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Products of Generalized Probabilistic Metric

Spaces

Ahmed Khan,

Department of General Studies, Jubail Industrial College, Royal Commission for Jubail, Jubail Industrial City, Saudi Arabia

Khan_a@jic.edu.sa

Abstract— In this paper we have studied and expanded the theory of generalized probabilistic metric spaces. Main emphasis is given on the results concerning the products of generalized probabilistic

metric spaces.

Keywords—Generalized probabilistic metric space PG – equilateral space PG – Simple space.

I. INTRODUCTION

Probabilistic generalization of metric space was first introduced by Karl Menger

 

8 in 1942 and almost at the same time was studied by A. Wald

 

15 . The study of Probabilistic metric space expanded rapidly by pioneering works of Schweizer and Sklar

 

12 along

with many other mathematicians (see the references). In 1963, S. Gaḧlar

 

6 , introduced the concept of

2-metric space which was supposed to be the generalization of usual notion of metric space. This concept was studied by many mathematicians and later generalized to probabilistic 2-metric space. Very recently, Mustafa and Sims

 

10 introduced a perfect deterministic generalization of metric space called generalized metric space.(G – Metric Space). On the pattern of Manger’s probabilistic metric space, Zhou, Wang, Ciric and Alsulami

 

16 introduced probabilistic version of G – metric space. Keeping in view that it is the frontier branch between probability theory and functional analysis, we have tried to extend the theory of Probabilistic G – Metric Space in the present work.

II. PRELIMINARIES

Definition 2.1: A distribution function is a real valued

non-decreasing function F defined on R, with F(−∞) = 0 andF(−∞) = 1. The set of all distribution functions that are left continuous on (−∞, ∞)may be denoted as Δ.

For any aєR, the family of distribution functions ɛa may be defined as

ɛa(x) = {0 ,1 , x ≤ ax > a

A distance distribution function (briefly, a d. d. f.) is a non-decreasing function F defined on R+ , that satisfies F(0) = 0 and F(∞) = 1 and is left continuous on (0, ∞).

For any distribution function F, and any x > 0, F ( x

0 ) = 1 and F ( 0 0 ) = 0.

Definition 2.2: A function T: [0,1] × [0,1]n to [0,1] is called continuous t-norm, if it satisfies the following:

(T1) T(a, 1) = a for all a є [0,1];

(T2)T is commutative and associative, i.e. T(a, b) = T(b, a) and T(T(a, b), c) = T(a, T(b, c)) , for all a, b and c є [0,1];

(T3) T is continuous ;

(T4) T(a, b) ≥ T(c, d) for a ≥ c and b ≥ d and for all a, b, c and d є [0,1].

Here, we list some t-norms in order of decreasing "strength", where T” is said to be stronger than T if T”(a, b) ≥ T(c, d)for all a, b, c and d є [0,1] with strict inequality for at least one (a, b).

1. M(a, b) = Min(a, b) . 2. P(a, b) = a × b .

3. W(a, b) = Max(a + b − 1,0) .

Recently, Mustafa and Sims

 

10 introduced a deterministic generalization of metric space called generalized metric space ( G – metric space) as follows:

Definition 2.3:Let S be a non-empty set and G: S × S × S → R+be a function satisfying the following conditions:

(G1) G(u, v, w) = 0, if u = v = w for all u, v, w є S ; (G2)0 < G(u, u, v) for all u, v є S with u ≠ v; (G3) G(u, u, v) ≤ G(u, v, for all u, v, w є S with w ≠ v;

(G4) G is invariant under all permutations of u, v and w;

(G5)G(u, u, w) ≤ G(u, a, a) + G(a, v, w) for all u, v, w, and a є S .

Then G is called generalized metric or G – metric On S and the ordered pair (S, G) is called

G – metric space.

On the pattern of Manger’s probabilistic metric

space, Zhou, Wang, Ciric and Alsulami

 

16

introduced probabilistic version of G – metric space.

Here, we discuss the probabilistic generalization generalized metric space in detail:

Consider the ordered pair (S, 𝒯)consisting of non-empty set S and a mapping 𝒯 fromS × S × S to ∆, collection of distribution functions. The value of 𝒯 at (u, u, v) belonging to S × S × S is represented

by Gu,v,w∗ . The function Gu,v,w∗ is assumed to satisfy

(2)

(P1) Gu,v,w(x) = 1 for all x > 0 and for all u, v, w є S if and only if u = v = w ;

(P2) Gu,u,v∗ (x) ≥ Gu,v,w∗ (x) for all x > 0 and for all

u, v, w є S with v ≠ w;

(P3) Gu,v,w(x) = G

u,w,v

(x) = G v,u,w

(x) = ⋯

(Symmetry in all three variables) (P4) If Gu,a,a(x) = 1 and G

a,v,w

(y) = 1 for all

y > 0, u, v, w and a є S .Then Gu,v,w(x + y) = 1.

In every G – metric space(S, G) , metric G induces a mapping 𝒯: S × S × S → ∆ such that

𝒯(u, v, w)(x) = Gu,v,w(x) = ɛ

0(x − G(u, v, w))for every triplet (u, u, v) in S × S × S.

Clearly:

(i) Gu,v,w∗ (x) = ɛ0(x − G(u, v, w))

= ɛ0(x) if and only if u = v = w ⇒ Gu,v,w∗ (x) = 1 , for all x > 0

if and only if u = v = w. (ii) Gu,u,v∗ (x) = ɛ0(x − G(u, u, v))

≥ ɛ0(x − G(u, v, w)) [Since G(u, u, v) ≤ G(u, v, w) when w ≠ v ] ⇒ Gu,u,v∗ (x) ≥ Gu,v,w∗ (x) , for all u, v, w є S with w ≠ v;

(iii) Proof of (P3) is straight forward, so we omit the details.

(iv) Gu,a,a∗ (x) = 1 ⇒ x − G(u, a, a) > 0 (v) ⇒ G(u, a, a) < x

Ga, v, w ∗ (y) = 1 ⇒ y − G(a, v, w) > 0 ⇒ G(a, v, w) < y

G(u, u, w) ≤ G(u, a, a) + G(a, v, w) ⟹ G(u, v, w) < x + y

Thus, Gu,v,w(x + y) = ɛ

0(x + y − G(u, v, w)) = 1 . In fact (P4) is a minimal generalization of triangle inequality, which may be interpreted as: “Perimeter of the triangle with vertices u, a, a, is less than x and perimeter of the triangle with vertices a, v, w is less than y then perimeter of the triangle with vertices u, v, w must be certainly less than x +y”. However the most appropriate probabilistic generalization of triangle inequality of G – metric space was introduced in

 

17 .

(P5) Gu,v,w(x + y) ≥ T(G u,a,a ∗ (x), G

a,v,w

(y))

for all x, y > 0 and u, v, w, a є S.

We call the ordered pair (S, 𝒯) as probabilistic G – metric space (briefly PGM space) if (P1), (P2) , (P3) and (P4) are satisfied. (S, 𝒯, T) is called Menger probabilistic G – metric space (briefly MPGM space) if (P1), (P2) , (P3) and (P5) are satisfied.

Concept of neighborhoods in MPGM space is defined as follows:

Definition 2.4: Let, (S, G∗, T) be a Menger probabilistic G − metric space and u0be any point in S . For any ɛ > 0 and λ with0 < λ < 1, an ( ɛ, λ ) neighborhood of u0 is the set of all points v in S for which

Gu0,v,v(ɛ) ≥ 1 − λ and G v,u0,u0

(ɛ) ≥ 1 − λ.

In

 

16 topology, convergence and completeness have been discussed.

III. PRODUCT SPACES

Products of probabilistic metric spaces was first discussed by V. Istratescu and I Vaduva

 

7 . Later A. Xavier

 

15 , S., Zertaj, and A., Khan

 

13 and many others studied the results concerning products of probabilistic metric as well as 2-metric spaces. In this section, we have tried to define and establish someresults regarding products of probabilistic G – metric spaces.

Definition 3.1: Let (S1 , 𝒯1) and (S2 , 𝒯2) be probabilistic G – metric spaces (briefly PGM spaces) and let T be left – continuous t – norm. Then T – product (S1 , 𝒯1) × (S2 , 𝒯2) of (S1 , 𝒯1) and (S2 , 𝒯2) Is the space (S1 × S2, T( 𝒯1 , 𝒯2)) where S1 × S2 is the Cartesian product of S1 and S2 and T( 𝒯1 , 𝒯2) is the mapping from (S1 × S2) × (S1 × S2) × (S1 × S2) into the set of distribution functions Δ defined as

T( 𝒯1 , 𝒯2) (u, v, w) = T(𝒯1(u1,v1,, w1), 𝒯2(u2,v2,, w2) for any u = (u1 , u2 ), v = (v1 , v2 ), w = (w1 , w2 ). We shall denote S1 × S2 by S and T( 𝒯1 , 𝒯2) by 𝒯T and finally we omit the reference to T and write 𝒯T(u, v, w) = Gu,v,w∗ .

Following theorem is the immediate consequence of theDefinition 3.1.

Theorem 3.1: T − product (S, 𝒯T) of two PGM space (S1 , 𝒯1) and (S2 , 𝒯2) is a PGM space.

Proof:

(P1 ) G u,v,w

(x) = 𝒯

T(u, v, w)(x)

= T(𝒯1(u1,v1,, w1), 𝒯2(u2,v2,, w2) = T(G1u1,v1,w1(x), G

2u2,v2,w2

(x) = 1

for all x > 0 and for all u, v, w ∊ S If and only if u1 = v1= w1 and u2 = u2 = w2

If and only if u = v = w ;

(P2 ) G u,u,v

(x) = T (G 1u1,u1,v1

(x), G 2u2,u2,v2

(x))

≥ T(G1u1,v1,w1

(x), G 2u2,v2,w2

(x)),

with v1 ≠ w1 and v2 ≠ w2 = Gv,v,w(x), for all x > 0

and for all u, v, w ∊ S with v ≠ w; (P3) G

u,v,w∗ (x) = Gu,w,v∗ (x) = Gv,u,w∗ (x) =. . . … … ..

(Symmetry in all three variables) We omit details of proof as it is easy and straight forward,

(P4∗) Gu,a,a∗ (x) = 1 and Ga,v,w∗ (y) = 1 for all x, y > 0

and u, v, w and a ∊ S. ⇒

T(G1u1,a1,a1

(x), (G 2u2,a2,a2

(x)) = 1

and T(G1a1,v1,w1

(y), (G 2a2,v2,w2

(y)) = 1

⇒ G1u1,a1,a1(x) = G 2u2,a2,a2

(x) =

(G1a1,v1,w1(y) = (G 2a2,v2,w2

(3)

⇒ G1u1,v1,w1(x + y) = 1 and G 2u2,v2,w2

(x + y) = 1

⇒ Gu,v,w∗ (x + y) = T(G1u1,u1,v1

(x + y), (G 2u2,u2,v2

(x + y)

= 1 .

Theorem 3.2: If (𝑆1 , 𝒯1, 𝑇) and (𝑆2 , 𝒯2, 𝑇)are Menger probabilistic 𝐺 – metric space under same left – continuous t – normT, then their T – product is a Menger probabilistic 𝐺 – metric space.

Proof: Here, we only need to prove (𝑃5) 𝐺

𝑢,𝑣,𝑤∗ (𝑥 + 𝑦) ≥ 𝑇(𝐺𝑢,𝑎,𝑎∗ (𝑥), 𝐺𝑎,𝑣,𝑤∗ (𝑦))

𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣, 𝑤, 𝑎 ∈ 𝑆 𝑎𝑛𝑑 𝑥, 𝑦 > 0.

𝐺𝑢,𝑣,𝑤(𝑥 + 𝑦) = 𝑇 (𝐺

1∗𝑢1,𝑣1,𝑤1(𝑥 + 𝑦), 𝐺2∗𝑢2,𝑣2,𝑤2(𝑥 + 𝑦)) ≥

𝑇(𝑇(𝐺1𝑢1,𝑎1,𝑎1(𝑥), 𝐺

1∗𝑎1,𝑣1,𝑤1(𝑥)) , 𝑇(𝐺2∗𝑢2,𝑎2,𝑎2(𝑦), 𝐺2∗𝑎2,𝑣2,𝑤2(𝑦)) (Using(P5) and properties of t – norm)

𝑇(𝑇(𝐺1𝑢1,𝑎1,𝑎1(𝑥), 𝐺 2𝑢2,𝑎2,𝑎2

(𝑦)), 𝑇(𝐺 1𝑎1,𝑣1,𝑤1

(𝑥), (𝐺 2𝑎2,𝑣2,𝑤2

(𝑦))

= 𝑇(𝐺𝑢,𝑎,𝑎∗ (𝑥), (𝐺𝑎,𝑣,𝑤∗ (𝑦)).

Corollary 3.1: If (𝑆1 , 𝒯1, 𝑇1) and (𝑆2 , 𝒯2, 𝑇2) are Menger probabilistic 𝐺 – metric spaces, then their T – product is a Menger probabilistic 𝐺 – metric space under t – norm T if T is left continuous norm weaker than 𝑇1and 𝑇2.

We start our discussion regarding some particular

spaces. The simplest 𝐺 − metric spaces are

equilateral spaces in which

𝑓(𝑥) = { 𝐺(𝑢, 𝑣, 𝑤) = 0, 𝑖𝑓 𝑢 = 𝑣 = 𝑤 𝑎 > 0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Accordingly, we define Probabilistic Generalized equilateral space (𝑃𝐺 – equilateral space).

Definition 3.2: A PGM space (𝑆, 𝒯) is said to be 𝑃𝐺 – equilateral space if and only if there exists a 𝐺 – metric on S and a distribution function D such that 𝐷(0) = 0 and 𝐷(𝑥) > 0 for some 𝑥 > 0 in which 𝒯 is defined on 𝑆 × 𝑆 × 𝑆 by

𝒯(𝑢, 𝑣, 𝑤)(𝑥) = 𝐺𝑢,𝑣,𝑤(𝑥) = 𝜀

0(𝑥), 𝑖𝑓 𝑢 = 𝑣 = 𝑤 𝒯(𝑢, 𝑣, 𝑤)(𝑥) = 𝐺𝑢,𝑣,𝑤∗ (𝑥) = 𝐷(𝑥) , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. Following result insures that product of 𝑃𝐺 – equilateral spaces is of same type under certain conditions.

Theorem 3.3: If (𝑆1 , 𝒯1) and (𝑆2 , 𝒯2) are 𝑃𝐺 – equilateral space generated by the same distribution function D, then their Min – product (𝑆1 × 𝑆2, 𝒯𝑀𝑖𝑛) is

𝑃𝐺 – equilateral space generated by the same

distribution function D.

Proof: 𝐺𝑢,𝑣,𝑤∗ (𝑥) = 𝒯𝑇(𝑢, 𝑣, 𝑤)(𝑥)

= 𝑀𝑖𝑛(𝒯1(𝑢1,𝑣1, 𝑤1), 𝒯2(𝑢2,𝑣2, 𝑤2)) = 𝑀𝑖𝑛(𝐺1𝑢1 ,𝑣1,𝑤1(𝑥), (𝐺

2𝑢2 ,𝑣2,𝑤2

(𝑥))

𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 > 0 = 𝑀𝑖𝑛(𝜀0(𝑥), 𝜀0(𝑥))

= 𝜀0(𝑥), 𝑖𝑓 𝑢 = 𝑣 = 𝑤 ∈ 𝑆

In all the cases of 𝑢 = (𝑢1 , 𝑢2 ), 𝑣 = (𝑣1 , 𝑣2 ), 𝑤 = (𝑤1 , 𝑤2 ) not all equal, the result follows with the remark that 𝑇 − Min is a must.

𝐺𝑢,𝑣,𝑤(𝑥) = 𝑀𝑖𝑛(𝐺 1𝑢1,𝑣1,𝑤1

(𝑥), 𝐺 2 𝑢2,𝑣2,𝑤2

(𝑥))

= Min(D(𝑥), D(𝑥)) = D(𝑥)

(If 𝑢1 , 𝑣1, 𝑤1 are not all equal as well as 𝑢2 , 𝑣2 , 𝑤2 are also not all equal)

𝐺𝑢,𝑣,𝑤(𝑥) = 𝑀𝑖𝑛(𝜀

0(𝑥), 𝐷(𝑥))

(If 𝑢1 = 𝑣1= 𝑤1 but 𝑢2 , 𝑣2 , 𝑤2 are not equal) = Min(D(𝑥), D(𝑥)) = D(𝑥) Also, 𝐺𝑢,𝑣,𝑤(𝑥) = 𝑀𝑖𝑛(𝐷(𝑥), 𝜀

0(𝑥))

(If 𝑢1 , 𝑣1, 𝑤1 are not all equal but 𝑢2 = 𝑣2= 𝑤2 ) = 𝑀𝑖𝑛 (𝐷(𝑥), 𝐷(𝑥)) = 𝐷(𝑥)

It’s also necessary that both 𝑃𝐺 – equilateral spaces must be generated by the same distribution function D.

Now, we introduce the concept of Probabilistic Generalized simple spaces (𝑃𝐺 – simple spaces), another class of particular spaces.

Definition 3.3: A PGM space (𝑆, 𝒯) is said to be 𝑃𝐺 – simple space if and only if there exists a 𝐺 – metric on S and a distribution function D such that 𝐷(0) = 0 and 𝐷(𝑥) > 0 for some 𝑥 > 0 in which 𝒯 is defined on 𝑆 × 𝑆 × 𝑆 by

𝒯(u, v, w)(𝑥) = 𝐺𝑢,𝑣,𝑤(𝑥) = 𝜀

0(𝑥) 𝑖𝑓 𝑢 = 𝑣 = 𝑤 , 𝒯(u, v, w)(𝑥) = 𝐺𝑢,𝑣,𝑤∗ (𝑥) = 𝐷 (

𝑥

𝐺(𝑢,𝑣,𝑤)) , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

Theorem 3.4: 𝑃𝐺 – simple space is Menger probabilistic 𝐺 – metric space for 𝑇 = Min.

Proof: (𝑃1) 𝐺𝑢,𝑣,𝑤(𝑥) = 𝜀

0(𝑥) 𝑖𝑓 𝑢 = 𝑣 = 𝑤

⇒ 𝐺𝑢,𝑣,𝑤(𝑥) = 1 𝑖𝑓 𝑢 = 𝑣 = 𝑤 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 > 0.

(𝑃2) 𝐺𝑢,𝑢,𝑣∗ (𝑥) = 𝐷 ( 𝑥 𝐺(𝑢, 𝑢, 𝑣)) if 𝑎𝑛𝑑𝑣 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 .

≥ 𝐷 ( 𝑥

𝐺(𝑢,𝑢,𝑤))

𝑆𝑖𝑛𝑐𝑒, 𝐺 (𝑢, 𝑢, 𝑣) ≤ 𝐺(𝑢, 𝑣, 𝑤)𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ≠ 𝑤 ∈ 𝑆 = 𝐺𝑢,𝑣,𝑤(𝑥)𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 > 0. (𝑃3) 𝐺𝑢,𝑣,𝑤 (𝑥) = 𝐺

𝑢,𝑤,𝑣∗ (𝑥) =

𝐺𝑣,𝑢,𝑤∗ (𝑥) =. … … … . . (𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦 𝑖𝑛 𝑎𝑙𝑙 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠)

𝐼𝑡𝑠 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑠𝑜 , 𝑤𝑒 𝑜𝑚𝑖𝑡 𝑡ℎ𝑒 𝑑𝑒𝑡𝑎𝑖𝑙𝑠. (𝑃4) 𝐹𝑜𝑟 𝐺𝑢,𝑣,𝑤 (𝑥 + 𝑦)

≥ 𝑇 (𝐺 𝑢,𝑎,𝑎∗ (𝑥), 𝐺 𝑎,𝑣,𝑤∗ (𝑦)) 𝑤𝑒 𝑒𝑠𝑡𝑎𝑏𝑙𝑖𝑠ℎ 𝑡ℎ𝑎𝑡

𝐷 ( (𝑥+𝑦)

𝐺(𝑢,𝑣,𝑤)) ≥ 𝑚𝑖𝑛 (𝐷 (

𝑥

𝐺(𝑢,𝑎,𝑎)) , 𝐷 (

𝑦

𝐺(𝑎,𝑣,𝑤)))

𝑇𝑟𝑎𝑖𝑛𝑔𝑙𝑒 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑜𝑓 𝐺 − 𝑚𝑒𝑡𝑟𝑖𝑐;

𝐺(𝑢, 𝑣, 𝑤) ≤ 𝐺(𝑢, 𝑎, 𝑎) + 𝐺(𝑎, 𝑣, 𝑤) ⇒ 𝑥+𝑦

𝐺(𝑢,𝑣,𝑤) ≥

𝑥+𝑦

𝐺(𝑢,𝑎,𝑎)+𝐺(𝑎,𝑣,𝑤) (1)

𝑆𝑖𝑛𝑐𝑒 𝐺(𝑢, 𝑎, 𝑎) 𝑎𝑛𝑑𝐺(𝑎, 𝑣, 𝑤) 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒,

Max (( 𝑥

𝐺(𝑢,𝑎,𝑎)) , (

𝑦

𝐺(𝑎,𝑣,𝑤))) ≥

𝑥+𝑦

𝐺(𝑢,𝑎,𝑎)+𝐺(𝑎,𝑣,𝑤)≥

Min (( 𝑥

𝐺(𝑢,𝑎,𝑎)) , (

𝑦

𝐺(𝑎,𝑣,𝑤))) (2)

Combining (1) and right hand inequality in (2) 𝑥 + 𝑦

𝐺(𝑢, 𝑣, 𝑤)≥ 𝑀𝑖𝑛 (( 𝑥 𝐺(𝑢, 𝑎, 𝑎)) , (

𝑦 𝐺(𝑎, 𝑣, 𝑤)))

⇒ 𝐷 ( (𝑥+𝑦)

𝐺(𝑢,𝑣,𝑤)) ≥ 𝑀𝑖𝑛 (𝐷 (

𝑥

𝐺(𝑢,𝑎,𝑎)) , 𝐷 (

𝑦

𝐺(𝑎,𝑣,𝑤)))

𝑆𝑖𝑛𝑐𝑒 𝐷 𝑖𝑠 𝑛𝑜𝑛 − 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔.

Following lemma determines the condition under which product of G - metric is G - metric space.

(4)

Proof: 𝐿𝑒𝑡 𝑢𝑠 𝑑𝑒𝑛𝑜𝑡𝑒 𝑆1 × 𝑆2 𝑏𝑦 𝑆 𝑎𝑛𝑑 𝑀𝑎𝑥 (𝐺1, 𝐺2) 𝑏𝑦 𝐺.

𝑇ℎ𝑒𝑛, 𝐺(𝑢, 𝑣, 𝑤) = 𝑀𝑎𝑥(𝐺1(𝑢1,𝑣1,𝑤1), 𝐺2(𝑢2,𝑣2,𝑤2) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢 = (𝑢1,𝑢2), 𝑣 = (𝑣1,𝑣2), 𝑤 = (𝑤1,𝑤2) ∈ 𝑆.

( 𝐺1) 𝐺(𝑢, 𝑣, 𝑤) = 0

⇔ 𝑀𝑎𝑥(𝐺1(𝑢1, 𝑣1, 𝑤1), 𝐺2(𝑢2, 𝑣2, 𝑤2)) = 0

⇔ 𝑢1= 𝑣1= 𝑤1 𝑎𝑛𝑑 𝑢2= 𝑣2= 𝑤2 ⇔ 𝑢 = 𝑣 = 𝑤, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣, 𝑤 ∈ 𝑆.

( 𝐺2) 𝐺1(𝑢1, 𝑢1, 𝑣1) > 0 𝑎𝑛𝑑 𝐺2(𝑢2, 𝑢2, 𝑣2) > 0 ⇒ Max(G1(u1, u1, v1), G2(u2, u2, v2)) > 0 ⇒ G(u, u, v)) > 0, for all u, v, ∈ S, with u ≠ v .

( 𝐺3) 𝑀𝑎𝑥(𝐺1(𝑢1, 𝑢1, 𝑣1), 𝐺2(𝑢2, 𝑢2, 𝑣2) < 𝑀𝑎𝑥(𝐺1(𝑢1, 𝑣1, 𝑤1), 𝐺2(𝑢2, 𝑣2, 𝑤2)

(When v1≠ w1and v2≠ w2) ⇒ 𝐺(𝑢, 𝑢, 𝑣) ≤ 𝐺(𝑢, 𝑣, 𝑤)𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣, 𝑤 ∈ 𝑆 𝑤𝑖𝑡ℎ 𝑤 ≠ 𝑣.

( 𝐺4) It’s easy to show that G is invariant under all permutations of u, v, and w so we omit the details.

( 𝐺5) 𝐺1(𝑢1, 𝑣1, 𝑤1) ≤ 𝐺1(𝑢1, 𝑎1, 𝑎1) + 𝐺1(𝑎1, 𝑣1, 𝑤1) 𝑎𝑛𝑑

(𝐺2(𝑢2, 𝑣2, 𝑤2) ≤ (𝐺2(𝑢2, 𝑎2, 𝑎2) + (𝐺2(𝑎2, 𝑣2, 𝑤2)

⇒ 𝑀𝑎𝑥(𝐺1(𝑢1, 𝑣1, 𝑤1), (𝐺2(𝑢2, 𝑣2, 𝑤2)) ≤ 𝑀𝑎𝑥(((𝐺1(𝑢1, 𝑎1, 𝑎1) + 𝐺1(𝑎1, 𝑣1, 𝑤1)), (𝐺2(𝑢2, 𝑎2, 𝑎2) +

(𝐺2(𝑎2, 𝑣2, 𝑤2))) .

=

𝑀𝑎𝑥((𝐺1(𝑢1, 𝑎1, 𝑎1), (𝐺2(𝑢2, 𝑎2, 𝑎2)) +

𝑀𝑎𝑥(𝐺1(𝑎1, 𝑣1, 𝑤1), (𝐺2(𝑎2, 𝑣2, 𝑤2) ⇒ 𝐺(𝑢, 𝑣, 𝑤) ≤ 𝐺(𝑢, 𝑎, 𝑎) + 𝐺(𝑎, 𝑣, 𝑤)

𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣, 𝑤, 𝑎 ∈ 𝑆.

Theorem 3.5: If (𝑆1 , 𝒯1) and (𝑆2 , 𝒯2) are 𝑃𝐺 –simple spaces generated by 𝐺 −metric spaces

(𝑆1 , 𝐺1) and (𝑆2 , 𝐺2) respectively, and the same distribution function D, then their Min – product (𝑆1 × 𝑆2, 𝒯𝑀𝑖𝑛) is a 𝑃𝐺 – simple space generated by 𝐺 – metric space (𝑆1 × 𝑆2 , 𝑀𝑎𝑥 (𝐺1, 𝐺2)) and D.

Proof: Let 𝑢 = (𝑢1,𝑢2), 𝑣 = (𝑣1,𝑣2), 𝑤 = (𝑤1,𝑤2) ∈ 𝑆1 × 𝑆2.

Using (𝑃1 ) of Theorem 3.1 it follows that

𝐺𝑢,𝑣,𝑤∗ (𝑥) = 𝒯𝑀𝑖𝑛(𝑢, 𝑣, 𝑤)(𝑥) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣, 𝑤 ∈ 𝑆

= Min (G1u1,v1,w1(x), G 2u2,v2,w2

(x))

= ε0(x), If and only if 𝑢1= 𝑣1= 𝑤1 If and only if

u = (u1, u2 ) = v = (v1, v2 ) = w = (w1, w2 ). If 𝑢, 𝑣, 𝑤 are not equal, we have the following alternatives:

(i) u ≠ v = w (ii) u = v ≠ w (iii) u = w ≠ v and (iv) u ≠ v ≠ w

For each alternative we have to prove that

Gu,v,w∗ (x) = 𝒯Min(u, v, w)(x) = D (

x

G(u,v,w)) for all x > 0.

Here, G(u, v, w) = Max(G1(u1, v1, w1), (G2(u2, v2, w2)) First let us consider (i) 𝑢 ≠ 𝑣 = 𝑤.

We have following three possibilities in this case:

(𝑎) 𝑢1 ≠ 𝑣1 = 𝑤1, 𝑢2= 𝑣2 = 𝑤2 (𝑏) 𝑢1= 𝑣1 = 𝑤1, 𝑢2 ≠ 𝑣2 = 𝑤2 (𝑐) 𝑢1 ≠ 𝑣1 = 𝑤1, 𝑢2 ≠ 𝑣2 = 𝑤2 .

Let us consider,

(𝑎) 𝑢1 ≠ 𝑣1 = 𝑤1, 𝑢2= 𝑣2 = 𝑤2 Gu,v,w(𝑥)

= Min (D ( 𝑥

G1(u1, v1, w1)) , D ( 𝑥

G2(u2, v2, w2)) )

= Min (D ( 𝑥

G1(u1, v1, w1)) , ɛ0(𝑥))

= D ( 𝑥 G1(u1, v1, w1)),

= D ( 𝑥

Max(G1( u1, v1 , w1 ), 0))

= D ( 𝑥

Max(G1( u1, v1 , w1 ), G2( u2, v2, w2 )))

= D ( 𝑥

G(u,v,w)) for all 𝑥 > 0 .

(𝑏) In case of 𝑢1= 𝑣1 = 𝑤1, 𝑢2 ≠ 𝑣2 = 𝑤2 , we proceed as in(𝑎).

(𝑐) 𝑊ℎ𝑒𝑛 𝑢1 ≠ 𝑣1 = 𝑤1, 𝑢2 ≠ 𝑣2 = 𝑤2

𝐺𝑢,𝑣,𝑤(𝑥)

= 𝑀𝑖𝑛 (𝐷 ( 𝑥

𝐺1(𝑢1, 𝑣1, 𝑤1)) , 𝐷 ( 𝑥

𝐺2(𝑢2, 𝑣2, 𝑤2)) )

= 𝐷 ( 𝑥

𝑀𝑎𝑥(𝐺1( 𝑢1, 𝑣1 , 𝑤1 ), 𝐺2( 𝑢2, 𝑣2, 𝑤2 )) )

= 𝐷 ( 𝑥

𝐺(𝑢, 𝑣, 𝑤)) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 > 0.

For the proof of other three cases,we proceed as in (i).

We will conclude the section with the discussion of topologies on product of Menger probabilistic G – metric spaces.

Theorem 3.6: Let (𝑆1 , 𝒯1, 𝑇) and (𝑆2 , 𝒯2, 𝑇) are Menger probabilistic 𝐺 – metric space under same left – continuous t – norm T. Let

’ denotes the 𝜀 − 𝜆 neighborhood system in(𝑆1× 𝑆2 , 𝒯𝑇 , 𝑇) and

be neighborhood system in (𝑆1× 𝑆2 , 𝒯𝑇 , 𝑇) consisting of cartesian product 𝑁𝑢1× 𝑁𝑢2, where 𝑁𝑢1𝑎𝑛𝑑 𝑁𝑢2 are 𝜀 − 𝜆 neighborhoods in component spaces (𝑆1 , 𝒯1, 𝑇) and (𝑆2 , 𝒯2, 𝑇) respectively. Then neighborhood

system

’ and

induces equivalent topologies

in (𝑆1× 𝑆2 , 𝒯𝑇 , 𝑇).

Proof: It has been established in[16] that system of 𝜀 − 𝜆 neighborhoods

and

’ are bases for their respective topologies.

(5)

Let, 𝜀 = 𝑀𝑖𝑛(𝜀1, 𝜀2), 𝜆 = 𝑀𝑖𝑛(𝜆1, 𝜆2) and 𝑢 = (𝑢1, 𝑢2). We will show that 𝑁𝑢(𝜀, 𝜆) ⊆ 𝐴1× 𝐴2. Let 𝑣 = (𝑣1, 𝑣2) belong to 𝑁𝑢(𝜀, 𝜆).Then , we have

𝐺∗ 𝑢

1, 𝑣1, 𝑣1(𝜀1) = 𝑇(𝐺∗ 𝑢1, 𝑣1, 𝑣1(𝜀1), 1) ≥ 𝑇(𝐺∗ 𝑢

1, 𝑣1, 𝑣1(𝜀1), 𝐺∗ 𝑢2, 𝑣2, 𝑣2(𝜀2 ≥ 𝑇(𝐺∗ 𝑢

1, 𝑣1, 𝑣1(𝜀), 𝐺∗ 𝑢2, 𝑣2, 𝑣2(𝜀)) = 𝐺∗ 𝑢, 𝑣, 𝑣(𝜀) > 1 − 𝜆 ≥ 1 − 𝜆1. Also, 𝐺∗ 𝑣

1, 𝑢1, 𝑢1(𝜀1) = 𝑇(𝐺∗ 𝑣1, 𝑢1, 𝑢1(𝜀1), 1 ) ≥ 𝑇(𝐺∗ 𝑣

1, 𝑢1, 𝑢1(𝜀1), 𝐺∗ 𝑣2, 𝑢2, 𝑢2(𝜀2)) ≥ 𝑇(𝐺∗ 𝑣

1, 𝑢1, 𝑢1(𝜀), 𝐺∗ 𝑣2, 𝑢2, 𝑢2(𝜀))

= 𝐺𝑣,𝑢,𝑢(𝜀) > 1 − 𝜆 ≥ 1 − 𝜆

1.

This implies that 𝑣1∈ 𝑁𝑢1(𝜀1, 𝜆1). Similarly, we can prove that 𝐺∗ 𝑢

2, 𝑣2, 𝑣2(𝜀2) ≥ 1 − 𝜆2 and 𝐺∗ 𝑣

2, 𝑢2, 𝑢2(𝜀2) ≥ 1 − 𝜆2 .

Hence, 𝑣2∈ 𝑁𝑢2(𝜀2, 𝜆2). Thus, 𝑁𝑢(𝜀, 𝜆) ⊆ 𝐴2× 𝐴2 and

consequently B’⊆ B.

For converse, let 𝑁𝑢(𝜀, 𝜆) ∈

’ . Left continuity of T insures that 𝑆𝑢𝑝𝑥 <1𝑇(𝑥, 𝑥) = 1.

This implies the existence of an η such that 𝑇(1 − η, 1 − η) > 1 − 𝜆.

Let 𝑣 = (𝑣1, 𝑣2) belong to 𝑁𝑢1(𝜀, η) × 𝑁𝑢2(𝜀, η)

Then,

𝐺𝑢,𝑣,𝑣(𝜀) = 𝑇 (𝐺

𝑢1,𝑣1,𝑣1

(𝜀) , 𝐺 𝑢2,𝑣2,𝑣2

(𝜀)),

≥ 𝑇(1 − η, 1 − η) > 1 − 𝜆 and

𝐺𝑣,𝑢,𝑢∗ (𝜀) = 𝑇 (𝐺𝑣1,𝑢1,𝑢1

(𝜀) , 𝐺 𝑣2,𝑢2,𝑢2

(𝜀))

≥ 𝑇(1 − η, 1 − η) > 1 − 𝜆. Thus, 𝑣 ∈ 𝑁𝑢(𝜀, 𝜆) and hence

𝑁𝑢1(𝜀, η) × 𝑁𝑢2(𝜀, η) ⊆ 𝑁𝑢(𝜀, 𝜆).

So, the proof is complete.

REFERENCES

[1] A. Khan, N. Parveen and S. Zertaj, “Random 2-Normed Spaces,” Bull. Cal. Math. Soc. Vol. 87, pp. 231-236, 1995.

[2] A. Khan, and S. Zertaj, “Results Concerning Common Fixed Point in Random Normed Spaces,” Fasc.4, New Series, 8, pp. 1 8, 1996.

[3] A. Wald: On a statistical generalization of a metric spaces, Proc. Nat. Acad. Sci. U.S.A.29, pp196-197 (1943).

[4] AFS Xavier: On the product of probabilistic metric spaces, Portugal. Math.27, pp137-147 (1968).

[5] B.C.Dhage: Generalized metric spaces and mappings with fixed point, Bull. Calcutta Math Soc, 84, pp 329-336 (1992).

[6] B. Schweizer, A. Sklar: Probabilistic Metric Spaces, Elsevier, New York (1983).

[7] C Zhou, S Wang, L, Ciric, S Alsulami,: Generalized probabilistic metric spaces and fixed point theorems, Zhou et al. Fixed Point Theory and Applications, pp 1-15 (2014).

[8] I. Istratescu, and I. Vaduva: Products of statistical metric spaces, Stud. Glas. Mat. 12, pp 567-574 (1961).

[9] K. Menger: Statistical Matrices, Proceeding of the National academy of sciences of the United states of America, 28, pp 535-537 (1942).

[10] KPR Sastry, GA Naidu, V Madhvi, SSA Sastri, I Laxmi: Products of Menger Probabilistic Normed Spaces, Gen. Math. Notes, 7, pp 15-23 (2011).

[11] RW Freese, YJ Cho: Geometry of Linear 2- Normed Spaces: Nova Science Publishers, New York (2001).

[12] SS Chang, YJ Cho, SM Kang,: Nonlinear Operator theory in Probabilistic Metric Spaces, nova Science Publishers, New York (2001).

[13] S Gahler: 2-metriche Raume and the topologische structure, Math. Nachr. 26, pp 115 – 148 (1963).

[14] S Zertaj, and A Khan: Results concerning 2-metric spaces, J. Indian Institute of Science, pp 191-197(1997).

[15] Z Mustafa, B Sims: Remarks concerning D metric spaces, Proceedings of the International Conferences on Fixed Point Theory and Applications, Valencia Spain, pp 189-198 (2003).

References

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