View of CERTAIN OUTCOMES OF COMMON FIXED POINTS OF MAPPINGS IN COMPLETE MENGER SPACES

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (ISSN NO. 2456-1037) Vol.03, Issue 09, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

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CERTAIN OUTCOMES OF COMMON FIXED POINTS OF MAPPINGS IN COMPLETE MENGER SPACES

Rajesh Vyas

Department of Mathematics, Christian Eminent College, Indore (MP) [email protected]

Abstract- It is well known in the field of computational mathematics that in real measurements, assigning a fixed number to the distance between two points is an over idealized way of thinking. Indeed, the distance between two points is an average of several measurementssometimes, it is found appropriate to assign the average of several measurements as a measure to ascertain the distance between two points.Inspired from this line of thinking, Menger [1] introduced the notion of probabilistic metric space as a generalization of core notion of metric space^2,3.

Keywords :menger space,mappings,common fixed point, real measurements.

1 INTRODUCTION

Main objective of this paper is to demonstrate Certain Outcomes of Common Fixed Points of Mappings in Complete Menger Spaces. Singh and Tiwari¹ offer some results on common fixed points for a family of mappings. In this section we generalize the result of Singh and Tiwari¹in Complete Menger Spaces and give some outcomes on common fixed points of mappings^4,5. Theorem 4.2.1 : Let {Sn}, n =

1,2,………… be a sequence of mappings of a complete Menger space (X, F, t) into

itself with t (u, v) = min (u, v) for every u, v ∈ [0,1]. Let T be a continuous mapping of X into X such that T and S_n commute and

Sn (X) ⊂ T (X), n = 1,2,……….

Suppose

F_S_i_x,S_j_y (αp) ≥ min { F_S_i_x,Tx (p), F_S_i_x,Ty (p),

F_S_j_x,Ty (p), F_S_j_y,Tx (p), F_Tx,Ty (p) }--- (1)

for all x, y ∈ X, p > 0 and 0 ≤ α <

1.

Then Tand sequence of mappings { Sn} have a unique common fixed point.

Proof : Let x0 be a point of X. Then Tx₀ is also a point of X.

Put Tx_n = S_nx_n−1, n = 1,2,…………--- (2)

We can do this since Sn (X) ⊂ T (X).

Then by given inequality (1) for each p >

0 and 0 ≤ α < 1,

F_Tx₁_,Tx₂ (αp) = F_S₁_x₀_,S₂_x₁ (αp)

≥min {F_S₁_x₀_,Tx₀ (p), F_S₁_x₀_,Tx₁ (p), FS₂x₁,Tx₁ (p), FS₂x₁,Tx₀ (p), FTx₀,Tx₁ (p) }

≥min {F_Tx₁_,Tx₀ (p), F_Tx₁_,Tx₁ (p), FTx₂,Tx₁ (p), FTx₂,Tx₀ (p), FTx₀,Tx₁ (p) }

≥ F_Tx₀_,Tx₁ (p)

So FTx₁,Tx₂ (αp)

≥ F_Tx₀_,Tx₁ (p)

Similarly F_Tx₂_,Tx₃ (αp)

≥ F_Tx₁_,Tx₂ (p) ≥ F_Tx₀_,Tx₁ (p)

In general, we have F_Tx_n_,Tx_{n +1} (αp) ≥ FTx₀,Tx₁ (p)

This means that the sequence {Tx_n} is a Cauchy sequence.

Hence by the completeness of X, {Tx_n} convergences to some point ξ in X.

So by (2) {S_nx_n−1} also convergences to ξ.

By the continuity of T, T (Txn) → Tξ and T (S_nx_n−1) → Tξ.

By (1),

F_S_m_(Tx_{m −1}_),S_n_ξ (αp) ≥ min { FS_mTx_{m −1},TT x_{m −1} (p), FS_mTx_{m −1},Tξ (p),

FS_nξ,Tξ (p), FS_nξ,TT x_{m −1} (p), FTT x_{m −1},Tξ (p) } Since T (Txm) = T (Smxm−1)

= S_m (Tx_m−1)

Thus F_{TT x}_m_,S_n_ξ (αp) ≥min {F_T(Tx_m_{),TT x}_{m −1} (p), F_{TT x}_m_,Tξ (p),

FS_nξ,Tξ (p), FS_nξ,TT x_{m −1} (p), FTT x_{m −1},Tξ (p) }

Taking limits on both sides, we get

F_Tξ,S_n_ξ (αp) ≥ min { F_Tξ,Tξ (p), F_Tξ,Tξ (p),

FS_nξ,Tξ (p), FS_nξ,Tξ (p), FTξ,Tξ (p) }

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2

≥ F_S_n_ξ,Tξ (p)

If Tξ≠ S_nξ, a contradiction since α < 1.

This implies that Tξ= S_nξ

Again by (1)

F_S_n_ξ,S_n_(S_n_ξ) (αp) ≥ min { F_S_n_ξ,Tξ (p), F_S_n_ξ,T(S_n_ξ) (p),

F_S_n_(S_n_ξ),T(S_n_ξ) (p), F_S_n_(S_n_ξ),Tξ (p), F_Tξ,T(S_n_ξ) (p) }

Since S_nξ = Tξ, therefore

F_S_n_ξ,S_n_(S_n_ξ) (αp) ≥ min { F_S_n_ξ,S_n_ξ (p), F_S_n_ξ,S_n_(S_n_ξ) (p),

F_S_n_(S_n_ξ),S_n_(S_n_ξ) (p), F_S_n_(S_n_ξ),S_n_(S_n_ξ) (p), F_S_n_ξ,S_n_(S_n_ξ) (p) }

≥ F_S_n_ξ,S_n_(S_n_ξ) (p)

If S_nξ≠ S_n(S_nξ), a contradiction since α

< 1.

Hence S_nξ= S_nS_nξ = S_nTξ= TS_nξ.

Therefore S_nξ is a common fixed point of S_nand T.

To see T and S_n have only one common fixed point let,

Tξ= ξ = Snξ and Tη = η=

S_nη, n = 1,2,………

where η is a another fixed point of T and S_n.

Then by (1)

F_ξ,η (αp) = F_S_i_ξ,S_j_η (αp)

≥min {FS_iξ,Tξ (p), FS_iξ,Tη (p), F_S_j_η,Tη (p), F_S_j_η,Tξ (p), F_Tξ,Tη (p) }

≥min {Fξ,ξ (p), Fξ,η (p), F_η,η (p), F_η,ξ (p), F_ξ,η (p) }

≥F_ξ,η (p)

which is imposible.

Therefore ξ = η.

Hence ξ is a unique common fixed point of T and Sn.

By taking S_n= S for each n in the above theorem, we have

Corollary 4.2.2 : Let S and T be two commuting mappings of X into X such that

S (X) ⊂ T (X).

Suppose the following condition

FSx,Sy (αp) ≥ min { FSx,Tx (p), FSx,Ty (p),

FSy,Ty (p), FSy,Tx (p), FTx,Ty (p) }

holds for all x, y ∈ X, p > 0 and 0 ≤ α < 1.

If T is continuous then S and Thave a unique common fixed point.

By taking {Sn} = {T1,T2,T1,T2,T1, … … … . . } in theorem 4.2.1, we have the following:

Corollary 4.2.3 : Let T1,T2, T be mappings from X into X such that T1(X) and T₂ (X) are subsets of T (X).

Suppose for all x, y ∈ X, p > 0 and 0

≤ α < 1,

FT₁x,T₂y (αp) ≥ min { FT₁x,Tx (p), FT₁x,Ty (p),

F_T₂_y,Ty (p), F_T₂_y,Tx (p), F_Tx,Ty (p) }

If T is continuous and commutes with each of T1 and T2 then T1,T2 and Thave a unique common fixed point.^6,7

Corollary 4.2.4 : With the hypotheses of Theorem 4.2.1, suppose there is a non negative integer m_i for each S_i such that for all x, y of X and for every pair i, j with i ≠ j, the condition (1) replaced by

F_S

i

mix,S_j^mjy (αp) ≥ min { F_S

i

mix,Tx (p), F_S

i

mix,Ty (p), F_S

j

mjy,Ty (p), F_S

j

mjy,Tx (p), FTx,Ty (p) } Then T and the sequence of mappings {Sn} have a unique common fixed point.

Proof : Clearly, S_i^mⁱ commutes with T and

S_i^mⁱ (X) ⊂Si (X) ⊂ T (X).

Thus Theorem 4.2.1 pertains to Sn

m_n and T.

So there is a common unique ξin X such that

ξ = T(ξ) = Sn m_n(ξ).

But then, since T and S_n commute, we can write

Sn(ξ) = T(Sn(ξ)) = Sn

m_n(Sn(ξ)), which says that Sn(ξ) is a common fixed point of T and S_n^mⁿ.

This uniqueness of ξ implies ξ

= T(ξ) = Sn(ξ).

Theorem 4.2.5 : Let T be a continuous mapping of X into X and

S_k( k = 1,2,………,n) a family of mappings of X into X.

Suppose that

(i) S_kS_t = S_tS_k ( k,t = 1,2,………,n),

(ii) S_kT = TS_k ( k = 1,2,………,n),

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3 (iii) Sk (X) ⊂ T (X) ( k =

1,2,………,n), and

(iv) there is system of positive integers m1,m2,………,mn such that

F_Ux,Uy (αp) ≥ min { F_Ux,Tx (p), F_Ux,Ty (p),

FUy,Ty (p), FUy,Tx (p), FTx,Ty (p) }

for every x, y ∈ X, p > 0 and 0 ≤ α <

1, and where

U = S₁^m¹S₂^m² ………… Sn m_n.

Then Tand Sk( k = 1,2,………,n) have a unique common fixed point.⁸

Proof : In view of (iii) U (X) ⊂ T (X)

Since Skand T commute, S_k^m^k and T also commute.

Thus corollary 4.2.2 pertains to U and T, so there is unique ξin X such that ξ = T(ξ) = U(ξ).

Hence Skξ = Sk (Tξ) = Sk

(Uξ).

By (ii) and the commutativity of

Sk, Skξ = T (Skξ) = U (Skξ) .

This says that S_kξ is a common fixed point of T and U.

The uniqueness of ξ implies ξ = S_kξ = Tξ.

This is true for every k = 1,2,………,n.

This completes the proof.

REFERENCES

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2. Singh S.L., Tiwari B.M.L. and Gupta V.K., Common fixed points of commuting mappings in 2-metric spaces and an application. Math. Nachr. 95 (1980), 293- 307.

3. Computers & Mathematics with Applications, Volume, December 2010, Pages 3152-3159

4. K. MengerStatistical metrics, Proc. Nat.

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6. Singh S.L. and Virendra, Relative Asymptotic Regularity and fixed points.Indian J. of Math. Vol. 31 (1), (1989).(158) Singh S.P., Lecture notes on fixed point theorems in metric and banachSpaces. Matscience, Madras, (1974).

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8. T.L. HicksFixed point theory in probabilistic metric spaces,Univ. u NovomSaduZb. Rad. Prirod.-Mat. Fak. Ser.

Mat., 13 (1983), pp. 63-72