixed point of T.
Definition 27. Let
U ,e S, Ee
be a soft S-metric space and T : Ue → Ue be a soft mapping. Then T is called a soft S-contraction if
S(Te (Pex1), T(Pex1), T(Pey2))≤e hS(Pe ex1, Pex1, Pey2),
for all Pex1, Pey2 ∈ SP(Ue), where 0 ≤e h <e 1 which is called a soft S-contraction constant.
Theorem 9. Let
U ,e S, Ee
be a complete soft S-metric space whereE is nonempty finite set and T be a soft S-contraction with soft S-contraction constanth. Then T has a unique soft fixed point Pαβ.
Proof. Let Pex be a soft point and Pe,0x =Pex. We define the sequence
Pe,nx n by Pe,nx =Tn(Pex). Using the soft S-contraction hypothesis, we have
S(Pe e,n+1x , Pe,n+1x , Pe,nx ) =S(T(Pe e,nx ), T(Pe,nx ), T(Pe,n−1x ))
≤e hS(Pe e,nx , Pe,nx , Pe,n−1x ) ≤e h2S(Pe e,n−1x , Pe,n−1x , Pe,n−2x )
≤ · · ·e
≤e hnS(Pe e,1x , Pe,1x , Pe,0x ) =hnS(Pe e,1x , Pe,1x , Pex).
(7)
For n > m, using the conditions (S3), (7) and Lemma 1, we gete
S(Pe e,nx , Pe,nx , Pe,mx ) ≤e 2S(Pe e,nx , Pe,nx , Pe,n−1x ) + 2S(Pe e,n−1x , Pe,n−1x , Pe,n−2x ) +· · ·+S(Pe e,m+1x , Pe,m+1x , Pe,mx )
≤e 2S(Pe e,nx , Pe,nx , Pe,n−1x ) + 2S(Pe e,n−1x , Pe,n−1x , Pe,n−2x ) +· · ·+ 2S(Pe e,m+1x , Pe,m+1x , Pe,mx )
≤e 2(hn−1+hn−2+· · ·+hm)S(Pe e,1x , Pe,1x , Pex)
≤e 2hm
1−hS(Pe e,1x , Pe,1x , Pex).
Now we show that the sequence
Pe,nx nis a softS-Cauchy sequence. Let us choose εe≥e 0. We can construct the parameter set
E={ei : 1≤i≤k},
since E is a nonempty finite set. Therefore for each i ∈ {1, . . . , k}, there exists a natural number ni such that
"
2hni
1−hS(Pe e,1x , Pe,1x , Pex)
#
(ei)<ε(ee i).
