Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 885–894
Research Article
A fixed point theorem on soft G-metric spaces
Aysegul Caksu Gulera,∗, Esra Dalan Yildirimb , Oya Bedre Ozbakira
a
Ege University, Faculty of Science, Department of Mathematics, 35100- İzmir, Turkey.
b
Yaşar University, Faculty of Science and Letters, Department of Mathematics, 35100- İzmir, Turkey.
Communicated by C. Alaca
Abstract
We introduce soft G-metric spaces via soft element. Then, we obtain soft convergence and soft continuity
by using soft G-metric. Also, we prove a fixed point theorem for mappings satisfying sufficient conditions
c
in soft G-metric spaces. 2016
All rights reserved.
Keywords: Soft set, soft metric space, generalized metric space, soft G-metric space, fixed point.
2010 MSC: 06D72, 54A05, 47H10.
1. Introduction and Preliminaries
There are uncertainties in many complicated problems in the fields of engineering, physics, computer
science, medical science, social science and economics. These problems can not be solved by classical methods
such as probability theory, fuzzy set theory, intuitionistic fuzzy sets, rough set etc. In [10], Molodtsov
introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties. Then, Maji
et al. [8] studied soft set theory in detail. Also, Ali et al. [1] established new algebraic operations on soft
sets. Shabir and Naz [15] introduced soft topological spaces and investigated their fundamental properties.
Zorlutuna [18] also studied those spaces. Das and Samanta [5] introduced the notions of soft real set and
soft real number and gave their properties. Recently, soft set theory have a high potential for applications
in several areas, see for example [3, 4, 9].
Fixed point theory plays an important role and has many applications in mathematics. A basic result
in fixed point theory is the Banach contraction principle. There has been a lot of activity in this area since
the appearance of this principle.
∗
Corresponding author
Email addresses: aysegul.caksu.guler@ege.edu.tr (Aysegul Caksu Guler), esra.dalan@yasar.edu.tr (Esra Dalan
Yildirim), oya.ozbakir@ege.edu.tr (Oya Bedre Ozbakir)
Received 2015-08-19
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
886
Many researchers obtained fixed point theorems for various mapping in different metric spaces. Among
them are G-metric spaces [13] which are a generalization of metric spaces. Several new fixed point theorems
were inferred in this new structure [2, 11, 12, 14, 16].
Recently, Wardowski [17] introduced a notion of soft mapping and its fixed points. Also, Das and
Samanta [6] gave the concept of soft metric spaces and Banach fixed point theorem in soft metric space
settings.
In the first section, we define the concept of soft G-metric, according to soft element, and describe
some of its properties. Later, we give a relation between the soft metric and soft G-metric. Further, we
introduce soft G-ball and get its basic features. In the second section, we investigate soft convergence and
soft continuity with the help of soft G-metric. Then, we prove existence and uniqueness of fixed point in
soft G-metric spaces.
Definition 1.1 ([10]). Let E be a set of parameters, a pair (F, E), where F is a mapping from E to P(X),
is called a soft set over X.
Definition 1.2 ([8]). Let (F1 , E) and (F2 , E) be two soft sets over a common universe X. Then, (F1 , E)
e 2 , E).
is said to be a soft subset of (F2 , E) if F1 (λ) ⊆ F2 (λ), for all λ ∈ E. This is denoted by (F1 , E)⊆(F
(F1 , E) is said to be soft equal to (F2 , E) if F1 (λ) = F2 (λ), for all λ ∈ E. This is denoted by (F1 , E) = (F2 , E).
Definition 1.3 ([1]). The complement of a soft set (F, E) is defined as (F, E)c = (F c , E), where F c : E →
P(X) is a mapping given by F c (λ) = X\F (λ), for all λ ∈ E.
Definition 1.4 ([8]). Let (F, E) be a soft set over X. Then
(a) (F, E) is said to be a null soft set if F (λ) = ∅, for all λ ∈ E. This is denoted by e
∅.
e
(b) (F, E) is said to be an absolute soft set if F (λ) = X, for all λ ∈ E. This is denoted by X.
e c=e
e
Clearly, we have (X)
∅ and (e
∅)c = X.
Definition 1.5 ([15]). The difference (H, E) of two soft sets (F1 , E) and (F2 , E) over X, denoted by
(F1 , E)\(F2 , E), is defined by H(λ) = F1 (λ)\F2 (λ), for all λ ∈ E.
Definition 1.6 ([8]). The union (H, E) of two soft sets (F, E) and (G, E) over a common universe X,
e (G, E) is defined by H(λ) = F (λ) ∪ G(λ), for all λ ∈ E.
denoted by (F, E)∪
The following definition of intersection of two soft sets is given in [7], where it was named bi-intersection.
Definition 1.7 ([7]). The intersection (H, E) of two soft sets (F, E) and (G, E) over a common universe
e (G, E), is defined by H(λ) = F (λ) ∩ G(λ), for all λ ∈ E.
X, denoted by (F, E)∩
Definition 1.8 ([18]). Let I be an index set and {(Fi , E)}i∈I a nonempty family of soft sets over a common
universe X. Then
S
(a) The union of these soft sets is the soft set (H, E), where H(λ) = i∈I Fi (λ) for all λ ∈ E. This is
S
denoted by e i∈I (Fi , E) = (H, E).
T
(b) The intersection of these soft sets is the soft set (H, E), where H(λ) = i∈I Fi (λ) for all λ ∈ E. This
T
is denoted by e i∈I (Fi , E) = (H, E).
Definition 1.9 ([5]). A function ε : E → X is said to be a soft element of X. A soft element ε of X is
b (F, E), if ε(e) ∈ F (e) for each e ∈ E. In that case,
said to belong to a soft set (F, E) of X, denoted by ε∈
ε is also said to be a soft element of the soft set (F, E). Thus, every singleton soft set (a soft set (F, E) for
which F (e) is a singleton set for each e ∈ E) can be identified with a soft element by simply identifying the
singleton set with the element that contains each e ∈ E.
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
887
Definition 1.10 ([5]). Let R be the set of real numbers, B(R) the collection of all nonempty bounded
subsets of R, and E a set of parameters. Then a mapping F : E → B(R) is called a soft real set. It
is denoted by (F, E) and R(E) denotes the set of all soft real sets. Also, R(E)∗ denotes the set of all
nonnegative soft real sets ((F, E) is said to be a nonnegative soft real set if F (λ) is a subset of the set of
nonnegative real numbers for each λ ∈ E).
If, in particular, (F, E) is a singleton soft set, then identifying (F, E) with the corresponding soft element,
we will call it a soft real number. R(E) denotes the set of all soft real numbers.
Definition 1.11 ([5]). Let (F, E), (G, E) ∈ R(E). We define
(a)
(b)
(c)
(d)
(e)
(F, E) = (G, E) if F (λ) = G(λ) for each λ ∈ E.
(F + G)(λ) = {a + b : a ∈ F (λ), b ∈ G(λ)} for each λ ∈ E.
(F − G)(λ) = {a − b : a ∈ F (λ), b ∈ G(λ)} for each λ ∈ E.
(F.G)(λ) = {a.b : a ∈ F (λ), b ∈ G(λ)} for each λ ∈ E.
(F/G)(λ) = {a/b : a ∈ F (λ), b ∈ G(λ)\{0}} provided 0 ∈
/ G(λ) for each λ ∈ E.
e denotes the set of soft sets (F, E) over X for which F (λ) 6= ∅ for all
In this paper, following [6], S(X)
e
λ ∈ E and SE(F, E) denotes the collection of all soft elements of (F, E) for any soft set (F, E) ∈ S(X).
e
Also, x
e, ye, ze denote soft elements of a soft set and re, se, t denote soft real numbers, while r, s, t will denote a
particular type of soft real numbers such that r(λ) = r for all λ ∈ E.
Definition 1.12 ([6]). For two soft real numbers re, se, we define
e s if re(λ) ≤ se(λ) for all λ ∈ E.
(a) re≤e
e s if re(λ) ≥ se(λ) for all λ ∈ E.
(b) re≥e
e s if re(λ) < se(λ) for all λ ∈ E.
(c) re<e
e s if re(λ) > se(λ) for all λ ∈ E
(d) re>e
Proposition 1.13 ([6]). Any collection of soft elements of a soft set can generate a soft subset of that soft
set.
Remark 1.14 ([6]). Let (F, E) be a soft set. Y generates a soft subset of (F, E) by considering for each
λ ∈ E. Considering the set {e
x(λ) : x
e ∈ Y } and associating a soft set with its λ-levels to be this set.
The soft set constructed from a collection B of soft elements is denoted by SS(B).
e we have (F, E)⊆(G,
e
Proposition 1.15 ([6]). For any soft subsets (F, E), (G, E) ∈ S(X),
E) if and only if
every soft element of (F, E) is also a soft element of (G, E).
e × SE(X)
e → R(E)∗ , is said to be a soft metric on X
e if d
Definition 1.16 ([6]). A mapping d : SE(X)
satisfies
e
e for all x
b X.
(M1) d(e
x, ye)≥0
e, ye∈
(M2) d(e
x, ye) = 0 if and only if x
e = ye.
e
b X.
(M3) d(e
x, ye) = d(e
y, x
e) for all x
e, ye∈
e
e x, ye) + d(e
b X.
(M4) d(e
x, ze)≤d(e
y , ze) for all x
e, ye, ze∈
e with a soft metric d on X
e is said to be a soft metric space and is denoted by (X,
e d, E).
The soft set X
Definition 1.17 ([13]). Let X be a nonempty set, and let G : X × X × X → R+ , be a function satisfying:
(G1)
(G2)
(G3)
(G4)
(G5)
G(x, y, z) = 0 if x = y = z,
0 < G(x, x, y) for all x, y ∈ X with x 6= y,
G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with z 6= y,
G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · (symmetry in all three variables),
G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X (rectangle inequality),
then the function G is called a generalized metric or a G-metric on X, and the pair (X, G) is a G-metric
space.
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
888
2. Soft G-Metric Spaces
Definition 2.1. Let X be a nonempty set and E be the nonempty set of parameters. A mapping
e : SE(X)
e × SE(X)
e × SE(X)
e → R(E)∗ is said to be a soft generalized metric or soft G-metric on X
e
G
e
if G satisfies the following conditions:
e1 )
(G
e2 )
(G
e3 )
(G
e4 )
(G
e5 )
(G
e x, ye, ze) = 0 if x
G(e
e = ye = ze.
e
e with x
e
0<G(e
x, x
e, ye) for all x
e, ye ∈ SE(X)
e 6= ye.
e x, x
e x, ye, ze) for all x
e with ye 6= ze.
e G(e
G(e
e, ye)≤
e, ye, ze ∈ SE(X)
e
e
e
G(e
x, ye, ze) = G(e
x, ze, ye) = G(e
y , ze, x
e) = · · · .
e
e
e
e
e
G(e
x, ye, ze)≤G(e
x, e
a, e
a) + G(e
a, ye, ze) for all x
e, ye, ze, e
a ∈ SE(X).
e with a soft G-metric G
e on X
e is said to be a soft G-metric space and is denoted by
The soft set X
e G,
e E).
(X,
e : SE(X)
e ×
Example 2.2. Let X be a nonempty set and E the nonempty set of parameters. We define G
∗
e × SE(X)
e → R(E) by
SE(X)

 0, if all of the variables are equal;
e
1, if two of the variables are equal, and the remaining one is distinct;
G(e
x, ye, ze) =

2, if all of the variables are distinct.
e Then G
e satisfies all the soft G-metric axioms.
for all x
e, ye, ze ∈ SE(X).
e G,
e E) is symmetric if
Definition 2.3. A soft G-metric space (X,
e 6 ) G(e
e x, ye, ye) = G(e
e x, x
e
(G
e, ye) for all x
e, ye ∈ SE(X).
e G,
e E) given in Example 2.2 is also symmetric.
Remark 2.4. A soft G-metric space (X,
Remark 2.5.
(a) Parameterized family of crisp G-metrics {Gλ : λ ∈ E} on a crisp set X may not be soft G-metric on
e
the soft set X.
(b) Any soft G-metric is not just a parameterized family of crisp G-metrics on a crisp set X.
The following example relates to Remark 2.5.
Example 2.6.
(a) Let X = {a, b}, G(a, a, a) = G(b, b, b) = 0, G(a, a, b) = 1, and G(a, b, b) = 2. Then, we know that
e = {e
G : X × X × X → R+ is a G-metric, from Example 1 of [13]. Let E = {e1 , e2 } and SE(X)
x, ye, ze, e
t}
be defined by x
e(e1 ) = a, x
e(e2 ) = a, ye(e1 ) = b, ye(e2 ) = b, ze(e1 ) = b, ze(e2 ) = a, e
t(e1 ) = a, e
t(e2 ) = b.
e : SE(X)
e × SE(X)
e × SE(X)
e → R(E)∗ where G(e
e x, ye, ze)(λ) = G(e
Then, G
x(λ), ye(λ), ze(λ)) for each
λ ∈ E, is not a soft G-metric.
e = {e
(b) Let X = {a, b}, E = {e1 , e2 } and SE(X)
x, ye, ze, e
t} with x
e(e1 ) = a, x
e(e2 ) = a, ye(e1 ) = b, ye(e2 ) = b,
e
e
ze(e1 ) = a, ze(e2 ) = b, t(e1 ) = b, t(e2 ) = a. Consider the soft G-metric of Example 2.2. We decompose
e Then,
e parameter wise as G(e
e x, ye, ze)(λ) = Gλ (e
G
x(λ), ye(λ), ze(λ)) for each λ ∈ E and x
e, ye, ze ∈ SE(X).
+
Gλ : X × X × X → R is not a well defined mapping. So, it is not G-metric on X.
e : SE(X)
e × SE(X)
e × SE(X)
e →
Theorem 2.7. If {Gλ : λ ∈ E} are G-metrics on X and if for λ ∈ E, G
∗
e
e
R(E) is defined by G(e
x, ye, ze)(λ) = Gλ (e
x(λ), ye(λ), ze(λ)) where x
e(λ), ye(λ), ze(λ) are constant, then G is a
e
soft G-metric on X.
Proof. The proof is obvious since {Gλ : λ ∈ E} are G-metrics on X.
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
889
e be a soft G-metric on X.
e If for (r, s, m) ∈ X × X × X and λ ∈ E, {G(e
e x, ye, ze)(λ) :
Theorem 2.8. Let G
+
x
e(λ) = r, ye(λ) = s, ze(λ) = m} is a singleton set and if for λ ∈ E, Gλ : X × X × X → R is defined by
e x, ye, ze)(λ) for x
e then Gλ is a G-metric on X.
Gλ (e
x(λ), ye(λ), ze(λ)) = G(e
e, ye, ze ∈ SE(X),
Proof. The proof is clear from the hypothesis and soft G-metric axioms.
e G,
e E) be a soft G-metric space; then the following hold for all x
e
Proposition 2.9. Let (X,
e, ye, ze, e
a ∈ SE(X).
e x, ye, ze) = 0, then x
(a) If G(e
e = ye = ze.
e
e
e x, x
e
(b) G(e
x, ye, ze)≤G(e
x, x
e, ye) + G(e
e, ze).
e
e
e
(c) G(e
x, ye, ye)≤2G(e
y, x
e, x
e).
e x, ye, ze)≤
e x, e
e a, ye, ze).
e G(e
(d) G(e
a, ze) + G(e
2
e x, ye, ze)≤
e x, ye, e
e x, e
e a, ye, ze)).
e (G(e
(e) G(e
a) + G(e
a, ze) + G(e
3
e x, ye, ze)≤
e x, e
e y, e
e z, e
e G(e
(f ) G(e
a, e
a) + G(e
a, e
a) + G(e
a, e
a).
e
e
e
e
e
(g) |G(e
x, ye, ze) − G(e
x, ye, e
a)|≤ max{G(e
a, ze, ze), G(e
z, e
a, e
a)}.
e x, ye, ze) − G(e
e x, ye, e
e x, e
e G(e
(h) |G(e
a)|≤
a, ze).
e x, ye, ze) − G(e
e y , ze, ze)|≤
e x, ze, ze), G(e
e z, x
e max{G(e
(i) |G(e
e, x
e)}.
e x, ye, ye) − G(e
e y, x
e y, x
e x, ye, ye)}.
e max{G(e
(j) |G(e
e, x
e)|≤
e, x
e), G(e
Proof.
e x, x
e x, ye, ze) from (G
e 2 ) and
e G(e
e G(e
(a) Case 1: Let all of the variables be distinct. Then, we have 0≤
e, ye)≤
e
(G3 ), respectively.
Case 2: Let two of the variables be equal and the remaining one be distinct. Thus, we have
e
e 4 ) and (G
e 2 ). From the two cases, we obtain G(e
e x, ye, ze) 6= 0.
e by (G
G(e
x, ye, ze)>0
e
e
e
e
e
e x, ye, ze) from (G
e 4 ) and (G
e 5 ).
e G(e
(b) We have G(e
x, x
e, ye) + G(e
x, x
e, ze) = G(e
y, x
e, x
e) + G(e
x, x
e, ze)≥
y, x
e, ze) = G(e
e x, ye, ye)≤
e x, x
e x, x
e x, x
e x, ye, ye)≤2
e y, x
e G(e
e G(e
(c) By using (b), we have G(e
e, ye) + G(e
e, ye) = 2G(e
e, ye). Thus, G(e
e, x
e)
e
from (G4 ).
e x, ye, ze)≤
e a, e
e a, ye, ze)≤
e a, x
e a, ye, ze) from
e G(e
e G(e
(d) Case 1: Let x
e 6= ze. Then, we have G(e
a, x
e) + G(e
e, ze) + G(e
e
e
e
e
e
e
e
e
(G5 ), (G4 ) and (G3 ), respectively. So, we obtain G(e
x, ye, ze)≤G(e
x, e
a, ze) + G(e
a, ye, ze) by (G4 ).
e x, ye, ze) = G(e
e x, x
e x, ye, e
e x, ye, e
e G(e
e G(e
Case 2: Let x
e = ze and ye 6= e
a. Thus, we get G(e
e, ye)≤
a) ≤
a) +
e
e
e
e
e
G(e
x, e
a, x
e) = G(e
a, ye, x
e) + G(e
x, e
a, x
e) from (G4 ) and (G3 ).
Case 3: Let x
e = ze and ye = e
a. It is obvious.
e 4 ), we have 3G(e
e x, ye, ze)≤2
e x, ye, e
e x, e
e a, ye, ze). Thus,
e G(e
(e) By (d) and (G
a) + 2G(e
a, ze) + 2G(e
e x, ye, ze)≤
e x, ye, e
e x, e
e a, ye, ze)).
e 2 (G(e
G(e
a) + G(e
a, ze) + G(e
3
e 5 ) and (b), we obtain 3G(e
e x, ye, ze)≤3
e x, e
e y, e
e z, e
e x, ye, ze)≤
e G(e
e
(f) By (G
a, e
a) + 3G(e
a, e
a) + 3G(e
a, e
a). Hence, G(e
e
e
e
G(e
x, e
a, e
a) + G(e
y, e
a, e
a) + G(e
z, e
a, e
a).
e
e
e a, x
e 5 ). Then, G(e
e x, ye, ze) − G(e
e a, x
e z, e
e
e G(e
e
(g) We have G(e
x, ye, ze)≤G(e
z, e
a, e
a) + G(e
e, ye) from (G
e, ye) ≤
a, e
a)≤
e
e
max{G(e
z, e
a, e
a), G(e
a, ze, ze)}. In a similar way, we get
e z, e
e a, ze, ze)}≤
e a, ze, ze)≤
e x, ye, ze) − G(e
e a, x
e − G(e
e G(e
− max{G(e
a, e
a), G(e
e, ye). Thus, we obtain
e x, ye, ze) − G(e
e x, ye, e
e a, ze, ze), G(e
e z, e
e max{G(e
|G(e
a)|≤
a, e
a)}.
e 4 ).
(h) The proof is clear, by (d) and (G
e 5 ) and (G
e 4 ).
(i) It is obvious, by (G
e
(j) It follows from (c) and (G4 ).
e G,
e E) be a soft G-metric space; then the following are equivalent.
Proposition 2.10. Let (X,
e G,
e E) is symmetric.
(a) (X,
e
e x, ye, e
e
e G(e
(b) G(e
x, ye, ye)≤
a) for all x
e, ye, e
a ∈ SE(X).
e x, ye, ze)≤
e x, ye, e
e z , ye, eb) for all x
e
e G(e
(c) G(e
a) + G(e
e, ye, ze, e
a, eb ∈ SE(X).
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
890
Proof.
e G,
e E) is symmetric, we have G(e
e x, ye, ye) = G(e
e x, x
(a) ⇒ (b) Since (X,
e, ye).
e
e
e
e
Case 1: Let x
e 6= e
a. We have G(e
x, ye, ye)≤G(e
x, ye, e
a) by (G3 ).
e 4 ).
Case 2: Let x
e=e
a. It is clear by (G
e
e
e y , ye, ze)≤
e x, ye, e
e z , ye, eb) by using Proposition 2.9(b)
e
e G(e
(b) ⇒ (c) We obtain G(e
x, ye, ze)≤G(e
y , ye, x
e) + G(e
a) + G(e
and the hypothesis.
e 4 ), we have G(e
e x, ye, ye)≤
e x, ye, x
e y , ye, ye) = G(e
e x, ye, x
e x, x
e G(e
(c) ⇒ (a) By hypothesis and (G
e) + G(e
e) = G(e
e, ye). In
e
e
e
a similar way, we get G(e
y, x
e, x
e)≤G(e
x, ye, ye).
e we can construct a soft G-metric by the following mappings
Proposition 2.11. For any soft metric d on X,
e s and G
em .
G
e s (d)(e
(a) G
x, ye, ze) = 13 (d(e
x, ye) + d(e
y , ze) + d(e
x, ze)),
e
(b) Gm (d)(e
x, ye, ze) = max{d(e
x, ye), d(e
y , ze), d(e
x, ze)}.
Proof.
e 1 ), (G
e 2 ), and (G
e 4 ) are obvious. The proof of (G
e 5 ) follows from
(a) Since d is a soft metric, the proofs of (G
(M 4).
e 3 ) Case 1: Let ye 6= ze and x
e s (d)(e
(G
e = ye. Since G
x, x
e, ye) = 0, the assertion is clear.
e
e s (d)(e
Case 2: Let x
e = ze, ze 6= ye, and x
e 6= ye. Then Gs (d)(e
x, x
e, ye) = G
x, ye, ze).
e x, ye) + d(e
x, ye)≤d(e
x, ze) + d(e
z , ye).
Case 3: Let x
e 6= ze, ze 6= ye, and x
e 6= ye. From (M4), we have 2d(e
2
1
e
e
e
Then, Gs (d)(e
x, x
e, ye) = 3 d(e
x, ye)≤ 3 [d(e
x, ye) + d(e
x, ze) + d(e
z , ye)] = Gs (d)(e
x, ye, ze).
e
e
e
(b) The proofs of (G1 ), (G2 ) and (G4 ) are obvious.
e 3 ) Let ze 6= ye. G
e m (d)(e
e m (d)(e
e max{d(e
(G
x, x
e, ye) = d(e
x, ye)≤
x, ye), d(e
x, ze), d(e
y , ze)} = G
x, ye, ze).
e
e
e m (d)(e
e
(G5 ) Case 1: Gm (d)(e
x, ye, ze) = d(e
x, ye). From (M4), we have d(e
x, ye)≤d(e
x, e
a) + d(e
a, ye) = G
x, e
a, e
a) +
e m (d)(e
e m (d)(e
eG
d(e
a, ye)≤
x, e
a, e
a) + G
a, ye, ze).
The other cases can be proved in a similar way.
e on X,
e we can construct a soft metric d0 G
e on X
e defined by
Proposition 2.12. For any soft G-metric G
e x, ye, ye) + G(e
e x, x
dGe (e
x, ye) = G(e
e, ye)
e 1 ) and (G
e 2 ), (G
e 4 ) and (G
e 5 ), respectively.
Proof. The proofs of (M1), (M3), and (M4) obviously follow from (G
e
e
e x, ye, ye)≤0
e by
(M2) Let dGe (e
x, ye) = 0. Assume x
e 6= ye. Since G(e
x, ye, ye) + G(e
x, x
e, ye) = 0, we would have G(e
e 2 ). Thus, x
Proposition 2.9 (c). This contradicts to (G
e = ye. The converse is clear.
e then the following hold.
Proposition 2.13. Let d be a soft metric on X;
(a) dGes (d) (e
x, ye) = 43 d(e
x, ye).
(b) dGem (d) (e
x, ye) = 2d(e
x, ye).
e m , and G
es .
Proof. The proofs follow from the definitions of dGe , G
e be a soft G-metric on X;
e then the following hold.
Proposition 2.14. Let G
e s (d e )(e
e x, ye, ze)≤
e x, ye, ze).
eG
e G(e
(a) G(e
x, ye, ze)≤2
G
e x, ye, ze)≤
e m (d e )(e
e x, ye, ze).
eG
e G(e
(b) G(e
x, ye, ze)≤3
G
e x, ye, ze)≤
e 1 [d e (e
(a) By Proposition 2.9 (b), we have G(e
e)+dGe (e
y , ze)+dGe (e
x, ze)]. On the other hand,
3 G x, y
1
e x, ye, ze).
e
e G(e
by Proposition 2.9 (c) and (G3 ), we obtain 3 [dGe (e
x, ye) + dGe (e
y , ze) + dGe (e
x, ze)]≤2
Proof.
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
891
e x, ye, ze)≤
e m (d e )(e
e 3 ),
eG
(b) By Proposition 2.9 (b), we obtain G(e
x, ye, ze). Also, from Proposition 2.9 (c) and (G
G
e m (d e )(e
e x, ye, ze).
e G(e
we get G
x, ye, ze)≤3
G
e G,
e E) be a soft G-metric space. For e
e and re>0,
e
e the G-ball
Definition 2.15. Let (X,
a ∈ SE(X)
with a center
e
a and radius re is
e | G(e
e a, x
e
e r} ⊆ SE(X).
BGe (e
a, re) = {e
x ∈ SE(X)
e, x
e)<e
e
SS(BGe (e
a, re)) will be called a soft G-ball
with center e
a and radius re.
e G,
e E) given in Example 2.2. We have
Example 2.16. Consider the soft G-metric space (X,
BGe (e
a, re) =
e
e
SE(X),
re>1
e
for any e
a ∈ SE(X).
e
{e
a},
re≤1
e G,
e E) be a soft G-metric space. For e
e and re>0,
e the following hold.
Proposition 2.17. Let (X,
a ∈ SE(X)
e a, x
e r, then x
(a) If G(e
e, ye)<e
e, ye ∈ BGe (e
a, re).
e ⊆ B e (e
e such that BGe (e
(b) If x
e ∈ BGe (e
a, re), then there exists a δe>0
x, δ)
a, re).
G
Proof.
e 3 ).
(a) It obviously follows from (G
e Then, we have G(e
e a, ye, ye)≤
e a, x
e x, ye, ye)<
e G(e
e
(b) Let x
e ∈ BGe (e
a, re). Assume that ye ∈ BGe (e
x, δ).
e, x
e) + G(e
e a, x
e 5 ). Say δe = re − G(e
e a, x
e a, ye, ye)<e
e r. Hence, ye ∈ BGe (e
G(e
e, x
e) + δe by (G
e, x
e). Thus, we get G(e
a, re).
e G,
e E) be a soft G-metric space. For x
e and re>0,
e we have
Proposition 2.18. Let (X,
f0 ∈ SE(X)
1
BGe (f
x0 , re) ⊆ BdGe (f
x0 , re) ⊆ BGe (f
x0 , re)
3
Proof. It obviously follows from Proposition 2.9 (c).
3. Soft G-convergence
e G,
e E) be a soft G-metric space and (f
e The
Definition 3.1. Let (X,
xn ) a sequence of soft elements in X.
e
e there exists a natural number
sequence (f
xn ) is said to be soft G-convergent at x
e in X if for every e
≥0,
e xn , x
e G(f
e whenever n ≥ N i.e., n ≥ N ⇒ (f
N=N(e
) such that 0≤
fn , x
e) <e
xn ) ∈ BGe (e
x, e
). We denote this by
x
fn → x
e as n → ∞ or by limn→∞ (f
xn ) = x
e.
e G,
e E) be a soft G-metric space. For a sequence (f
e and a soft element x
Proposition 3.2. Let (X,
xn ) in X
e
the following are equivalent:
(a) (f
xn ) is soft G-convergent to x
e.
(b) dGe (f
xn , x
e) → 0 as n→ ∞.
e
(c) G(f
xn , x
fn , x
e) → 0 as n→ ∞.
e
(d) G(f
xn , x
e, x
e) → 0 as n→ ∞.
e xn , xf
(e) G(f
e) → 0 as n, m → ∞.
m, x
Proof. That (a) implies (b) follows from Proposition 2.18. The other implications can be proved by using
Propositions 2.9 and 2.12.
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
892
f0 , G
f0 , E 0 ) be two soft G-metric spaces. Then a function f: X
f0 is
e G,
e E), (X
e →X
Definition 3.3. Let (X,
e
e
e
e there exists δ ≥0 such that
a soft G-continuous at a soft element e
a ∈ SE(X) if and only if for every e
>0,
g
g
g
0
f
e
e
e
e
e
e
x
e, ye∈X and G(e
a, x
e, ye)<δ implies that G (f (a), f (x), f (y))<e
.
e
A function f is soft G-continuous if and only if it is soft G-continuous at all e
a ∈ SE(X).
f0 , G
f0 , E 0 ) be two soft G-metric spaces. Then a function f: X
f0 is a
e G,
e E), (X
e →X
Proposition 3.4. Let (X,
e if and only if it is soft G-sequentially continuous at a soft
soft G-continuous at a soft element e
a ∈ SE(X)
e i.e., whenever (f
element e
a ∈ SE(X),
xn ) is soft G-convergent to e
a, (f^
(xn )) is soft G’-convergent to fg
(a).
Proof. Necessity. Assume that f is soft G-continuous. Given x
fn → e
a, we wish to show that f^
(xn ) → fg
(a).
e and
e such that x
e By hypothesis, there exists a δe≥0
eX
Let e
>0.
e, ye∈
f0 (fg
e a, x
e δe ⇒ G
e .
G(e
e, ye)<
(a), fg
(x), fg
(y))<e
e
e where δ(λ)
= δλ , there exists a natural number N such that
Since x
fn → e
a corresponds to δe>0,
e a, x
e a, x
e δe ⇒ G(e
n ≥ N ⇒ G(e
fn , x
fn )<
fn , x
fn )(λ) < δλ .
Hence we have
f0 (fg
e fg
e ⇒ G(
n≥N ⇒G
(a), f^
(xn ), f]
(xn ))<e
(a), f^
(xn ), f]
(xn ))(λ) < λ .
This implies that f^
(xn ) → fg
(a).
Sufficiency. This can be clearly proved assuming that f is not soft G-continuous at a soft element e
a.
e G,
e E) be a soft G-metric space having at least two soft elements. Then (X,
e G,
e E)
Definition 3.5. Let (X,
is said to poses soft G-Hausdorff property if for any two soft elements x
e, ye such that x(λ) 6= y(λ) for all λ
∈ E, there are two soft open balls BGe (e
x, re) and BGe (e
y , re) with centers x
e and ye respectively, such that
BGe (e
x, re) ∩ BGe (e
y , re) = e
∅.
Theorem 3.6. Every soft G-metric space poses soft G-Hausdorff property.
e G,
e E) be a soft G-metric space having at least two soft elements. Let x
Proof. Let (X,
e, ye be two soft
e
e x, x
e x, x
e i.e., G(e
elements in X such that x(λ) 6= y(λ) for all λ ∈ E. Then G(e
e, ye)>0
e, ye)(λ) > 0 for all λ ∈ E.
e x, x
e such that
eX
Let us choose a real number e
(λ) = 12 G(e
e, ye)(λ) for each λ ∈ E. Suppose that there exists ze∈
e z , ze, x
e z , ze, ye)<e
e and G(e
e . By (b) of Proposition 2.9, we have
ze ∈ BGe (e
x, e
) and ze ∈ BGe (e
y, e
). Then G(e
e)<e
e x, ye, ze)≤
e z , ze, x
e z , ze, ye)<e
e G(e
e + e
G(e
e) + G(e
= 2e
.
It is a contradiction. So, X poses the soft G-Hausdorff property.
Theorem 3.7. If a sequence in a soft G-metric space has a limit, then it is unique.
Proof. the assertion is similar to that of Theorem 3.6.
e G,
e E) be a soft G-metric space. Let T : (X,
e G,
e E) → (X,
e G,
e E) be a mapping. If
Definition 3.8. Let (X,
e such that T(f
there exists a soft element x
f0 ∈ SE(X)
x0 ) = x
f0 , then x
f0 is called a fixed point of T .
e G,
e E) be a soft G-metric space and T : (X,
e G,
e E) → (X,
e G,
e E) a mapping. For
Definition 3.9. Let (X,
e we can construct the sequence x
every x0 ∈ SE(X),
fn of soft elements by choosing x
f0 and continuing by:
2
n
x
f1 = T (f
x0 ), x
f2 = T (f
x1 ) = T (f
x0 ),. . . , x
fn = T (]
xn−1 ) = T (f
x0 ). We say that the sequence is constructed by
iteration method.
e G,
e E) be a soft G-metric space and T : (X,
e G,
e E) → (X,
e G,
e E) a mapping satisfying
Theorem 3.10. Let (X,
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
893
e x
e x, T x
e y , T ye, T ye) + cG(e
e z , T ze, T ze) for all x
e where 0 <
e aG(e
ea+
(i) G(T
e, T ye, T ze) ≤
e, T x
e) + bG(e
e, ye, ze ∈ SE(X)
e1
b + c<
e
(ii) T is soft G-continuous at a soft element u
e ∈ SE(X).
n
e
(iii) There is an x
e ∈ SE(X) such that T (e
x) has a subsequence T ni (e
x) soft G-converging to u
e.
Then u
e is a unique fixed point. (i.e., T u
e=u
e).
Proof. T ni+1 (e
x) is soft G-convergent to T (e
u) by (ii) and Proposition 3.4. Assume contrary: that T u
e 6= u
e i.e.,
e
Tu
e(λ0 ) 6= u
e(λ0 ) for a λ0 ∈ E. Consider the two soft G-balls B1 = SS(BGe (e
u, e
)) and B2 = SS(BGe (T u
e, e
))
1
ni
ni+1
e
e
u, T u
e, T u
e), G(e
u, u
e, T u
e)}. We would have T (e
x) → u
e and T
(e
x) → T u
e, so that there
where e
< 6 min{G(e
would exist an N such that T ni (e
x) ∈ B1 and T ni+1 (e
x) ∈ B2 . Hence we would have
e ni (e
G(T
x), T ni+1 (e
x), T ni+1 (e
x)) > e
, ∀i > N.
(3.1)
We would have from (i),
e ni (e
e ni+1 (e
e aG(T
x), T ni+1 (e
x), T ni+1 (e
x)) +
G(T
x), T ni+2 (e
x), T ni+3 (e
x)) ≤
ni+1
ni+2
ni+2
e
bG(T
(e
x), T
(e
x), T
(e
x)) +
(3.2)
e ni+2 (e
cG(T
x), T ni+3 (e
x), T ni+3 (e
x))
e 3 ) of the definition of the soft G-metric, we would obtain
but, by axiom (G
e ni+1 (e
e ni+1 (e
e G(T
G(T
x), T ni+2 (e
x), T ni+2 (e
x))≤
x), T ni+2 (e
x), T ni+3 (e
x))
(3.3)
e ni+2 (e
e ni+1 (e
e G(T
G(T
x), T ni+3 (e
x), T ni+3 (e
x))≤
x), T ni+2 (e
x), T ni+3 (e
x)).
(3.4)
e ni+1 (e
e ni (e
e G(T
G(T
x), T ni+2 (e
x), T ni+3 (e
x))≤q
x), T ni+1 (e
x), T ni+1 (e
x))
(3.5)
Then
e
where q = a/(1 − (b + c)) and q <1.
Hence by inequality(3.3) and (3.5), we would get
e ni+1 (e
e ni (e
e G(T
G(T
x), T ni+2 (e
x), T ni+2 (e
x))≤q
x), T ni+1 (e
x), T ni+1 (e
x))
(3.6)
For l > j > N and by (3.5), we would have
e nl (e
e nl −1 (e
e q G(T
x), T nl +1 (e
x), T nl +1 (e
x)) ≤
x), T nl (e
x))
G(T
x), T nl (e
nl −2
nl −1
2
e
e
≤ q G(T
(e
x), T
(e
x), T nl −1 (e
x))
nj
nj +1
n
−n
e
j
e
e
l
≤ . . . ≤q
x)).
G(T (e
x), T
(e
x), T nj +1 (e
(3.7)
e nl (e
e which contradicts (3.1), hence T u
So, as l → ∞ we would have lim G(T
x), T nl +1 (e
x), T nl +1 (e
x))≤0
e=u
e.
To prove uniqueness, suppose that for a ve 6= u
e we have T ve = ve. Then
e u, ve, ve) = G(T
e (e
e u, T u
e v , T ve, T ve) = 0,
e aG(e
G(e
u), T (e
v ), T (e
v )) ≤
e, T u
e) + (b + c)G(e
which implies that u
e=e
v.
4. Conclusion
We introduced soft G-metric spaces as generalizations of G-metric spaces and soft G-metric spaces, and
discussed the result about existence and uniqueness of fixed point in the setting of these spaces. We believe
that this study will help researchers to upgrade and support the further studies on soft topology, soft metric
and fixed point theory to carry out a general framework for their applications in real life.
Acknowledgements
The authors are grateful to the referees for their valuable comments and suggestions in improving the
paper.
A. C. Guler, E. D. Yildirim, O. B. Ozbakir, J. Nonlinear Sci. Appl. 9 (2016), 885–894
894
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