INTRODUCTION:
The notion of a 2-metric and its basic properties were initially introduce by Gahler [4]. Then after a decade Iseki [5] started investigation of fixed points of mappings in a setting of 2-metric space. Then many researchers like Rhodes [10], De sarkar and K. Tiwary [3], G. Jungck {[6],[7],[8]} N.S. Simoniya [12] worked on several generalization version of fixed point theorems for different types of contractive type mappings in 2-metric spaces. It is interesting to note that every convergent sequences in a 2-metric space need not be a Cauchy sequence[9]. A 2-metric space d is said to be continous when it is continous in two of its arguments[10]. Cho et al. [2] introduced the notion of semi compatible maps in d-topological space. B.Singh and S.Jain {[13],[14]} introduced the concept of semi compatibility in fuzzy metric space, D metric space, 2-metric space and proved fixed point results using implicit relations in these spaces.
In this paper we generalize the result of V.H. Badshah et al. [1] by using weak compatibility and semi compatibility mappings.
PRELIMANARIES DEFINATIONS:
Let X be a non-empty set and let d: X×X×Xà[0,∞) be such that
(i) To each pair of point x, y in X with x≠y there exists a point z in X such that d(x,y,z)≠0.
(ii) d(x,y,z)=0 when at least two of the three points are equal.
(iii) For any x,y,z in X , d(x,y,z)=d(x,z,y)=d(y,z,x).
(iv) For any x,y,z,w in X , d(x,y,z)≤ d(x,y,w)+d(x,w,z)+d(w,y,z),
Then d is called a 2-metric [4] and (X,d) is called a 2-metric space[4].
In this note X will denote a complete 2-metric space unless or otherwise stated instead of (X,d).
A sequence {xn} in X is called a Cauchy sequence[10] when d(xn,xm,a)→0 as n,m →∞
A sequence {xn} in X is said to be converge[10] to an element x in X when d(xn,x,a)→0 as n→∞
A 2-metric space (X,d) is said to be complete if every Cauchy sequence in X converges to a point of X.
Definition 1: Two self maps A and B of a 2-metric space (X,d) are
said to be compatible if 
d( ABxn,BAxn,a) =0
for all a in X , where {xn} is a sequence in X such that
if Axn= Bxn=x for some x in X.
Definition 2: Let A and B be mappings from a metric space (X,d) intoitself.A and B are said to be weakly compatible if they commute at their coincidence point i.e, Ax=Bx for some x in X implies ABx=BAx
Definition 3: Two self maps A and B of a 2-metric space (X,d) are
said to be semi-compatible if d( ABxn,Bxn,a) =0 for
all a in X , where {xn} is a sequence in X such that
if Axn= Bxn=x for some x in X.
MAIN RESULTS:
Theorem: Let, P,Q,S and T be mappings from a complete 2-metric space (X,d) to itself satisfying the conditions:
(A) S(X) 
Q(X),
T(X) P(X)
(B) d(Sx,Ty,a) ≤ α [{ d(Px,Sx,a)3 + d(Qy,Ty,a)3} / { d(Px,Sx,a)2 + d(Qy,Ty,a)2}] + β d(Px,Qy,a) + γ d(Sx,Px,a) + δ d(Sx,Qy,a) + μ d(Ty,Px,a) for all x,y in X, where α,β,γ,δ,μ ≥ 0 and 2α+β+γ+δ+2μ < 1
(C) One of S and Pare continuous function
(D) The pair (S,P) semi compatible and (T,Q) weak compatible on X.
Then, P,Q,S and T have a unique common fixed point in X.
Proof: Let x0 be any point in X and as S(X) Q(X),
T(X) P(X)
then there exists x1,x2 in X such that Sx0=Qx1
, Tx1=Px2
Inductively, we can construct sequences {xn} and {yn} in X such that yn=Sxn=Qxn+1 and yn+1=Txn+1=Pxn+2, n=0,1,2,…. ………(1)
Now, we will prove that {yn} is a cauchy sequence.
Now, d( y1,y2,a)= d(Sx1,Tx2,a) ≤ α [{ d(Px1,Sx1,a)3 + d(Qx2,Tx2,a)3} / { d(Px1,Sx1,a)2 + d(Qx2,Tx2,a)2}] + β d(Px1,Qx2,a) + γ d(Sx1,Px1,a) + δ d(Sx1,Qx2,a) + μ d(Tx2,Px1,a)
= α[ d(y0,y1,a) + d(y1,y2,a)] +β d(y0,y1,a) + γ d(y1,y0,a) + δ d(y1,y1,a) + μ d(y2,y0,a)
≤ α[ d(y0,y1,a) + d(y1,y2,a)] +β d(y0,y1,a) + γ d(y1,y0,a) + δ d(y1,y1,a) + μ d(y2,y0,y1) + μ d(y2,y1,a) + μ d(y1,y0,a) ..(2)
Now, put a=y0 in (2) and get, d( y1,y2,y0) ≤ α d( y1,y2,y0) +2 μ d( y1,y2,y0)
Or, (1-α-2μ) d( y1,y2,y0) ≤ 0
Or, d( y1,y2,y0) ≤ 0 , i.e., d( y1,y2,y0) =0
Using d( y1,y2,y0) =0 , from (2) we get,
d( y1,y2,a) ≤
d(y0,y1,a) = K d(y0,y1,a)
where K= < 1 as
2α+β+γ+δ+2μ < 1 ……..(3)
now, d( y2,y3,a)= d(Sx2,Tx3,a) ≤ α [{ d(Px2,Sx2,a)3 + d(Qx3,Tx3,a)3} / { d(Px2,Sx2,a)2 + d(Qx3,Tx3,a)2}] + β d(Px2,Qx3,a) + γ d(Sx2,Px2,a) + δ d(Sx2,Qx3,a) + μ d(Tx3,Px2,a)
= α[ d(y1,y2,a) + d(y2,y3,a)] +β d(y1,y2,a) + γ d(y2,y1,a) + δ d(y2,y2,a) + μ d(y3,y1,a)
≤ α[ d(y1,y2,a) + d(y2,y3,a)] +β d(y1,y2,a) + γ d(y2,y1,a) + δ d(y2,y2,a) + μ d(y3,y1,y2) + μ d(y3,y2,a) + μ d(y2,y1,a) ..(4)
Put, a= y1 in (4) and we get, d( y2,y3,y1) ≤ α d(y2,y3,y1)] + 2 μ d(y3,y1,y2)
Or, (1-α-2μ) d( y2,y3,y1) ≤ 0 or, d( y2,y3,y1) ≤ 0 i.e., d( y2,y3,y1) =0
So, from (4) we get, d(y2,y3,a)
≤ d(y1,y2,a) = K d(y1,y2,a)
≤ K2 d(y0,y1,a) ( by
3) where K= < 1 as
2α+β+γ+δ+2μ <
1 …………….(5)
Similarly, if we proced we will get
d(yn,yn+1,a) ≤ K d(yn-1,yn,a) ≤……….≤ Kn d(y0,y1,a) where K < 1 ……………..(6)
and, d(yn,yn+1,yn-1) =0 ………….(7)
Now, taking lim n→∞ we get from (6), 
d(yn,yn+1,a) =0 as K
< 1
d(yn,yn+2,a) ≤ d(yn,yn+2,yn+1) + d(yn,yn+1,a) + d(yn+1,yn+2,a)
= d(yn,yn+2,yn+1)
+ 
d(yn+r,yn+r+1,a)
Using (7) on the above inequality we get, d(yn,yn+2,a)
≤ d(yn+r,yn+r+1,a)
Similarlyproceding as above we will get,
d(yn,yn+p,a) ≤ 
d(yn+r,yn+r+1,a)
=(Kn + Kn+1 +…+ Kn+p-1 ) d(yn+r,yn+r+1,a) = Kn(1-Kp)/ (1-K) d(yn+r,yn+r+1,a) ≤ Kn/(1-K) d(yn+r,yn+r+1,a)
Taking lim n→∞ on the above inequlity we
get, d(yn,yn+p,a)
→0
Which shows that {yn} is a cauchy sequence and hence converges to some point u in X.
Consequently the sub sequences {Sxn} , {Pxn+1}, { Txn+1}, {Qxn+2} of sequence {yn} are also converges to u.
i.e., Sxn = Pxn+1 =
xn+1 =
xn+2 =u.
Let, P is continuous and (S,P) is semi compatible pair, then we have PSxn→Pu, PPxn→Pu and SPxn→PSxn→Pu
Now, d(SPxn,Txn+1,a) ≤ α [{ d(PPxn,SPxn,a)3 + d(Qxn+1,Txn+1,a)3} / {d(PPxn,SPxn,a)2 + d(Qxn+1,Txn+1,a)2}] + β d(PPxn,Qxn+1,a) + γ d(SPxn,PPxn,a) + δ d(SPxn,Qxn+1,a) + μ d(Txn+1,PPxn,a)
Taking on both sides of the above inequality we
get,
d(Pu,u,a) ≤ α { d(Pu,Pu,a) + d(u,u,a)} + β d(Pu,u,a) + γ d(Pu,Pu,a) + δ d(Pu,u,a) + μ d(u,Pu,a)
Or, (1-β-δ-μ) d(Pu,u,a) ≤ 0, So, Pu=u …………….(8)
Now, from (B) we get, d(Su,Txn+1,a) ≤ α [{ d(Pu,Su,a)3 + d(Qxn+1,Txn+1,a)3} / {d(Pu,Su,a)2 + d(Qxn+1,Txn+1,a)2}] + β d(Pu,Qxn+1,a) + γ d(Su,Pu,a) + δ d(Su,Qxn+1,a) + μ d(Txn+1,Pu,a)
Taking lim n→∞ on both sides of the above inequality we get,
d(Su,u,a) ≤ α { d(u,Su,a) + d(u,u,a)} + β d(u,u,a) + γ d(Su,u,a) + δ d(Su,u,a) + μ d(u,u,a)
Or, (1-α-γ-δ) d(Su,u,a) ≤ 0 So, Su=u …………..(9)
So, from (8) and (9) we get, Pu=Su=u ……………(10)
As, S(X) Q(X)
, then there exists v in X such that Su=Qv i.e.,
Pu=Su=Qv(=u) ………..(11)
From (B) we get, d(Su,Tv,a) ≤ α [{ d(Pu,Su,a)3 + d(Qv,Tv,a)3} / {d(Pu,Su,a)2 + d(Qv,Tv,a)2}] + β d(Pu,Qv,a) + γ d(Su,Pu,a) + δ d(Su,Qv,a) + μ d(Tv,Pu,a)
Or, d(u,Tv,a) ≤ α { d(u,u,a) + d(u,Tv,a)} + β d(u,u,a) + γ d(u,u,a) + δ d(u,u,a) + μ d(Tv,u,a)
OR, (1-α-μ) d(u,Tv,a) ≤ 0, Or, d(u,Tv,a)≤ 0, i.e., Tv=u …………(12)
Since, (T,Q) are weakly compatible, then we have
TQv=QTv i.e., Tu=Qu [ by (11) and (12) ] …………..(13)
Now, from (B) we get, d(Sxn,Tu,a) ≤ α [{ d(Pxn,Sxn,a)3 + d(Qu,Tu,a)3} / {d(Pxn,Sxn,a)2 + d(Qu,Tu,a)2}] + β d(Sxn,Qu,a) + γ d(Sxn,Pxn,a) + δ d(Sxn,Qu,a) + μ d(Tu,Pxn,a)
Taking lim n→∞ on both sides of the above inequality we get,
d(u,Tu,a) ≤ α { d(u,u,a) + d(Tu,Tu,a)} + β d(u,Tu,a) + γ d(u,u,a) + δ d(u,Tu,a) + μ d(Tu,u,a)
Or, (1-β-δ-μ) d(u,Tu,a) ≤ 0
So, Tu=u which implies that Tu=Qu=u …………..(14)
From, (10) and (14) we can say that u is a common fixed point of S,P,Q and T
Again, if we take S is continous and (S,P) is semi compatible pair, we have SPxn→Su, SSxn→Su and PSxn→SPxn→Su
Now, d(SSxn,Txn+1,a) ≤ α [{ d(PSxn,SSxn,a)3 + d(Qxn+1,Txn+1,a)3} / {d(PSxn,SSxn,a)2 + d(Qxn+1,Txn+1,a)2}] + β d(PSxn,Qxn+1,a) + γ d(SSxn,PSxn,a) + δ d(SSxn,Qxn+1,a) + μ d(Txn+1,PSxn,a)
Taking lim n→∞ on both sides of the above inequality we get,
d(Su,u,a) ≤ α { d(Su,Su,a) + d(u,u,a)} + β d(Su,u,a) + γ d(Su,Su,a) + δ d(Su,u,a) + μ d(u,Su,a)
Or, (1-β-δ-μ) d(Su,u,a)≤ 0, i.e., d(Su,u,a)≤0 So, Su=u ……………(15)
As, (S,P) are semi compatible so, PSxn→SPxn→Su=u [by (15)]
Taking lim n→∞ on both sides we get, Pu→u i.e., Pu=u ……….(16)
As, S(X) Q(X)
, then there exists w in X such that Su=Qw i.e.,
Pu=Su=Qw(=u) …………..(17)
Now, from (B) we get, d(Sxn,Tw,a) ≤ α [{ d(Pxn,Sxn,a)3 + d(Qw,Tw,a)3} / {d(Pxn,Sxn,a)2 + d(Qw,Tw,a)2}] + β d(Sxn,Qw,a) + γ d(Sxn,Pxn,a) + δ d(Sxn,Qw,a) + μ d(Tw,Pxn,a)
Taking lim n→∞ on both sides of the above inequality we get,
d(u,Tw,a) ≤ α { d(u,u,a) + d(u,Tw,a)} + β d(u,u,a) + γ d(u,u,a) + δ d(u,u,a) + μ d(Tw,u,a)
Or, (1-α-μ) d(u,Tw,a)≤ 0 i.e., d(u,Tw,a) ≤ 0 So, Tw=u
So, from (17) we get, Tw=Qw=u=Su=Pu ………….(18)
Since, (T,Q) are weak compatible , then we have TQw=QTw i.e., Tu=Qu ……….(19)
Now, d(Sxn,Tu,a) ≤ α [{ d(Pxn,Sxn,a)3 + d(Qu,Tu,a)3} / {d(Pxn,Sxn,a)2 + d(Qu,Tu,a)2}] + β d(Sxn,Qu,a) + γ d(Sxn,Pxn,a) + δ d(Sxn,Qu,a) + μ d(Tu,Pxn,a)
Taking on both sides of the above inequality we
get,
d(u,Tu,a) ≤ α { d(u,u,a) + d(Tu,Tu,a)} + β d(u,Tu,a) + γ d(u,u,a) + δ d(u,Tu,a) + μ d(Tu,u,a)
Or, (1-β-δ-μ) d(u,Tu,a) ≤ 0
So, Tu=u which implies that Tu=Qu=u …………..(20)
So, from (18) and (20) we get that Su=Pu=u=Tu=Qu, i.e., u is a common fixed point of S,P,Q and T
To prove uniqueness of the fixed point, let, z be another fixed point of S,P,Q and T such that u≠z
Using (B) we get, d(u,z,a)= d(Su,Tz,a) ≤ α [{ d(Pu,Su,a)3 + d(Qz,Tz,a)3} / { d(Pu,Su,a)2 + d(Qz,Tz,a)2}] +
β d(Pu,Qz,a) + γ d(Su,Pu,a) + δ d(Su,Qz,a) + μ d(Tz,Pu,a)
Or, d(u,z,a) ≤ α { d(u,u,a) + d(z,z,a)} + β d(u,z,a) + γ d(u,u,a) + δ d(u,z,a) + μ d(z,u,a)
Or, (1-β-δ-μ) d(u,z,a) ≤ 0 i.e, d(u,z,a) ≤0 So, u=z
i.e., the fixed point u is unique.
So, u is a unique common fixed point of S,P,Q and T.
CONCLUSION:
The main object of this paper is to generalize some fixed point theorems in 2-metric spaces using weak compatibility and semi comtability mappings. If we put γ=δ=μ=0 in our main result then we get the result of Badshah et al.[1], which is also the generalization of the result of Sharma D. [11]
ACKNOWLADGEMENT:
The authors are thankful to the editor and the referees for their valuable suggestions.
REFERENCES
1. V.H.Badshah et al., Common fixed point theorem in 2-metric space, Int. J. of Inovative Research and Development, 2(5)(2013), 802-808.
2. Y.J.Cho, B.K.Sharma and D.R.Sahu, Semi Compatiblity and Fixed Points, Math.Japon,42(1),(1995),91-98.
3. S. De Sarkar and KalishankarTiwary, Some Common Fixed Point Theorems For Contractive Type Compatible Mappings in 2-metric Spaces, Journal of Mathematics, II, 2(2009),181-194.
4. S.Gahler,2-metrische Raume and IherTopogosche Struktur,Math.nache.,26(1963), 115-148.
5. K.Iseki, Fixed point theorems in 2-metric Spaces. Math.Sem.Notes. Kobe. Univ., 3(1975), 133-136.
6. G.Jungck,Compatible Mappings and Common Fixed Points, Internat. J. Math. and Math. Sci., 9(1986), 771-779.
7. G.Jungck, Common Fixed Points for Commuting and Compatible Mappings on Compacts, Proc. Amer. Math. Soc., 103(1988), 977-985.
8. G.Jungck and B.E.Rhodes, Fixed Point Theorems for Compatible Mappings, Internat. J. Matha and Math. Sci., (3) 16(1993), 417-428.
9. S.V.R.Naidu and J.Rajendra Prasad, Fixed Point in2-metric Spaces, Ind. J. Pure and Appl. Math., (8)17(1986), 974-993.
10. B,E.Rhoades, Contraction Type Mapping On 2-metric Spaces, Math.Nacher,91(1979), 151-155.
11. D. Sharma,Common Fixed Point theorems for Compatible Mappings, Int. J. of Theoretical and Applied Sciences, 1(2),(2009),79-82.
12. N.S.Simoniya , An Approach to Fixed point Theorem in a 2-metric apace, The Math. Education, 27(1993), 8-10.
13. B.Singh and S.Jain, Semi Compatiblity and Fixed Point Theorems in Fuzzy Metric Spaces using implicit relation, Int. Math.Sci., 16(2005), 2617-2629.
14. B.Singh and S.Jain, Semi Compatiblity and Fixed Point Theorems in Menger Spaces. Journal of the Chungcheong Mathematica Socity, 17(1)(2004),1-17.
|
Received on 22.03.2018 Accepted on 09.09.2018 © EnggResearch.net All Right Reserved Int. J. Tech. 2018; 8(2): 100-103. DOI:10.5958/2231-3915.2018.00013.5 |
|