SciELO - Scientific Electronic Library Online

 
vol.18 número1Note on cost minimization for a multi-product fabrication- distribution problem with commonality, postponement and quality assuranceUser-Acceptance instrument development: a content validity study in the e-participation context índice de autoresíndice de assuntospesquisa de artigos
Home Pagelista alfabética de periódicos  

Serviços Personalizados

Journal

Artigo

Indicadores

Links relacionados

  • Não possue artigos similaresSimilares em SciELO

Compartilhar


Journal of applied research and technology

versão On-line ISSN 2448-6736versão impressa ISSN 1665-6423

J. appl. res. technol vol.18 no.1 Ciudad de México Jan./Fev. 2020  Epub 05-Jan-2021

 

Articles

Common fixed points using (ψ,ϕ ) - type contractive maps in fuzzy metric spaces

Vishal Gupta*  1 

Manu Verma1 

1Department of Mathematics, Maharishi Markandeshwar, Deemed to be University, Mullana, Haryana, India


Abstract

This In this paper, we define new control functions to give unique fixed point in fuzzy metric space. A fruitful contractive condition of (ψ, ϕ)- type is used to obtain common fixed point theorem for two maps in fuzzy metric spaces. We extend the existing results in metric space to fuzzy metric space using these control functions. The first theorem is the extension of the result of Zhang and Song (2009) under the required contractive conditions. Second result is analogous to the result of Doric (2009) in metric spaces.

Keywords: ψ-contractive mappings; fuzzy metric space; control functions; weak contraction 2000 MSC: 54H25; 47H10

1. Introduction and preliminaries

Zadeh (1965) investigated fuzzy set theory. Many authors utilized the concept of fuzzy set theory in metric space in number of ways. Banach contraction principal is the elate result of fixed point theory. Several authors have developed different contractive conditions to find fixed point in metric space (Dutta & Choudhury, 2008; Gupta & Mani, 2014; Gupta, Mani & Tripathi, 2012; Gupta, Saini, Mani & Tripathi, 2015; Song, 2007; Song & Xu, 2007). Kramosil and Michalek (1975) defined fuzzy metric space using the concept of t-norm. George and Veeramani (1994) modified the notion of fuzzy metric spaces by using continuous t-norm. Gregori and Sapena (2002) also explored the Banach contraction principal to fuzzy contractive mapping on complete fuzzy metric space. Particularly, Mihet (2008) introduced the concept of fuzzy contractive mappings, which is one of the weak contractions in fuzzy metric space. In 1997, the concept of weak contraction was defined (Alber & Guerre-Delabriere, 1997) for single valued maps on Hilbert spaces.

Rhoades (2001) introduced weakly contractive mapping in metric space by defining a map T:X→X, which satisfy the condition d(Tx,Ty)d(x,y)-ϕ(d(x,y)), where x,yX and ϕ:[0,)[0,) is a continuous and nondecreasing function such that ϕ(t)=0 if and only if t=0 .

Zhang and Song (2009) proved a unique common fixedpoint theorem of hybrid generalized ϕ-weak contraction for two maps T,S on complete metric space X. The result is given below.

Theorem 1.1 (Zhang & Song, 2009) Let (X,d) be a complete metric space and T,S:X→X be two mappings such that for all x,yX,

d(Tx,Sy)M(x,y)-ϕ(M(x,y)), 1

where ϕ:[0,)[0,) is a lower semi-continuous function with ϕ(t)>0 for t(0,) and

ϕ(0)=0,

Mx,y=maxdx,y,dTx,x,dSy,y,12dy,Tx+d(x,Sy) 2

Then there exists a unique uX such that u=Tu=Su .

Doric (2009) established a fixed point theorem which generalized the result of Zhang and Song (2009) using control functions, which is given below;

Theorem 1.2 (Doric, 2009) Let (X,d) be a complete metric space and T,S:X→X be two mappings such that for all x,yX,

ψ(d(Tx,Sy))ψ(M(x,y))-ϕ(M(x,y)), 3

where

  1. ψ:[0,)[0,) is a continuous monotone non-decreasing function with ψ(t)=0 if and only if t=0,

  2. ϕ:[0,)[0,) is a lower semi-continuous function with ϕ(t)=0 if and only if t=0, and

     Mx,y=maxdx,y,dTx,x,dSy,y,12dy,Tx+d(x,Sy) 4

Then there exists a unique uX such that u=Tu=Su . The aim of our work is to prove above results in fuzzy metric spaces. The first theorem is the extension of the result of Zhang and Song (2009) under the different contractive conditions using control functions. Second result is analogous to the result of Doric (2009) in metric spaces.

Definition 1.1 (Schweizer & Sklar, 1960) A binary operation *:[0,1]×[0,1][0,1] is continuous t- norm if * satisfies the following conditions

  • (T-1) * is commutative and associative;

  • (T-2) * is continuous;

  • (T-3) a*1=a for all a[0,1] ;

  • (T-4) a*bc*d whenever ac and bd for all a,b,c,d[0,1] .

Definition 1.2 (George & Veeramani, 1994) The 3-tuple (X,M,*) is said to be fuzzy metric space if X is an arbitrary set, * is continuous t -norm and M is fuzzy set on X2×[0,) satisfying the following conditions for all x,y,zX and s,t>0,

(FM-1) M(x,y,t)>0

(FM-2)M(x,y,t)=1 , t>0 iff x=y ;

(FM-3)M(x,y,t)=M(y,x,t);

(FM-4)M(x,y,t)*M(y,z,s)M(x,z,t+s);

(FM-5)  M(x,y,.):[0,)[0,1] is continuous.

The triplet M(x,y,t) can be taken as the degree of nearness between x and y with respect to t0 .

Remark 1.1 (Shen, Qiu & Chen, 2013) Since * is continuous, it follows from (FM-4) that the limit of the sequence in fuzzy metric space is uniquely determined.

Let (X,M,*) is a fuzzy metric space then the following condition also holds:

(FM-6) limtM(x,y,t)=1.

Lemma 1.1 (Grabiec, 1988) In fuzzy metric space (X,M,*) , M(x,y,) is non-decreasing for all x,yX.

Definition 1.3 (George & Veeramani, 1994) Let (X,M,*) be a fuzzy metric space. Then a sequence xn X is said to be

  1. Convergent to a point xX if for all t>0 , limnM(xn,x,t)=1;

  2. Cauchy sequence if for all t>0 and p>0 , limnM(xn+p,xn,t)=1;

  3. A fuzzy metric space (X,M,*) is said to be complete if and only if every Cauchy sequence in X is convergent.

2. Main results

Theorem 2.1 Let (X,M,*) be a complete fuzzy metric space and T,S:XX be two mappings such that for all x,yX ,

M(Tx,Sy,t)N(x,y,t)+ϕ(N(x,y,t)), 5

where ϕ:[0,1][0,1] is a upper semi-continuous function such that ϕ(t)>0 for t(0,1) and ϕ(1)=0, and

      Nx,y,t=min{Mx,y,t,MTx,x,t,

MSy,y,t,My,Tx,t *Mx,Sy,t} 6

then there exists a unique uX such that u=Tu=Su .

Proof. To prove our result, we follow the following steps.

Step- I We show that N(x,y,t)=1 if and only if x=y , is a common fixed point of T and S .

Infact, if x=y=Tx=Ty=Sx=Sy, then N(x,y,t)=1. Let N(x,y,t)=1 , then

Mx,y,tNx,y,t,MTx,x,tNx,y,t,M(Sy,y,t)N(x,y,t)

we have x=y=Tx=Ty=Sx=Sy.

Step- II Let x0 be a given point in X . We construct the sequence xn for n0 inductively as follows.

Choose a sequence xnX so that x2n+1=Sx2n , x2n+2=Tx2n+1 and prove that M(xn+1,xn,t)1 as n .

Suppose that n is an odd number. Substituting x=xn and y=xn-1 in (5), we obtain

M(xn+1,xn,t)=M(Txn,Sxn-1,t)N(xn,xn-1,t)+ϕ(N(xn,xn-1,t))min{M(xn,xn-1,t),M(Txn,xn,t),M(Sxn-1,xn-1,t),M(xn-1,Txn,t)*M(xn,Txn-1,t)}min{M(xn,xn-1,t),M(xn+1,xn,t),M(xn,xn-1,t),M(xn-1,xn,t)*M(xn,xn-1,t)},

which implies that M(xn+1,xn,t)≥M(xn,xn-1,t),

If we take M(xn+1,xn,t)<M(xn,xn-1,t),

Then N(xn,xn-1,t)=M(xn+1,xn,t) and moreover

Mxn+1,xn,tMxn+1,xn,t+(Mxn+1,xn,t),

this is a contradiction. Therefore we have

Mxn+1,xn,tNxn,xn-1,tMxn,xn-1,t) 7

Similarly, we can also obtain inequalities (7) in case when n is an even number.

So, the sequence {Mxn+1,xn,t} is non-decreasing sequence and bounded above. So there exists r>0, such that limnM(xn+1,xn,t)=limnM(xn,xn-1,t)=r, then upper continuity of ϕ implies that ϕrlimnsupϕMxn,xn-1,t.

We claim that r=1 In fact, taking upper limit as n→∞ on either side of the following inequality, we have

Mxn+1,xn,tNxn,xn-1,t+ϕ(Nxn,xn-1,t)

rr+limnsupϕMxn,xn-1,t,rr+ϕ(r).

this is a contradiction unless ϕ(r)=0 at r=1.

Hence

limnM(xn+1,xn,t)=1. 8

Step -III Next we prove that xn is a Cauchy sequence. To prove this, it is sufficient to prove that the sub-sequence x2n of xn is a Cauchy sequence. Suppose opposite that the sequence xn is not a Cauchy sequence. Then there exists ϵ>0 such that n(k) is the smallest index for which n(k)>m(k)>k , we have M(x2m(k),x2n(k),t)ϵ.

Therefore,

M(x2m(k),x2n(k),t)=M(Tx2m-1,Sx2n-1,t)N(x2m-1,x2n-1,t)+ϕ(N(x2m-1,x2n-1,t),

this gives ϵϵ+ϕ(ϵ),

this gives a contradiction with ∈>0.

Thus x2n is a Cauchy equence and hence xn is a Cauchy sequence.

In complete fuzzy metric space (X,M,*) , there exists uX such that sequence xnu as n .

Step - IV Now we prove that u is a fixed point of T and S . For this suppose that uTu , then for M(u,Tu,t)<1 , there exist N1N such that for any nN1

M(x2n+1,u,t)>M(u,Tu,t),M(x2n,u,t)>M(u,Tu,t),M(x2n,x2n+1,t)>M(u,Tu,t).

Under this consideration, we have

M(u,Tu,t)N(u,xn,t)=min{M(u,x2n,t),M(Tu,u,t),M(Sx2n,x2n,t), M(x2n,Tu,t)*M(u,Sx2n,t)}=min{M(u,x2n,t),M(Tu,u,t),M(x2n+1,x2n,t), M(x2n,Tu,t)*M(u,x2n+1,t)}.

Letting n , we have

M(u,Tu,t)>min{M(u,u,t),M(Tu,u,t),

M(u,u,t), M(u,Tu,t)*1}=M(u,Tu,t),

That is

N(u,x2n,t)=M(u,Tu,t) as n . 9

Since

M(Tu,x2n+1,t)=M(Tu,Sx2n,t)N(u,x2n,t)+ϕ(N(u,x2n,t)),

then letting n , we have

M(Tu,u,t)M(Tu,u,t)+ϕ(M(Tu,u,t)),

we obtain a contradiction. Hence Tu=u .

Also, Mu,Su,t=MTu,Su,tNu,u,t +ϕ(N(u,u,t))=M(u,u,t)+ϕ(M(u,u,t))

implies Su=u

Suppose there exists another fixed point vX such that v=Tv=Sv , then using an argument similar to the above, we get

M(u,v,t)=M(Tu,Sv,t)N(u,v,t)+ϕ(N(u,v,t))M(u,v,t)+ϕ(M(u,v,t)).

Hence u=v . The proof is completed.

Corollary 2.1 Let (X,M,*) be a complete fuzzy metric space and T:XX be mapping such that for all x,yX ,

M(Tx,Ty,t)N(x,y,t)+ϕ(N(x,y,t)),

where ϕ:[0,1][0,1] is a upper semi-continuous function such that ϕ(t)>0 for t(0,1) and ϕ(1)=0 , and

Nx,y,t=min{Mx,y,t, MTx,x,t, MTy,y,t, My,Tx,t*Mx,Ty,t} 10

then there exists a unique uX such that u=Tu .

Example 2.1 Let (X,M,*) be a complete fuzzy metric space with metric d(x,y)=x-y and X=[0,1] . Let

Tx=x2andSx=x16

for each x[0,1] . Then

N(x,y,t)=mintt+x-y,tt+x2,tt+y,tt+x-y*tt+x,=tt+x-y x2yxtt+yyx.=mintt+x-y,tt+x2-x,tt+y16-y,tt+y-x2*tt+x-y16,=tt+y-x2*tt+x-y160yx.

For ϕ(t)=t2-1 , it is easy to show that

M(Tx,Sy,t)N(x,y,t)+ϕ(N(x,y,t))

for all x,yX . One can show that all the condition of Theorem 2.1 fulfill and T,S satisfy the Theorem 2.1.

Theorem 2.2 Let (X,M,*) be a complete fuzzy metric space and T,S:XX be two mappings such that for all x,yX ,

ψ(M(Tx,Sy,t))ψ(N(x,y,t))+ϕ(N(x,y,t)), 11

where,

1. ψ:[0,1][0,1] is a continuous monotone non-decreasing function with ψ(t)=1 iff t=1 ,

2. ϕ:[0,1][0,1] is a upper semi-continuous function ϕ(t)>0 for t(0,1) and ϕ(1)=0 , and

N(x,y,t)=minMx,y,t,MTx,x,t,MSy,y,t,M(y,Tx,t)*M(x,Sy,t),

then there exists a unique uX such that u=Tu=Su .

Proof. For any x0X , we construct a sequence xn for n0 as x2n+1=Sx2n, x2n=Tx2n+1 and will prove that M(xn,xn-1,t)1 as n. Suppose that n is an odd number, substituting x=xn and y=xn-1 in Equation (11), we obtain

ψ(M(xn+1,xn,t))=ψ(M(Txn,Sxn-1,t))

                    ψ(N(xn,xn-1,t))+ϕ(N(xn,xn-1,t)).

ψ(M(xn+1,xn,t))ψ(N(xn,xn-1,t)),

this implies that

M(xn+1,xn,t)N(xn,xn-1,t). 12

From triangle inequality, we have

   Nxn,xn-1,t=min{Mxn,xn-1,t,Mxn+1,xn,t,Mxn,xn-1,t,M(xn-1,xn+1,t)*M(xn,xn,t)}

= Mxn,xn-1,t,Mxn+1,xn,t,M(xn-1,xn+1,t) min{Mxn,xn-1,t,Mxn+1,xn,t, M(xn-1,xn,t)*M(xn,xn+1,t)}.

If we consider

Nxn,xn-1,tMxn+1,xn,tN(xn,xn-1,t)M(xn+1,xn,t),

it further implies that

ψMxn+1,xn,tψMxn+1,xn,t+ϕ(M(xn+1,xn,t)),

this is a contradiction. So we have

Mxn+1,xn,tNxn,xn-1,tMxn,xn-1,t                                          

this implies

Mxn+1,xn,tMxn,xn-1,t. 13

Similarly we can prove that Equation (13) is true when n is even number.

Therefore the sequence {Mxn+1,xn,t} is monotonically non decreasing and bounded sequence. Therefore we can write

limnM(xn+1,xn,t)=limnM(xn,xn-1,t)=r,

Letting n→∞ in the given inequality (11), we get

ѱMxn+1,xn,tѱMxn,xn-1,t+Mxn,xn-1,t,ѱrѱr+r,

which is contradiction, unless r=1 and at Ǿ(r) =0 Hence

limnM(xn+1,xn,t)=1. 14

Next we prove that {x n } is a Cauchy sequence. To prove this it is sufficient to prove that sub-sequence {x 2n } of {x n } is a Cauchy sequence. Suppose {x 2n } is not a Cauchy sequence, then there existsor which we can find sub-sequence {x 2m(k) } and {x 2n(k) } such that n(k) is the smallest index for which, n(k)>m(k)>k we have

M(x2mk,x2nk,t)ò 15

This means that Mx2mk,x2nk-2,t>ò . From Equation (15) and the triangle inequality, we have òMx2mk,x2nk,t

Mx2mk,x2nk-2,t *Mx2n(k)-2),x2nk-1,t*Mx2nk-1,x2nk,t,>ò*Mx2nk-2,x2nk-1,t*Mx2nk-1,x2nk,t,

Moreover

Mx2mk,x2nk+1,tMx2nk,x2nk+1,t*Mx2mk,x2nk,t,Mx2mk-1,x2nk,tMx2mk,x2nk,t*Mx2mk,x2mk-1,t.                         

Using (13) and (14), we have

limnM(x2m(k)-1,x2n(k),t)=1* ò=limnM(x2m(k),x2n(k)+1,t). 16

Also,

       Mx2mk-1,x2nk+1,tMx2nk,x2nk+1,t*Mx2mk-1,x2mk,t. 17

Using (13) and (16), we have

limnM(x2m(k)-1,x2nk+1,t)=ò 18

Again, using (14)-(18), we have limnM(x2m(k)-1,x2nk,t)=ò

Putting x= x2m(k)-1 , y=x 2n(k) in (11), one can get

ѱMx2mk,x2nk+1,t=ѱMTx2mk-1,Sx2nk,tѱMx2mk-1,x2nk,t+ϕMx2mk-1,x2nk,t,

letting k→ ∞ and using (16) and (18), we get

ѱòѱò+ϕò,

this is contradiction with ò > 0. Thus {x 2n } is a Cauchy sequence and hence {x n } is a Cauchy sequence. In complete metric space x, there exists z such that xn→z as n→∞.

Let us now prove that z is a fixed point for T and S.

ѱMTz,x2nk+1,t=ѱMTz,Sx2nk,t ѱNz,x2nk,t+ϕNz,x2nk,t,

using the same argument as in Theorem 2.1 (eq-9)and letting n→ ∞, we obtain

ѱMTz,z,t ѱMTz,z,t+ϕMTz,z,t,

this implies ѱ(M(Tz,z,t))=1. Hence M(Tz,z,t)=1 gives that z is a fixed point of T.

Thus, we have

ѱMz,Sz,t= ѱ(MTz,Sz,t)ѱNz,z,t+ϕ(Nz,z,t)=ѱMz,Sz,t+ϕMz,Sz,t,

this implies ѱ(M(Sz,z,t))=1. Hence M(Sz,z,t)=1 or Sz = z.

To prove uniqueness, we consider another fixed point wX then

ѱMz,w,t=ѱ(MTz,Sw,t)ѱMz,w,t+ϕ(Nz,w,t)=ѱMz,w,t+ϕ(Mz,w,t)

Thus ѱ(M(z,w,t))=1

Hence M(z,w,t)=1 or z=w This completes the proof.

Corollary 2.2 Let (X,M,*) be a complete fuzzy metric space and T:X→X be a mapping such that for all x,y, ε X,

ѱMTx,Ty,tѱNx,y,t+ϕNx,y,t, 19

where

  1. ѱ:[0,1]→[0,1] is a continuous monotone non-decreasing function with ѱ(t)=1 iff t=1,

  2. ϕ:[0,1]→[0,1] is a upper semi-continuous function ϕ(t)>0 for t ∊ (0,1) and ϕ(1)=0, and

N(x, y, t) = min {M (x, y, t), M(Tx, x, t), M(Ty, y, t),

M(y, Tx, t) ∗ M(x, Ty, t)},

then there exists a unique u ∊ X such that u = Tu.

Example 2.2 Let (X,M,*) be a complete fuzzy metric space with metric dx,y=|x-y| and x=[0,1]. Let Tx=x2 and Sx=0 for each x ∊ [0,1] Then

Nx,y,t=mintt+x-y,tt+x2,tt+y,tt+x-y*tt+x,={tt+|x-y| x2yxtt+y yx.

For ϕ(t)=t and ѱ(t)=3t, it is easy to show that

ѱMTx,Sy,tѱNx,y,t+ϕNx,y,t,

All the conditions of Theorem 2.2 are satisfied.

3. Conclusions

In this paper, the use of control functions has renewed the possibility of establishing new results in fuzzy metric fixed point theory. We extend the existing result in metric space towards fuzzy metric space with a new approach of using control functions.

References

Alber, Y. I., & Guerre-Delabriere, S. (1997). Principles of weakly contractive maps in Hilbert spaces. In I. Gohberg & Yu. Lyubich (Eds.), New results in operator theory and its applications (pp. 7-22). Basel: Birkhäuser. [ Links ]

Al-Thagafi, M.A., & Shahzad, N. (2006). Non-commuting selfmaps and invariant approximations. Nonlinear Analysis, 64(12), 2778-2786. [ Links ]

Doric, D. (2009). Common fixed points for generalized (ѱ,Ø) -weak contractions. Applied Mathematics Letters, 22(12), 1896-1900. [ Links ]

Dutta, P.N., & Choudhury, B.S. (2008). A generalization of contraction principle in metric spaces. Fixed Point Theory and Applications, 8. Article ID 406368. [ Links ]

George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets Syst., 64(3), 395-399. [ Links ]

Grabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. , 27(3), 385-389. [ Links ]

Gregori, V., & Sapena, V. (2002). On fixed point theorem in fuzzy metric spaces. Fuzzy Sets Syst, 125(2), 245-252. [ Links ]

Gupta, V., & Mani, N. (2013). Existence and uniqueness of fixed point for contractive mapping of integral type. International Journal of Computing Science and Mathematics, 4(1), 72 - 83. [ Links ]

Gupta, V., & Mani, N. (2014). Existence and uniqueness of fixed point in fuzzy metric spaces and its applications. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012 (pp. 217-223). Springer, New Delhi. [ Links ]

Gupta, V., Mani, N., Tripathi, AK. (2012). A fixed point theorem satisfying a generalized weak contractive condition of integral type. International Journal of Mathematical Analysis, 6(38), 1883-1889. [ Links ]

Gupta, V., Saini, R.K., Mani, N., & Tripathi, A.K. (2015). Fixed points theorems by using control functions in fuzzy metric space. Cogent mathematics, 2(1), 1-7. [ Links ]

Kramosil, I., & Michalek, J. (1975). Fuzzy metric and statistical metric spaces. Kybernetica, 11(5), 336-344. [ Links ]

Mihet, D. (2008). Fuzzy ѱ-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst . , 159(6), 739-744. [ Links ]

Rhoades, B.E. (2001). Some theorems on weakly contractive maps. Nonlinear Analysis , 47(4), 2683-2693. [ Links ]

Schweizer, B. & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10, 314-334. [ Links ]

Shen, Y., Qiu, D., & Chen, W. (2013). On convergence of fixed points in fuzzy metric spaces. In Abstract and Applied Analysis (Vol. 2013). Hindawi. Article ID 135202, 1-6. [ Links ]

Song, Y. (2007). Common fixed points and invariant approximations for generalized (f,g)- nonexpansive mappings. Communications in Mathematical Analysis, 2(24), 17-26. [ Links ]

Song, Y., & Xu, S. (2007). A note on Common fixed points for Banach operator pairs. Int. J. Contemp. Math sci., 2, 1163-1166. [ Links ]

Zadeh, L.A. (1965). Fuzzy sets. Information and control, 8, 338-353. [ Links ]

Zhang, Q., & Song, Y. (2009). Fixed point theory for generalized.ϕ-weak contractions, Applied Mathematics Letters, 22(1), 75 - 78. [ Links ]

Peer Review under the responsibility of Universidad Nacional Autónoma de México

Published: February 22, 2021

Corresponding author. E-mail address: vishal.gmn@gmail.com (Vishal Gupta).

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License