Fixed Point Theorems in Non - Archim edean Intuitionistic ... · It is also of fundamental...

Fixed Point Theorems in Non-

Archimedean Intuitionistic Menger Pm-

Spaces 1V. Malliga Devi,

2M. Jeyaraman and

3L. Muthulakshmi

1Department of Mathematics,

University V.O.C College of Engineering,

Tuticorin Cambus, Tuticorin, India.

[email protected]

2PG and Research Department of Mathematics,

Raja Doraisingam Govt. Arts College,

Sivagangai, Tamil Nadu, India.

[email protected] 3PG and Research Department of Mathematics,

Raja Doraisingam Govt. Arts College,

Sivagangai, Tamil Nadu, India.

[email protected]

Abstract In this paper, we introduce two types of compatible maps in non-

Archimedean intuitionistic Menger PM-spaces and obtain common

fixed point theorems for six maps.

Key Words:Common fixed point theorem, intuitionistic menger

PM-spaces, non-archimedean fuzzy metric spaces.

AMS Mathematics Subject Classification (2010): 47H10,

54H25

International Journal of Pure and Applied MathematicsVolume 119 No. 12 2018, 14687-14704ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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1. Introduction

In 1942 Menger [8] introduced the notion of probabilistic metric spaces

(briefly, PM-spaces) as a generalization of a metric space. Such a

probabilistic generalization of metric spaces appears to be well adapted

for the investigation of physical quantities and physiological thresholds.

It is also of fundamental importance in the probabilistic functional

analysis.

In 1997, Cho et al., [2] introduced the concepts of compatible maps and

compatible maps of type (A) in non-Archimedean Menger probabilistic

metric space and gave some fixed point theorems for these maps. In this

paper, we introduce he concept of compatible maps of type (J-1) and

type (J-2), show that they are equivalent to compatible maps under

certain conditions and illustrating wih an example, prove common fixed

point theorems for such maps in the spaces which generalizes extends

and fuzzifies several fixed point theorems for contractive type maps on

metric spaces and intuitionistic fuzzy metric spaces.

2. Preliminaries

Definition 2.1.

A triple (X, F, G) is said to be a Non-Archimedean Intuitionistic

Probabilistic Metric Space (shortly NAI PM-space) if X is a nonempty

set, and F is a probabilistic distance and G is a probabilistic non-

distance on X satisfying the following conditions: for all x, y, z ∈ X and

t, s ≥ 0,

(1) ,

(2) ,

(3) ,

(4) ,

(5) , ⇒

(6) ,

(7)

(8) ,

(9) , ⇒

A 5-tuple (X, F, G, , ) is said to be a Non-Archimedean Intuitionistic

Menger Probabilistic Metric Space (shorty NAIM PM-space) if (X, F, G)

is an NAI PM-space, and in addition, the following inequalities hold for

all for all x, y, z, X and it, s > 0,

(10) ,

(11) ,

International Journal of Pure and Applied Mathematics Special Issue

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where * is a continuous t-norm and is a continuous t-conorm. Definition 2.2.

A IPM-space (X, F, G) is said to be of type if there exists a g, h

such that and

for all x, y, z X and where

is continuous, strictly decreasing, g(1) = 0 , g(0) < and

h(1) = 1 , h(0) > }.

Definition 2.3.

A NAI Menger PM- space (X, F, G, *, ) is said to be of type if

there exists a g, h such that and

for all s, t .

Lemma 2.4

If the function satisfies the condition ( ), then we

have

i) For all where is the n-th iteration of

ϕ(t),

ii) If is a non- decreasing sequence of real numbers and

then In particular, if

for all then .

Lemma 2.5

Let { be a sequence in X, such that and

for all If { is not a Cauchy sequence in X,

then there exist and two sequence { of

positive integers such that

i) and as ,

ii) and

iii) and

Definition 2.6

Self maps A and B of a non-Archimedean intuitionistic Menger

probabilistic metric space (X,F,G, , ) are said to be compatible if

and for all ,

whenever is a sequence in X such that for some z in

X as

Definition 2.7

Self maps A and B of a non-Archimedean intuitionistic Menger

probabilistic metric space (X,F,G , ) are said to be compatible of type

(J) if and ,


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and t for all , whenever is a sequence

in X such that for some z in X as

Now we introduce the concept of compatible mappings of type (J-1) and

type (J-2) in NAI Menger PM-space and show that they are equivalent

to compatible mappings under certain conditions.

Definition 2.8

Self maps A and B of a NAI Menger PM-space (X, F, G, , ) are said to

be compatible type (J-1) if and

for all whenever is a sequence in X

such that for some z in X as

Definition 2.9.

Self maps A and B of a NAI Menger PM-space (X, F, G, , ) are said to

be compatible type (J-2) if and

for all whenever is a sequence in X

such that for some z in X as

Proposition 2.10

Let A and B be self maps of a NAI Menger PM-space (X, F, G, , ).

(a) If B is continuous then the pair (A, B) is compatible of type (J-1)

iff A and B are compatible.

(b) If A is continuous then the pair (A, B) is compatible of type (J-2)

iff A and B are compatible.

Proof.

(a) Let be a sequence in X such that for some z in X

as and let the pair (A, B) be compatible of type (J-1). Since B is

continuous, we have and and so

and

.

Hence the mappings A and B are compatible.

Now, let A and B be compatible. Therefore, using the continuity of B,

we have

and

.


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Hence the mappings A and B are compatible of type (J-1). (b) The proof

is similar with (a).

3. Main Results

Theorem 3.1

Let A, B, P, Q, S and T be self maps on a complete NAI Menger PM-

space (X, F, G, , ) satisfying:

(a)

(b) and

,

(c)

(d)

for all and where a function

satisfies the condition ( ) and ( ).

(e) AB = BA, ST = TS, PB = BP, QT = TQ,

(f) Either P or AB is continuous,

(g) The pairs (P, AB) and (Q, ST) are mutually compatible of type (J).

Then A, B, P, Q, S and T have a unique common fixed point.

Proof.

Let be an arbitrary point of X. By (a), there exists such

that and . Inductively, we can

construct sequences and in X such that

and for n = 0,1,2…

Step1.

We shall show that the sequence is a Cauchy sequence.

Since using (b), (c) and (d), we have


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and

and

since , we also have

and

.

Thus

and

for …. Hence

and

for …. Therefore, from

lemma 2.4.

and .

(3.1.1)

Suppose is a not Cauchy sequence. Since g is strictly decreasing

from lemma 2.5, there exists and two sequences

of positive integers such that

(i) and ,

(ii) and

,

and for k = 1, 2

…

Therefore

and


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and letting , we have

and .

(3.1.2)

On the other hand, we have

and

.

(3.1.3)

Without loss of generality assume that both and are even.using

(c) and (d)

we have

and


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.

Substituting this in this (3.1.3), letting and using (3.1.1) and

(3.1.2), we have and

,

which is a contradiction. Hence } is a Cauchy sequence. Since (X, F,

G, , ) is complete, it converges to a point z in X. Also its subsequences

converges as follows:

and .

Case I. AB is continuous, and (P, AB) and (Q, ST) are compatible of

type (J-1).

Since AB is continuous, and ( .

Since (P, AB) is compatible of type (J-1) ,

Step 2. By taking in (c) and (d), we have

,


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this implies that, as

and

,

which means that, by lemma 2.4,

for all and it follows that z = ABz.

By taking in (c) and (d), we have

,


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this implies that as

, which means that z = Pz. Therefore z

= ABz = Pz

Step 4. By taking in (c) and (d) using (e),we have


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and

, which means that z = Bz.

Since z = ABz ,we have z = Az. Therefore z = Az = Bz = Pz.

Step 5. Since there exists such that z = Pz =

STw.


,



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and

, which means that z = Qw. Hence, STw = z = Qw.

Since (Q, ST) is compatible of type (J-1), we have Q(ST)w = ST(ST)w.

Thus, STz = Qz.

Step 6. By taking in (c) and (d), and using step 5, we

have

,



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and

,which means that z = Qz.

Since, STz = Qz, we have z = STz. Therefore z = Az = Bz = Pz = Qz =

STz.

Step 7. By taking in (c) ,(d) and using (e), we have

,

,


and

= , which means that z = Tz.

Since, z = STz, we have z = Sz. Therefore z = Az = Bz = Pz = Qz = Sz

= Tz, that is the commom fixed point of A, B, P, Q, S and T.


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Simiary, it is clear that z is also the commom fixed point of A, B, P, Q, S

and T in the case AB is continuous, and (P, AB) and (Q, ST) are

compatible of type (J-2).

Case II. P is continuous, and (P, AB) and (Q,ST) are compatible of type

(J-1).

Since P is continuous, and .

Since (P, AB) is compatible of type (J-1),

Step 8. By taking in (c) and (d), we have

≥


and

, which means that z = Pz.


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Now using step 5-7, we have, z = Qz = STz = Sz =Tz.

Since Q(X) ⊆ AB(X), There exists such that z = Qz =

ABw.


,

,


and

, which means that z = Pw.

Snce z = Qz = ABw, Pw = ABw. Since (P, AB) is compatible of type (J-

1), we have Pz = ABz. Also z = Bz follows from step 4. Thus z = Az =

Bz = Pz. Hence, z is the common fixed point of the six maps in this case

also.

Similarly, it is clear that z is also the common fied point of A, B, P, Q, S,

and T in the case P is continuous, and (P, AB) and (Q, ST) are

compatible of type (J-2).


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Step 10. For uniqueness, let be another common fixed point

of A, B, P, Q, S, and T. Taking in (c) and (d), we have

,

,

which implies that

and

, so we have

This complete the proof of the theorem.

If we take A = B = S = T = ( the identity map on X ) in Theorem (3.1)

we have the following.

Corollary 3.2. Let P and Q be self maps on a compete NAI. Menger PM-space

(X, F, G, *, ). If and

,

and

for all and , where a function

satisfies the condition ( ) and ( ), then P and Q have a unique common

fixed point.


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References

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[2] Cho Y.J., Ha K.S., Chang S.S, common fixed point theorems for compatible maps of type (A) in non-Archimedean Menger P.M-spaces , Math. Japon. 46(1) (1997), 169-179.

[3] George A. and Veeramani P., On some resuls in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395-399.

[4] Hadzic O, A note on Istradescu’s fied point heorem in non-Archimedean Menger spaces, Bull. Mah.soc. sci. Math. R .S. Roumanie 24(72) (1980), 277-280

[5] Kramosil O. and Michelek J., Fuzzy metric and statiscal metric spaces, Kybernetica 11(1975), 36-344.

[6] Kutukcu S., Yildiz. C., A common fixed point theorem of compatible and weak compatible map on Merger spaces, Kochi Math.J. (2008).

[7] Menger.K, Statistical Metrics, Pro. Nat. Acad. Sci. USA (28)(3) (1942), 535-537

[8] Pant R.P., Common fied points non commuing mapping J.Math.Anal.APPL. 18 (1994), 436-440.

[9] Park J. H., Intuitionisic fuzy meic spaces, Chaos, Soliions and Fractals 22 (2004), 1039-1046.

[10] Sehgal V.M., Bharucha–Reid A.T, Fixed point of contraction mapping on PM spaces, Math. Systems Theory 6 (1972), 97-100.

[11] Turkoglu D., Alaca C., Cho Y.J., Yildiz C., common fxed point theorems intuiionistic fuzzy metric space, J.Appl Math. And Computing 22 (2006), 411-424.

[12] Zadeh L.A., Fuzzy Sets, Inform and control 89 (1965), 338-353.


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