1. Introduction

Maurice Frechet introduced the metric space in 1906. Since then it has been generalized by many researchers. The notion of b metric was introduced by ^{Bakhtin (1989)} which was further used by ^{Czerwik (1993}, ¹⁹⁹⁸⁾ to prove many results.

Definition 1.1. (^{Czerwik, 1993}) Let M be a non empty set and λ≥1 be a given real number. A function d:M×M→[0,∞) is a b -metric on M if the following conditions hold for all x,y,z∈M,

The triplet (M,d,λ) is called b -metric space.

Examples and fixed point theorems related to b metric space are mentioned in (^{Ansari, Gupta, & Mani 2020}; ^{Boriceanu, 2009}; ^{Boriceanu Bota, & Petrusel 2010}; ^{Shatanawi, Pitea, & Lazovic, 2014}).

The concept of b-metric space is broader than concept of metric space, when we take λ=1 in b -metric space then we get metric space.

^{Huang and Zhang (2007)} generalized the concept of metric space by introducing the concept of cone metric space. In the research by ^{Huang and Zhang (2007)} real numbers are replaced with an ordered Banach space and some fixed point theorems for non linear mappings are proved. After the work of ^{Huang and Zhang (2007)}, lot of literature appeared related to the study of cone metric spaces. Details are available (^{Jankovi’c, Kadelburg, & Radenovi’c, 2011}; ^{Latif, Hussain, & Ahmad, 2016}; ^{Mehmood, Azam, & Ko’cinac, 2015}; ^{Shatanawi, Karapinar, & Aydi,2012}).

^{Zadeh (1965)} introduced the concept of fuzzy set theory. After his work many researchers started applying this new concept to classical theories. In particular, ^{Kramosil and Michalek (1975)} introduced the new concept fuzzy metric space and proved many results. ^{George and Veeramani (1994)} introduced a stronger form of fuzzy metric space. Afterwards, many mathematicians studied fixed point theorems in the related spaces (^{Chauhan & Utreja, 2013}; ^{Chauhan & Kant, 2015}; ^{Gupta, Saini, & Verma, 2020}; ^{Gupta, & Verma, 2020}). ^{Czerwik (1998)} introduced b metric space and proved some results. The concept of b metric space is the extension to metric space.

^{Hussain and Shah (2011)} introduced the concept of cone b metric space, which generalizes both b-metric space and cone metric space.

^{Oner, Kandemire, and Tanay (2015)} applied the concept of fuzziness to cone metric space and introduced fuzzy cone metric space as a generalized form of fuzzy metric space given by ^{George and Veeramani (1994)}. They proved some basic properties and fixed point theorems under this space. We can see related work in (^{Oner, 2016a}, ^2016b; ^{Priyobarta, Rohen, & Upadhyay, 2016}).

In this paper, we have introduced the concept of fuzzy cone b metric space in the sense of ^{George and Veeramani (1994).} Here we combine the notion of cone b metric space with the concept of fuzziness in the sense of ^{George and Veeramani (1994)} and proved new version of Banach contraction principle using this concept. We have defined fuzzy cone b contractive mapping and proved the fuzzy cone b -Banach contraction theorem for single mappind as well as the pair of mappings. Other important results which are helpful in this study are (^{Abbas, Khan & Radenovic 2010}; ^{Ali & Kanna 2017}: ^{Boriceanu, Bota & Petrusel 2010}: ^{Li & Jiang 2014}; ^{Turkoglu & Abuloha 2010}).

Some more basic definitions which are used directly or indirectly are mentioned below:

Definition 1.2. (^{Schweizer & Sklar, 1960}) The binary operation *:[0,1]×[0,1]→[0,1] is called continuous t-norm if * satisfies the following conditions for al a,b,c,d∈0,1,

Example 1.1. Some examples of continuous t-norms are ∧ , and *L, which are defined by c∧d=min⁡c,d,c.d=cd (usual multiplication in [0,1]) and c*Ld=max⁡c+d-1,0.

Definition 1.3. (^{George & Veeramani, 1994}) The triple Y,N,* is said to be fuzzy metric space if Y is an arbitrary set, * is a continuous t -norm and N is a fuzzy set on Y×Y×0,∞ such that for all a,b,c∈Y and s,t>0, we have

Definition 1.4. (^{Sedghi & Shobe, 2012}) Let Y be a non empty set, * a continuous t-norm and let k≥1 be a given real number. A fuzzy set N in Y×Y×0,∞ is called b -fuzzy metric if for any a,b,c∈Y, and t,s>0, the following conditions hold:

Throughout this paper B denotes a real a Banach space and θ denotes the zero of B.

Definition 1.5. (^{Huang & Zhang, 2007}) Let Qbe the subset of B. Then Q is called a cone if

For a given cone Q⊂B a partial ordering ≺ on B via Q is defined by u≺v if and only if v-u∈Q. u≺v stands for u<v and u≠v, while u<<v stands for v-u∈int(Q), where int(Q) is the set of all interior points of Q. In this paper, we assume that all cones has non empty interior.

Definition 1.6. (^{Oner et al., 2015}) A three tuple (Y,N,*) is said to be a fuzzy cone metric space if Q is a cone of B, Yis an arbitrary set, * is a continuous t-norm and Nis a fuzzy set on Y2×int(Q) satisfying the following conditions for a,b,c∈Y and t,s∈int(P),

Definition 1.7. (^{Oner et al., 2015}) Consider a fuzzy cone metric space (Y,N,*),y∈Y and yn be a sequence in Y, then

We denote it by limn→∞⁡yn=y or yn→y as n→∞;

2. Main results

Definition 2.1 Let Y be a non empty arbitrary set, * is a continuous t-norm, N is a fuzzy set on Y×Y×Int(Q), Q is a cone of B (Real Banach space). A quadruple (Y,N,*,λ) is said to be fuzzy cone b -metric space if following conditions are satisfied for all a,b,c∈Yand t,s∈IntQ,λ≥1,

Example 2.1: LetB=R2. Then Q=(r1,r2):r1,r2≥0 subset of B, is a normal cone with normal constant k=1.

Let Y=R,a*b=ab and M:X2×int⁡Q→0,1, defined by Mx,y,t=1ex-yλt for all x,y∈Xand t≥0.

This gives, s≤λ(t+s) and t≤λ(t+s), we have λ(t+s)s≥1 and λ(t+s)t≥1.

Now, x-z≤x-y+y-z, we can write, x-z≤x-yλ(t+s)t+y-zλ(t+s)s, this implies, x-zλ(t+s)≤x-yt+y-zs, so, one can get ex-zλ(t+s)≤ex-yt+ey-zs or 1ex-zλ(t+s)≥1ex-yt+1ey-zs.

Thus, M(x,z,λ(t+s))≥M(x,y,t)*M(y,z,s).

Then Mx,y,.:IntQ→0,1 is composite function of f1 and f2.

Both f1 and f2 are continuous, hence M(x,y,.) is also continuous and limt→∞M(x,y,t)=limt→∞1ex-yt=1.

Definition 2.2: Let(Y,N,*,λ) be a fuzzy cone b -metric space.R:Y→Y be a self mapping. Then R is said to be fuzzy cone b -contractive if there exist α∈0,1 such that 1N(Rx,Ry,t)-1 ≤ α (1N(x,y,t)-1) for x,y∈Y and t≥θ, where θ denotes the zero of B and α is known as contraction constant of R.

Definition 2.3: Let (Y,M,*,λ) be a fuzzy cone b -metric space and xn be a sequence in Y. Then xn is said to be fuzzy cone b -contractive if 1M(xn+1,xn+2,t)-1 ≤ α (1M(xn,xn+1,t)-1) for all t≥θ, nis natural number,α∈0,1.

Definition 2.4: Let (Y,M,*,λ) be a fuzzy cone b -metric space.A,B:Y→Y are self mappings. Then mappings AandBare known as fuzzy cone b -contractive ifα∈0,1 such that 1N(Ax,By,t)-1 ≤ α (1N(x,y,t)-1) for t≥θ, α is called contraction constant of Aand B.

Fuzzy Cone b -Banach Contraction Theorem

Theorem 2.1: Let X,M,*,λbe complete fuzzy cone b -metric space in which fuzzy cone b -contractive sequence is Cauchy andS,T→R be fuzzy cone contractive mappings and SX⊆TX. Then S and Thave unique common fixed point.

Proof: Let x0∈X, define a sequence xn such that x2n+1=Sx2n,x2n+2=Tx2n+1 for n=0,1,2...

First we show that sub sequence x2n is a Cauchy sequence 1M(x2n+1,x2n+2,t)-1=1M(Sx2n,Tx2n+1,t)-1≤α1M(x2n,x2n+1,t)-1=α1M(Sx2n-1,Tx2n,t)-1≤α21M(x2n-1,x2n,t)-1,

Continue in this way, we get 1M(x2n+1,x2n+2,t)-1≤α2n+11M(xo,x1,t)-1.

Then x2n is a Cauchy sequence in Xand Xis complete .

Therefore x2n converges to y for some y∈X. Then using Theorem 2.10 (^{Nadaban, 2016}) we have, 1M(Sx2n,Tx2n+1,t)-1≤α1M(x2n+1,x2n+2,t)-1≤α1M(y,y,t)-1 this gives, 1M(Sx2n,Tx2n+1,t)=1. Thus, Sx2n=Tx2n+1 and therefore Sy=Ty as n→∞.

Hence y is a coincidence point of Sand T.

Now we will prove that yis a fixed point of Sand T.

Since, 1M(Sy,Tx2n,t)-1≤α1M(y,x2n,t)-1, this gives, 1M(Sy,x2n+1,t)-1 ≤ α(1M(y,y,t)-1) as x2n→y and Tx2n=x2n+1, thus 1M(Sy,y,t)-1≤0. Hence Sy=y.

For uniqueness, let uis also a fixed point of Sand T, ie. Su=Tu=u.

Therefore, 1M(y,u,t)-1=1M(Sy,Tu,t)-1≤α1M(y,u,t)-1 ie. (1-α)1M(y,u,t)-1≤0, this gives,M(y,u,t)=1.

Thus y=u. and henceyis a unique common fixed point of Sand T.

Corollary 2.1 (Fuzzy Cone b -Banach Contraction Theorem) Let (X,M,*λ) be a complete fuzzy cone b -metric space in which fuzzy cone b -contractive sequence is Cauchy and S:X→Xbe a fuzzy cone contractive mapping. Then S has unique fixed point.

Proof. If we put S=Tin Theorem 2.1, we get the result.

Definition 2.5: Let (X,M,*λ) be fuzzy cone b -metric space. A self mapping TonX is called Chauhan-Gupta contraction if it satisfies the following condition for allx,y∈X,k≥0 and a,b∈0,1 suchthat a+b<1,a<1-k, 1M(Tx,Ty,t)-1≤a1M(x,Tx,t)*M(y,Tx,t)-1+b1M(y,Ty,t)-1+k1minM(x,Ty,t),M(y,Tx,t)-1.

Theorem 2.2: Let(X,M,t,λ) be a complete fuzzy cone b -metric space. Let T:X→X is a Chauhan-Gupta contraction given by (1). ThenThas a unique fixed point inX.

Proof: Let x0∈X and define a sequence xn by xn=Txn-1 for n≥0. Then by (1), for t≥0,n≥1, 1M(Tx,Ty,t)-1≤a1M(x,Tx,t)*M(y,Tx,t)-1+b1M(y,Ty,t)-1+k1minM(x,Ty,t),M(y,Tx,t)-1. this gives, 1M(xn,xn+1,t)-1=1MTxn-1,Txn,t-1≤a1Mxn-1,Txn-1,t*Mxn,Txn-1,t-1+b1Mxn,Txn,t-1+k1minMxn-1,Txn,t,Mxn,Txn,t-1 ≤a1M(xn-1,xn,t)*M(xn,xn,t)-1+b1M(xn,xn+1,t)-1+k1minM(xn-1,xn+1,t),M(xn,xn,t)-1≤a1M(xn-1,xn,t)-1+b1M(xn,xn+1,t)-1+k(0), we get, 1M(xn,xn+1,t)-1 ≤ a(1-b)1M(xn-1,xn,t)-1

This implies,1M(xn,xn+1,t)-1≤h1M(xn-1,xn,t)-1≤hn1M(x0,x1,t)-1, where h=a(1-b)<1 as a+b<1. xn is a fuzzy cone b -contractive sequence and therfore one can getlimn→∞M(xn,xn+1,t)=1 for t≥0.

Now, for m>n≥n0, 1M(xn,xm,t)-1≤1Mxn,xn+1,t-1+1Mxn+1,xn+2,t-1+...+1Mxm-1,xm,t-1≤hn1Mx0,x1,t-1+hn+11Mx0,x1,t-1+...+hm-11Mx0,x1,t-1 =(hn+hn+1+...+hm-1)1M(x0,x1,t)-1→0 as n→∞.

Thus limn→∞M(xn,xm,t)=1, which gives xn is a Cauchy sequence. The completenes of X, one can say limn→∞xn=u.

Now, 1M(xn+1,Tu,t)-1=1MTxn,Tu,t-1≤a1Mxn,Txn,t*Mu,Txn,t-1+b1Mu,Tu,t-1+k1minMxn,Tu,t,Mu,Txn,t-1=a1Mxn,xn+1,t*Mu,xn+1,t-1+b1Mu,Tu,t-1+k1minMxn,Tu,t,Mu,xn+1,t-1=a1Mu,Tu,t*Mu,u,t-1+b1Mu,Tu,t-1 +k1minM(u,Tu,t),M(u,u,t)-1, we get,1M(u,u,t)-1≤b1M(u,Tu,t)-1 for t≥0and b<1.

Hence u is a fixed point of T.

For uniqueness, let v is another fixed point of T.

1M(u,v,t)-1=1M(Tu,Tv,t)-1≤a1M(u,Tu,t)*M(v,Tu,t)-1+b1M(v,Tv,t)-1+k1min⁡{Mu,Tv,t,M(v,Tu,t)}-1=a1Mu,u,t*Mv,u,t-1+b1Mv,v,t-1+k1minM(u,v,t),M(v,u,t)-1,implies, 1M(u,v,t)-1≤a+k1M(v,u,t)-1, where a+k<1.

Thus M(u,v,t)=1 and this gives u=v.

Hence, uis a unique fixed point of T.

3. Conclusions

In this article, we Introduced the idea of fuzzy cone b -metric space and the fuzzy cone b-contractive mapping has been defined. Also, the Banach contraction theorem has been proved in the setting of fuzzy cone b metric space. Based on the results in this paper, interesting researches may be prospective. In the future study, one can establish the integral version of fixed point theorem in this space and can also think of establishing some new fixed point results in fuzzy cone b -metric space. The work presented here is likely to provide a ground to the researchers to do work in different structures by using these conditions.